رياضيات تخصصية by sabrigooda

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									                   ‫א‬‫א‬‫א‬
                ã¹]<gè…‚jÖ]æ<ËÖ]<Üé×Ãj×Ö<íÚ^ÃÖ]<퉉ö¹]
                    ‫א‬‫א‬‫א‬‫א‬




  


    ١٧١
                                            ١٧١                                   ‫א‬
                                                                           



                                               
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                W،،‫א‬‫א‬،‫א‬
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‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
  ‫א‬ ‫א‬  ،‫א‬   ‫א‬ ‫א‬  ‫א‬ ‫א‬ ‫א‬ 
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
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                                                              K‫א‬‫א‬‫א‬
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‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
  ‫א‬        ،‫א‬ ‫א‬‫א‬  
‫א‬  ‫א‬ ‫א‬ ‫א‬ ‫א‬ ‫א‬   ‫ א‬    ،  
‫א‬،‫א‬‫א‬‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬
     ‫א‬  ‫א‬   ،‫א‬   
                                     K‫א‬‫א‬،‫א‬‫א‬
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‫א‬????‫א‬‫א‬
                             K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
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‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
،‫א‬،‫א‬‫א‬‫א‬‫א‬
                           K‫א‬‫א‬‫א‬‫א‬‫א‬
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   W    ‫א‬ ‫א‬  ‫א‬    ‫א‬
                                                                                                K‫א‬
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  ‫א‬‫א‬‫א‬‫א‬                                                     
                                           ١٧١                                   ‫א‬
                                                                         



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                                             
‫א‬‫א‬ ‫א‬  ،‫א‬  ‫א‬ ‫א‬   ‫א‬ ‫א‬ ،‫א‬   ‫א‬
 K،‫א‬‫א‬‫א‬،‫א‬‫א‬،
‫א‬ ‫א‬         ١J     
‫א‬‫א‬   ‫א‬ ‫א‬ ‫א‬‫א‬   ‫א‬ ‫א‬‫א‬ ‫א‬  ‫א‬
‫א‬‫א‬‫א‬‫א‬‫א‬K‫א‬
‫א‬     ‫א‬ ‫א‬  ‫א‬  ‫א‬ ‫א‬  ‫א‬ 
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
‫א‬  ‫א‬      K    ‫א‬‫א‬  ‫א‬
 ‫א‬     ‫א‬ ‫א‬‫א‬ ‫א‬ ‫א‬   ‫א‬
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
                                                    W‫א‬‫א‬‫א‬‫א‬K‫א‬
                           ‫א‬‫א‬‫א‬‫• א‬
        ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ •
                                               ‫א‬‫א‬‫א‬‫א‬‫• א‬
                                  ‫א‬‫א‬‫א‬‫א‬‫א‬ •
                     ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫• א‬
                                               ‫א‬‫א‬‫א‬‫• א‬
                                         ‫א‬‫א‬‫א‬‫א‬ •
                         ‫א‬‫א‬‫א‬‫א‬ •
                ‫א‬‫א‬‫א‬‫א‬‫א‬ •
                                                                   
     W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬‫א‬
                 W‫א‬‫א‬‫א‬K‫א‬‫א‬
                                            ١٧١                                 ‫א‬
                                                                         



        ‫א‬ ‫א‬ ‫א‬  ‫א‬ ‫א‬ 
                                        K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
    K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
  ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
                                        K‫א‬‫א‬‫א‬‫א‬
 ‫א‬‫א‬   ‫א‬ ‫א‬ ‫א‬  ‫א‬   ‫א‬ ‫א‬ 
                           E‫א‬‫א‬‫א‬‫א‬J ‫א‬‫א‬‫א‬J ‫א‬‫א‬F‫א‬
 ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
                                                 K‫א‬‫א‬‫א‬
  ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
                                         K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
 ‫א‬  ‫א‬    ،‫א‬‫א‬ ‫א‬ ‫א‬   ‫א‬ ‫א‬
  ‫א‬ ،E‫א‬‫א‬ ‫،א‬‫א‬ ‫א‬F ‫א‬ ‫א‬ ‫א‬  ‫א‬
  J ‫א‬‫א‬F‫א‬‫א‬‫א‬‫א‬K‫א‬‫א‬‫א‬
 ‫א‬   ‫א‬ ‫א‬  ‫א‬ ،E‫א‬J ‫א‬J ‫א‬J ‫א‬‫א‬
                                                                                        K‫א‬
 ‫א‬‫א‬‫א‬‫א‬‫א‬
                                                    ‫א‬‫א‬‫א‬‫א‬
 ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
 ‫א‬ ‫א‬‫א‬ ‫א‬ ‫א‬   ‫א‬‫א‬   ‫א‬‫א‬ ‫א‬ 
                                                                           ‫א‬‫א‬‫א‬
 ‫א‬‫א‬
                                   ‫א‬‫א‬‫א‬
                                ã¹]<gè…‚jÖ]æ<ËÖ]<Üé×Ãj×Ö<íÚ^ÃÖ]<퉉ö¹]
                                    ‫א‬‫א‬‫א‬‫א‬




                
                ‫א‬‫א‬
 ‫א‬‫א‬




١
       ‫א‬‫א‬                         ١٧١                            ‫א‬
       ‫א‬‫א‬                                            



                                 K‫א‬‫א‬‫א‬‫א‬‫א‬W‫א‬‫א‬
                                                                                                
                                                                                W‫א‬‫א‬
                                       W‫א‬‫א‬‫א‬
                                                  K‫א‬‫א‬‫א‬‫א‬      •

           E‫א‬‫א‬،‫א‬‫א‬F‫א‬‫א‬            •

                                  K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬        •

                       ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬                 •

                                                                                                
                                                                    
‫א‬،‫א‬‫א‬W‫א‬‫א‬
                                                                         K‫א‬




                                            -١-
           ‫א‬‫א‬                                 ١٧١                                   ‫א‬
           ‫א‬‫א‬                                                          



                                       ‫א‬‫א‬W‫א‬‫א‬
                                                                                                               
                                                                     ‫א‬‫א‬K١
                                                                
     ‫א‬‫א‬‫א‬‫א‬
     ‫א‬‫א‬‫א‬‫א‬‫א‬K‫א‬‫א‬
           K 3  − 3  − 3x 2 y ‫א‬‫א‬،‫א‬‫א‬
     ‫א‬K‫א‬‫א‬‫א‬
        ‫א‬ − 9x 2 12x 2 ،‫א‬‫א‬E‫א‬F‫א‬‫א‬
    ‫א‬‫א‬‫א‬ K 7 x 3 y 2  − 2 x 3 y ‫א‬
      K (2 = 2 x 0 ) ‫א‬‫א‬‫א‬‫א‬K 4 x 2 + x   4 x 2 + 3x − 2 x 
                                                               W‫א‬‫א‬‫א‬‫א‬
    an ‫א‬K n  an ≠ 0  an x n + an−1 x n−1 + .......... + a2 x 2 + a1 x + a0
                                                                   K‫א‬‫א‬ a0 ‫א‬‫א‬
                                                                                     
           W‫א‬‫א‬‫א‬،‫א‬،‫א‬‫א‬‫א‬‫א‬
          ‫א‬‫א‬       ‫א‬             ‫א‬            ‫א‬             ‫א‬
               9              9 , − 1, 5             2              9x2 , − x , 5       9x2 − x + 5
              −2               − 2 , 11              1             − 2x , 11           11 − 2 x
               1              1, 5 , − 3             3              x3 , 5x , − 3       x3 + 5x − 3
                                                                                                               
                                                                         ‫א‬‫א‬‫א‬‫א‬K٢
                                                                                   ‫א‬‫א‬K١{٢
                                                                                        W‫א‬W١
 1) (3x 2 + 6 x − 4) + (4 x 2 − 5 x + 3) 2) ( x 2 − 5 x + 7) − (2 x 2 + 3x − 2) 
                                                                                                        W‫א‬
    1) (3 x 2 + 6 x − 4) + (4 x 2 − 5 x + 3) = (3 x 2 + 4 x 2 ) + (6 x − 5 x) + ((−4) + 1) = 7 x 2 + x − 3

    2) ( x 2 − 5 x + 7) − (2 x 2 + 3 x − 2) = ( x 2 − 2 x 2 ) + [(−5 x) − (+3 x)] + (7 − (−2)) = − x 2 − 8 x + 9
                                                                                                               
                                                        -٢-
            ‫א‬‫א‬                                       ١٧١                                          ‫א‬
           ‫א‬‫א‬                                                                         



                                                                     ‫א‬‫א‬K٢{٢
               K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
                                                           (2 x − 3)(3x 2 − x + 1) W‫א‬‫א‬W٢
                                                                                                                      W‫א‬
        (2 x − 3)(3x 2 − x + 1) = (2 x)(3x 2 ) + (2 x)(− x) + (2 x)(1) + (−3)(3 x 2 ) + (−3)(− x) + (−3)(1)
                                                                                                                                    
                                             3    2                 2                       3       2
                                    = 6 x − 2 x + 2 x − 9 x + 3x − 3 = 6 x − 11x + 5 x − 3
                                                                                     W‫א‬‫א‬‫א‬‫א‬
        ( x + y )( x − y ) = x 2 − y 2

        ( x + y )( x + y ) = ( x + y ) 2 = x 2 + 2 xy + y 2

 ( x − y )( x − y ) = ( x − y ) 2 = x 2 − 2 xy + y 2                                  
        ( x + y )( x + y )( x + y ) = ( x + y ) 3 = x 3 + 3 x 2 y + 3xy 2 + y 3

        ( x − y )( x − y )( x − y ) = ( x − y ) 3 = x 3 − 3 x 2 y + 3 xy 2 − y 3
                                                  W‫א‬‫א‬‫א‬‫א‬‫א‬W٣
                                                                                                                       W‫א‬
        1) (7 x + 10)(7 x − 10)          2) (2 y 2 + 11z 2 ) 2          3) (2 x − 3 y ) 3

        1) (7 x + 10)(7 x − 10) = (7 x) 2 − (10) 2 = 49 x 2 − 100

        2) (2 y 2 + 11z 2 ) 2 = (2 y 2 ) 2 + 2(2 y 2 )(11z 2 ) + (11z 2 ) 2 = 4 y 4 + 44 y 2 z 2 + 121z 4

        3) ( 2 x − 3 y ) 3 = (2 x) 3 − 3(2 x) 2 (3 y ) + 3(2 x)(3 y ) 2 − (3 y ) 3 = 8 x 3 − 36 x 2 y + 54 xy 2 − 27 y 3
                                                                  ‫א‬K٣{٢
                               1) x = −4          2) x = 2 W 2 x 3 − 6 x 2 + 7 ‫א‬W٤
                                                                                                                           W‫א‬
                                                       W‫א‬ x ‫א‬
        a ) 2 x 3 − 6 x 2 + 7 = 2(−4) 3 − 6(−4) 2 + 7 = 2(−64) − 6(16) + 7 = −128 − 96 + 7 = −217
                                                                                                                                  
              3       2                  3            2
        b) 2 x − 6 x + 7 = 2( 2 ) − 6( 2 ) + 7 = 2(2 2 ) − 6(2) + 7 = 4 2 − 12 + 7 = 4 2 − 5



                                                              -٣-
                 ‫א‬‫א‬                                       ١٧١                                            ‫א‬
                ‫א‬‫א‬                                                                           



                                                           E‫א‬‫א‬F‫א‬‫א‬K٤{٢
 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
                                                       W‫א‬‫א‬ x − 3  x 2 + 9 x − 16 
                                                                   x + 12
                                                      x−3          x 2 + 9 x − 16
                                                                   x 2 − 3x

                                                                   12 x − 16

                                                                   12 x − 36

                                                                    20
                       20 ‫א‬ x + 12  x − 3  x 2 + 9 x − 16 ‫א‬‫א‬‫א‬
               W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬،‫א‬،‫א‬‫א‬W١
           1) x 2 + 2 x − 7                  2) 2              3) 4 x 2 y 2 − 5 x 3 y 2 + 17 xy 3
                                                                                                         
                   4          3       2          3                        5           4           2   2
            4) 2 x + 3 x + 4 x + 5 5) x − 1                    6) − 9 x y + 10 xy − 11x y
                                                                                            W‫א‬‫א‬W٢
           1) (3 x 2 + 4 x + 5) + (2 x 2 + 7 x − 2)                5) (5 x − 7)(3x 2 − 8 x − 5)

           2) (4 w 2 − 2 w + 7) + (5w3 + 8w 2 − 1)                 6) (3 x 2 − 2 x + 5)(2 x 2 − 5 x + 2)
                                                                                                                 
           3) (7 s 2 − 4 s + 11) − (−2s 2 + 11s − 9)               7) (3c − 2)(4c + 1)(5c − 2)

           4) (u 3 − 3u 2 −4u + 8) − (u 2 − 2u + 4)                8) (4u − 5)(2u − 1)(3u − 4)
                                                       W‫א‬‫א‬‫א‬‫א‬‫א‬W٣
           1) (3 x + 5)(3 x − 5)              7) [( x + 5) + y ][( x + 5) − y ]
           2) (4 x 2 − 3 y )(4 x 2 + 3 y )     8) [( x − 2 y ) + 7][( x − 2 y ) − 7]
           3) (3x 2 − y ) 2                    9) ( x − 1) 3
                                                                                         
           4) ( 4 w + z ) 2                  10) (2 x + y ) 3
           5) [( x − 2) + y ]                11) [( x − 1) + 2 y ]
                                  2                                  3


           6) [( x + 3) − y ]                12) [4 − (1 − 2 y )]
                                  2                                  3



                                                                    -٤-
          ‫א‬‫א‬                                  ١٧١                                           ‫א‬
          ‫א‬‫א‬                                                                    



                                           W‫א‬‫א‬‫א‬‫א‬‫א‬W٤
           1) x 2 + 7 x − 1 ; x = 3

          2) − x 2 − 5 x + 4 ; x = −5
                                            
                 3      2
          3) 5 x − x + 5 x − 3 ; x = −1

          4 )1 − x 3 − x 5 ; x = 2
                         W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W٥
           1) 5 x 3 + 6 x − 17 x + 20 , x + 3

          2) 6 x 4 + 3 x 2 − 11x 2 − 3 x + 9 , 2 x − 3
                                                               
                 4      3       2                  2
          3) 2 x − x − 23x + 9 x + 45 , 2 x − x − 5

          4) 24 x 5 + 20 x 3 − 16 x 2 − 15 , 6 x 2 + 5
                                                              ‫א‬‫א‬K٣
  K‫א‬
      ‫א‬‫א‬K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
                                                                      K‫א‬‫א‬‫א‬‫א‬
                                               EKKF‫א‬‫א‬‫א‬K٣{١
 ‫א‬‫א‬‫א‬‫א‬
                                                                                                            K‫א‬‫א‬
                                     W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W5
 1)10 x 3 + 6 x       2)12 x 2 y − 6 xy − 30 xy 2      3) ( x − 4)(2a − b) + ( x + 4)(2a − b) 
                                                                                                                     W‫א‬
       W‫א‬ 2 x  6 x  10x 3  EKKF‫א‬‫א‬E١

                                                               10 x 3 + 6 x = 2 x(5 x 2 ) + 2 x(3) = 2 x(5 x 2 + 3)
     W‫א‬ 6 xy ‫א‬‫א‬EKKF‫א‬E٢
 12 x 2 y − 6 xy − 30 xy 2 = 6 xy(2 x) − 6 xy(1) − 6 xy(5 y ) = 6 xy(2 x − 1 − 5 y ) = 6 xy(2 x − 5 y − 1)
                W‫א‬ 2a − b ‫א‬EKKFE٣

                                                         -٥-
                     ‫א‬‫א‬                                ١٧١                                   ‫א‬
                 ‫א‬‫א‬                                                              


          ( x − 4)(2a − b) + ( x + 4)(2a − b) = (2a − b)[( x − 4) + ( x + 4)] = (2a − b)( x − 4 + x + 4)
    
                                                      = (2a − b)(2 x) = 2 x(2a − b)
                                    K‫א‬‫א‬‫א‬
                                              6 y 3 − 21y 2 − 4 y + 14 W‫א‬‫א‬W6
                                                                                                                     W‫א‬
                                                W‫א‬‫א‬‫א‬‫א‬
                            6 y 3 − 21y 2 − 4 y + 14 = (6 y 3 − 21y 2 ) − (4 y − 14)
                                                                       W‫א‬EKKF
         6 y 3 − 21y 2 − 4 y + 14 = (6 y 3 − 21y 2 ) − (4 y − 14) = 3 y 2 (2 y − 7) − 2(2 y − 7) 
             W‫א‬‫א‬‫א‬ 2 y − 7 ‫א‬
              6 y 3 − 21 y 2 − 4 y + 14 = (6 y 3 − 21 y 2 ) − (4 y − 14) = 3 y 2 (2 y − 7) − 2(2 y − 7)
                                                                                                           
                                                           2
                                           = (2 y − 7)(3 y − 2)
                                                                             ax 2 + bx + c ‫א‬K٣{٢
                                                                      a = 1 W‫א‬‫א‬
         x 2 ‫א‬‫א‬
                                K‫א‬‫א‬‫א‬K b‫א‬ c ‫א‬
                                                          x 2 + 7 x − 18 W‫א‬‫א‬W7
                                                                                                                    W‫א‬
                                                                 
                − 18 ‫א‬‫א‬ c = −18  b = 7 ‫א‬
       K (−2) ⋅ 9 = −18 , (−2) + 9 = 7  9  − 2 ‫א‬‫א‬K 7 ‫א‬
                                                                                     W‫א‬‫א‬
                 K‫א‬‫א‬‫א‬‫א‬‫א‬ x 2 + 7 x − 18 = ( x − 2)( x + 9) 
        ‫א‬‫א‬‫א‬‫א‬‫א‬
                                                                                                         K‫א‬‫א‬
                                                                                           a ≠ 1 W‫א‬‫א‬
                          W‫א‬‫א‬‫א‬ m , n , p , q ‫א‬‫א‬
     1) mn = a , 2) pq = c , 3) mq + np = b 
                                                                -٦-
            ‫א‬‫א‬                                      ١٧١                                        ‫א‬
           ‫א‬‫א‬                                                                     



                                                                W‫א‬‫א‬‫א‬
  ax 2 + bx + c = (mx + p)(nx + q) 
   K c < 0 ‫א‬ c > 0 ‫א‬ b  q  p ‫א‬
    K n  m ‫א‬‫א‬ 3) ‫א‬ 2) 1) ‫א‬ n  m ‫א‬
                                                   6 x 2 + 11x + 3 W‫א‬‫א‬W8
                                                                                                                      W‫א‬
                   mn = 6 , pq = 4 , mq + np = −11 W m , n , p , q ‫א‬‫א‬‫א‬
W‫א‬‫א‬‫א‬ b > 0  c > 0  q  p ‫א‬
                                                                               
                                                 W‫א‬‫א‬ m = 2, n = 3, p = 3, q = 1
   6 x 2 + 11x + 3 = (2 x + 3)(3x + 1) 
     b 2 − 4ac ‫א‬ ax 2 + bx + c W
     K b 2 − 4ac = (−5) 2 − 4(6)(−4) = 121 = 112  6 x 2 − 5 x − 4 K
          (mn + p) 2  ax 2 + bx + c  m = n  p = q ‫א‬
                                                                      K٣{٣
                     W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
                                             x 2 − y 2 = ( x + y )( x − y )
                                                                                             49 x 2 − 144 W9
                                                                                                                      W‫א‬
  ‫א‬‫א‬ 49 x 2 − 144 = (7 x) 2 − (12) 2  49 x 2 − 144 
  49 x 2 − 144 = (7 x) 2 − (12) 2 = (7 x + 12)(7 x − 12) W‫א‬‫א‬
                                                                 K٤{٣
                           W
  a) x 3 − y 3 = ( x − y )( x 2 + xy + y 2 )                b) x 3 + y 3 = ( x + y )( x 2 − xy + y 2 ) 
                                                 1) 8a 3 + b 3         2) a 3 − 64 WW10
                                                                                                                     W‫א‬
  b) ‫א‬‫א‬ 8a 3 + b 3 = (2a) 3 − (b) 3  8a 3 + b 3 E١
                                                                         W‫א‬
                                                               -٧-
            ‫א‬‫א‬                                  ١٧١                                        ‫א‬
           ‫א‬‫א‬                                                                  



    8a 3 + b 3 = (2a) 3 + (b) 3 = (2a + b)[(2a ) 2 − (2a)(b) + (b) 2 ]
                                                                               
                                   = (2a + b)(4a 2 − 2ab + b 2 )
     b) ‫א‬‫א‬ a 3 − 64 = (a) 3 − (4) 3  a 3 − 64 E٢
                                                                    W‫א‬
      a 3 − 64 = (a ) 3 − (4) 3 = (a − 4)[(a ) 2 + (a)(4) + (4) 2 ] = (a − 4)(a 2 + 4a + 16) 
                                                                 ‫א‬‫א‬ K٥{٣
                                                                        
                                   K‫א‬‫א‬‫א‬‫א‬
 1) 2m 2 + 6mn − 15m − 5n                2) p 2 + p − q − q 2 W‫א‬W11
                                                                                                                    W‫א‬
                                                                                  W‫א‬‫א‬
                                                                       
              1) 2m 2 + 6mn − 15n − 5m = (2m 2 + 6mn) + (−15n − 5m)
                                                                                                             
                                              = 2m(m + 3n) − 5(3n + m) = (m + 3n)(2m − 5)
              2) p 2 + p − q − q 2 = p 2 − q 2 + p − q = ( p 2 − q 2 ) + ( p − q )
                                                                                         
                                  = ( p + q )( p − q ) + ( p − q ) = ( p − q)( p + q + 1)




                                                          -٨-
   ‫א‬‫א‬                             ١٧١                                   ‫א‬
   ‫א‬‫א‬                                                       



                                                                                       
                                                          W‫א‬‫א‬‫א‬‫א‬
1) − 15 x 2 − 12 x              13) x 4 + 11x 2 + 18          27)1 + y12

2) 6a 3b 2 − 12a 2b + 72ab 3    14) 9 x 4 + 10 x 2 + 1        28) 8 − x 6
3) ( x − 4)(m + 2n) + n( x − 4) 15) 6 x 4 + 23x 2 + 15        29) ( x − 2) 3 − 1
4) x( y − 3) − 5(3 − y )        18) x 2 − 9                   31)(a + b) 3 + (a − b) 3
5) 3x 3 + x 2 + 6 x + 2         19) 81b 2 − 16c 2             33) 27 − ( x + 1) 6
6) 2 x 2 − 2 xy + x − y          20) x 4 − 9                  34) 5 xy + 20 y − 15 x − 60
7)10 z 3 − 15 z 2 − 4 z + 6      21)16 y 4 − 196              35) 4 x 2 + 2 x − y − y 2

8) 6m 3 + 4m 2 − 15m − 10        22)1 − 121n 2                36) x 2 + 6 x + 9 − y 2

9) x 2 + 9 x + 20                23) x 2 − ( y + z ) 2        37) 4 x 2 − 9 y 2 + 4 x + 1

10) b 2 + 12b − 28               24) x 3 − 8                  38) 27 x 3 − 6 x 2 + 2 x − 1

11) 8a 2 − 26a + 15              25) p 3 + 64                 39) 8 x 3 + 12 x 2 + 6 x + 1

12) 6 x 2 − 23x + 20             26) 64u 3 − 27 w3            40) a 2 + a + b − b 2




                                                                                                    




                                            -٩-
                  ‫א‬‫א‬                                 ١٧١                                    ‫א‬
                  ‫א‬‫א‬                                                            



                                                          E‫א‬‫א‬‫א‬F‫א‬‫א‬K٤
         ‫א‬‫א‬‫א‬‫א‬
                                                                            x 2 − 3x + 4     3x + 1
                ‫א‬‫א‬‫א‬‫א‬K                                         K
                                                                           x 2 + 7 x + 12 2 x − 5
          ،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
                                                                                          2x
         ‫א‬ x = 0 , x = 3 ‫א‬‫א‬‫א‬                 2
                                                                                               ‫א‬
                                                                                        x − 3x
                                                   W‫א‬‫א‬K‫א‬
          P R            P PR        P −P    P

           = ⇔ PS = QR ,  =   R≠0 , − =   =     
          Q S            Q QR        Q  Q   −Q
          P R P+R     P R P−R    P R PR   P R PS
          + =      ,  − =     ,  ⋅ =   ,   ÷ =        R ≠ 0 
          Q Q  Q      Q Q   Q    Q S QS   Q S QR
                                                                          ‫א‬‫א‬‫א‬K١{٤
              ‫א‬‫א‬K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
                                           K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
                                                                          x 2 + 2x − 3
                                                                                      W‫א‬W12
                                                                          x 2 + 5x + 6
                                                                                                                   W‫א‬
                                            W‫א‬‫א‬‫א‬
          x 2 + 2 x − 3 = ( x − 1)( x + 3) , x 2 + 5 x + 6 = ( x + 2)( x + 3) 
          x 2 + 2 x − 3 ( x − 1)( x + 3) x − 1
                       =                 =        , x + 3 ≠ 0 ⇒ x ≠ −3 W‫א‬‫א‬
          x 2 + 5 x + 6 ( x + 2)( x + 3) x + 2
                                                       x+6 x 6 x                       x+6
                                                            = + = + 3              ≠ x + 3 
                                                        2      2 2 2                    2
        x 2 − 5x + 6 x − 1           x 2 + 6 x + 9 x 2 + 7 x + 12
     1) 2            ⋅           2) 3             ÷ 3               W‫א‬W13
         x +x−2 x−2                     x + 27        x − 3x 2 + 9 x
                                                                                                                   W‫א‬
            x 2 − 5 x + 6 x − 1 ( x − 2)( x − 3) x − 1 ( x − 2)( x − 3)( x − 1) x − 3
         1) 2           ⋅     =                ⋅     =                        =      , x ≠ 1 x ≠ 2 
             x + x − 2 x − 2 ( x − 1)( x + 2) x − 2 ( x − 1)( x + 2)( x − 2) x + 2




                                                               -١٠-
       ‫א‬‫א‬                                 ١٧١                                  ‫א‬
       ‫א‬‫א‬                                                          



       x 2 + 6 x + 9 x 2 + 7 x + 12     ( x + 3)( x + 3)     ( x + 4)( x + 3)
  2)                ÷ 3             =                      ÷
          x 3 + 27   x − 3x 2 + 9 x ( x + 3)( x 2 − 3x + 9) x( x 2 − 3 x + 9)

                                               ( x + 3)( x + 3)      x ( x 2 − 3 x + 9)
                                       =                           ⋅
                                           ( x + 3)( x 2 − 3 x + 9) ( x + 4)( x + 3)

                                               ( x + 3)( x + 3) x( x 2 − 3 x + 9)   x
                                      =                                           =     , x ≠ 0 x ≠ −3
                                           ( x + 3)( x − 3 x + 9)( x + 4)( x + 3) x + 4
                                                       2


                                                                W‫א‬‫א‬W14
        2mn + m mn + m      x     3         x     2x − 1
 1)           −       2)      +       3) 2   − 2           
         m+n     m+n      x + 1 2x − 1   x − 4 x − 3 x − 10
                                                                                                     W‫א‬
 KE‫א‬F‫א‬‫א‬

         2mn + m mn + m (2mn + m) − (mn + m) 2mn + m − mn − m    mn
   1)           −      =                    =                 =
          m+n     m+n          m+n                m+n           m+n
           x     3      x(2 x − 1) + 3( x + 1) 2 x 2 − x + 3 x + 3 2 x 2 + 2 x + 3
    2)        +       =                       =                   =
         x + 1 2x − 1     ( x + 1)(2 x − 1)     ( x + 1)(2 x − 1)   ( x + 1)(2 x − 1)
            x       2x − 1          x              2x − 1        x( x − 5) − (2 x − 1)( x − 2)
    3)          − 2         =                −                 =
         x 2 − 4 x − 3x − 10 ( x − 2)( x + 2) ( x − 5)( x + 2)      ( x − 2)( x + 2)( x − 5)

                                    ( x 2 − 5 x ) − ( 2 x 2 − 4 x − x + 2) x 2 − 5 x − 2 x 2 + 4 x + x − 2
                                =                                         =
                                           ( x − 2)( x + 2)( x − 5)            ( x − 2)( x + 2)( x − 5)

                                         − x2 − 2                     x2 + 2
                                =                          =−
                                  ( x − 2)( x + 2)( x − 5)    ( x − 2)( x + 2)( x − 5)
                                         
       ‫א‬K‫א‬‫א‬
 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
                            2     1
                                +
                                                            x − y −1
                      1) x − 2 x                         2) −1      W‫א‬W15
                          3x      2                         x −y
                              −
                         x−5 x−5
                                                                                                          

                                                   -١١-
             ‫א‬‫א‬                                       ١٧١                                       ‫א‬
             ‫א‬‫א‬                                                                    



                                                                                                                       W‫א‬
              2     1 2 x + 1( x − 2) 2 x + x − 2    3x − 2
                  +
                         x ( x − 2)    x ( x − 2)   x ( x − 2)    3x − 2 x − 5
         1) x − 2 x =                =            =            =           ⋅
            3x      2     3x − 2        3x − 2       3x − 2      x ( x − 2) 3 x − 2
                −
           x−5 x−5         x−5           x−5          x−5
                                   (3x − 2)( x − 5)   x−5
                               =                    =
                                   x( x − 2)(3x − 2) x( x − 2)
                                                                                                                 
                         1 xy − 1
                 −1   x−
            x− y         y   y      xy − 1   x   ( xy − 1)( x)
          2) −1     =      =      =        ⋅   =               =
            x − y 1 − y 1 − xy        y 1 − xy ( y )(1 − xy )
                      x      x
                            (−1)(1 − xy )( x) (−1)( x)    x
                        =                    =         =−
                              ( y )(1 − xy )    ( y)      y

                                                     x −1 + y −1              2x 2 + y
                                         W                               ‫א‬‫א‬
                                                          z −1                   3x 2
          x −1 + y −1      z            2x2 + y 2 + y
                       =                =       
              z −1        x+ y            3x 2     3
                                                                                                                           
                                                                                W‫א‬‫א‬‫א‬‫א‬W١
             x2 − 4                         x 2 − x − 20                        x3 − 9x                           a3 + 8
    1)                                 2)                               3)                                   4)
         (x − 2)(x + 3)                        3 x − 15                      x3 + x 2 − 6x                        a2 − 8

        x 2 + 3 x − 40                    10 x 2 − 3 x − 1                 2 x 3 − 6 x 2 + 5 x − 15             x3 − x 2 + x
    5)                                  6) 2                            7)                                   8)
       − x 2 + 3 x + 10                   2 x + 5x − 3                             9 − x2                          x3 + 1
                                                                   W‫א‬‫א‬W٢
       x 2 + x 3x 2 + 19 x + 28                                          x 2 − 16    x 2 − 4 x − 21
    1)        ⋅                                                    2) 2            ⋅
       2x + 3 x 2 + 5x + 4                                            x + 7 x + 12      x2 − 4x
      12m 2 + 28m + 15 2m 2 − m − 3                                   6u 2 − 5u + 1 2u 2 + 3u − 2
 3)                   ⋅                                          4) 2               ÷              
       6m 2 + 35m + 25 3m 2 + 11m − 20                               3u + 11u − 4 u 2 + 3u − 4
      z 2 − 81 z 2 − z − 20                                          2a 2 − 5a + 3 3a 2 − 8a − 3
    5) 2       ÷                                                  6) 2             ÷ 2
      z − 16 z 2 + 5 z − 36                                           a +a−2           a −a−6


                                                                -١٢-
                ‫א‬‫א‬                                   ١٧١                                                 ‫א‬
               ‫א‬‫א‬                                                                            



                                                                                              W‫א‬‫ א‬W٣

                    9 x + 1 3x + 4                            x + 1 2x − 1                                  x      3x − 1
               1)          −                            2)         +                                 3)        − 2
                    2x − 1 2x − 1                            2x + 3 2x − 3                                 2
                                                                                                          x − 9 x + 7 x + 12
          
                 1   2      3x 2 + 11x − 4                   x +1   2x   x+5                            ⎛       2 ⎞⎛    1⎞
               4) +       ⋅                             5)        −    ÷                             6) ⎜ 1 +     ⎟⎜ 3 − ⎟
                 x 3x − 1        x−5                         x−3 x−3 x−3                                ⎝       x ⎠⎝    x⎠
                                                                            W‫א‬‫א‬W٤
            2      3       4                                                  1           1     1
    1)          −      +                                             2)               + 2   + 2
         2 x + 1 3x + 1 4 x + 1                                               2
                                                                          x + 7 x + 12 x − 9 x − 16
                                                                                                                          
          2          3      5                                                      1           2x  2x − 1
    3) 2         + 2   − 2                                            4)                    +     − 2
      x − 3 x + 2 x − 1 x + 3 x − 10                                          (x − 2)2        x+2 x −4
                                                                                              W‫א‬‫א‬W٥
         1     2                          x+2      1                    z 2 + 3 z − 10                       2 y 2 + 11y + 15
            −                                   +
                                                                                                               y 2 − 4 y − 21
                                      2) x − 1 x + 1
                                            2                               2
   1) x + 3 x − 1                                                   3) z2 + z − 6                         4) 2                
         x    3                               x      1                   z − z − 30                          6 y + 11 y − 10
            +                                     +
       x −1 x + 3                       2x − x − 1 x − 1
                                          2
                                                                       2 z 2 − 15 z + 18                     3 y 2 − 23 y + 14
                                                                                              W‫א‬‫א‬W٦
                                           1     1                                                               1                  1
                                               −                                                               1+                   4+
         1)
            (x + h )   −2
                            −x   −2
                                      2) x + h x      3) −1
                                                           a    −1
                                                                                  4)
                                                                                       −1
                                                                                     a b − ab   −1
                                                                                                          5)
                                                                                                                p−2
                                                                                                                              6)    x
                      h                     h          a + a −2                      a2 + b2                    1                  1
                                                                                                             1−                  1−
                                                                                                                p+3                 x


                                                                                                                                      




                                                               -١٣-
        ‫א‬‫א‬                                     ١٧١                                 ‫א‬
        ‫א‬‫א‬                                                             



                                        ‫א‬W‫א‬‫א‬
                                                    
                                                                         ‫א‬K١
‫א‬K‫א‬K‫א‬‫א‬
 x = 3  2 x + 1 = 7 ‫א‬K
     3  x  x = 3 ‫א‬K x 
                                                                      
                                                                         K‫א‬ 2(3) + 1 = 7 ‫א‬
    ‫א‬،‫א‬‫א‬‫א‬‫א‬
                       K x 2 − 5 x + 6 = 0  x = 2 , x = 3 K‫א‬
                                                                               
    x ‫א‬‫א‬‫א‬‫א‬‫א‬
        K x = ‫ﺛﺎﺑﺖ‬        W‫א‬‫א‬‫א‬‫א‬
                                                  W‫א‬‫א‬‫א‬‫א‬
     ،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫• א‬
                      K 7 x + 3 = −11  2 x + 3 + 5 x = −11 K‫א‬‫א‬
             K 3 x = 9  3 x − 7 = 2 K‫א‬‫א‬ •
                K‫א‬‫א‬‫א‬‫א‬ •
                                                                                                     5
                                                                    x = 12  x = 10 
                                                                                                     6
                                                                        ‫א‬‫א‬K٢
                                 W x ‫א‬
       a ≠ 0 ‫א‬ b  a  ax + b = 0 
                          3
    1) 2x + 5 = 9      2) x − 6 = 0        3) ( x + 2)(5x + 1) = 5x( x + 1)     W‫א‬‫א‬W1
                          4
                                                                                                            W‫א‬
                     K٣‫א‬‫א‬٥‫א‬E١
   2x + 5 − 5 = 9 − 5 , 2x = 4 , x = 4                                               
                               3                           4
                           K ‫א‬ ‫א‬‘٦E٢
                               4                           3


                                                        -١٤-
             ‫א‬‫א‬                                 ١٧١                                  ‫א‬
            ‫א‬‫א‬                                                           


    3            3                   3           ⎛ 4 ⎞⎛ 3 ⎞ ⎛ 4 ⎞
     x − 6 = 0 , x − 6 + 6 = 0 + 6 , x = 6 , ⎜ ⎟⎜ x ⎟ = ⎜ ⎟(6) , x = 8 
    4            4                   4           ⎝ 3 ⎠⎝ 4 ⎠ ⎝ 3 ⎠
                                                                                  
 K٦‫א‬‫א‬ 5 x , 5 x , 2 ‫א‬‫א‬‫א‬‫א‬E٣
                                         2


        ( x + 2)(5 x + 1) = 5 x( x + 1) , 5 x 2 + 11x + 2 = 5 x 2 + 5 x , 11x + 2 = 5 x
                                                        2   1        1                   
                           6 x + 2 = 0 , 6 x = −2 , x = − = − , x = −
                                                         6   3        3
                                               K‫א‬‫א‬‫א‬‫א‬W
           ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
        ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬K‫א‬
                      K‫א‬‫א‬‫א‬‫א‬
               x    24 − 5 x                      x   5
     1)        =                    2) 1 +       =    W‫א‬‫א‬W٢
              x−3    x−3                         x−5 x−5
                                                                                                              W‫א‬
         ‫א‬K‫א‬‫א‬ x ≠ 3 E١
                                      K‫א‬‫א‬‫א‬‫א‬ ( x − 3)

                                            x             24 − 5 x
                                ( x − 3)       = ( x − 3)          , x = 24 − 5 x , x + 5 x = 24 − 5 x + 5 x
                                           x−3             x−3
                           
                                                                                6 x 24
                                                                   6 x = 24 ,      =   , x=4
                                                                                 6   6
                                      K‫א‬‫א‬‫א‬ 3 ‫א‬‫א‬
    ‫א‬K‫א‬‫א‬ x ≠ 5 E٢
                                             K‫א‬‫א‬‫א‬ ( x − 5) ‫א‬

                              ⎛      x ⎞            ⎛ 5 ⎞                          ⎛ x ⎞              ⎛ 5 ⎞
                      ( x − 5)⎜1 +      ⎟ = ( x − 5)⎜      ⎟ , ( x − 5)1 + ( x − 5)⎜      ⎟ = ( x − 5)⎜      ⎟
                             ⎝    x − 5⎠           ⎝ x − 5⎠                       ⎝ x − 5⎠           ⎝ x − 5⎠
                                     x − 5 + x = 5 , 2 x − 5 = 5 , 2 x − 5 + 5 = 5 + 5 , 2 x = 10 , x = 5
     x = 5 ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
                                                             K‫א‬‫א‬‫א‬



                                                         -١٥-
      ‫א‬‫א‬                                 ١٧١                                ‫א‬
      ‫א‬‫א‬                                                       



                                               
                                                                 W‫א‬‫א‬‫א‬
1) 2 x + 10 = 40                             3     5                    3x      12
                                     9)         =                14)       =2−
                                           x + 2 2x − 7                x+4     x+4
2) − 3 y + 20 = 2
                                            x   x+4                     5   3   4
3) 4 x − 11 = 7 x + 20               10)      =                  15)      −   =
                                           x−3 x+2                     x−3 x−2 x−3
4) 4(2 x − 17) + 5(3 x − 8) = 0            x+3 x−3
                                     11)      =                  17) 5[x − (4 x − 5)] = 3 − 2 x
  3   1 2                                  x+5 x−4
5) x + =                                                         18) 6[3 y − 2( y − 1)] − 2 + 7 y = 0
  4   2 3                                     9   3m
                                     12) 2 +    =
  1       1    19                            m−3 m−3             19) 4[2 + ( y + 1) 2 ] = ( y + 2) 2
6) x + 7 − x =
  2       4     2                          4x − 3 2x − 4
                                     13)         =               19) ( y + 3) = ( y + 4) 2 + 1
                                                                              2
7) 5( x + 3)( x − 3) = 5 x( x − 1)          2x     x−2

   40 − 3 x 6 x + 7                        12 + x 5 x − 7         21) ( z − 7) 2 = ( z − 2) 2 + 9
8)         =                         14)         =        +2
     5x        8                            −4       3


                                                                                                       




                                                  -١٦-
          ‫א‬‫א‬                                   ١٧١                                   ‫א‬
          ‫א‬‫א‬                                                             



                                                                  ‫א‬‫א‬‫א‬K٣
    K ax 2 + bx + c = 0 , a ≠ 0 ‫א‬‫א‬‫א‬‫א‬
 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
                                                                        ‫א‬K١{٣
    ‫א‬‫א‬‫א‬ ax 2 + bx + c ‫א‬‫א‬‫א‬
                          AB = 0 ⇔ A = 0 ‫ أو‬B = 0 W‫א‬
              1) x 2 + 10 x + 25 = 0           2) 2 x 2 + x − 6 = 0 W‫א‬‫א‬W٣
                                                                                                            W‫א‬
  x 2 + 8 x + 15 = ( x + 3)( x + 5) ‫א‬‫א‬‫א‬‫א‬‫א‬E١
                                        W‫א‬‫א‬‫א‬
    x 2 + 8 x + 15 = ( x + 3)( x + 5) = 0 , x + 3 = 0 ⇒ x = −3 ‫ أو‬x + 5 = 0 ⇒ x = −5 
    ‫א‬‫א‬‫א‬K x = −5  x = −3  x 2 + 8 x + 15 = 0 ‫א‬
                                                            K‫א‬‫א‬‫א‬‫א‬
                             W‫א‬ a ≠ 1  m , n , p , q ‫א‬E٢
                                                                                    3
      2 x 2 + x − 6 = (2 x − 3)( x + 2) = 0 , 2 x − 3 = 0 ⇒ 2 x = 3 ⇒ x = , x + 2 = 0 ⇒ x = −2
                                                                                    2
                                                                     3
                                                           x =  x = −2  2 x 2 + x − 6 = 0 ‫א‬
                                                                     2
                                                                              ‫א‬‫א‬K٢{٣
                            K A = ± B ‫א‬ B > 0  A 2 = B W B  A ‫א‬
                                    1) x 2 − 5 = 0       2) ( x + 1) 2 = 49 W‫א‬‫א‬W٤
                                                                                                            W‫א‬
                         W‫א‬‫א‬‫א‬٥E١
                    x 2 − 5 + 5 = 0 + 5 , x 2 = 5 ⇒ x = ± 5 
                                                             x = − 5  x = 5 W‫א‬‫א‬
                                                             W‫א‬E٢
         ( x + 1) 2 = 49 , x + 1 = ± 49 , x + 1 = ±7 , x = ±7 − 1 ⇒ x = 7 − 1 = 6 ‫ أو‬x = −7 − 1 = −8
                                                                            K x = −8  x = 6 W‫א‬‫א‬

                                                        -١٧-
          ‫א‬‫א‬                                      ١٧١                                  ‫א‬
          ‫א‬‫א‬                                                              



                                                                             ‫א‬K٣{٣
         a ≠ 1 ‫א‬ a ‫א‬‫א‬‫א‬‫א‬‫א‬
                                                                                                      2
  ‫א‬‫א‬‫א‬‫א‬K‫א‬ ⎛ ⎞ ‫א‬
                                                                           b
                                                                          ⎜ ⎟
                                                                          ⎝2⎠
                                                                   2
                                                          ⎛     b⎞
  K‫א‬‫א‬‫א‬‫א‬‫א‬K ⎜ x ± ⎟ 
                                                          ⎝     2⎠
                 1) x 2 − 2 x + 6 = 0                2) 2 x 2 + 8 x − 15 = 0 W‫א‬‫א‬W٥
                                                                                                               ‫א‬
                        2
                −2
        ⎛ ⎞ = 1‫א‬ a = 1 ‫א‬‫א‬‫א‬‫א‬
              ⎜    ⎟
              ⎝ 2 ⎠
                                                                                                W‫א‬
    x 2 − 2 x − 8 = 0 , x 2 − 2 x = 8 , x 2 − 2 x + 1 = 8 + 1 , ( x − 1) 2 = 9 
                     W‫א‬‫א‬‫א‬‫א‬‫א‬
‫א‬٢‫א‬‫א‬ a ≠ 1  a = 2 ‫א‬‫א‬‫א‬E٢
                             W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬K‫א‬
                                                     1                  1                    15
    2 x 2 + 8 x − 15 = 0 , 2 x 2 + 8 x = 15 , (2 x 2 + 8 x) = (15) , x 2 + 4 x =               
                                                     2                  2                     2
                                                 2            2
         W‫א‬ ⎛ ⎞ = ⎛ ⎞ = 4 ‫א‬‫א‬‫א‬‫א‬
                                         b       4
                                       ⎜ ⎟ ⎜ ⎟
                                       ⎝2⎠ ⎝2⎠
                        15                  23         23       23         ± 46 − 4
         x 2 + 4 x + 4 = + 4 , ( x + 2) 2 =    , x+2=±    , x=±    −2, x=
                         2                  2          2        2              2
                                                                                                                   
                                                     46 − 4        − 46 − 4
                                            x=              ‫ أو‬x =
                                                      2               2




                                                           -١٨-
        ‫א‬‫א‬                                     ١٧١                                     ‫א‬
        ‫א‬‫א‬                                                                



                                                                                ‫א‬K٤{٣
          ‫א‬K‫א‬‫א‬‫א‬
                                                             − b ± b 2 − 4ac
    ‫א‬ b 2 − 4ac ‫א‬K x =                              ax 2 + bx + c = 0 , a ≠ 0
                                                                   2a
                                                                              W
               − b ± b 2 − 4ac
          x =                 W b 2 − 4ac ‫א‬‫א‬ •
                     2a
                    −b
               x =    W‫א‬ b 2 − 4ac ‫א‬‫א‬ •
                    2a
                                     K b 2 − 4ac ‫א‬‫א‬ •
 1) 2 x 2 − 5 x + 2 = 0       2) x 2 + 6 x + 9 = 0       3) 3 x 2 + 6 x + 7 = 0 W‫א‬‫א‬W٦
                                                                                                             W‫א‬
                                                                 W‫א‬ a = 2 b = −5 c = 2 ‫א‬E١
  b 2 − 4ac = (−5) 2 − 4(2)(2) = 25 − 16 = 9 > 0 
                                                           W‫א‬
         − b ± b 2 − 4ac − (−5) ± 9 5 ± 3      5+3 8           5−3 2 1
 x=                    =          =      , x=    = = 2 ‫ أو‬x =    = = 
               2a            2( 2)    4         4  4            4  4 2
                                                                   W‫א‬ a = 1 b = 6 c = 9 ‫א‬E٢
  b 2 − 4ac = (6) 2 − 4(1)(9) = 36 − 36 = 0 
                                             − b − ( 6)
                                      x =      =       = −3 W‫א‬
                                             2a 2(1)
  b 2 − 4ac = (6) 2 − 4(3)(7) = 36 − 84 = −48 < 0 W‫א‬ a = 3 b = 6 c = 7 ‫א‬E٣
                                                    .‫א‬‫א‬‫א‬‫א‬‫א‬




                                                        -١٩-
         ‫א‬‫א‬                                  ١٧١                                        ‫א‬
         ‫א‬‫א‬                                                                 



                                                                                                                
                                                               ‫א‬‫א‬‫א‬W١
      1) x 2 − 2 x − 15 = 0          2) 8 y 2 + 189 y − 72 = 0                 3) 3 x 2 − 7 x = 0
                                                                                                                           
                       2                         2                                            2                    2
      4) 8 + 14t − 15t = 0           5) ( x − 5) − 9 = 0                       6) (2 x − 5) − (4 x − 11) = 0
                                                       ‫א‬‫א‬‫א‬‫א‬W٢
      1) x 2 = 81                    2) 2 x 2 − 48 = 0               3) ( x − 5) 2 = 36
                                                                                                 
                 2             2         2             2                   2                  2
      4) ( x − 8) = ( x + 1)         5) x = ( x + 1)                 6) 4 x = (2 x + 3)
                                                        ‫א‬‫א‬‫א‬W٣
       1) x 2 + 8 x − 10 = 0          2) x 2 − 6 x = 0               3) x 2 + 7 x − 2 = 0
                                                                                                    
             2                               2                                            2
       4) 2 x + 10 x − 3 = 0          5) 4 x − 4 x + 15 = 0          6) 2 + 10 x − 5 x = 0
                                                                 ‫א‬‫א‬‫א‬W٤
       1) x 2 − 2 x − 15 = 0          2) x 2 + x − 1 = 0             3) 2 x 2 + 4 x + 1 = 0

                                        1     3                        2           1
     4) 3 x 2 − 5 x + 3 = 0          5) x 2 + x − 1 = 0             6) x 2 − 5 x + = 0 
                                        2     4                        3           2
       7) − x 2 = 7 x − 1             8) 2 x 2 + 3 x + 2 = 0         9) 2 x 2 + 5 x − 3 = 0
                                ‫א‬‫א‬ k W٥
 1)16 x 2 + kx + 9 = 0 2) x 2 + kx + 81 = 0 3) y 2 − 3 y + k = 0 4) x 2 + 15 x + k = 0 




                                                      -٢٠-
                                                ‫א‬‫א‬‫א‬
                                             ã¹]<gè…‚jÖ]æ<ËÖ]<Üé×Ãj×Ö<íÚ^ÃÖ]<퉉ö¹]
                                                  ‫א‬‫א‬‫א‬‫א‬




                            
                          ‫א‬‫א‬
 ‫א‬‫א‬




٢
   ‫א‬‫א‬                          ١٧١                                ‫א‬
‫א‬‫א‬                                            



   K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W‫א‬‫א‬
                                                                                                   
                              :‫א‬‫א‬‫א‬W‫א‬‫א‬
                                                    ‫א‬‫א‬‫א‬           •

                                                                   ‫א‬‫א‬‫א‬       •

                                                            ‫א‬‫א‬‫א‬‫א‬          •

                                                                  K‫א‬‫א‬‫א‬‫א‬      •

                                                               ‫א‬‫א‬‫א‬‫א‬      •

                                                                 K‫א‬‫א‬         •

                                                                                                   
                                                                                                   
                                                           KW‫א‬‫א‬




                                            -١٩-
      ‫א‬‫א‬                                  ١٧١                                           ‫א‬
   ‫א‬‫א‬                                                              



                                          ‫א‬‫א‬
                                                                                                                      
                               K‫א‬‫א‬‫א‬‫א‬
‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W
                                                                                 K‫א‬‫א‬
                                                                                                    ‫א‬‫א‬K١
 X ‫א‬‫א‬ Y  X  f W١
 X  x2  x1 . Y ‫א‬‫א‬
                                                                                          W Y 
                                       x1 = x2 ⇒ f ( x1 ) = f ( x2 )
                                                 K x ‫א‬‫א‬ f (x) 
‫א‬ x  f (x) ‫א‬‫א‬ Y ‫א‬‫א‬ X ‫א‬
 x ‫א‬ Y  f (x)  f ‫א‬‫א‬‫א‬ y = f (x)  x ‫א‬ f ‫א‬‫א‬
                                        Y  f (x)  Y 
                                                                       f : X →Y            W‫א‬‫א‬
،‫א‬‫א‬‫א‬‫א‬
                                                           K‫א‬
                               f                             
                                                           W‫א‬‫א‬W١
                0                                2          f ‫א‬ Y = {2,4,6,8} ‫ و‬X = {0,1,2,3}
               1                                                                                    W Y  X
                                                 4
                                                                   f (1) = 2, f (2) = 4, f (3) = 6
               2                                            X ‫א‬ f ‫א‬
                                                 6
                                                                          K Y ‫א‬‫א‬
               3                                 8       W‫א‬‫א‬‫א‬‫א‬
                                                                       W‫א‬‫א‬
                                                                       f ( x) = 2 x W f : X → Y
                   X                         Y
                                                                                                                       

                                                       -٢٠-
      ‫א‬‫א‬                                    ١٧١                                         ‫א‬
   ‫א‬‫א‬                                                               



                                                              
 f ‫א‬ Y = {0,1,4,9,−2}  X = {0,1,−1,2,−2,5} W‫א‬‫א‬W٢
                                                                                                  W Y  X
                              f (0) = 0, f (1) = f (−1) = 1, f (2) = f (−2) = 4

                               f
                                                              X ‫א‬ f ‫א‬
               0                             0                             W Y ‫א‬‫א‬
                                                              − 2  2  − 1 1  0 ‫א‬
              1                              1                5 ‫א‬ Y ‫א‬
              -1                                              f (5)  Y 
                                                   4                                                           K Y 
              2                                                W‫א‬‫א‬‫א‬‫א‬
                                               9
                                                                      W‫א‬‫א‬
              -2
                                                                                       f : X →Y
                                            -2
              5                                                                     f ( x) = x 2 W
                                                             ‫א‬ f ‫א‬
                                                                      W١‫א‬‫א‬‫א‬‫א‬
                X                          Y
                                                              x1 = x2 ⇒ ( x1 )2 = (x2 )2 ⇒ f ( x1 ) = f ( x2 )
                                                            
 Countries ،‫א‬ Cities W‫א‬‫א‬W٣
            K f (x)  x W Countries  Cities  f ‫א‬‫א‬‫א‬
‫א‬‫א‬ Cities  x ‫א‬ f ‫א‬
                                                                                           K Countries ‫א‬
                                                                             
 x ‫א‬،‫א‬‫א‬‫א‬‫א‬ x ‫א‬
                                                                    WK Countries 
                                       f ( Riyadh) = Saudi Arabia
                                       f ( A lg iers) = A lg eria
                                   
                                       f (Cairo) = Egypt
                                       f ( London) = United Kingdom
                            K Abha  Countries  f ( Abha ) 


                                                         -٢١-
          ‫א‬‫א‬                                      ١٧١                                         ‫א‬
       ‫א‬‫א‬                                                                 



                                                                  
     Cities  g ‫א‬ Countries  Cities W‫א‬‫א‬W٤
                                                     K x ‫א‬‫א‬‫א‬ g (x) W Countries
     
     ‫א‬‫א‬ Cities  x ‫א‬ g ‫א‬
                                                                                         WK Countries ‫א‬
                                              g ( Riyadh) = Saudi Arabia
                                              g ( A lg iers ) = A lg eria
                                           g (Cairo) = Egypt
                                              g ( London) = United Kingdom
                                              g ( Abha ) = Saudi Arabia
     Cities  Countries  f ‫א‬ Countries ‫א‬‫א‬W٥
                                                                                 K x ‫א‬ f (x) W
                 K‫א‬K Saudi Arabia ‫א‬W‫א‬‫א‬
                                                     
    ‫א‬‫א‬‫א‬‫א‬EF f ‫א‬‫א‬W٢
    ‫א‬‫א‬‫א‬‫א‬ f ‫א‬‫א‬،‫א‬‫א‬‫א‬
                                                                                                    K‫א‬‫א‬‫א‬
                                                              K R f ‫א‬ D f  f ‫א‬‫א‬
                                                      K‫א‬٤١‫א‬‫א‬‫א‬‫א‬W٦
                                                                                                                      W‫א‬
      1) D f = {1,2,3}          R f = {2,4,6}

      2) D f = {0,1,−1,2,−2}              R f = {0,1,4}
                                                                   W Capitals ‫א‬‫א‬E٣
                                           D f = Capitals             R f = Countries
                                         4) D g = Cities             R g = Countries




                                                               -٢٢-
     ‫א‬‫א‬                                           ١٧١                                             ‫א‬
  ‫א‬‫א‬                                                                          



  K f ( x) = 2 x W Ν  Ν  f ‫א‬ Ν ‫א‬‫א‬‫א‬W٧
        K f (14)  f (5) ‫א‬E٣                    K‫א‬ f E٢                        K‫א‬ f E١
                                                                                                                           W‫א‬
 x1 = x2 ⇒ 2 x1 = 2 x2 ⇒ f ( x1 ) = f ( x2 )                                                                    W‫א‬ f E١
                     D f = Ν        
                                                                         W‫א‬ f ‫א‬E٢
                 R f = {2,4,6,8,10,L}                  W Ν ‫א‬‫א‬‫א‬‫א‬
           . f (14) = 2 × 14 = 28 ‫ و‬f (5) = 2 × 5 = 10                            E٣
  K f ( x) = 2 x W R  R  g ‫א‬ R ‫א‬‫א‬‫א‬W٨
         . g (5)  g (2.5) ‫א‬E٣                      K‫א‬ g E٢                     K‫א‬ g E١
                                                                                                                           W‫א‬
         x1 = x 2 ⇒ 2 x1 = 2 x 2 ⇒ g ( x1 ) = g ( x 2 ) W‫א‬ g E١
        D g = R            
                                                                         W‫א‬ g ‫א‬E٢
 y = 2 × = 2⎛ ⎞ = g ⎛ ⎞ 
         y    y       y
             ⎜ ⎟     ⎜ ⎟                                        
                                               W R g = R W‫א‬ R ‫א‬‫א‬‫א‬
            2          ⎝2⎠        ⎝2⎠
                                                                                      0.6 = g (0.3) ‫ = 3 و‬g (1.5) WK
  . g (5) = 2 × 5 = 10 ‫ و‬g (2.5) = 2 × 2.5 = 5                         E٣
                                                     W‫א‬ g  f ‫א‬W٣
                      1) D f = Dg
                
                      2) x ∈ D f ⇒ f ( x ) = g ( x )
                                                                                             K f = g W
                                ‫؟‬‫א‬٤٣‫א‬‫א‬ g  f ‫א‬‫א‬W٩
                                                                                                                           W‫א‬
                           W f ≠ g W٦‫א‬٤٣‫א‬‫א‬
                       D f = Capitals ≠ Dg = Cities
                               ‫؟‬‫א‬٨٧‫א‬‫א‬ g  f ‫א‬‫א‬W١٠
                                                                                                                           W‫א‬
                                          W f ≠ g W٨٧‫א‬‫א‬
                        D f = Ν ≠ Dg = R
                                                               -٢٣-
      ‫א‬‫א‬                                     ١٧١                                     ‫א‬
   ‫א‬‫א‬                                                            



                                                                              W‫א‬‫א‬‫א‬W١١
                    g:Ν → Ν
                        g ( x) = x 2 W
                                                               ‫؟‬ f = g K٧‫א‬‫א‬ f ‫א‬‫א‬
                                                                                                             W‫א‬
                                D f = Ν = Dg       WE٣F‫א‬‫א‬
              f (3) = 2 × 3 = 6 ≠ g (3) = 3 2 = 9  3 ∈ D f = Ν W‫א‬‫א‬
                                                                                                K f ≠ g W
                                                                                                    ‫א‬‫א‬‫א‬K٢
‫א‬‫א‬‫א‬ f : X → Y ‫א‬W٤
                                                         
                                                         K D f = X W‫א‬،
                     ‫؟‬‫א‬‫א‬‫א‬،١١٨٧٤١‫א‬W١٢
                                                                                                             W‫א‬
                                     K f ( x) = 2 x  f : {0,1,2,3} → {2,4,6,8} ‫א‬‫א‬W١‫א‬
                          
     K‫א‬‫א‬‫א‬E‫א‬ f (0) F D f = {1,2,3} ≠ {0,1,2,3}
                            K f ( x) = x 2  f : {0,1,−1,2,−2,5} → {0,1,4,9,−2} ‫א‬‫א‬W٢‫א‬
               
‫א‬‫א‬‫א‬E‫א‬ f (5) F D f = {0,1,−1,2,−2} ≠ {0,1,−1,2,−2,5}
                                                                                                           K
    K x ‫א‬‫א‬‫א‬ f (x)  f : Cities → Countries ‫א‬‫א‬W٣‫א‬
                  
‫א‬‫א‬‫א‬E‫א‬ f ( Abha ) F D f = Capitals ≠ Cities
                                                                                                           K
   K x ‫א‬‫א‬‫א‬ g ( x)  g : Cities → Countries ‫א‬‫א‬W٤‫א‬
                                                                                                 
                                                                                 K‫א‬‫א‬‫א‬ Dg = Cities
                                                    K f ( x) = 2 x  f : Ν → Ν ‫א‬‫א‬W٧‫א‬
                                                                                                      
                                                                                      K‫א‬‫א‬‫א‬ D f = Ν
                                                     K g ( x) = 2 x  g : R → R ‫א‬‫א‬W٨‫א‬
                                                                                                      
                                                                                      K‫א‬‫א‬‫א‬ Dg = R

                                                          -٢٤-
      ‫א‬‫א‬                                    ١٧١                                    ‫א‬
   ‫א‬‫א‬                                                          



                                                    K g ( x) = x 2  g:Ν → Ν ‫א‬‫א‬W١١‫א‬
                                                                                                   
                                                                                   K‫א‬‫א‬‫א‬ D g = N
                                             K f : D f → Y ‫א‬ f : X → Y ‫א‬W١
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
                        KE‫א‬‫א‬‫א‬‫א‬F
                                                                                           
                                          K١٢‫א‬‫א‬‫א‬‫א‬ W١٣
                                                                                                           W‫א‬
                                  KK f ( x) = 2 x  f : {1,2,3} → {2,4,6,8} ‫א‬‫א‬W١‫א‬
                        K f ( x) = x 2  f : {0,1,−1,2,−2} → {0,1,4,9,−2} ‫א‬‫א‬W٢‫א‬
                                 
K x ‫א‬‫א‬‫א‬ f (x)  f : Capitals → Countries ‫א‬‫א‬W٣‫א‬
                                                                                                         KK
                                 
‫א‬EF f : X → Y ‫א‬W٥
 x2  x1 ،‫א‬‫א‬‫א‬
                                                             K D f = X  f ( x1 ) = f ( x 2 ) ⇒ x1 = x 2 W
                                         ‫؟‬K١٢١١‫א‬W١٣
                                                                                                           W‫א‬
                                      K f ( x) = 2 x  f : {1,2,3} → {2,4,6,8} ‫א‬W١‫א‬
                                   KK f ( x1 ) = f ( x2 ) ⇒ 2 x1 = 2 x2 ⇒ x1 = x2      W‫א‬‫א‬
                             K f ( x) = x 2  f : {0,1,−1,2,−2} → {0,1,4,9,−2} ‫א‬W٢‫א‬
                                      KK f (1) = 12 = 1 = (−1) 2 = f (−1)  W‫א‬‫א‬K
 KKK x ‫א‬‫א‬‫א‬ f (x)  f : Capitals → Countries ‫א‬W٣‫א‬
                KK f ( x1 ) = f ( x 2 ) ⇒ x1 = x 2  W‫א‬‫א‬‫א‬
   K x ‫א‬‫א‬‫א‬ g (x)  g : Cities → Countries ‫א‬W٤‫א‬
                            K g ( Riyadh) = Saudi Arabia = g ( Abha)  W‫א‬‫א‬K
                                                 K f ( x) = 2 x  f : Ν → Ν ‫א‬W٧‫א‬
                                   K f ( x1 ) = f ( x2 ) ⇒ 2 x1 = 2 x2 ⇒ x1 = x2      W‫א‬‫א‬K
                                                   K g ( x) = 2 x  g : R → R ‫א‬W٨‫א‬
                                                         -٢٥-
       ‫א‬‫א‬                                   ١٧١                                      ‫א‬
    ‫א‬‫א‬                                                           



                                  K g ( x1 ) = g ( x 2 ) ⇒ 2 x1 = 2 x 2 ⇒ x1 = x 2      W‫א‬‫א‬K
                                                   K g ( x) = x 2  g:Ν → Ν ‫א‬W١١‫א‬
                                                                                          W‫א‬‫א‬K
            K g ( x1 ) = g ( x 2 ) ⇒ ( x1 ) 2 = ( x 2 ) 2 ⇒ x1 2 = x 2 2 ⇒ x1 = x 2 ⇒ x1 = x 2
 ‫א‬‫א‬‫א‬ x2  x1 ‫א‬‫א‬‫א‬
                         
                                                                                                        K
                                  
 ‫א‬EF f : X → Y ‫א‬W٦
 R f = Y W،‫א‬‫א‬‫א‬‫א‬‫א‬
             
                                                                                                    K D f = X 
                                         ‫؟‬‫א‬K١٢١١‫א‬W١٤
                                                                                                             W‫א‬
                                      K f ( x) = 2 x  f : {1,2,3} → {2,4,6,8} ‫א‬W١‫א‬
                                            KK R f = {2,4,6} ≠ {2,4,6,8}                 
                                                                                      W‫א‬‫א‬‫א‬
                                     KKEF‫א‬‫א‬٨‫א‬W
                             K f ( x) = x 2  f : {0,1,−1,2,−2} → {0,1,4,9,−2} ‫א‬W٢‫א‬
                                                                                      
                                                KK R f = {0,1,4} ≠ {0,1,4,9,−2} W‫א‬‫א‬‫א‬
                                     
                                  EF‫א‬‫א‬٩‫א‬W
 KKK x ‫א‬‫א‬‫א‬ f (x)  f : Capitals → Countries ‫א‬W٣‫א‬
                                             KK R f = Countries  W‫א‬‫א‬
   K x ‫א‬‫א‬‫א‬ g (x)  g : Cities → Countries ‫א‬W٤‫א‬
                      KK R g = Countries          W‫א‬‫א‬‫א‬‫א‬K
                                                  K f ( x) = 2 x  f : Ν → Ν ‫א‬W٧‫א‬
                                                  KK R f = {2,4,6,8,10, L} ≠ N  W‫א‬‫א‬‫א‬
                                     KK‫א‬‫א‬١‫א‬W
                                                   K g ( x) = 2 x  g : R → R ‫א‬W٨‫א‬
                                                                       K R f = R          W‫א‬‫א‬K


                                                          -٢٦-
      ‫א‬‫א‬                                    ١٧١                                   ‫א‬
   ‫א‬‫א‬                                                         



                                                   K g ( x) = x 2  g:Ν → Ν ‫א‬W١١‫א‬
                                                 K R f = {1,4,9,16,25,L} ≠ N W‫א‬‫א‬‫א‬K
                                                                                   
                                        
                                    KK‫א‬‫א‬٢‫א‬
                                             K f : D f → R f ‫א‬ f : X → Y ‫א‬W٢
‫א‬‫א‬‫א‬‫א‬‫א‬
                                            K‫א‬‫א‬‫א‬‫א‬
                                            K‫א‬١٤‫א‬‫א‬‫א‬‫א‬‫א‬W١٥
                                                                                                          W‫א‬
                                      K f ( x) = 2 x  f : {1,2,3} → {2,4,6} ‫א‬W١‫א‬
                                K f ( x) = x 2  f : {0,1,−1,2,−2} → {0,1,4} ‫א‬W٢‫א‬
                                KK f ( x) = 2 x  f : Ν → {2,4,6,8,10, L} ‫א‬W٧‫א‬
                                  K g ( x) = x 2  g:Ν → {1,4,9,16,25, L} ‫א‬W١١‫א‬
                                
‫א‬EF f : X → Y ‫א‬W٧K
‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬
                                                                                            K‫א‬
       KK‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬،١٥١٤١٣‫א‬‫א‬W١٦
                                                                                                          W‫א‬
                                                                  W‫א‬‫א‬‫א‬‫א‬‫א‬
                                                                  K f ( x) = 2 x  f : {1,2,3} → {2,4,6} E١
                              KKK x ‫א‬‫א‬‫א‬ f (x)  f : Capitals → Countries E٣
                                                           K f ( x) = 2 x  f : Ν → {2,4,6,8,10, L} E٧
                                                                           K g ( x) = 2 x  g : R → R E٨
                                                          K g ( x) = x 2  g:Ν → {1,4,9,16,25, L} E١١
                                                                      K‫א‬‫א‬
                            ‫؟‬‫א‬‫א‬K f ( x) = x 2 W f : R → R ‫א‬‫א‬W١٧
                                                                                                          W‫א‬
                     K f (1) = 12 = 1 = (−1) 2 = f (−1)  W‫א‬‫א‬
                                                         -٢٧-
      ‫א‬‫א‬                                 ١٧١                                     ‫א‬
   ‫א‬‫א‬                                                        



                                                                  W Y  X  f W
 K x 2  x1  x1 = x 2 ⇒ f ( x1 ) = f ( x 2 )            W‫א‬‫א‬E١
 K x 2  x1  x1 = x 2 ⇒ f ( x1 ) = f ( x 2 )  W‫א‬E٢
                                                                  D f = X                           
 K x 2  x1  x1 = x 2 ⇒ f ( x1 ) = f ( x 2 )  W‫א‬E٣
                                                                  D f = X                           
 K x 2  x1  f ( x1 ) = f ( x2 ) ⇒ x1 = x2                             
 K x 2  x1  x1 = x 2 ⇒ f ( x1 ) = f ( x 2 )  W‫א‬‫א‬E٤
                                                                  D f = X                           
                                                                   R f = Y                          
 K x 2  x1  x1 = x 2 ⇒ f ( x1 ) = f ( x 2 )  W‫א‬E٥
                                                                  D f = X                           
 K x 2  x1  f ( x1 ) = f ( x2 ) ⇒ x1 = x2                             
                                                                   R f = Y                          
                                                                                              ‫א‬‫א‬‫א‬K٤
                            K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W٩
                                                 K‫א‬١٨‫א‬‫א‬‫א‬‫א‬W٢٠
                                                                           ‫א‬‫א‬K١{٤
‫א‬‫א‬‫א‬‫א‬‫א‬
                                                                                     W‫א‬‫א‬‫א‬
             K‫א‬‫א‬ y = f (x) E‫א‬F x E١
                                                    K‫א‬‫א‬‫א‬ ( x, y ) ‫א‬E٢
 x ‫א‬‫א‬‫א‬‫א‬‫א‬E٣
                                                                                                       KKK




                                                      -٢٨-
    ‫א‬‫א‬                                     ١٧١                                   ‫א‬
 ‫א‬‫א‬                                                          



                                            K f ( x) = x  f : R → R  W‫א‬‫א‬‫א‬W٢١
                                                                                                         W‫א‬
                                                                             W‫א‬W‫א‬‫א‬
x           0.00         0.25       0.50    0.75        1.00        1.25     1.50     1.75    1.50        2.25
y= x        0.00         0.50       0.71    0.87        1.00        1.12     1.22     1.32    1.41        1.50

                                                                            W‫א‬‫א‬W‫א‬‫א‬
                            1.6
                            1.4
                            1.2
                                1
                            0.8
                            0.6
                            0.4
                            0.2

                                           0.5          1           1.5         2
                                                                                                              
                                                                 W‫א‬W‫א‬‫א‬
                            1.6
                            1.4
                            1.2
                                1
                            0.8
                            0.6
                            0.4
                            0.2

                                 0.5       1       1.5       2
       KK‫א‬‫א‬‫א‬
                                                       K‫א‬‫א‬‫א‬‫א‬‫א‬
                                                                                 W‫א‬W١٠
   K − x ∈ D f  x ∈ D f  f (− x) + f ( x) = 0  f (− x) = − f ( x) W‫א‬E١
     K − x ∈ D f  x ∈ D f  f (− x) − f ( x) = 0  f (− x) = f ( x) W‫א‬E٢

                                                        -٢٩-
      ‫א‬‫א‬                                     ١٧١                                     ‫א‬
   ‫א‬‫א‬                                                            



                                          ‫؟‬١٨‫א‬‫א‬‫א‬‫א‬W٢٢
                                                                                                             W‫א‬
                                       W f ( x) = x + 1  f : R → R ‫א‬‫א‬E١
                           f (− x) + f ( x) = (− x) + 1 + x + 1 = 2 ≠ 0
                       
                           f (− x) − f ( x) = (− x) + 1 − ( x + 1) = − x + 1 − x − 1 = −2 x ≠ 0
                                                              W g ( x) = x 2  g : R → R ‫א‬‫א‬
                            g (− x) − g ( x) = (− x) 2 − x 2 = x 2 − x 2 = 0
 x ∈ D f = N ‫א‬ f ( x) = x + 1  f : N → N ‫א‬‫א‬E٢
                                                                                                      − x∉Df
                  K‫א‬ g ( x) = x 2  g : N → N ‫א‬‫א‬
 x ∈ D f = N ‫א‬ f ( x) = x  f : N → N ‫א‬‫א‬E٣
                                                                                                       − x∉Df
                                             
‫א‬‫א‬ f ( x) = x  f : R → R ‫א‬‫א‬E٤
                                                                                       KE‫א‬‫א‬F‫א‬
                                                                                               ‫א‬‫א‬‫א‬K٥
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W١١
                                                                            E‫א‬FK‫א‬‫א‬‫א‬
                                                                              
‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W
                                                      K‫א‬‫א‬‫א‬‫א‬
    K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W‫א‬‫א‬‫א‬
                                                                                       ‫א‬‫א‬‫א‬K١{٥
                                KK a  y = f ( x) = a  f : R → R W‫א‬
                                                                                                   W‫א‬
                                                                  K D f = R E١
                                                                   K‫א‬ R f = {a} E٢
                                                                            K f (− x) = f ( x) E٣
                                                        K‫א‬‫א‬E٤

                                                          -٣٠-
   ‫א‬‫א‬                                      ١٧١                                       ‫א‬
‫א‬‫א‬                                                              



                                                                                                
                                             K f ( x) = − 2  f : R → R W‫א‬‫א‬‫א‬ W٢٣
                                                                                                                 
                                                                                                            W‫א‬
                                                        1
                                                      0.5

                                 -4           -2                        2         4
                                                     -0.5
                                                       -1
                                                     -1.5
                                                       -2
                                                     -2.5

                                                      -3

                                                                                   ‫א‬‫א‬‫א‬K٢{٥
                     KK a ≠ 0  y = f ( x) = a x  f : R → R W‫א‬W
                                                                                                   W‫א‬
                                                                K D f = R E١
                                                                KK‫א‬‫א‬ R f = R E٢
                                                                             K f (− x) = − f ( x) E٣
                                                    KK‫א‬E٤
                                                                                                 
                                            K f ( x) = − 2 x  f : R → R W‫א‬‫א‬‫א‬ W٢٤
                                                                                                            W‫א‬
                                                        3

                                                        2

                                                        1


                            -3        -2        -1                  1         2        3
                                                       -1

                                                       -2

                                                      -3




                                                        -٣١-
      ‫א‬‫א‬                                 ١٧١                                        ‫א‬
   ‫א‬‫א‬                                                          



                                                                                     ‫א‬‫א‬‫א‬K٣{٥
                                                                                         
‫א‬ b ≠ 0  a ≠ 0  y = f ( x) = a x + b  f : R → R W‫א‬
                                                                          KK‫א‬‫א‬
                                                                                                  W‫א‬
                                                                K D f = R E١
                                                                KK‫א‬‫א‬ R f = R E٢
                                                                                    KE٣
                                                                      E٤
                               3

                               2
                                                                                           KK‫א‬
                               1
                                                                       f : R → R W‫א‬‫א‬‫א‬  W٢٥
                                                                                                      
        -2             -1                1          2             3             K f ( x) = − 2 x + 1.5 
                               -1                                                                          W‫א‬
                               -2                                                        ‫א‬‫א‬‫א‬K٤{٥
                               -3                                                          
                                                                       f : R → R W‫א‬
‫א‬‫א‬ c  b  a ≠ 0  y = f ( x) = a x 2 + b x + c
                                                                                                          K‫א‬
                                                                                                  W‫א‬
                                                                K D f = R E١
                                                          KK‫א‬‫א‬ R f ≠ R E٢
                                                K b = 0 ‫א‬E٣
                             K b = c = 0 ‫א‬‫א‬‫א‬E٤




                                                        -٣٢-
      ‫א‬‫א‬                                      ١٧١                                  ‫א‬
   ‫א‬‫א‬                                                          



                                                                                                     
                                        K f ( x) = − 2 x 2 + 1.5  f : R → R W‫א‬‫א‬‫א‬ W٢٦
                                                                                                           W‫א‬
                                                            2


                                                            1


                                  -2           -1                         1            2

                                                           -1


                                                           -2


                                                          -3

                                                                                            ‫א‬‫א‬‫א‬K٦
                                 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W‫א‬‫א‬‫א‬
E‫א‬‫א‬F‫א‬‫א‬‫א‬‫א‬K‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
                                                                                     
                            π = 180° W‫א‬‫א‬E ° ‫א‬F
                                                                                         
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W‫א‬‫א‬‫א‬
                                                                 KK‫א‬‫א‬‫א‬‫א‬‫א‬
  f ( x + p ) = f ( p ) W‫א‬ p > 0 ‫א‬ f : R → R ‫א‬W١٢
              K‫א‬‫א‬،E p F x ∈ D f 
          W‫א‬‫א‬ x ‫א‬‫א‬‫א‬W١٣
               ،‫א‬‫א‬‫א‬‫א‬‫א‬ sin x
             K‫א‬‫א‬‫א‬‫א‬‫א‬ cos x 
                                                                                 ‫א‬‫א‬K١{٦
      KK‫א‬K y = sin x  sin : R → R W‫א‬ sin W
                                                                                                  W‫א‬
                                                                   K D f = R E١
                                 KKK − 1 ≤ sin x ≤ 1 W‫א‬‫א‬ R f = [−1,1] E٢
                                                                           K sin(− x) = − sin x E٣
                                                          K 2π ‫א‬ sin( x + 2π ) = sin x E٤
                                                           -٣٣-
   ‫א‬‫א‬                                          ١٧١                                     ‫א‬
‫א‬‫א‬                                                                 



                                                                         K‫א‬E٥
                                                                1


                                                            0.5



                              -2p     3p      -p       p                  p    p     3p     2p
                                    - €€€€
                                        €
                                       €€€€            €€
                                                    - €€€                €€
                                                                          €€
                                                                           €         €€€€
                                                                                        €
                                                                                    €€€€€
                                       2              2                   2          2
                                                          -0.5



                                                           -1

                                                                                                                    
                                                                        ‫א‬‫א‬K٢{٦
 KK‫א‬K y = cos x  cos : R → R W‫א‬ cos W
                                                                                                     W‫א‬
                                                                        K D f = R E١
                             KKK − 1 ≤ cos x ≤ 1W‫א‬‫א‬ R f = [−1,1] E٢
                                                                                K cos(− x) = cos x E٣
                                                        K 2π ‫א‬ cos( x + 2π ) = cos x E٤
                                                                        K‫א‬E٥
                                                                1


                                                            0.5



                              -2p     3p      -p       p                  p    p     3p     2p
                                       €€€€
                                        €
                                    - €€€€             €€
                                                    - €€€                €€
                                                                          €€
                                                                           €         €€€€
                                                                                        €
                                                                                    €€€€€
                                       2              2                   2          2
                                                          -0.5



                                                           -1

                                                            
                                                                                ‫א‬‫א‬‫א‬K٣{٦
              KK b ≠ 1  y = f ( x) = b x  f : R → R W‫א‬
                                                                                                     W‫א‬
                                                                -٣٤-
      ‫א‬‫א‬                                       ١٧١                                      ‫א‬
   ‫א‬‫א‬                                                              



                                                                    K D f = R E١
                                     KK b x > 0 W‫א‬‫א‬ R f = [0, ∞) E٢
                                                                                        KE٣
K e ≅ 2.71828  y = e x ‫א‬‫א‬‫א‬W‫א‬‫א‬E٤
                K x ‫א‬،‫א‬
                                                                         KK b E٥
                                                                              K y = e x W‫א‬‫א‬‫א‬W٢٧
                                                                                                               W‫א‬
                                        50

                                        40

                                        30

                                        20

                                        10


                              -1                     1             2         3          4

                                                                              ‫א‬‫א‬‫א‬K٤{٦
 b ≠ 1  x = b y ‫א‬ y = log b x  log b : R → R W‫א‬ log b W
                                                                                        
                                                                                         KK
                                                                                                      W‫א‬
     K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ b logb x = x  log b (b x ) = x E٠
                                                   K‫א‬‫א‬ D f = (0, ∞) E١
                                                                KK‫א‬‫א‬ R f = R E٢
                                                                                        KE٣
 y = ln x = log e x ‫א‬‫א‬‫א‬W‫א‬‫א‬E٤
                                                                                                 
K‫א‬‫א‬‫א‬‫א‬K e ≅ 2.71828
                                K x ‫א‬‫א‬‫א‬‫א‬‫א‬
                                                                    ln x
                                                 K log b x =           W‫א‬‫א‬‫א‬E٥
                                                                    ln b
                                                            -٣٥-
   ‫א‬‫א‬                          ١٧١                                   ‫א‬
‫א‬‫א‬                                              



                                             K b x = e x ln b W‫א‬‫א‬‫א‬E٦
                                                         KK b E٧
                                                           K y = ln x W‫א‬‫א‬‫א‬W٢٨
                                                                                            W‫א‬
                            3


                            2


                            1



                                 5            10           15           20

                            -1

                        




                                            -٣٦-
         ‫א‬‫א‬                                  ١٧١                                      ‫א‬
      ‫א‬‫א‬                                                          



                                                      
                                                                                         
                                                               W‫א‬‫א‬‫א‬W١
      1) f : N → N, f ( x) = x 3              2) f : N → N, f ( x) = 3 x

      3) f : R → R , f ( x) = x 3 + 1          4) f : R → R , f ( x ) = 3 x − 1
                                                           ‫א‬١‫א‬‫א‬W٢
                                            WE،،،F‫א‬W٣
       1) f : N → N, f ( x) = x 3              2) f : N → N , f ( x ) = 3 x
     3) f : R → R , f ( x) = x 3 + 1           4) f : R → R , f ( x ) = 3 x − 1
                                     x3 + 1                                   3
                                                                                x −1
       5) f : R → R , f ( x) =                  6) f : R → R , f ( x ) =
                                      x2                                       x2 + 1
                                                   ‫؟‬‫א‬W4
                                                                          1
      1) f : R → R , f ( x) = sin 2 x          2) f : R → R , f ( x ) =
                                                                         x
      3) f : R → R , f ( x) = e                4) f : R → R , f ( x ) = x
                                     x


                               ‫؟‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W5
                                                                            W‫א‬‫א‬‫א‬W6
 1) f : R → R , f ( x) = sin 2 x             2) f : R → R , f ( x ) = e
                                                                           x
                                                                                  3) f : R → R , f ( x) = x
                                   ‫؟‬‫א‬‫א‬‫א‬‫א‬‫א‬W7




                                                          -٣٧-
                                ‫א‬‫א‬‫א‬
                             ã¹]<gè…‚jÖ]æ<ËÖ]<Üé×Ãj×Ö<íÚ^ÃÖ]<퉉ö¹]
                                 ‫א‬‫א‬‫א‬‫א‬




            
               ‫א‬
 ‫א‬




٣


                   - ٢١ -
‫א‬‫א‬                        ١٧١                                ‫א‬
  ‫א‬                                                    



                              ‫א‬‫א‬‫א‬‫א‬W‫א‬‫א‬
                                                                                              
                                                                            W‫א‬‫א‬
                                   W‫א‬‫א‬‫א‬
                                                            K‫א‬‫א‬       •

                                                            K‫א‬‫א‬        •

  ‫א‬E‫א‬‫،א‬‫א‬‫א‬‫א‬J ‫א‬‫א‬F‫א‬‫א‬‫א‬          •

                                                                           K‫א‬
                                                 K‫א‬‫א‬‫א‬‫א‬       •

                                                                                              
                                                KW‫א‬‫א‬




                                       - ٣٨ -
     ‫א‬‫א‬                                   ١٧١                                               ‫א‬
       ‫א‬                                                                               



                                                  ‫א‬

                                                                                                                   ١
                               f : I → ℜ  I ≠ {x0 }، I  x0 ، ℜ I 
                                 b ‫א‬ x0  f ‫א‬‫א‬
                                                                                                       f ( x ) − f ( x0 )
                          f ′( x0 ) ‫ﺒ‬ x0  f  b        lim                              =b
                                                                                      x→ x0 , x ≠ x0        x − x0
 I  x0        ‫א‬ I      f    
                                                                                                           ‫א‬‫א‬
                                                                          f , : I → ℜ                           
                                                                                  x → f , ( x)   
                                                                                       f ‫א‬‫א‬
 ε  b  ‫א‬‫א‬ x0   f  W١
                                                                               ( x0 + h) 
                                         f ( x0 + h) = f ( x0 ) + bh + hε (h) , lim ε (h) = 0        
                                                                                      h→0

                                                                                     f ′( x0 ) ≠ ( f ( x0 ))′ W٢
                                                                            ‫א‬‫א‬‫א‬K١
                                                   f ‫ ﻠ‬‫א‬ C f ‫א‬ x0  f 
                                                                       ( x0 , f ( x0 )) ‫א‬‫א‬‫א‬ M 0 ‫א‬
                                                                                                                          
                                                                         f ( x) − f ( x 0 ) NM
                                               ( D) ‫א‬ =                       =     
                                                                              x − x0         M0N
                                                                                                                          
                                                                                                                          
                                                                                                                          
                       M 0  C f ‫ ﻠ‬‫א‬ M 0T  (D) ‫א‬ x0  x 



                                                       - ٣٩ -
      ‫א‬‫א‬                                    ١٧١                                      ‫א‬
        ‫א‬                                                                       



                                                                                       ‫א‬K٢
 y = f (x ) ‫א‬ ‫א‬ ‫א‬  ℜ  I  ‫א‬    y = f ( x )  ‫א‬‫א‬  ‫א‬
                                                                                                   :
                                                          f ( x + ∆x) − f ( x)
                                       f ′( x) = lim
                                                 ∆x → 0            ∆x
                                                                                       W‫א‬‫א‬
                            y′  f ′( x) 
                                                       d
                                                          [ f (x)]  df  dy                          
                                                       dx                 dx     dx
                                      f ( x ) = x 2 + 2 ‫א‬‫א‬‫א‬‫א‬‫א‬W١ 
                                                                                                                W‫א‬

       f ′( x) = lim
                         f ( x + ∆x) − f ( x)
                                              = lim
                                                     ( x + ∆x ) + 2 − x 2 + 2
                                                                           2
                                                                               (   )
                 ∆x → 0           ∆x            ∆x→0             ∆x
             = lim
                       (2                   2
                                                 ) (
                      x + 2 x∆x + (∆x) + 2 − x + 2     2
                                                                   )
                                                               = lim
                                                                        2 x∆x + (∆x) 2
                                                                                       = lim 2 x + ∆x = 2 x
               ∆x→0                     ∆x                       ∆x → 0      ∆x          ∆x → 0

                                         f ( x ) = 1 − x 2 ‫א‬‫א‬‫א‬‫א‬‫א‬W٢
                                                                                                             W‫א‬
                        f ( x + ∆x) − f ( x)           1 − ( x + ∆x) 2 − (1 − x 2 )
        f ′( x) = lim                         = lim
                 ∆x → 0          ∆x             ∆x → 0               ∆x
                         2                  2          2
                   1 − x − 2 x∆x − (∆x) − 1 + x                      − 2 x∆x − (∆x) 2
            = lim                                         = lim                       = lim − 2 x − ∆x = −2 x
              ∆x→0                  ∆x                        ∆x → 0       ∆x           ∆x → 0

                                     w = 1.2 − 0.3m 2 ‫א‬‫א‬‫א‬‫א‬‫א‬W٣
                                                                                                                    W‫א‬

        w′ =
              dw
                   = lim
                           f (m + ∆m) − f (m)
                                              = lim
                                                                       (
                                                       1.2 − 0.3(m + ∆m) 2 − 1.2 − 0.3m 2 ) (               )
              dm ∆m → 0            ∆m           ∆m → 0                ∆m
                              2                     2             2
                  1.2 − 0.3m − 0.6m∆m − 0.3(∆m) − 1.2 + 0.3m
       = lim
           ∆m → 0                        ∆m
                    − 0.6m∆m − 0.3(∆m) 2
        = lim                             = lim − 0.6m − 0.3∆m = −0.6m
             ∆m → 0           ∆m            ∆m → 0

                                                                                                                      




                                                          - ٤٠ -
‫א‬‫א‬                                  ١٧١                                     ‫א‬
  ‫א‬                                                                    



                                      s = 2 + 3t 2 ‫א‬‫א‬‫א‬‫א‬‫א‬W٤
                                                                                                   W‫א‬
        ds           f (t + ∆t ) − f (t )         2 + 3(t + ∆t ) 2 − 2 − 3t 2
 s′ =      = lim                          = lim
        dt ∆t → 0            ∆t             ∆t →0             ∆t
                          2               2          2
                   2 + 3t + 6t∆t + 3(∆t ) − 2 − 3t               6t∆t + 3(∆t ) 2
          = lim                                         = lim                    = lim 6t + 3∆t = 6t
            ∆t → 0                   ∆t                   ∆t → 0       ∆t          ∆t → 0

                                  f ( x ) = 2 x + 5 ‫א‬‫א‬‫א‬‫א‬‫א‬ W٥
                                                                                                       W‫א‬
                    f ( x + ∆x) − f ( x)            2( x + ∆x ) + 5 − (2 x + 5)
  f ′( x) = lim                           = lim
             ∆x → 0          ∆x              ∆x → 0             ∆x
                    2 x + 2 ∆x + 5 − 2 x − 5           2 ∆x
            = lim                            = lim          = lim 2 = 2
              ∆x→0             ∆x               ∆x →0 ∆x       ∆x→0

                              f ( x ) = 3x − 7 ‫א‬‫א‬‫א‬‫א‬‫א‬W٦
                                                                                                       W‫א‬
                  f ( x + ∆x) − f ( x)              3( x + ∆x ) − 7 − 3 x − 7
  f ′( x) = lim                         = lim
          ∆x → 0            ∆x            ∆x → 0                ∆x
          3( x + ∆x ) − 7 − 3 x − 7 3( x + ∆x ) − 7 + 3 x − 7
= lim
  ∆x → 0                ∆x                 3( x + ∆x ) − 7 + 3 x − 7
                       3( x + ∆x ) − 7 − (3 x − 7 )                  3 x + 3∆x − 7 − 3 x + 7
         = lim                                           = lim
                       (
           ∆x→0 ∆x 3( x + ∆x ) − 7 + 3 x − 7         )             (
                                                           ∆x → 0 ∆x 3( x + ∆x ) − 7 + 3 x − 7    )
                                   3∆x                                       3
         = lim                                           = lim
                       (
           ∆x → 0 ∆x 3( x + ∆x ) − 7 + 3 x − 7       )     ∆x → 0 3 x + 3∆x − 7 + 3 x − 7

                        3                   3
      =                             =
              3x − 7 + 3x − 7 2 3x − 7
                                                                               ‫א‬‫א‬‫א‬K٣
                                                                   n ‫א‬‫א‬‫א‬‫א‬‫א‬ W١‫א‬
                                                                           y = f ( x) = x n  W‫א‬‫א‬
                                                                              y′ = nx n−1            
                                                                              y = x 3 ‫א‬W٧
                                                                        y′ = 3x 3−1 = 3x 2           


                                                 - ٤١ -
       ‫א‬‫א‬                                              ١٧١                                                 ‫א‬
         ‫א‬                                                                                            



                                                                                                        y = x −4 ‫א‬W٨
                                                                                          y′ = −4 x −4 −1 = −4 x −5               
                                                                                          ١‫א‬ y = x 
                                                                                             y = x ⇒ y ′ = 1x1−1 = x 0 = 0 
                                  y ′ = 0  c  y = c ‫א‬‫א‬‫א‬:٢‫א‬
                                          y′ = 0  y = −5 ‫א‬ y′ = 0  y = 7 ‫א‬W٩
                                                                        y ′ = nax n−1  y = ax n ‫א‬‫א‬W٣‫א‬
                                                                                                     y = 3x 6 ‫א‬W١٠
                                                                                          y′ = 3 × 6 x 6−1 = 18 x 5               
                                                                                           y = 53 x ‫א‬‫א‬W١١
                                                                                                                                   W‫א‬
                                                                                                                          1
                                                                                               y=5 x =       3
                                                                                                                    5( x) 3        
                                                                                                              1         −2
                                                                                                1       −1 5
                                                                                           y′ = × 5 x 3 = x 3                       
                                                                                                                                    ‫א‬
                                                                                                3          3
                                                                                           ‫א‬‫א‬W٤‫א‬
 f1 ,..., f n ( x)  F ( x) =   f1 ( x) ± f 2 ( x) ± ... ± f n −1 ( x) ± f n ( x) ‫א‬ F ( x) ‫א‬‫א‬

                                         F ′( x) = f1′( x) ± f 2′ ( x) ± ... ± f n −1 ( x) ± f n ( x n )        ‫א‬
                                                                      y = 4 x −3 − 5 x 2 + 7 x − 12 ‫א‬‫א‬W١٢
                               y′ = −3 × 4 x −3−1 − 5 × 2 x 2−1 + 7 x1−1 = −12 x −4 − 10 x + 7                                    
                                                                                                     ‫א‬‫א‬W٥‫א‬
‫א‬ f1 ( x), f 2 ( x)   F ( x) = f1 ( x). f 2 ( x)  ‫א‬   F (x)  ‫א‬‫א‬ 
                         F ′( x) = f1′( x) f 2 ( x) + f 2′ ( x) f1 ( x)  K ℜ  I ‫א‬
                                                                         F ( x) = (3 x − 2)(4 x + 1) ‫א‬‫א‬W١٣
                                                                          F ′( x) = 3(4 x + 1) + 4(3x − 2)                        
                                                                                                   ‫א‬W٦‫א‬


                                                                   - ٤٢ -
      ‫א‬‫א‬                                         ١٧١                                     ‫א‬
        ‫א‬                                                                           


                                                            f1 ( x )
 f1 ( x), f 2 ( x)  F ( x) =                     ‫א‬ F (x) ‫א‬‫א‬
                                                            f 2 ( x)
                                                                              f 2 ( x) ≠ 0  ℜ  I ‫א‬
                                                      f1′( x) f 2 ( x) − f 2′ ( x) f1 ( x)
                                     F ′( x) =
                                                                 ( f 2 ( x) )2
                                                                1            8x 7
                                                            x ≠  y =        ‫א‬‫א‬W١٤
                                                                2           2x − 1
                                                                                                                 W‫א‬
                 f 2 ( x) = 2 x − 1 ⇒ f 2′ ( x) = 2  f1 ( x) = 8 x 7 ⇒ f1′( x) = 7 × 8 x 7−1 = 56 x 6 
               f1′( x) f 2 ( x) − f 2′( x) f1 ( x) 56 x 6 (2 x − 1) − 2 × 8 x 7 56 x 6 (2 x − 1) − 16 x 7
   F ′( x) =                                     =                              =                                
                                                                                                          ‫א‬
                           ( f 2 ( x)) 2                     (2 x − 1) 2                (2 x − 1) 2
              112 x 7 − 56 x 6 − 16 x 7 96 x 7 − 56 x 6 2 x 6 (48 x − 28)
           =                                  =                 =                                       
                       (2 x − 1) 2                (2 x − 1) 2          (2 x − 1) 2
                                           F ( x) = ( f ( x)) n ‫א‬‫א‬‫א‬‫א‬W٧‫א‬
                                          f (x)  F ( x) = ( f ( x)) n ‫א‬
                                           F ′( x) = n( f ( x)) n −1 f ′( x)
                                                                 dF ( x) dF ( x) df ( x)
                                                    F ′( x) =          =                = n( f ( x)) n −1 f ′( x) 
                                                                  dx      df ( x) dx
                                                 y = F ( x) = 4(2 x 2 + 3x − 2) 2 ‫א‬‫א‬W١٥
                                                                                                                W‫א‬
                                                       f ( x) = (2 x 2 + 3 x − 2) ⇒ f ′( x) = 4 x + 3  
                                                                                                             
                          y ′ = F ′( x) = 2 × 4(2 x 2 + 3x − 2) 2−1 (4 x + 3) = 8(2 x 2 + 3x − 2)(4 x + 3) ‫א‬




                                                             - ٤٣ -
     ‫א‬‫א‬                                ١٧١                                     ‫א‬
         ‫א‬                                                                



                                                   
                                                   W‫א‬‫א‬‫א‬‫א‬‫א‬ E١
1) y = x 2 + 4 x − 3      3) y = 2 t + 3               5) y = x 3 − 1                7) y = 5 − +3t + 2t 2
2) y = x − 5              4) y = − x 2 + 5 x − 7       6) y = 2 x − 7                8) y = 3t + 7
                                                                                                     
                                                                                                        
                                                                       W‫א‬‫א‬E2

1) y = (2 x 3 − 7)(3x 2 ) 6) y = (3 x − 2)( x + 7)                   ⎛ 2x − 7 ⎞
                                                                                    −1                        4

                                       3x − 1                11) y = ⎜        ⎟                    (3 −  2 x) 3
                                                                     ⎜ x2 ⎟              16) y =
                                                                     ⎝        ⎠                        x2
                                                                                                                     −2
       2x2 − 5                          3x 2                           x −1 ⎞ 2
                                                                               3
2) y =                    7) y =                             12) y = ⎛
                                                                                                    3        2
         3x                      (5 x + 7)(2 x − 1)                  ⎜      ⎟            17) y = x (5 x          + 1) 3
                                                                     ⎝ x + 2⎠
3) y =
             2            8) y = (2 x 4 − 1.9) 3             13) y = 2
                                                                          2x             18) y = 2 x −3 + 7 x −5
         2
      3x − 7 x + 4                                                      x + 1.8
           5                       x3 + 4
                                                                       2
                                                             14) y = x x − 1             19) y = (4 x 2 x 3 )
                                                                                                                   1
4) y = 3                  9) y = 2
                                                                                                                       4
      4 x − 3x 2                x ( x − 2) 3
5) y =
          1
             −x           10) y = (2 x 3 − 4 x + 7) −2 15) y =            1 .9
                                                                                         20) y =
                                                                                                     x−2
         x+2                                                         ( 2 x + 4) 3                    x2 + 5




                                                   - ٤٤ -
‫א‬‫א‬                                      ١٧١                                              ‫א‬
  ‫א‬                                                                                 



                                                                                    W‫א‬‫א‬‫א‬‫א‬K٤
                        ℜ I‫א‬ x ‫א‬ u  y = sin u ‫א‬‫א‬ E١
                                           y ′ = dy = d (sin u ) du = cos u du                                
                                                        dx          du     dx dx
                       ℜ I‫א‬ x ‫א‬ u  y = cos u ‫א‬‫א‬ E٢
                                        y ′ = dy = d (cos u ) du = − sin u du                               
                                                   dx          du        dx                  dx
                                                                 y = sin(2 x 3 − 3) ‫א‬‫א‬W١٦
                                     y′ = 3 × 2 x 2 cos(2 x 3 − 3) = 6 x 2 cos(2 x 3 − 3) 
                                       y = cos(2θ 3 − 3θ −2 ) ‫א‬‫א‬W١٧
                                                                                                                W‫א‬
                                                                        du
                                                    u = 2θ 3 − 3θ −2 ⇒      = 6θ 2 + 6θ −3                      
                                                                        dθ
                                                    y ′ = −(6θ + 6θ ) sin( 2θ 3 − 3θ −2 ) 
                                                               2    −3
                                                                                                                       
                                                                                                                       ‫א‬
                   ℜ I‫א‬ x ‫א‬ u  y = tan u ‫א‬‫א‬E٣
                                y ′ = dy = dy du = d (tan u ) du = sec 2 u du                                 
                                        dx     du dx            du dx             dx
                                                                 dy
                                                                    y = tan x −2 ‫א‬W١٨
                                                                 dx
                                                                                                                W‫א‬
                                                                                             du
                                                                      u = x−2 ⇒                = −2 x − 3 
                                                                                             dx
                                                      y ′ = d (tan u ) = du sec 2 u = −2 x −3 sec 2 x −2 ‫א‬
                                                                                                           
                                                                      dx            dx
                    ℜ I‫א‬ x ‫א‬ u  y = cot u ‫א‬‫א‬E٤
                         y ′ = dy = dy du = d (cot u ) du = − csc 2 u du
                                dx    du dx             du     dx                       dx
                                                                     y = cot 3x ‫א‬‫א‬‫א‬W١٩
                                                                                                                W‫א‬
                                                                                                  du
                                                                                u = 3x ⇒            = 3 
                                                                                                  dx
                                                    y ′ = d (cot u ) = − du csc 2 u = −3 csc 2 3 x 
                                                                 dx                dx
                                                    - ٤٥ -
     ‫א‬‫א‬                                     ١٧١                                       ‫א‬
       ‫א‬                                                                         



                    ℜ I‫א‬ x ‫א‬ u  y = sec u ‫א‬‫א‬E٥
                            y ′ = dy = dy du = d (sec u ) du = du tan u sec u
                                    dx     du dx         du        dx      dx
                                                                       y = sec xθ 2 ‫א‬‫א‬‫א‬W٢٠
                                                                                                              W‫א‬
                                                                                             du
                                                                                 u =θ 2 ⇒      = 2θ 
                                                                                             dθ
                                                   y ′ = d (csc u ) = du tan u sec u = 2θ tan θ 2 sec θ 2 
                                                                 dθ           dθ
                        ℜ I‫א‬ x ‫א‬ u  y = csc u ‫א‬‫א‬E٦
                            y ′ = dy = dy du = d (csc u ) du = − du cot u csc u
                                    dx    du dx          du        dx         dx
                                                                         y = csc x 3 ‫א‬‫א‬‫א‬W٢١
                                                                                              du
                                                                                 u = x3 ⇒       = 3x 2  W‫א‬
                                                                                              dx
                                                     y ′ = d (csc u ) = − du cot u csc u = −3 x 2 cot x 3 csc x 3
                                                                   dx              dx
                                                            y = csc(2 x 5 − 3) ‫א‬‫א‬‫א‬W٢٢
                                                                                                              W‫א‬
                                                                                            du
                                                                         u = 2x5 − 3 ⇒        = 10 x 4 
                                                                                            dx
                                   y ′ = d (csc u ) = − du cot u csc u = −10 x 4 cot( 2 x 5 − 3) csc( 2 x 5 − 3)
                                               dx             dx
                                                                         W‫א‬‫א‬‫א‬‫א‬‫א‬W
1) y = sin 5 3x 2             5) y = csc 3 (−7 x 4 )               sin x 2                   13) y = sin (cos 2 x )
                                                           9) y =
                                                                  cos 2 x 2
           1                                                                            5
                                                                                             14) y = 1 + sin x
2) y = x tan                  6) y = 1 + cos 2 x           10) y = sec(2 x + 1)2
           x
3) y = x cos 2 x              7) y = tan 2 ( x 2 + 1)                  1                     15) y = x cot (− 4 x )
                                                           11) y =        2
                                                                   x sin 3 x
                                                           12) y = (sin x − cos x )          16) y = x csc x
                                                                                   2
4) y = csc x 3                8) y = ( x 4 − cot x) 3

                                                                                                                    


                                                        - ٤٦ -
       ‫א‬‫א‬                                             ١٧١                                               ‫א‬
         ‫א‬                                                                                         



                                                                                                ‫א‬ ‫א‬‫א‬‫א‬‫א‬K٥
                                                                                                      ‫א‬‫א‬‫א‬‫א‬‫א‬K١{٥
                                 x ‫א‬ u  y = ba u ‫א‬‫א‬‫א‬W١‫א‬
                                                        dy
                                                          = ba u ln a u ′                                                       
                                                        dt
                                                         y = 8 2 (3 x                      )W‫א‬‫א‬‫א‬‫א‬W٢٢
                                                                             2
                                                                                 + 4 x +5


                                                                           ln 2 (6 x + 4 ) = (48 x + 32) ln 2 23 x
                                                             2                                                          2
                                           y′ = 8 23 x          + 4x +5                                                    + 4 x +5
                                                                                                                                       
                                                 e ≅ 2,718 ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W٢‫א‬
                                          x ‫א‬ u  y = b e u ‫א‬‫א‬‫א‬
                                                            dy
                                                y′ =          = b e u u ′                                                       
                                                            dt
                                                           y = 8 e 2 x+1 W‫א‬‫א‬‫א‬‫א‬‫א‬W٢٣
                                                                                                                                  W‫א‬
                                 y′ = 8 × 2 e 2 x +1 = 16e 2 x +1 W‫א‬‫א‬‫א‬‫א‬
                                                                                                 y = −5 esin x ‫א‬W٢٤
                                                                                                y′ = −5 cos x e sin x             
                                                                                               ‫א‬‫א‬‫א‬‫א‬‫א‬K٢{٥
 x ‫א‬ u  a > 0, a ≠ 1  y = b log a u ‫א‬‫א‬‫א‬W١‫א‬
                                                                                                 W‫א‬‫א‬
                                                      dy dy du    u′
                                                       =      = b log a e                                                         
                                                      dx du dx    u
                                                       y = 3 log(6 x 5 ) W‫א‬‫א‬‫א‬‫א‬‫א‬W٢٥
                                                                                                                                  W‫א‬

     y′ =
                 (         )
               3 30 x 4 log e 15 x 4    15
                             = 5 log e = log e. W‫א‬‫א‬‫א‬‫א‬
                   6x5         x         x
                          x ‫א‬ u  y = ln u  ‫א‬‫א‬‫א‬W٢‫א‬
                                                                   dy   u′
                                                        y′ =         =b                                                          
                                                                   dx   u
                                                                               y = e − x ln x 2 W‫א‬‫א‬‫א‬‫א‬W٢٦
                                                                   - ٤٧ -
     ‫א‬‫א‬                                        ١٧١                                       ‫א‬
       ‫א‬                                                                           



                                                                                                                W‫א‬
                                                                              2 x −x ⎛ 2         ⎞
                        y′ = −e − x ln x 2 +     2
                                                                                  e = ⎜ − ln x 2 ⎟e − x        
                                                                              x       ⎝x         ⎠
                                                                       W‫א‬‫א‬‫א‬‫א‬‫א‬W
                 (
1) y = log 3 3x 2 − 5       )       5) y = ln
                                                   x4                    9) y = e x
                                                                                      2
                                                                                                 13) y = e − x ln x
                                               (3 x − 4) 2
2) y = ln( x + 3)                   6) f ( x) = ln sin 3 x                                       14) y = e −2 x sin 3 x
                       2                                                                  2
                                                                         10) y = 53 x

3) y = ln 2 ( x + 3)                                 (
                                    7) f ( x) = ln x + 1 + x 2       )   11) y = x 2 3 x         15) f ( x) = ln tan e x
                                                                                                                           2




            (          )(
4) y = ln x 2 + 2 x 2 + 3        ) 8) y = e     1
                                               − x
                                                2
                                                                                                 16) f ( x) = ln 1 − 2 x

                                                                                                                      




                                                            - ٤٨ -
    ‫א‬‫א‬                                ١٧١                                       ‫א‬
      ‫א‬                                                                    



                                                                f (x) ‫א‬‫א‬‫א‬‫א‬‫א‬K٦
                                                                                         
 x0  U ‫א‬ x0 ‫א‬ f ( x) ‫א‬‫א‬
                                                                                               W
                                 f ( x) < f ( x0 ) ∀x ∈U
  t 0  I ‫א‬ t 0 ‫א‬ f (x) ‫א‬‫א‬
                                                                                               W
                                       f ( x) > f (t 0 )       ∀x ∈ I
                                                                                            
  ‫א‬ x0 ‫א‬ f (x) ‫א‬‫א‬‫א‬
                                                                                 f ′( x0 ) = 0 
                                                                                 ‫א‬‫א‬K١{٦
       K f ( x) ‫א‬‫א‬‫א‬‫א‬‫א‬ f ( x) ‫א‬‫א‬‫א‬ 
     ، x ‫א‬‫א‬‫א‬ f (x) ‫א‬‫א‬‫א‬
 K x ‫א‬‫א‬‫א‬ x ‫א‬‫א‬‫א‬‫א‬
                                              1         1
                                       y = x 3 − x 2 − 2 x + 2 ‫א‬‫א‬‫א‬W٢٧
                                              3         2
                                                                                                       W‫א‬
   x ‫א‬‫א‬‫א‬‫א‬، x ‫א‬‫א‬‫א‬
                                 y ′ = x 2 − x − 2 = ( x − 2)( x + 1) = 0 ⇒ x = 2  x = −1  
                                                                          x ‫ﻟ‬‫א‬‫א‬
                                                            1            1        4
                                                 y = (2) 3 − (2) 2 − 2(2) + 2 = −  x = 2 
                                                            3            2        3
                                                 1         1                   19
                                            y = (−1) 3 − (−1) 2 − 2(−1) + 2 =  x = −1 
                                                 3         2                    6
                                                          4      19
                                                  (2, − ), (−1, ) W‫א‬‫א‬
                                                          3       6
                                                                                                              


                                                   - ٤٩ -
     ‫א‬‫א‬                                     ١٧١                                       ‫א‬
        ‫א‬                                                                        



                                                         W‫א‬‫א‬‫א‬‫א‬‫א‬K٧
 x < x0   f (x)    x0 ‫א‬ f (x) ‫א‬‫א‬ ‫א‬
K‫א‬‫א‬ x0 ‫א‬‫א‬‫א‬، 
‫א‬‫א‬، x > x0   f (x)  
                                                                                K‫א‬‫א‬ x0 ‫א‬
 x < x0  f ′( x) < 0  x0 ‫א‬ f ( x ) ‫א‬‫א‬‫א‬
                      K x0  x > x0  f ′( x) > 0 ، x0
                                y = 16 − x 2 ‫א‬‫א‬‫א‬‫א‬‫א‬W٢٨
                                                                                                            W‫א‬
 4   [− 4,+4]‫א‬  y = 16 − x 2 ‫א‬‫א‬
                                                                                                W x = 0 ‫א‬
                                                  − 4 ≤ x < 0  0 < x ≤ 4  f ( x) < 4 
                                                  x
                               f ( x) =          ‫א‬‫א‬‫א‬‫א‬‫א‬W٢٩
                                               1 + x2
                                                                                                              W‫א‬
         W ℜ  f (x) ‫א‬‫א‬
                                                                     1 − x2
                                                       f ′( x) =                                         
                                                                    (1 + x )
                                                                         2 2


K  ‫א‬  ‫א‬     x = 1  x = −1  ‫א‬   ‫א‬ 
                                                                                       W‫א‬‫א‬
          x                −∞                      −1                     1                +∞
       1− x                        +                       +                         -
       1+ x                         -             +                                  +
       1 − x2                       -             +                                  -
‫א‬  − 1 < x < 1  f ′( x) > 0  x < −1  f ′( x) < 0 
  f (x) ‫א‬ x = −1 ‫א‬‫א‬‫א‬‫א‬
                                                                                                             1
                                                                                                    (−1,− ) 
                                                                                                             2



                                                        - ٥٠ -
        ‫א‬‫א‬                                 ١٧١                                        ‫א‬
          ‫א‬                                                                      



  ‫א‬    x > 1     f ′( x) < 0  − 1 < x < 1     f ′( x) > 0   
    f ( x) ‫א‬ x = 1 ‫א‬‫א‬‫א‬‫א‬
                                                                                                      1
                                                                                              K (1, ) 
                                                                                                      2
                                                  W‫א‬‫א‬‫א‬‫א‬‫א‬K٨
   ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
 ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
                                1      1
                          y = x 3 − x 2 − 2 x + 2 ‫א‬‫א‬‫א‬‫א‬‫א‬W٣٠
                                3      2
                                                                                                             :‫א‬
                                                                                           ‫א‬‫א‬‫א‬
                                                                                   y ′ = x 2 − x − 2,
                                                                                                            
                                                                                   y ′′ = 2 x − 1
                                                                4         19
                                                      (2, − ), (−1, ) ‫א‬‫א‬١‫א‬
                                                                3          6
                                                                                         4
                                                                                 (2, − ) ‫א‬
                                                                                         3
                                                         y ′′          = 2(2) − 1 = 3 > 0                  
                                                                 x=2
                                                                                                             4
                                                                           (2, − ) 
                                                                                                             3
                                                                                       19
                                                                              (−1, ) ‫א‬
                                                                                        6
                                                    y ′′            = 2(−1) − 1 = −3 < 0                  
                                                            x = −1
                                                                                                            19
                                                                          (−1, ) 
                                                                                                             6
                                                                                 ‫א‬K٩
              K‫א‬‫א‬‫א‬‫א‬‫א‬
                                                                    1      1
                                                         y = x 3 − x 2 − 2 x + 2 ‫א‬ W٣٠
                                                                    3      2
                                                                                             1
                                                             y ′′ = 2 x − 1 = 0 ⇒ x =                      
                                                                                             2

                                                       - ٥١ -
     ‫א‬‫א‬                                ١٧١                                   ‫א‬
       ‫א‬                                                                


                                                                 1                               1
                                             y ′′ > 0  x > ‫א‬ y ′′ < 0  x < ‫א‬
                                                                 2                               2
                                                                                    1
                                                   ‫א‬ x = 
                                                                                    2
                                                                     1
                                             ‫א‬‫א‬ x = ‫א‬‫א‬‫א‬
                                                                     2
                                                      3          2
                                               1⎛ 1 ⎞ 1⎛ 1 ⎞  1       11
                                           y = ⎜ ⎟ − ⎜ ⎟ − 2( ) + 2 =                             
                                               3⎝ 2 ⎠ 2⎝ 2⎠   2       12

                                                                  K‫א‬ ⎛ ,
                                                                                          1 11 ⎞
                                                                                    ⎜                    
                                                                                               ⎟ ‫א‬‫א‬
                                                                                        ⎝ 2 12 ⎠
                                                                                     W
                           W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
1) y = x3                 2) y = − x 3                 3) y = x                4) y = 1 − x 2
               16                  x2 + 1                      1   x           8) y = 3x 5 − 10 x 3 + 15 x
5) y = x 2 +              6) y =                       7) y =    +
               x2                  x2 − 1                     x−2 4




                                                    - ٥٢ -
                                           ‫א‬‫א‬‫א‬
                                        ã¹]<gè…‚jÖ]æ<ËÖ]<Üé×Ãj×Ö<íÚ^ÃÖ]<퉉ö¹]
                                            ‫א‬‫א‬‫א‬‫א‬




                     
                     ‫א‬‫א‬
 ‫א‬‫א‬




٤


                             - ٤٠ -
 ‫א‬‫א‬‫א‬                ١٧١                                  ‫א‬
‫א‬‫א‬                                         



                                              ‫אא‬‫א‬‫א‬W‫א‬‫א‬
                                                                                          
                                                                     W‫א‬‫א‬
                            W‫א‬‫א‬‫א‬
                                                     ‫א‬‫א‬       •

                                                      ‫א‬            •

                                               ‫א‬‫א‬           •

                   ‫א‬‫א‬‫א‬‫א‬                 •

                                                                                          
                                                     W‫א‬‫א‬




                                                                                          
                                                                                          




                               - ٥٣ -
    ‫א‬‫א‬‫א‬                              ١٧١                                ‫א‬
   ‫א‬‫א‬                                                     



                                  ‫א‬‫א‬
                                                                                                     
                                                          ‫א‬‫א‬K١
 K‫א‬‫א‬
  xy ‫א‬‫א‬K‫א‬
 a  (a , b) ‫א‬‫א‬‫א‬K‫א‬‫א‬‫א‬‫א‬‫א‬
  ‫א‬‫א‬ b  x ‫א‬‫א‬
 x ‫א‬‫א‬‫א‬‫א‬K y 
 1 ‫א‬K‫א‬‫א‬ (0 , 0) ‫א‬‫א‬K y ‫א‬
  ‫א‬ x ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
    ‫א‬‫א‬‫א‬‫א‬‫א‬K y 
                             K‫א‬‫א‬‫א‬‫א‬‫א‬K‫א‬
                                                                                                     

                         x<0                                           x>0
                         y>0                                           y>0
                                                 4
                                                      ‫א‬‫א‬

                        ‫א‬‫א‬              2       ‫א‬‫א‬


                            -4         -2                      2       4       x
                                                      ‫א‬         ‫א‬‫א‬
                                                 -2
                        ‫א‬‫א‬                          ‫א‬‫א‬‫א‬

                         x<0                     -4                    x>0
                         y<0                                           y<0

                                                ‫א‬
                                                1




                                                - ٥٤ -
      ‫א‬‫א‬‫א‬                                     ١٧١                                          ‫א‬
     ‫א‬‫א‬                                                                      



        ‫א‬٢‫א‬K‫א‬‫א‬ P(a , b) 
   ‫א‬‫א‬‫א‬K (4 , 3) , (−3,1) , (−2 , − 3) , (3 , − 2) , (0 ,1) , (1, 3) , (3,1)
                                             K‫א‬‫א‬ (1, 3) , (3 ,1) ‫א‬
                                                        y

                                                        5

                                                         4
                                                              (1,3)              (4,3)
                                                         3       •                 •

                                                        2

                                 (-3,1) •               1 • (0,1)            • (3,1)

                           -5    -4    -3    -2    -1            1     2     3     4     5      x
                                                        -1

                                                        -2            (3,-2) •

                                      (-2,-3) •         -3

                                                        -4

                                                        -5



                                                    2 ‫א‬
                                                                       ‫א‬K٢
    ‫א‬K‫א‬ x ‫א‬‫א‬‫א‬‫א‬
        K‫א‬ y ‫א‬‫א‬‫א‬
‫א‬‫א‬K d =| 2 − (−3) |= 5 W (1, − 3) ‫א‬ (1, 2) ‫א‬ d  3 ‫א‬
   4 ‫א‬ P2 ( x2 , y 2 )  P1 ( x1 , y1 )
                                       W ( y1 , y2 )  ( x2 − x1 ) ‫א‬‫א‬‫א‬‫א‬‫א‬
                                                                                                W
                d 2 =| x2 − x1 |2 + | y 2 − y1 |2 ⇒ d = | x2 − x1 |2 + | y 2 − y1 |2 
                                 | y2 − y1 |2 = ( y 2 − y1 )2  | x2 − x1 |2 = ( x2 − x1 ) 2 
                                                         W P2 ( x2 , y 2 )  P1 ( x1 , y1 ) ‫א‬
                                  d = ( x2 − x1 ) 2 + ( y 2 − y1 ) 2 
                                                        - ٥٥ -
     ‫א‬‫א‬‫א‬                                                    ١٧١                                          ‫א‬
    ‫א‬‫א‬                                                                                   


                                  y                                                                y
                                                                                                         P ( x1 , y1 )
                                                                                                          1
                                                                                                   y2   •
                                    4
                                                                                                                   d
                                                                                     | y2 − y1 |
                                            (1,2)
                                    2        •
                                                                                                   y1                      • P2 ( x2 , y2 )
               -4          -2                        2        4         x                               x1                 x2       x
                                                 5
                                                                                                             | x2 − x1 |
                                 -2
                                             •
                                            (1,-3)
                                    -4



                             4 ‫א‬ 3 ‫א‬
                                                                                                                                        
                                                                   P2 (7 , 2)  P1 (−3 , 4) ‫א‬‫א‬W١
                                                                                                                                 W‫א‬
                        W y 2 = 2  y1 = 4 ، x2 = 7 ، x1 = −3 ‫א‬
 d ( P1 , P2 ) = ( x2 − x1 ) 2 + ( y 2 − y1 ) 2 = (7 − (−3)) 2 + (2 − 4) 2 = 100 + 4 = 104 ≈ 10.2
                                                                                                
                                                                   ‫א‬‫א‬K٣
       E 5 ‫א‬F ( xm , y m ) ‫א‬‫א‬
        ‫א‬K‫א‬‫א‬ y ‫א‬‫א‬‫א‬ x ‫א‬
                                                                                                                            W
                                                                                              ⎛ x + x y + y2 ⎞
                       y                      P ( x1 , y1 )
                                               1                                ( xm , ym ) = ⎜ 1 2 , 1      ⎟
                                                     •                                        ⎝ 2       2 ⎠

                                           •M ( x , y )
                                                 m   m

                                •
                            P2 ( x2 , y2 )

                                                              x
 
                                           5 ‫א‬

                                                                       - ٥٦ -
      ‫א‬‫א‬‫א‬                                         ١٧١                                              ‫א‬
    ‫א‬‫א‬                                                                              



                        P2 (7 , 2)  P1 (−3 , 4) ‫א‬‫א‬‫א‬W٢
                                                                                                                           W‫א‬
                 W y 2 = 2  y1 = 4 ، x2 = 7 ، x1 = −3 ‫א‬
                                                     ⎛ x + x2 y1 + y 2 ⎞ ⎛ − 3 + 7 4 + 2 ⎞
                                      ( xm , y m ) = ⎜ 1     ,         ⎟=⎜        ,      ⎟ = (2 , 3)
                                                     ⎝ 2         2 ⎠ ⎝ 2             2 ⎠


                                                                                                                          
                                                         ‫א‬‫א‬‫א‬W١
              1) (2 , 4)          2) (0 , − 3)        3) (−2 ,1)      4) (−5 , − 3)
                                                                                     
              5) (−3 , − 5)        6) (−4 , 3)        7) (0 , 2)       8) (−2 , 0)
                                                                                                                                 
                                                                             ‫א‬‫א‬‫א‬W٢
             1) (6 , 4) , (−8 ,11)                          2) (−4 , − 20) , (−10 ,15)

             3) (5 , − 8) , (0 , 0)                         4) ( 3 , 8 ) , ( 12 , 27 )
                                                                                                
             5) (a , b) , (− a , − b)                       3) (a − b , b) , (a , a + b)
             7 ) ( x , 4 x ) , ( −2 x , 3 x ) x < 0         8) ( x , 4 x) , (−2 x , 3x) x > 0
                                                                                                                                 
                                                ‫א‬‫א‬‫א‬‫א‬W٣
          1) (1, − 1) , (5 , 5)            2) (4 , 7) , (6 ,10)      3) (6 , − 3) , (6 ,11)      4) (2a , 0) , (0 , 2b) 
                                                                          ‫א‬‫א‬K٤
   ‫א‬EF‫א‬‫א‬‫א‬‫א‬ (m) ‫א‬‫א‬
‫א‬‫א‬ ( x2 , y 2 )  ( x1 , y 2 ) ‫א‬K‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬K 6 ‫א‬
 K (∆x = x2 − x1 )  x ‫א‬ (∆y = y 2 − y1 )  y ‫א‬‫א‬،‫א‬‫א‬
                                                                             W‫א‬‫א‬‫א‬
                                                                                                                                 



                                                             - ٥٧ -
     ‫א‬‫א‬‫א‬                                                        ١٧١                                               ‫א‬
   ‫א‬‫א‬                                                                                               


        y
                                                                                              ∆y y2 − y1
                                      ( x2 , y2 )                                   m=          =
        y2                                      •                                             ∆x x2 − x1
                                                          y2 − y1
                    ( x1 , y1 )
         y1                  •
                                      x2 − x1
                             x1                  x2              x


                                              6 ‫א‬
                                                                                                                                               
                                                      y 2 − y1 − ( y 2 − y1 ) y1 − y 2
                                                            =              =         W‫א‬‫א‬‫א‬
                                                      x2 − x1 − ( x2 − x1 ) x1 − x2
                                                                                                                                               
    ،‫א‬‫א‬،‫א‬‫א‬W 7 ‫א‬
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬K‫א‬‫א‬‫א‬‫א‬‫א‬
                                                                                                                K‫א‬‫א‬

                                                                                                                                               
            y                     1                   y                             y                             y
                        m1 =                                   m2 = 0                               m3 = −4                       m 4 
                                  5
                3
                                             (-1,2)
                                                          3
                                                               (2,2)                    3 • (0,3)                     3         • (2,3)
                2
                          (3,1)                    •            •
    (-2,0)      1           •                             1                             1                             1         • (2,1)
        •
       -2 -1         1 2 3             x        -2 -1         1 2 3        x    -2 -1            2 3      x   -2 -1        1       3      x
               -1                                       -1                              -1   • (1,-1)                 -1

 m > 0 ‫א‬                             
                                                 m = 0 ‫א‬                    m < 0 ‫א‬  m ‫א‬
                                                                                             
          ‫א‬‫א‬
                                                   
                                                                      ‫א‬
                                                                                    
                                                                                     ‫א‬    
                                                                                                          
                                                                       
                                                                     7 ‫א‬
                                                                       



                                                                                                                                               
                                                                                                                                               
                                                                           - ٥٨ -
        ‫א‬‫א‬‫א‬                                         ١٧١                                           ‫א‬
      ‫א‬‫א‬                                                                            



                                                                                                     ‫א‬‫א‬K٥
                                                                     ‫א‬K١{٥
  K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
 W‫א‬‫א‬‫א‬ ( x , y ) ‫א‬K ( x1 , y1 ) ‫א‬‫א‬ m 
                                                                                                                          y − y1
                                     W‫א‬‫א‬‫א‬‫א‬ m =
                                                                                                                          x − x1
  y = m( x − x1 ) + y1 
                                        (1, − 2) ٣‫א‬‫א‬‫א‬W٣
                                                                                                                         W‫א‬
                            W‫א‬‫א‬‫א‬ ( x1 , y1 ) = (1, − 2)  m = 3 ‫א‬‫א‬
                                                   y = 3( x − 1) + (−2) = 3 x − 3 − 2 = 3 x − 5 
                                                                     y = 3x − 5 W‫א‬‫א‬‫א‬
                                                                                                                               
                                      (2 , 5)  (4 , 3) ‫א‬‫א‬‫א‬W٤
                                                                                                                     W‫א‬
        ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
                                                          K٣‫א‬‫א‬‫א‬‫א‬
                                 y 2 − y1 3 − 5 − 2
                          m=            =     =    = −1 
                                 x2 − x1 4 − 2   2
                                     W‫א‬ (4 , 3) ‫א‬ m = −1 ‫א‬‫א‬
                                            y = −1( x − 4) + 3 = − x + 4 + 3 = − x + 7 
                                                              ‫א‬‫א‬‫א‬K٢{٥
     K‫א‬‫א‬‫א‬‫א‬‫א‬
                                                                           W‫א‬‫א‬
                                               y = mx + b 
       K (0 , b) ‫א‬ y ‫א‬EF‫א‬E‫א‬F‫א‬ b ‫א‬ m 
      ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
                                                                                                                  ‫א‬‫א‬
                                                              - ٥٩ -
   ‫א‬‫א‬‫א‬                                        ١٧١                                ‫א‬
  ‫א‬‫א‬                                                               



                             (1, 3)  2 ‫א‬‫א‬‫א‬W٥
                                                                                                         W‫א‬
                      y = 2 x + b W‫א‬‫א‬‫א‬
   W‫א‬، b ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
     y = 2 x + 1 W‫א‬‫א‬‫א‬ b = 3 − 2 = 1  3 = 2 × 1 + b
                                                                                   ‫א‬‫א‬
                                y = m( x − x1 ) + y1 WE‫א‬F‫א‬ •
                              y = mx + b WE‫א‬‫א‬‫א‬F‫א‬ •
                                y = mx WE‫א‬‫א‬F‫א‬ •
                              y = b WE‫א‬F‫א‬‫• א‬
                           x = a WE‫א‬F‫א‬‫• א‬
                                                   ‫א‬‫א‬‫א‬‫א‬‫א‬K٦
  ‫א‬‫א‬‫א‬‫א‬‫א‬
 (m1 = m2 ) ‫א‬‫א‬‫א‬،K 8 ‫א‬
   ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
                                                                                                              1
                                                                                             K (m1 = −         )
                                                                                                              m2
           y                                                           y
                          ‫א‬                                            


                           m1                 m2                                    m1         m2



                                    m1 = m2          x                                   m1 ⋅ m2 = −1           x
                                                            
                                                          8 ‫א‬

                                                                                                               
 W‫א‬‫א‬ (2 , − 1) ‫א‬‫א‬‫א‬‫א‬W٦
                                                    2 x − 3 y = 5 ‫א‬‫א‬‫א‬ (a 


                                                         - ٦٠ -
        ‫א‬‫א‬‫א‬                                 ١٧١                                             ‫א‬
      ‫א‬‫א‬                                                                      



                                                   2 x − 3 y = 5 ‫א‬‫א‬‫א‬ (b 
                                                                                                                   W‫א‬
                       W y = mx + b ‫א‬‫א‬‫א‬‫א‬
                                                              2       5
              2 x − 3 y = 5 ⇒ −3 y = −2 x + 5 ⇒ y = x − 
                                                              3       3
                                                                                      2
                                                                   W ‫א‬‫א‬‫א‬‫א‬
                                                                                      3
                                                                                  2
 ‫א‬‫א‬K‫א‬‫א‬‫א‬ m = ‫א‬‫א‬‫א‬ (a 
                                                                                  3
                                      W‫א‬‫א‬‫א‬‫א‬
                                     2                   2        4           2           7
       y = m( x − x1 ) + y1 = ( x − 2) + (−1) = x − − 1 = x − 
                                     3                   3        3           3           3
          W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ (b
                                                                                                                  1          3
                                              W (a ‫א‬‫א‬ m = − = −
                                                                                                                  2          2
                                                                                                                  3
                              3                    3      6               3       4           3
                   y = − ( x − 2) + (−1) = − x + − 1 = − x + = − x + 2 
                              2                    2      2               2       2           2
                                                          ‫א‬‫א‬‫א‬K٧
   ‫א‬ y ‫א‬K‫א‬‫א‬‫א‬
 K y ‫א‬‫א‬‫א‬ x ‫א‬ x ‫א‬‫א‬
         ‫א‬ y = 0  y ‫א‬‫א‬‫א‬ x = 0 ‫א‬
                                                                                                   K x ‫א‬‫א‬
                          K‫א‬ y = 2 x + 3 W‫א‬‫א‬‫א‬‫א‬W٧
                                                                                                                   W‫א‬
    (0 , 3) W y ‫א‬‫א‬‫א‬  x = 0 → y = 2 × 0 + 3 = 3 W y ‫א‬‫א‬
     ⎛ − , 0 ⎞ W y ‫א‬‫א‬‫א‬ y = 0 → 2 x + 3 = 0 → x = − W x ‫א‬‫א‬
         3                                                                  3
      ⎜       ⎟
       ⎝ 2        ⎠                                                                                2
                                                                                                                         




                                                       - ٦١ -
 ‫א‬‫א‬‫א‬                                           ١٧١                                            ‫א‬
‫א‬‫א‬                                                                              



                                                   
                      (d )  (a ) ‫א‬‫א‬‫א‬‫א‬W١
  1) (2 , 3) :       (a) 0      (b)1       (c ) − 2        (d ) 
                                                      1
   2) (−4 ,1) : (a ) 3           (b) − 3      (c )          (d ) 0 
                                                      3
                                         W‫א‬‫א‬‫א‬‫א‬W٢
                                                                                          ⎛7 3⎞ ⎛5 1⎞
   1) (3 , − 4) , (5 , 2)       2) (2 ,1) , (2 , 5)        3) (1, 2) , (−2 , 4)       4) ⎜ , ⎟ , ⎜ , − ⎟ 
                                                                                          ⎝8 4⎠ ⎝4 4⎠
               ‫א‬E‫א‬F‫א‬‫א‬W٣
   1) x + 5 y = 20             2) 6 x − 5 y = 15             3) x = 4          4) y = −1 
                                          W‫א‬‫א‬‫א‬‫א‬W٤
     1) (2 ,1) , (0 , − 3)        2) (−3 , − 4) , (1, 4)            3) (0 , 0) , (−1, 3)
                                                                      ⎛ 7 3 ⎞ ⎛ 5 1 ⎞ 
      4) (−3 , 6) , (1, 2)         5) (1, − 2) , (3 , − 2)          6) ⎜ , ⎟ , ⎜ , − ⎟
                                                                       ⎝8 4⎠ ⎝4 4⎠
                              ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W٥
                     3                          2                            3
   1) (0 , 3) , m =           2) (0 , 0) , m =           3) (−2 , 4) , m = − 
                     4                          3                            5
   4) (0 , 2) , m = 4         5) (0 , 4) , m = 0         6) (−1, 2) , m 
                        3 ‫א‬ x ‫א‬‫א‬‫א‬‫א‬W٦
                    W‫א‬‫א‬‫א‬‫א‬‫א‬W٧
                                                        ‫א‬‫א‬‫א‬ (a ) 
                                               ‫א‬‫א‬‫א‬ (b) 
                                                                                        ⎛7 3⎞
      1) (2 ,1) , 4 x − 2 y = 3                   2) (−3 , 2) , x + y = 7            3) ⎜ , ⎟ , 5 x + 3 y = 0
                                                                                        ⎝8 4⎠                 
      4) (−6 , 4) , 3 x + 4 y − 7 = 0             5) (2 , 5) , x = 4                6) (−1, 0) , y = −3




                                                          - ٦٢ -
                                              ‫א‬‫א‬‫א‬
                                           ã¹]<gè…‚jÖ]æ<ËÖ]<Üé×Ãj×Ö<íÚ^ÃÖ]<퉉ö¹]
                                               ‫א‬‫א‬‫א‬‫א‬




                         
                        ‫א‬
 ‫א‬




٥


                                  - ٥٤ -
  ‫א‬‫א‬                 ١٧١                                  ‫א‬
‫א‬                                     




                                                      ‫א‬‫א‬:‫اﻟﺠﺪارة‬
                                                                                          
                                                                        W‫א‬‫א‬
                                     W‫א‬‫א‬‫א‬
                                                           ‫א‬‫א‬        •

                      ‫א‬‫א‬‫א‬‫א‬          •

                                   ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬    •

                                              ‫א‬‫א‬‫א‬‫א‬      •

                                                                                          
                                                                                          
                                                       W‫א‬‫א‬




                                  - ٦٣ -
           ‫א‬‫א‬                                      ١٧١                                        ‫א‬
         ‫א‬                                                                 



                                                   ‫א‬
                                                                                                          ‫א‬‫א‬K١
                                                                      
    K‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬                         
                             
    ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬K ( o ) ‫א‬‫א‬?‫?א‬
     θ ‫א‬K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
                                
                   1
           
                       ‫א‬1o  360o  ‫א‬‫א‬
                                                                           
                  360
                                                                                                                    K
                                            K1 ‫א‬‫א‬‫א‬
                                                                                                                             
                  ‫א‬
                                              
                                               ‫א‬                   ‫א‬                  ‫א‬



                                                       90o                                                      180o
                           30o                                                      135o


                        0o < θ < 90o                  θ = 90o                        90o < θ < 180o                 θ = 180o


                                                                1 ‫א‬

                                                                                                                             
       E 60′ F 60 E1o F‫א‬‫א‬‫א‬، 60 ‫א‬ 60 ‫א‬
                                                                                              WE 60′′ F‫א‬ 60 ‫א‬
                                              E 60′ F 60 = E1o F‫א‬
                                             E 60′′ F 60 = E1′ F‫א‬
           θ = 61o35′47′′ W‫א‬ 47  35 ، 61 ‫א‬
                                                               ‫א‬ θ = 43o 74′89′′ W١
                                                                                                                        W‫א‬
                                        89′′ = 60′′ + 29′′ = 1′ 29′′  74′ = 60′ + 14′ = 1o14′ W‫א‬‫א‬
                                  θ = 43o 74′89′′ = (43o + 1o )(14′ + 1′)(29′′) = 44o15′29′′ W‫א‬
                                                                                                    

                                                                - ٦٤ -
         ‫א‬‫א‬                                           ١٧١                                                ‫א‬
    ‫א‬                                                                                



                                  1) 360o − 75o18′48′′                      2) 75o 23′41′′ + 34o 47′25′′ W٢
                                                                                                                                   W‫א‬
           1) 360 o − 75o18′48′′ = (359 o 59′60′′) − (75o18′48′′) = (359 o − 75o )(59′ − 18′)(60′′ − 48′′)

                                                                                      = 284 o 41′ 12′′
                                                                                                                                             
            2) 75 23′41′′ + 34 47′25′′ = (75 + 34 )(23′ + 47′)(41′′ + 25′′) = 109 70′66′′
                  o                  o                  o            o                                           o



                                             = (109o + 1o )(10′ + 1′)(6′′) = 110o11′ 6′′
         ‫א‬ 47 o30′ ‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
‫א‬‫א‬‫א‬K 23.75o  23o 45′ ‫א‬‫א‬ 47.5o 
                                                                                    K‫א‬‫א‬‫א‬‫א‬
    ‫א‬‫א‬ θ = 23.456o (2 ‫א‬‫א‬θ = 43o 25′51′′  (1 W٣
                                                                                                                                   W‫א‬
                                                                                              o                            o
                                                                      W‫א‬1′ = ⎛ ⎞                          ⎛ 1 ⎞
                                                                                    1
                                                                                  ⎜ ⎟             ‫و‬      1′′ = ⎜      ⎟ W‫א‬
                                                                                  ⎝ 60 ⎠                       ⎝ 3600 ⎠
                                                                         o
                                     ⎛ 25  51 ⎞
            1) θ = 43 25′51′′ = 43 + ⎜ +
                         o               o
                                               ⎟ = 43.431
                                                         o

                                     ⎝ 60 3600 ⎠

      2) θ = 23.456o = 23o + 0.456o = 23o + 0.456 × 60′ = 23o + 27.36′                                                  
                                   = 23o + 27′ + 0.36 × 60′′ = 23o + 27′ + 21.6′′ ≈ 23o 27′22′′
(radians)‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
    360o ‫א‬K rd ‫א‬
                                      W 2π rd  360o  2π rd ‫א‬‫א‬‫א‬
            W 2π rd = 360o ⇒ π rd = 180o 
                                                    o
                                                                                       π
                                 1rd = ⎛
                                            180 ⎞
                                         ⎜     ⎟               ‫أو‬             1o =         rd 
                                          ⎝ π ⎠                                       180
                    1rd = 57.29577951o                     ‫و‬            1o = 0.017453293 rd ‫א‬‫ א‬
                                                                                                                                          




                                                                     - ٦٥ -
         ‫א‬‫א‬                             ١٧١                                ‫א‬
       ‫א‬                                                


                                                   π
 1)150o , 548o 23′15′′  (rd ) ‫א‬، 2)       rd , − 5 rd  ( o ) WW٤
                                                   4
                                                                                                    W‫א‬
                             o                                     o
         π    π ⎛ 180 ⎞                                ⎛ 180 ⎞
       1) rd = ⎜      ⎟ = 45                − 5 rd = −5⎜     ⎟ = −5 × 57.296 = −286 28′48′′
                             o                                              o      o
                                       ,
         4    4⎝ π ⎠                                   ⎝ π ⎠
   
                   ⎛ π ⎞        5π                                            ⎛ π ⎞
       2)150o = 150⎜     ⎟ rd =        ,   548o 23′15′′ = 548.3875o = 548.3875⎜     ⎟ ≈ 9.5958 rd
                   ⎝ 180 ⎠       6                                            ⎝ 180 ⎠
                                                                                                        
                                                                                                        
                                      ‫א‬‫א‬‫א‬‫א‬‫א‬
                                 (o )         (rd )            (o )        (rd )                        
                                                                           2π                           
                                  0            0             120
                                                                            3                           
                                               π
                                 30                          180           π                            
                                               6
                                               π                           4π                           
                                 45                          240
                                               4                            3                           
                                               π                           3π
                                 60                          270                                        
                                               3                            2
                                               π                                                        
                                 90                          360           2π
                                               2                                                        




                                                    - ٦٦ -
    ‫א‬‫א‬                                          ١٧١                                             ‫א‬
‫א‬                                                                           



                                                       
                                                ‫א‬‫א‬‫א‬W١
           1) 225o              2)15o 4′9′′      3) 0 o      4) π rd 
                                                                                          W‫א‬W٢
                  1)15o 25′35′′ + 43o 35′27′′                 1)109o 47′38′′ + 43o 35′27′′
                                                                                                          
                  3) 57 43′28′′ − 27 31′ 49′′
                         o                o
                                                                4)123 13′20′′ − 27 31′ 49′′
                                                                           o                o


                                  ‫א‬‫،א‬‫א‬‫א‬‫א‬‫א‬‫א‬W٣
 1) 40.25o                  2) 75.2 o        3)17.45o              4) 96.6 o 
                                                      ‫א‬‫א‬‫א‬‫א‬‫א‬W٤
          π              3π               π          3π            5π                  4π
     1)             2)            3) −          4)            5)                6) −
         3               2               6           4             6                   5 
     7) 3π         8h) − 4π        9) 2          10) 5.3        11) − 6.4          12) − 8
                                                      ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W٥
 1) 45o             2) 225o         3) 75o      4) − 135o         5) 7 o 30′          6) − 270o 




                                                            - ٦٧ -
             ‫א‬‫א‬                                   ١٧١                                      ‫א‬
           ‫א‬                                                          



                                                                                    ‫א‬K٢
            K C ‫א‬‫א‬‫א‬K 2 ‫א‬ ABC ‫א‬‫א‬‫א‬‫א‬
     ‫א‬،Eθ F‫א‬ AC ‫א‬‫א‬‫א‬، θ ‫א‬‫ א‬A ‫א‬‫א‬‫א‬
W‫א‬‫א‬ θ ‫א‬‫א‬KE θ F‫א‬ BC
        sine , cosine , tan gent , cot angent , cosecant and secant 
                                                                                                              
                                              B
                        ‫א‬
                                                      KKKK sin θ , cosθ , tan θ Wθ ‫א‬
                                             ‫א‬
                                                          W‫א‬‫א‬‫א‬
       A            θ                        C                                                                
                         
                         ‫א‬                                                                                  
                        2 ‫א‬
                                                                                                              
                                           ‫اﻟﻤﻘﺎﺑﻞ‬              BC                     ‫اﻟﻮﺗﺮ‬              AB
                                  sin θ =                     =             cscθ =                      =   
                                              ‫اﻟﻮﺗﺮ‬             AB                  ‫اﻟﻤﻘﺎﺑﻞ‬               BC
                                           ‫اﻟﻤﺠﺎور‬              AC                    ‫اﻟﻮﺗﺮ‬              AB
                                  cosθ =                     =             secθ =                     =       
                                             ‫اﻟﻮﺗﺮ‬              AB                  ‫اﻟﻤﺠﺎور‬              AC
                                           ‫اﻟﻤﻘﺎﺑﻞ‬              BC                  ‫اﻟﻤﺠﺎور‬               AC
                                  tan θ =                    =             cot θ =                     =    
                                           ‫اﻟﻤﺠﺎور‬              AC                  ‫اﻟﻤﻘﺎﺑﻞ‬               BC
                                                                                                                
                                                                                                                
                                      B
                                                                              θ ‫א‬‫א‬‫א‬‫א‬W٥
                                                                                                                     W‫א‬
                                       5
                                                       ‫א‬‫א‬‫א‬
      A         θ
                        12            C                                       W ABC ‫א‬‫א‬‫א‬
                                           AB 2 = AC 2 + BC 2 ⇒ AB 2 = 5 2 + 12 2 = 169 ⇒ AB = 169 = 13
                                                                        θ ‫א‬‫א‬
                                       5        12          5                       13              13                12
                           sin θ =      cosθ =    tan θ =                 cscθ =          secθ =            cot =       
                                      13        13         12                        5              12                 5
                                                                                                                          

                                                              - ٦٨ -
        ‫א‬‫א‬                                         ١٧١                                            ‫א‬
      ‫א‬                                                                       


                                                                          7
                                                          K sin θ =          tan θ  cosθ W٦
                                                                          25
                                                                                                                            W‫א‬
                                                                                                                              7
 KE‫א‬F AB = 25  BC = 7 ‫א‬‫א‬ sin θ =                                              
                                                                                                                              25
                               AC 2 = AB 2 − BC 2 = 252 − 7 2 = 576 ⇒ AC = 576 = 24 W
                                         B                                                                              W
                    25                                                         24                    7
                                          7
                                                                   cosθ =              tan θ =         
                                                                               25                    24
  A             θ
                                         C                                                                                       
                                                              ‫א‬‫א‬‫א‬K١{٢
                                    
          ‫א‬‫א‬‫א‬‫א‬
            ‫א‬‫א‬ 45o ‫א‬‫א‬‫א‬K
                                                                        K‫א‬
          ‫א‬K 45o  A ‫א‬‫א‬ 3 ‫א‬ ABC ‫א‬‫א‬‫א‬‫א‬
K BC  AC ‫א‬ ABC ‫א‬ 45o  B ‫א‬‫א‬‫א‬
                      AC = AB = 1 ‫א‬
                         B                           W AB 2 = AC 2 + BC 2 = 12 + 12 = 2 ⇒ AB = 2 
   
 3 ‫א‬
                                                   BC   1     2                     AC   1     2                     1
            2                      sin θ =      =    =              cosθ =       =    =                 tan θ = = 1 
                             1                     AB    2   2                      AB    2   2                      1
 A      θ
            1            C

                                                                              ‫א‬
                                                        θ ( o , rd )       sin θ             cosθ               tan θ
                                                              o π              1                 3                    3
                                                          30 ,
                                                                   6           2                2                    3
                                                                   π            2                 2
                                                          45o ,                                                    1
                                                                   4           2                 2
                                                                   π            3                1
                                                          60o ,                                                       3
                                                                   3           2                 2
                                                                 - ٦٩ -
      ‫א‬‫א‬                                   ١٧١                                 ‫א‬
    ‫א‬                                                       



                                                             ‫א‬‫א‬‫א‬‫א‬K٢{٢
K‫א‬‫א‬‫א‬K‫א‬‫א‬
      ‫א‬‫א‬K‫א‬‫א‬‫א‬‫א‬‫א‬
                                                           K‫א‬‫א‬‫א‬‫א‬‫א‬
                K 40o  A ‫א‬‫א‬‫א‬ BC = 5  ABC ‫א‬‫א‬‫א‬‫א‬W٧
                                                                                                        W‫א‬
                       W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
              C ‫א‬‫א‬‫א‬‫א‬ AC  AB ‫א‬ B ‫א‬‫א‬‫א‬‫א‬
                                                         
                                                        W‫א‬ 180o ‫א‬‫א‬K 90o 
                     B                  B = 180o − ( A + C ) = 180 o − (40o + 90o ) = 50o
                 ?                        W‫א‬‫א‬E A , B , C ‫א‬‫א‬‫א‬F
AB = ?               BC = 5                   BC     5             5        5
                                 sin 40 o =      =     ⇒ AB =          =       ⇒ AB ≈ 7.8 
         40o                                  AB AB             sin 40 o 0.6427
A                        C
       AC = ?                           ‫א‬‫א‬ AC ‫א‬
                                                     WK‫א‬‫א‬
          AC 2 = AB 2 − BC 2 = (7.8) 2 − 5 2 = 60.84 − 25 = 35.84 ⇒ AC = 35.84 ≈ 6 
                                            K‫א‬‫א‬‫א‬
                                                                                                            
                                                K AC = 12  BC = 10 ‫א‬‫א‬‫א‬‫א‬W٨
                                                                                                        W‫א‬
                      ‫א‬‫א‬ AB ‫א‬K‫א‬‫א‬
                         B                AB 2 = 12 2 + 10 2 = 244 ⇒ AB ≈ 15.6 W
                 ?                                               W A ‫א‬‫א‬ tan ‫א‬‫א‬
AB = ?               BC = 10
                                                      10
                                             tan A =  ≈ 0.8333 ⇒ A ≈ 39o 48′ 
A      ?                 C
                                                      12
      AC = 12                           W180 ‫א‬‫א‬‫א‬ B ‫א‬‫א‬
                                             o


                               B = 180o − ( A + C ) = 180o − (39o 48′ + 90o ) = 180o − 129o 48′ = 50o12′
                                                                                                        

                                                       - ٧٠ -
             ‫א‬‫א‬                                                         ١٧١                                            ‫א‬
         ‫א‬                                                                                       



                                                                                   
                                                                           W‫א‬‫א‬ sin θ , cosθ , tan θ ‫א‬W١
    a)                       b)                          c)                              d)                   e)                       f)



         5                                                                                                         θ
                     4                          8                                                             30            34                         6
                                                                      13                      26    θ
         θ                                               5                                                                                   θ
                                  θ
                                                                              θ
             3                        6                                                        24                                                3

                                   cos 30o  sin 30o ، cos 45o ، sin 45o ‫א‬‫א‬‫א‬W٢
                                                                  B

                                                                                                                        B
                                                                      a
                                                                                                        2a
                                                     o                                                                  a
                                                45                                                      30o
                                                                                              A                         C
                                      A             a             C
                                                                                  12
                                                                    sin θ =         tan θ  cosθ  (1 W٣
                                                                                  13
                                                                                    1
                                                                           cosθ =  tan θ  sin θ  (2 
                                                                                    2
                                                                                    2
                                                                           tan θ =  cosθ  sin θ  (3 
                                                                                    3
                                                           W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W٤
    a)                       b)                              c)                          d)                        e)                       f)



                                          40o                                                                                   10               15
                                                                                     6        12         o
                                                                                                    51
          60o                                                                                                                                    43o
           4                          8                                   6
                                                                                                                           6




                                                                                    - ٧١ -
       ‫א‬‫א‬                                       ١٧١                                                   ‫א‬
    ‫א‬                                                                              



                                                                          ‫א‬K٣
   ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
‫א‬‫א‬F‫א‬ θ ‫א‬‫א‬‫א‬‫א‬K‫א‬‫א‬
 Kθ ‫א‬‫א‬E‫א‬F P( x , y ) ‫א‬E1 ‫א‬FE‫א‬
   r ‫א‬ ( x , y ) ‫א‬‫א‬‫א‬ θ  r = x 2 + y 2 W
                                             y                  x                        y
                             sin θ =  cosθ =  tan θ = W
                                             r                  r                        x
                   Kθ Kθ ‫א‬‫א‬E 2 ‫א‬F P(3 , 4) ‫א‬W٩
                                                                                                                               W‫א‬
                       y    4                     x 3                      y 4
      sin θ = =                         cosθ =    =        tan θ =         = W‫א‬ r = 32 + 4 2 = 5 W‫א‬
                       r    5                     r 5                      x 3
                                                                                      Kθ = 0o ‫א‬‫א‬W١٠
                            r = OP = 1 ‫א‬‫א‬E 3 ‫א‬F P(1, 0) ‫א‬‫א‬W‫א‬
                                                 y 0                            x 1                                y 0
                                     sin θ =      = =0              cosθ =       = =1                  tan θ =      = = 0 W
                                                 r 1                            r 1                                x 1
                      y                                         y                                              y


                           P( x , y )                                    P(3 , 4)
                                     •                                           •
                             r                                          r =5
                                         y                                           4
                                 θ                                          θ                                              P(0 ,1)
                                                                    o                                                      •
                       o         x           x                              3                x                 o      r =1       x
                              
                            1 ‫א‬                                      2 ‫א‬
                                                                                                                        
                                                                                                                       3 ‫א‬
                 y
                                                  ‫א‬ P ‫א‬‫א‬‫א‬‫א‬‫א‬
                                             E1 ‫א‬‫א‬‫א‬‫א‬F‫א‬
 P( x , y ) •
          r =1         θ                                         P ‫א‬،‫א‬‫א‬‫א‬
                  o                      x                                       WE 4 ‫א‬F r = OP = 1 
                                                                                              y
                                                   sin θ = y             cosθ = x              
                                                                                                     tan θ =
                                                                                              x
                  
                4 ‫א‬                                              W1  − 1  y  x 
                                                          − 1 ≤ sin θ ≤ 1                     ‫و‬             − 1 ≤ cosθ ≤ 1 

                                                             - ٧٢ -
  ‫א‬‫א‬                                ١٧١                                ‫א‬
‫א‬                                                   


                                                y
                                                                                                    
                                                                                                    
                                         sin θ > 0   sin θ > 0                                      
                                         cosθ < 0    cosθ > 0
                                         tan θ < 0   tan θ > 0                                      
                                         sin θ < 0   sin θ < 0
                                         cosθ < 0    cosθ > 0
                                                                    x
                                                                                                    
                                         tan θ > 0   tan θ < 0
                                                                                                    
                                                                                                    
                                                                                                    
                W sin e , cosine , tangent ‫א‬‫א‬‫א‬
                            sin θ , cosθ , tan θ ‫א‬‫א‬ θ ‫א‬‫א‬‫א‬ •
                       cosθ , tan θ  sin θ ‫א‬‫א‬ θ ‫א‬‫א‬‫א‬ •
               tan θ  sin θ , cosθ ‫א‬‫א‬ θ ‫א‬‫א‬‫א‬ •
                cosθ  sin θ , tan θ ‫א‬‫א‬‫א‬ θ ‫א‬‫א‬‫א‬ •




                                                 - ٧٣ -
      ‫א‬‫א‬                                       ١٧١                                                     ‫א‬
    ‫א‬                                                                            



                                                         
                                           W P ‫א‬‫א‬ θ ‫א‬‫א‬W١
          1) P(2 , 3)             2) P (3 , − 4)       3) P(1, 3 )               4) P(1, 2 2 )          5) P (2 , 0) 
                                     Wθ ‫א‬‫א‬ tan θ  cosθ ، sin θ W٢
                                3π
          1)180 o         2)                3) 360o           4) − π 
                                 2
                                              W‫א‬ tan θ  cosθ  sin θ W٣
                           5          12                            3 34           5                              4         4
         1) sin θ =        , cosθ =                2) cosθ =           , tan θ =                    3) sin θ = − , tan θ = 
                          13          13                             34            3                              5         3
                                                                                 W‫א‬‫א‬W٤
                               5π
 1) cos 50o         2) sin            3) tan 125o      4) sin 247 o              5) cos(−119o ) 
                                6
                                                                         W‫א‬‫א‬‫א‬W٥
                                                   5π              7π                                 π
 1)120o         2)135o        3) 210o       4)              5)              6) − 30o         7) −         8) − 150o 
                                                    3               4                                 4
                                                                             W [sin θ = (sin θ ) 2 ] ‫א‬W٦
                                                                                       2


                                                              π                π                                 π           π
     1) sin 2 0o + cos 2 0o                        2) sin 2        + cos 2                            3) 2 sin       cos
                                                              2                2                                 2           2
                                                                                                                                
             π        π         π        π                π             π          π       π                         2   π
    4) cos cos − sin sin                           5) cos cos + sin sin                               6)1 + tan
          4   2     4    2                               4   2     4    2                                                2
                                                                                                                                     




                                                               - ٧٤ -
      ‫א‬‫א‬                                     ١٧١                                       ‫א‬
    ‫א‬                                                               



                                                                                  ‫א‬‫א‬K٤
                                                      P( x, y ) ‫א‬‫א‬‫א‬‫א‬
                y                                                  WE 5 ‫א‬F
                                               W sin θ = y                cosθ = x 
                              •   P( x , y)
                      1                                          sin 2 θ + cos 2 θ = x 2 + y 2 
                          θ
                o                      x       W‫א‬E‫א‬‫א‬F x 2 + y 2 = r 2 = 1 

                                                                         
                                                                      (θ ‫א‬‫א‬) sin 2 θ + cos2 θ = 1
                                                          
               
             5 ‫א‬                                                2                           2
                                                              π  ⎛1⎞ ⎛ 3⎞     π
                                                                             1 3
                                                    sin + cos = ⎜ ⎟ + ⎜ ⎟ = + = 1 W
                                                          2               2
                                                                       ⎜ 2 ⎟
                                                        6     6 ⎝2⎠ ⎝      ⎠ 4 4
   W cos 2 θ ‫א‬،‫א‬‫א‬
                    sin 2 θ cos 2 θ   1    sin 2 θ                                        ⎛ 1           ⎞
                           +        =    ⇒         + 1 = sec 2 θ                          ⎜      = secθ ⎟
                    cos θ cos θ cos θ      cos θ                                          ⎝ cosθ
                        2       2      2       2
                                                                                                        ⎠
                                                                                                           
                                                                                         ⎛ sin θ         ⎞
                                                   ⇒ tan 2 θ + 1 = sec 2 θ               ⎜       = tan θ ⎟
                                                                                         ⎝ cosθ          ⎠
                                                      cot 2 θ + 1 = csc 2 θ W‫א‬‫א‬
                                                                                                                       
                                                                                                     2
 K cosθ ‫א‬‫א‬‫א‬K‫א‬‫א‬ θ ‫א‬‫א‬ sin θ = W١١
                                                                                                     3
                                                                                                                W‫א‬
                                                                                  2
                        W‫א‬ cos θ = 1 − sin θ = 1 − ⎛ ⎞ = 1 − = W‫א‬‫א‬
                                               2           2      2    4 5
                                                         ⎜ ⎟
                                                         ⎝3⎠           9 9
                     5                                                              5     5
          cosθ = − ‫א‬‫א‬‫א‬ cosθ ‫א‬ cosθ =   =±    
                    3                                                               9    3
             ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
                                                      (1 + sin θ )(1 − sin θ ) W‫א‬‫א‬‫א‬W١٢
                                                                                                                W‫א‬
                (1 + sin θ )(1 − sin θ ) = 12 − sin 2 θ = 1 − sin 2 θ W‫א‬‫א‬
               (1 + sin θ )(1 − sin θ ) = 1 − sin 2 θ = cos 2 θ W‫א‬‫א‬‫א‬

                                                          - ٧٥ -
      ‫א‬‫א‬                                  ١٧١                                         ‫א‬
    ‫א‬                                                              



                                                             ‫א‬‫א‬K٤{١
 K‫א‬‫א‬‫א‬‫א‬
                                                     cos(α + β ) = cos α cos β − sin α sin β  E١
                                                     cos(α − β ) = cos α cos β + sin α sin β  E٢
                                                    sin(α + β ) = sin α cos β + cosα sin β  E٣
                                                   sin(α − β ) = sin α cos β − cos α sin β  E٤
                                                                  tan α − tan β
                                                   tan(α + β ) =                    E٥
                                                                 1 + tan α tan β
                                                                  tan α + tan β
                                                   tan(α − β ) =                    E٦
                                                                 1 − tan α tan β
                      π             5π          π            5π        π              7π
            a ) cos        b) cos        c) sin      d ) sin    e) tan       f ) tan        WW١٣
                      12            12          12           12        12             12
                                                                                                               W‫א‬
   ‫א‬‫א‬‫א‬‫א‬
                                                                                                                   
                                                               π       π       π               π
                  WE٢‫א‬‫א‬             (a
                                                                   =       −       W
                                                 6          1212       4
        π       π π      π   π       π    π                          6+ 2
    cos = cos⎛ − ⎞ = cos cos + sin sin =
                                                2 3         2 1
              ⎜     ⎟                              ⋅   +     ⋅ =            ≈ 0.97
        12    ⎝4 6⎠      4   6       4    6    2 2         2 2         4
                                WE١‫א‬‫א‬‫א‬ (b
           5π      ⎛π π ⎞     π   π     π   π   2 3    2 1  6− 2
   cos       = cos⎜ + ⎟ = cos cos − sin sin =   ⋅  −   ⋅ =      ≈ 0.26
           12      ⎝4 6⎠      4   6     4   6  2 2    2 2    4
                                                                        WE٤‫א‬‫א‬ (c
            π       ⎛π π ⎞     π   π     π   π    2 3        2 1      6− 2
    sin      = sin ⎜ − ⎟ = sin cos − cos sin =      ⋅    −    ⋅ =            ≈ 0.26 
           12       ⎝4 6⎠      4   6     4   6   2 2        2 2          4
                                                WE٣‫א‬‫א‬ (d
            5π       ⎛π π ⎞     π   π     π   π   2 3    2 1  6+ 2
    sin       = sin ⎜ + ⎟ = sin cos + cos sin =   ⋅  +   ⋅ =      ≈ 0.97
            12       ⎝4 6⎠      4   6     4   6  2 2    2 2    4
                                          WE٦‫א‬‫א‬ (e



                                                      - ٧٦ -
      ‫א‬‫א‬                                    ١٧١                                    ‫א‬
    ‫א‬                                                           



                                    π          3   π
                              tan        − tan                1−
         π     ⎛π π ⎞       4      6 =        3 = 2 − 3 ≈ 0.27 
     tan = tan⎜ − ⎟ =
         12                   π
               ⎝ 4 6 ⎠ 1 + tan tan π            3
                                       1 + 1⋅
                              4    6           2
                                                  WE٥‫א‬‫א‬ ( f
                                     π              π
                               tan          + tan
            7π      ⎛π π ⎞              3              4 =       3 +1
 tan          = tan⎜ + ⎟ =                                            = −2 − 3 ≈ −3.73 
            12      ⎝3 4⎠                   π          π      1 − 3 ⋅1
                              1 − tan           tan
                                            3          4




                                                           - ٧٧ -
            ‫א‬‫א‬                                        ١٧١                                            ‫א‬
          ‫א‬                                                                      



                                                                
                                                                                    a
                                            tan θ  cosθ K sin θ = ‫א‬ θ W١
                                                                                    b
                                                    b
                cosθ  sin θ K tan = −         , a > 0 b > 0 ‫א‬‫א‬ θ ‫א‬‫א‬W٢
                                                    a
                                                                                                                                    
                                                                                            W‫א‬W٣
     1 − sin 2 θ                                1         1                        1          1                      1 − cos 2 t
  a)                                     b)           +                     c)           −                        d)
      1 + sin θ                             1 + sin x 1 − sin x                cos x − 1 cos x + 1                     tan 2 t
       sin x      1 + cos x                    sin x     cos x                           1                                         1
  e)            +                        f)           +                     g ) cos t −                          h) tan α +
    1 + cos x      sin x                    sin x + 1 sin x − 1                        cos t                                    tan α
                                              π            π                                 π       π       π
                                     sin          , cos        ‫א‬‫א‬          =       −       ‫א‬W٤
                                              12           12                               12       3       4
                                                                         W‫א‬‫א‬‫א‬W٥
                       3π                2π             4π          7π      7π
            a) cos            b) sin         c) cos             d ) sin  f ) sin 240o g ) cos 210o
                                                                                 e) tan
                        4                 3              3           6      12
                                                                    3                5
   WK‫א‬‫א‬ β ‫א‬‫א‬ α  cos β =  sin α = − W٦
                                                                    5               13
             a ) sin(α + β )        b) tan(α + β ) 
                                                                                                                   WW٧
           ⎛π    ⎞              ⎛π    ⎞                ⎛π    ⎞                                              ⎛π    ⎞
   a ) sin ⎜ − θ ⎟ = cosθ b) cos⎜ − θ ⎟ = sin θ c) sin ⎜ + θ ⎟ = cosθ                                d ) cos⎜ + θ ⎟ = − sin θ
           ⎝2    ⎠              ⎝2    ⎠                ⎝2    ⎠                                              ⎝2    ⎠
   d ) cos(π + θ ) = − cosθ e) sin(π + θ ) = − sin θ f ) cos(π − θ ) = − cosθ c) sin(π − θ ) = sin θ




                                                                  - ٧٨ -
                                                     ‫א‬‫א‬‫א‬
                                                  ã¹]<gè…‚jÖ]æ<ËÖ]<Üé×Ãj×Ö<íÚ^ÃÖ]<퉉ö¹]
                                                      ‫א‬‫א‬‫א‬‫א‬




                                 
                             ‫א‬‫א‬‫א‬‫א‬
 ‫א‬‫א‬‫א‬‫א‬




٦


                                         - ٦٤ -
     ‫א‬‫א‬                       ١٧١                              ‫א‬
‫א‬‫א‬‫א‬‫א‬                                        




                                              ‫א‬‫א‬‫א‬‫א‬‫א‬W‫א‬‫א‬
                                                                                                 
                                                                                 W‫א‬‫א‬
                                        W‫א‬‫א‬‫א‬
                           E‫א‬‫א‬J ‫א‬J ‫א‬‫א‬F‫א‬‫א‬‫א‬          •

                                 ‫א‬‫א‬‫א‬‫א‬‫א‬              •

                 E‫א‬J ‫א‬J ‫א‬J ‫א‬‫א‬J ‫א‬F‫א‬‫א‬‫א‬‫א‬              •

                            ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬           •

                                        ‫א‬‫א‬‫א‬‫א‬‫א‬          •

                                                                                                 
                                                                                                 
                                                            W‫א‬‫א‬




                                           - ٧٩ -
                 ‫א‬‫א‬                                                      ١٧١                                                          ‫א‬
           ‫א‬‫א‬‫א‬‫א‬                                                                                                   



                                                                ‫א‬‫א‬‫א‬‫א‬
                                                                                     
                                                                                                                ‫א‬‫א‬K١
                                                                               W‫א‬‫א‬‫א‬
                                                                                                                                                 K‫• א‬
                                                                                                                                                     K‫א‬‫• א‬
                                                                                                                                            ‫א‬‫א‬K١{١
         ‫א‬‫א‬ ،‫א‬‫א‬
                                                                                                                              W‫א‬
                                                                    KEF‫א‬ Ea
                                                                                            K‫א‬ Eb
                                                                                                   K‫א‬‫א‬‫ א‬Ec
                                                                                                                                            ‫א‬‫א‬K١{١{١
                                                                            W‫א‬‫א‬‫א‬
                              D                                                          A                      P = 2 EF
                                                                                                           = 2( AB + CD) 
                                                                                                  
                                                                              F
                                                                                                   W‫א‬
          C                   H                                       B                      A = ‫א‬‫א‬ × ‫א‬

 A = DH × CB    or          A = DF × AB 
                                                                                                                     
                                                                        K‫א‬‫א‬‫א‬ •
           K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫• א‬
                            K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫• א‬



                                                                                         - ٨٠ -
          ‫א‬‫א‬                                ١٧١                                    ‫א‬
    ‫א‬‫א‬‫א‬‫א‬                                                        



            ‫א‬‫א‬K 8 cm, 14cm ‫א‬W١
                                                                                          K 5cm ‫א‬‫א‬
                                                                                                             W‫א‬
                              P = 2 EFW‫א‬‫א‬
                                 P = 2(8 + 14) = 44 cm       
            W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W‫א‬
       A = ‫א‬ × ‫א‬ = 5 × 14 = 70 cm 2                               
D                                              A                                                   ‫א‬K٢{١{١
                                                                    K‫א‬‫א‬
     o                                        ow                                                    ‫• א‬
                                                     P = 2 [ (w) ‫א‬H (l ) ‫]א‬ = 2( AB + CD ) 
                                                                                             ‫• א‬
C                          l                   B
                                                            A = (w) ‫א‬ × (l ) ‫א‬ = AB × CD 
                                K‫א‬K11cm 17 cm W٢
                                                                                                                     W‫א‬
                                 P = 2 [ (w) ‫א‬H (l ) ‫]א‬ = 2(11 + 17) = 56 cm           W‫א‬
                                       A = (w) ‫א‬ × (l ) ‫א‬ = 11 × 17 = 187 cm 2         W‫א‬
                                     K‫א‬K16 ‫א‬، 320cm 2 W٣
                                                                                                                W‫א‬
                                                    (l ) ‫א‬ = ‫א‬ ( A) ÷ (w) ‫א‬
                                                                      320
                                                               l=        = 20cm                              
                                                                      16
                                                                                                 (P) ‫א‬
                                              P = 2 [ (w) ‫( + א‬l ) ‫ 27 = )02 + 61(2 = א‬cm ]
          K‫א‬KK 7 cm W٤




                                                          - ٨١ -
     ‫א‬‫א‬                                             ١٧١                                   ‫א‬
‫א‬‫א‬‫א‬‫א‬                                                                  



                                                                                                                   W‫א‬
             L = 3 × 7 = 21cm W (L) ‫א‬‫א‬
                                             
                                     P = 2 [ (w) ‫א‬H (l ) ‫]א‬ = 2(7 + 21) = 56 cm                W‫א‬
                                         A = (w) ‫א‬ × (l ) ‫א‬ = 7 × 21 = 147 cm 2              W‫א‬
                                                                                                    ‫א‬K٣{١{١
                                  o
                                                                          ‫א‬
                                                        P = 4 ×  (l ) ‫א‬ = 4 l         W (P) ‫א‬ •
       o                                        o            A = (l ) ‫ × א‬ (l ) ‫ = א‬l 2 W ( A) ‫א‬ •
                                                                  ‫א‬K 9cm W٥
                                                                                                              K
                                  o                                                                                W‫א‬
                              l
                                                              P = 4 ×  (l ) ‫א‬ = 4 l = 4 × 9 = 36 cm W‫א‬
                                                             A = (l ) ‫ × א‬ (l ) ‫ = א‬l 2  = 9 2 = 81cm 2 W‫א‬
                                                                           K‫א‬ 48 cm W٦
                                                                                                                   W‫א‬
                                                                 l = P ÷ 4 = 48 ÷ 4 = 12cm W‫א‬
                                                                            A = l 2 = (12) 2 = 144 cm 2 W‫א‬
                                                                        K‫א‬ 49cm 2 W٧
                                                                                                                   W‫א‬
                                                                       l = A = 49 = 7 cm  W‫א‬
                                            P = 4 × (l ) ‫א‬ = 4 l = 4 × 7 = 28 cm               W‫א‬
                                                                                                    ‫א‬K٤{١{١
                              A                o                     B
                                                                        W‫א‬‫א‬‫א‬
                                                   •                             K‫א‬ Ea
              o                                        o
                                                        l                                 K‫א‬ Eb
                      •
                                                                              ‫א‬ Ec
                          o
D                                 H        C                                         K
                                                                 - ٨٢ -
         ‫א‬‫א‬                                ١٧١                                ‫א‬
    ‫א‬‫א‬‫א‬‫א‬                                                   



         K‫א‬‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬ Ed
                                                              P = 4 ×  (l ) ‫א‬     ‫א‬ •
                                                               A = DC × AH                 ‫א‬ •
                                                              ‫א‬ × ‫א‬ = ‫א‬
                                                  ‫א‬‫א‬‫א‬
                                        1
                                A = E‫א‬‫א‬ × ‫א‬‫א‬F                                  
                                        2
    ‫א‬K 5cm ‫א‬13cm W٨
                                                                                                     K
                                                                                                     W‫א‬
                                        P = 4 × (l ) ‫א‬ = 4 l = 4 × 13 = 52 cm       W‫א‬
                                 A = ‫א‬ × ‫א‬ = 13 × 6 = 78 cm 2       W‫א‬
 ‫א‬‫א‬‫א‬ 4m, 7 m W٩
                                                                                  ‫א‬‫א‬،15
                                                                                                     W‫א‬
                          1                                           1
              A = E‫א‬‫א‬ × ‫א‬‫א‬F = (4 × 7) = 14 m 2 ‫א‬
                          2                                           2
                                                                     14 × 15 = 210     W‫א‬
                                                                                     ‫א‬K٥{١{١
             
                                        ‫א‬
    o                              •                K‫א‬‫א‬‫א‬
             
                                                P = ‫א‬‫א‬W‫א‬ •
o                                      •
                                                                       ‫א‬ •
                                        A = ‫א‬ × ‫א‬‫א‬
                                                          A = ‫א‬ × ‫א‬             W
                       K‫א‬‫א‬‫א‬‫א‬



                                                        - ٨٣ -
      ‫א‬‫א‬                             ١٧١                                   ‫א‬
 ‫א‬‫א‬‫א‬‫א‬                                                   



          K‫ א‬K11cm ‫א‬17cm ‫א‬W١٠
                                                                                                           
                                                                                                         W‫א‬
                                A = ‫א‬ × ‫א‬ = 17 × 11 = 187cm 2 
                                                                                      ‫א‬K٢{١
                               180o ‫א‬‫א‬‫א‬‫א‬
                       A                                                                           W‫א‬
                                                                                 K‫א‬ Ea
                                                                                 K‫א‬ Eb
           o                   o                                                 K‫א‬ Ec
                                                                                K‫א‬‫א‬‫א‬‫ א‬Ed
                                             ‫א‬‫א‬‫א‬‫א‬
                                                ∠CAB ‫א‬‫א‬ CB ‫א‬ A ‫א‬
C                      H                 B
                                                                             ‫א‬ •
                                        P = AB + BC + CA ‫א‬
                                                                                     ‫א‬ •
                                                             W‫א‬‫א‬
                                   1                                                1
                             A = × ‫א‬ × ‫א‬‫א‬ = × CB × AH 
                                   2                                                2
                              K 8 cm ‫א‬12 cm ‫א‬‫א‬W١١
                                                                                                     W‫א‬
            1                                                  1
   A = × ‫א‬ × ‫א‬‫א‬ = × 12 × 8 = 48 cm 2 W‫א‬
            2                                                  2
 K 8 cm ‫א‬‫א‬‫א‬K 7cm ‫א‬W١٢
                                                                                         K
                                                                                                     W‫א‬
                                                                     P = 3 × 7 = 21cm W‫א‬
                                                                          7×8
                                                                    A=       = 28 cm 2 W‫א‬
                                                                           2
                                                   - ٨٤ -
     ‫א‬‫א‬                             ١٧١                                                ‫א‬
‫א‬‫א‬‫א‬‫א‬                                                                



                                                                                ‫א‬‫א‬K٣{١
   ‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬
                                                                    ‫א‬‫א‬‫א‬‫א‬
                                                                       
 ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W‫א‬ -
                                                                                    K
 ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W‫א‬‫א‬J
                            ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W‫א‬‫א‬J
                                             ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬J
                                          K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬J
       r   ‫א‬                                                           ‫א‬‫א‬ •
                                           P = 2πr       W r ‫א‬‫א‬‫א‬
           d   ‫א‬                     E‫א‬‫א‬F‫א‬‫א‬ π 
                 ‫א‬
                                                                                                               
                                                                     22
                                                       ≈ 3,142                                   
                                                                     7
                                                                           W‫א‬‫א‬ •
                                          A = πr 2  W r ‫א‬‫א‬‫א‬
        K‫א‬ 2.8 m ‫א‬‫א‬W١٣
                                                                                                                    W‫א‬
                                                        2.8
                                                   r=       = 1.4 m  W‫א‬
                                                         2
                                               P = 2πr = 2 × 3.14 × 1.4 ≈ 8.8 m           W‫א‬
                                             A = πr 2 = 3.14 × (1.4) 2 ≈ 6.2 m 2                      W‫א‬
                                   ‫א‬ 66 m ‫א‬‫א‬W١٤
                                                                                                                    W‫א‬
                                            r = 66 ÷ 3.14 ≈ 21 m  W‫א‬
                                            A = πr 2 = 3.14 × (21) 2 ≈ 1385 m 2                       W‫א‬

                                                  - ٨٥ -
       ‫א‬‫א‬                                  ١٧١                                  ‫א‬
 ‫א‬‫א‬‫א‬‫א‬                                                        



                                                      
      ‫؟‬، 6 cm ‫א‬15 cm ‫א‬‫א‬ E١
‫א‬‫א‬‫א‬‫א‬،12 cm ‫א‬‫א‬ E٢
                                                                             K10 cm ‫א‬
،10 cm ‫א‬، 50 cm ،‫א‬‫א‬ E٣
                        A
                                              ‫؟‬12.35 m 2 ‫א‬‫א‬
                                    ‫א‬K BCDE ، ABCDE ‫א‬‫א‬‫ א‬E٤
 E                                   CD   8 cm   AC    ‫א‬ ABCDE 
                         B
                               8m                                                              K 6 cm 
                                                                                                               
                                                                                                               
D          6m           C                                                                                      
                                    ‫א‬ K       ‫א‬ ‫ א‬E٥
                    5m              4m
                                                                                         K‫א‬‫א‬
                                                                                                               
       4m                                                                                                      
                                             6m
                                                                                                               
       4m
                                                                                                               

                                                                        W‫א‬‫א‬ E٦
                                             2.5cm
             5cm
                                    1cm                  2.5cm                    70m

                                                                  20m
 6cm
             4cm                                                                                      25m
                             3cm                                               20m
                                             2cm                                                              25m
                                                                                        40m
       2cm               4cm           3cm              4cm

                                                                                                               

                                                         - ٨٦ -
       ‫א‬‫א‬                           ١٧١                               ‫א‬
  ‫א‬‫א‬‫א‬‫א‬                                             



 560 ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬، 42 cm ‫א‬‫א‬ E٧
                8cm                                                                             K
                                                      ‫א‬‫א‬‫א‬‫א‬‫ א‬E٨
                                                                                                        
                               8cm
                                                                                                        
                                                                                                        

                8cm
                                         ، 60 m   ‫א‬   E٩
                                               ‫؟‬‫א‬،
          K140 cm ‫א‬ 220 cm ،             E١٠
                                                              ‫؟‬‫؟‬‫א‬
                                         W‫א‬‫א‬‫א‬‫א‬‫א‬              E١١




                                                    5cm                              6cm
                                                                                              4cm
                                                          1cm

                                                                3cm
                                           
 ،‫א‬ ‫א‬      ، 27 m    ‫א‬             E١٢
                                                  ‫؟‬‫א‬‫א‬‫א‬K10 m 
K127 cm ‫א‬‫א‬، 27 m , 39m ‫א‬،‫א‬                 E١٣
                                                                        ‫؟‬‫א‬‫א‬‫א‬
                             ‫؟‬ 31.4 cm ،‫א‬‫א‬          E١٤
‫א‬‫א‬‫א‬K 81cm 2 ‫א‬                   E١٥
                                                                 K‫א‬
K12 m  17 m ‫א‬‫א‬،‫א‬                      E١٦
                                                                                              K‫א‬

                                                 - ٨٧ -
             ‫א‬‫א‬                                     ١٧١                                          ‫א‬
        ‫א‬‫א‬‫א‬‫א‬                                                                 



                                                                                                          ‫א‬‫א‬‫א‬K٢
                                                                                      
                  K‫א‬‫א‬‫א‬‫א‬‫א‬W‫א‬‫• א‬
     ‫א‬‫א‬‫א‬ W‫א‬‫• א‬
                                                                                        K‫א‬
     ‫א‬‫א‬WE‫א‬F‫א‬‫• א‬
      ‫א‬    ‫א‬ ‫א‬        ‫א‬ ‫א‬ 
                                                                                                     K‫א‬‫א‬
                     K‫א‬‫א‬‫א‬‫א‬‫א‬W‫א‬ •
                                                                        ‫א‬‫א‬K١{٢
          ‫א‬، 
                                                            K‫א‬‫א‬‫א‬
                                               K h ‫א‬ w ‫א‬، l ‫א‬W‫א‬‫א‬
                                                                              W‫א‬‫א‬‫א‬‫• א‬
                                                           A = 2(l × w + l × h + w × h) 
                                  h                                        V = l × w × h ‫א‬‫א‬ •
                                                   K 7cm, 9cm, 11cm ‫א‬‫א‬W١٥ 
                                                                                           K‫א‬‫א‬
                                  ‫א‬                                                                                   W‫א‬
                                      w
                   ‫א‬                                                     w = 7, l = 9, h = 11         ‫א‬
                    l
                                                                                  W‫א‬‫א‬
                  A = 2(l × w + l × h + w × h) = 2(9 × 7 + 9 × 11 + 7 × 11) = 2(63 + 99 + 77) = 478 cm 2
                                                                          W‫א‬
 V = l × w × h = 9 × 7 × 11 = 693 cm 3 
                                                                                                ‫א‬K٢{٢
                                           ،
                                                       K‫א‬‫א‬
                                                                  - ٨٨ -

                              l
         ‫א‬‫א‬                                   ١٧١                                     ‫א‬
    ‫א‬‫א‬‫א‬‫א‬                                                           



                                                                        l EF‫ א‬‫א‬
                                                                  A1 = 4l 2        ‫א‬ •
                                                                           A2 = 6l 2        ‫א‬ •
                                                             V = l 3             W
     ‫א‬‫א‬‫א‬K 7cm ‫א‬W١٦
                                                                                                              K
                                                                                                                W‫א‬
                                                    A1 = 4l 2 = 4(7) 2 = 196 cm 2  W‫א‬‫א‬
                                                   A2 = 6l 2 = 6(7) 2 = 294 cm 2  W‫א‬‫א‬
                                                V = l 3 = 7 3 = 343 cm 3               W‫א‬
                                                                                 ‫א‬‫א‬K٣{٢
                K‫א‬‫א‬
         K‫א‬‫א‬‫א‬‫א‬‫א‬
                                    K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
                                                                               ‫א‬‫א‬‫• א‬
                   W h ‫א‬ r ‫א‬‫א‬‫א‬‫א‬
                                                        A = 2πr 2 + 2πrh = 2π (r 2 + h)                          
                                                                                              ‫א‬‫א‬ •
                                 v = πr 2 h W r ‫א‬‫א‬‫א‬
 K‫א‬K11cm ‫א‬ 9cm ‫א‬‫א‬W١٧
                                                                                                                  W‫א‬
   A = 2π (r 2 + h) = 2 × 3.14 × ((9) 2 + 11) = 6.28 × 92 = 577.76 cm 2  W‫א‬‫א‬‫א‬
                                   v = πr 2 h = 3.14 × (9) 2 × 11 = 2797.74 cm 3                   W‫א‬‫א‬
                                                                                                                          
                                                                                   ‫א‬K٤{٢
         ‫א‬، r ‫א‬‫א‬
                                                                                       K‫א‬‫א‬‫א‬‫א‬
                                                            - ٨٩ -
     ‫א‬‫א‬                                   ١٧١                                       ‫א‬
‫א‬‫א‬‫א‬‫א‬                                                             



                                                                      ‫א‬‫• א‬
                                W h ‫א‬ r ‫א‬‫א‬‫א‬
                                                                       2
                                                            Al = πrh 
                                                                       3
                                                                              ‫א‬‫• א‬
                      W h ‫א‬ r ‫א‬‫א‬‫א‬
           h
                                                                       2
                                                          AT = πrh + πr 2 
                                                                       3
                                                                                  W‫א‬ •
                       r        W h ‫א‬ r ‫א‬‫א‬
                                                     1
                                          V = πr 2 h                                                           
                                                     3
       ‫א‬ h = 11cm ‫א‬  r = 14 cm ‫א‬W١٨
                                                                                       K‫א‬‫א‬
                                                                                                              W‫א‬
                                     2         2
                              Al = πrh = × 3.14 × 14 × 11 = 322.38 cm 2 W‫א‬‫א‬
                                     3         3
                                                                                        W‫א‬‫א‬
 AT = πrh + πr 2 = πr ⎛ h + r ⎞ = 3.14 × 14 × ⎛ × 11 + 14 ⎞ = 43.96 × 21.34 = 938.11cm 2
               2                   2                               2
                       ⎜       ⎟               ⎜           ⎟
               3                  ⎝3    ⎠                         ⎝3⎠
                                        1        1
                                    V = πr 2 h = × 3.14 × (14) 2 × 11 = 2256.61cm 3 W‫א‬
                                        3        3
                                                                           ‫א‬K٥{٢
 ‫א‬‫א‬‫א‬
‫א‬‫א‬‫א‬
                                               KKKKK‫א‬‫א‬‫א‬
                            K‫א‬‫א‬‫א‬‫ א‬
                                                    ‫א‬‫א‬‫א‬‫א‬‫א‬




                                                         - ٩٠ -
     ‫א‬‫א‬                               ١٧١                               ‫א‬
‫א‬‫א‬‫א‬‫א‬                                                 



                                                                              ‫א‬‫• א‬
                                   W h ‫א‬ l ‫א‬‫א‬‫א‬
                      h
                                                                    A = l 2 + 2lh 
                                                                                        ‫א‬ •
                                            W h ‫א‬ l ‫א‬‫א‬
                                                                         1
                                                                  V = l 2h 
                                                                         3
      ‫א‬K 7 cm 15 cm ‫א‬W١٩
                                                                                              K
                                                                                                  W‫א‬
                                     Ab = 7 × 7 = 49 cm 2 W‫א‬‫א‬
                                                   1         1
                                           V = l 2 h = × (7) 2 × 15 = 245 cm 3        W‫א‬
                                                   3         3
                                                                  ‫א‬K٦{٢ •
  ‫א‬‫א‬‫א‬
                                K‫א‬‫א‬‫א‬‫א‬‫א‬
                                                                        ‫א‬‫• א‬
                                           A = 4π r  W r ‫א‬‫א‬
                                                                                       ‫א‬ •
                                                        4
                           r                     V = π r 3  W r ‫א‬‫א‬
                                                        3
                                              ‫א‬K17 cm W٢٠
                                                                                  K‫א‬
                                                                                                  W‫א‬
                                A = 4π r = 4 × 3.14 × 17 = 213.52 cm 2 W‫א‬‫א‬
                                4         4
                        V = π r 3 = × 3.14 × (17) 3 = 20569.09 cm 3                 W‫א‬
                                3         3




                                                    - ٩١ -
       ‫א‬‫א‬                             ١٧١                                 ‫א‬
  ‫א‬‫א‬‫א‬‫א‬                                                 



                                                 
K 8 m ‫א‬،13 m ،17 m ،‫א‬‫א‬ E١
                                                                             ‫؟‬‫؟‬‫א‬‫א‬
،‫א‬‫א‬،، 850cm 3 ، E٢
‫א‬K11cm ،13 cm ،‫א‬‫א‬
                                                                                            ‫؟‬‫א‬‫א‬
‫א‬، 25 cm  84 cm ،‫א‬ E٣
                        ‫؟‬‫א‬‫א‬K،‫א‬‫א‬، 5 cm ،
‫א‬     9 cm  ‫א‬     19 cm        E٤
                                                                K‫א‬K13 cm ‫א‬‫א‬
‫א‬K 27 cm  ‫א‬ E٥
                                    2
               ‫؟‬‫א‬   ‫א‬13 m ‫א‬
                                   17
K12 m ‫א‬ 360m 2  E٦
                                                                                       K‫א‬
‫א‬‫א‬ H ‫א‬‫א‬K S  SABCD ‫א‬ ABCD ‫א‬ E٧
                                                   ‫א‬K‫א‬ SH ‫א‬
                             S
                                                                            ‫א‬‫א‬‫א‬
                                                             AB = 6.4 cm, BC = 4.8 cm, SH = 7.5 cm


                   D                      C
                                                                                                         

      A                            B           ،V ‫א‬ h ‫א‬ E٨
                             K‫א‬‫א‬17 cm ‫א‬‫א‬
‫א‬13 cm ‫א‬‫א‬‫ א‬E٩
                                                                                          K


                                                    - ٩٢ -
       ‫א‬‫א‬                       ١٧١                               ‫א‬
  ‫א‬‫א‬‫א‬‫א‬                                         



 7 cm ‫א‬‫א‬K‫א‬‫א‬‫א‬                    E١٠
                                               K12 cm ‫א‬‫א‬‫א‬‫א‬
 C2 ‫א‬‫א‬‫א‬K C1 ,C2 ‫א‬               E١١
V1 K C1  ‫א‬‫א‬ h ‫א‬‫א‬ C1 ‫א‬
 C2 ‫א‬‫א‬‫א‬‫א‬K‫א‬ V3  C2 ‫א‬‫א‬ V2  C1 ‫א‬‫א‬
                                                           KV2 = V1 + V3 ‫א‬ h 



                                       C1


                                             C2


                                                                                                  
                                                                                                  
                                                                               ‫א‬‫א‬‫א‬
                                                                W‫א‬‫א‬‫א‬K١{٣
  ،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
           ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
                                                                          ‫א‬‫א‬
                                                                       W‫א‬‫א‬‫א‬J a
                                                   (MKSsystem) J ‫א‬J 
                                                                      W‫א‬‫א‬‫א‬J b
                                                       (CGS system)–‫א‬–
                                                                          W‫א‬‫א‬J c
                                                       (FPS system)–J 




                                             - ٩٣ -
       ‫א‬‫א‬                                    ١٧١                                        ‫א‬
  ‫א‬‫א‬‫א‬‫א‬                                                               



                                                                                 W‫א‬‫א‬‫א‬K٢{٣
                         W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
                                                                
                                                          Ampere‫א‬J                                Meter‫א‬J 
                                                         Kelvin‫א‬J           Kilogram‫א‬‫א‬-
                                                         Candela‫א‬J                            Second‫א‬-
                                                  K‫א‬‫א‬‫א‬‫ א‬‫א‬
                             ‫א‬                ‫א‬     ‫א‬                 ‫א‬
                Quantity                     Symbol Unit                    Dimensions
                                                                                    
                   Length              ‫א‬        L         Meter ( m )                        L

                Area                ‫א‬         A         m2                    L2

                  Volume             ‫א‬         V          m3                    L3

                   Speed             ‫א‬         v         m s                   LT

                  Acceleration‫א‬                         m s2                           2
                                                     a                               LT

                   Mass                ‫א‬    m, M     Kilogram ( Kg )                    M

                  Density            ‫א‬        ρ         kg m 3                M L3

                  time          ‫א‬      t      Second ( S )                   t

                                  ‫א‬         -         Candela ( od )             -
                 Luminous Dentensity
                                  ‫א‬‫א‬      T       Degree Kelvin                  T
                temperature                                    (oK )
                  Force                 ‫א‬      F              N                 ML T 2

‫א‬‫א‬‫א‬‫א‬Wmeter‫א‬K١J J J J J J J J J J J
 E‫א‬ ‫א‬ F ٨٦J   ‫א‬ ‫א‬  ‫א‬ ‫א‬ ‫א‬  
        K‫א‬1650763.73 ‫א‬‫א‬‫א‬،‫א‬
 90 0 0 ‫א‬‫א‬W kilogram ‫א‬‫א‬K٢
   ‫א‬ ‫א‬ ‫א‬   ‫א‬  10 0 0  ، ‫א‬ 
                                            K 39mm ‫א‬‫א‬‫א‬

                                                          - ٩٤ -
       ‫א‬‫א‬                              ١٧١                                     ‫א‬
  ‫א‬‫א‬‫א‬‫א‬                                                     


                                    1
‫א‬‫א‬‫א‬                    ‫א‬WSecond ‫א‬K٣
                               31556925.975
 ‫א‬   ‫א‬ ‫א‬  ‫א‬‫א‬ ‫א‬   ١٩٠٠  ‫א‬‫א‬ ‫א‬
                                                  K‫א‬‫א‬‫א‬
                               1
       ‫א‬‫א‬              ‫א‬‫א‬‫א‬WKelvin‫א‬K٤
                             273.16
                                                                                                     K‫א‬
                                                                   ‫א‬‫א‬K٣{٣
                                                  
   ‫א‬‫א‬‫א‬‫א‬
‫א‬‫א‬K 20mi ‫א‬‫א‬‫א‬،‫א‬‫א‬K‫א‬‫א‬
                              K‫א‬‫א‬K‫א‬‫א‬‫א‬
                                                                              ،‫א‬،
                                                                      100cm = 1m 
                                                                          100cm 
                                                                       1m
                                                           1 =         = 0.010m cm 
                                                                     100cm
K‫א‬‫א‬K‫א‬‫א‬‫א‬‫א‬‫א‬
‫א‬K‫א‬‫א‬‫א‬
                                                  K‫א‬‫א‬‫א‬
‫א‬  ‫א‬ ‫א‬   ‫א‬  ‫03 ؟‬mi   ‫א‬ ‫א‬   J ١
‫א‬‫א‬K1.60 km mi ‫א‬ 1.600km = 1mi
                                              ،K‫א‬
                                                            km
                                       30mi = (30mi)(1.60     ) = 48km 
                                                            mi
      60 s = 1 min 1h = 60 min F‫ 000,002 ؟‬s ‫א‬‫א‬J ٢
                                                    1min 20.000
                             200.000s = (200.000s )(      )=          min 
                                                     60s          6
                                                                 20.000          1h
                                                  = (        min)(           ) = 55.6h 
                                                                   6          60 min
                                                in  5 cm ‫א‬ cm  30 in ‫א‬J ٣

                                                     - ٩٥ -
       ‫א‬‫א‬                                      ١٧١                                          ‫א‬
  ‫א‬‫א‬‫א‬‫א‬                                                                  


                                                                                          cm
                                                                       30 in = (30 in)(2.54  ) = 76.2 cm  
                                                                                           in
                                                                                            in
                                                                       5 cm = (5cm)(0.3937 ) = 1.97in  
                                                                                            cm
                                                               ‫א‬‫א‬‫א‬‫א‬
                                                                                   ‫א‬‫א‬ ‫א‬ ‫א‬K١{٣{٣
                                                                                   
                                 o                 /                      //             RADIAN                     Rev
 1degree =   1                               60                   3600                 1.745 × 10 −2        2.778 × 10 −3
 1minute =   1.667 × 10 −2                   1                    60                   2.909 × 10 −4        4.630 × 10 −5
 1second =   2.778 × 10 −4                   1.667 × 10 −2        1                    4.848 × 10 −6        7.716 × 10 −7
 1RADIAN = 57.30                               3438               2.063 × 10 5         1                    0.1592
 1revolution 360                             2.16 × 10 4          1.296 × 10 6         6.283                1
 =


                                                                                     ‫א‬‫א‬‫א‬K٢{٣{٣
                              cm           Meter              Km               In.                 Ft                  Mi
1centimeter =           1                10−2
                                                        10   −5
                                                                           0.3937              3.281 × 10   −2
                                                                                                                 6.214 × 10 −6
1Meter =                100              1              10 −3              39.37               3.281             6.214 × 10 −4
1Kilometer=             10 5             1000           1                  3.937 × 10 4        3281              0.6214
1inch =                 2.540            2.540 × 10 −   2.540 × 10 −5      1                   8.333 × 10 −2     1.578 × 10 −5
1foot =                 30.48            0.3048         3.048 × 10 −4      12                  1                 1.894 × 10 −4
1mile =                 1.609 × 105      1609           1.609              6.336 × 10 4        5280              1
                                                                                       ‫א‬‫א‬‫א‬K٣{٣{٣
                                     METER 2                 cm 2                       ft 2                      in. 2
 1SQUARE                     1                      10 4                       10.76                    1550
 METER =
 1square                     10 −4                  1                          1.076 × 10 −3             0.1550
 centimeter =
 1square foot =              9.290 × 10 −2          929.0                      1                        144
 1square inch =              6.452 × 10 −4          6.452                      6.944 × 10 −3            1




                                                             - ٩٦ -
      ‫א‬‫א‬                                      ١٧١                                          ‫א‬
 ‫א‬‫א‬‫א‬‫א‬                                                                   



                                                                                        ‫א‬‫א‬‫א‬K٤{٣{٣
                             Meter3                    cm 3                   L               ft 3                 in. 3
1CUBIC                      1                  10 6                        1000          35.31                6.102 × 10 4
METER =
1cubic                      10 −6              1                       1.000 × 10 −3     3.531 × 10 −5        6.102 × 10 −2
centimeter =
1liter =                    1.000 × 10 −3      1000                    1                   3.531 × 10 −2      61.02
1cubic foot =               2.832 × 10 −2      2.832 × 10 4 28.32         1                                   1728
1cubic inch =               1.639 × 10 −5      16.39        1.639 × 10 −2
                                                                           5.787 × 10 −4                      1
                                                                                         ‫א‬‫א‬‫א‬K٥{٣{٣
                       g             Kilogram                 u                 oz              lb                 ton
1gram       1                       0.001              6.022 × 10 2 3.527 × 10 −2 2.205 × 10 −3 1.102 × 10 −6
1Kilo-      1000                    1                  6.022 × 10 2 35.27              2.205 1.102 × 10 −3
gram
1u          1.661 × 10 −24          1.661 × 10 −27 1                       5.857 × 10 −26 3.662 × 10 −27 1.830 × 10 −30
                                                                                                              1
1ounce 28.35                        2.835 × 10   −2
                                                       1.718 × 10 1    2
                                                                                           6.250 × 10     3.125 × 10 −5
                                                                                                         −2

1pound 453.6                        0.4536             2.732 × 10 2 16                                  1 0.0005
1ton   9.072 × 10 5                 907.2              5.463 × 10 2 3.2 × 10 4                       2000 1



                                                                                      ‫א‬‫א‬‫א‬K٦{٣{٣
                                y                d             h            min         SECOND
1year                  1                    365.25        8.766 × 10 3
                                                                        5.259 × 10 5
                                                                                      3.156 × 10 7
1day                   2.738 × 10 −3        1             24            1440          8.640 × 10 4
1hour                  1.141 × 10 −4        4.167 × 10 −2 1             60            3600
1minute                1.901 × 10 −6        6.944 × 10 −4 1.667 × 10 −2 1             60
1Second                3.169 × 10 −8        1.157 × 10 −5
                                                          2.778 × 10 −4
                                                                        1.667 × 10 −2
                                                                                      1
                                                                                                                           




                                                              - ٩٧ -
      ‫א‬‫א‬                             ١٧١                                        ‫א‬
 ‫א‬‫א‬‫א‬‫א‬                                                        



                                                                            ‫א‬‫א‬‫א‬K٧{٣{٣
                            atm      dyne/cm 2        cm Hg         PASCA          lb/in. 2          lb/ft 2
                                                                      L

1atmosphere 1                       1.013 × 10 6 76                 1.013 × 10 14.70             2116

1dyne per              9.869 × 10 −7 1            7.501 × 10 −5 0.1           1.405 × 10 − 2.089 × 10 −3
cm 2
1centimeter            1.316 × 10 −2 1.333 × 10 4 1                 1333      0.1934             27.85
of mercury a
at 0 0 C
1PASCAL                9.869 × 10 −6 10           7.501 × 10 −4 1             1.450 × 10 − 2.089 × 10 −2

1pound per             6.805 × 10 −2 6.895 × 10 4 5.171             6.895 × 10 1                144
in 2
1pound per             4.725 × 10 −4 478.8                          47.88     6.944 × 10 − 1
ft2                                               3.591 × 10   −2




                                                      - ٩٨ -
                                                  ‫א‬‫א‬‫א‬
                                               ã¹]<gè…‚jÖ]æ<ËÖ]<Üé×Ãj×Ö<íÚ^ÃÖ]<퉉ö¹]
                                                   ‫א‬‫א‬‫א‬‫א‬




                              
                           ‫א‬‫א‬
 ‫א‬‫א‬




٧


                                      - ٧٨ -
   ‫א‬‫א‬                        ١٧١                            ‫א‬
‫א‬‫א‬                                       




                          ‫א‬‫א‬‫א‬‫א‬W‫א‬‫א‬
                                                                                          
                                                                             W‫א‬‫א‬
                                          W‫א‬‫א‬‫א‬
                                    ‫א‬‫א‬‫א‬‫א‬          •

                                      ‫א‬‫א‬‫א‬       •

                                    ‫א‬‫א‬‫א‬         •

                                                                                          
                                                       W‫א‬‫א‬




                                       - ٩٩ -
     ‫א‬‫א‬                                                               ١٧١                                            ‫א‬
 ‫א‬‫א‬                                                                                               



                                                           ‫א‬‫א‬
                                                                                                                                                   
                                                                                                                       ‫א‬‫א‬K١
                                                                                K١{١
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
                                                                                                                           K‫א‬‫א‬
 ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
                                                                                    ‫א‬ n ‫א‬
                                       z
                                                                                K xi , yi ‫א‬ i = 1,2,3,..., n Wi 
                                                                                                 ‫א‬‫א‬‫א‬‫א‬
                                                                                                                    W‫א‬
                                       W
                                           •                                                          W = ∑Wi
                           Wi                                                                                  i

                                                                                            ‫א‬‫א‬‫א‬
                                               xi                         y
                 yi                •
 x                                                                                                                   xc , yc ‫א‬
                      t                                                                     ∑ xiWi           x1W1 + x2W2 + x3W3 + ... + x3W3
                                                                               xc =          i
                                                                                                         =
                                                                                             ∑Wi                W1 + W2 + W3 + ... + Wn
                                                                                                 i

                                                                                            ∑ yiWi           y1W1 + y 2W2 + y3W3 + ... + y3W3
                                                                               yc =          i
                                                                                                         =
                                                                                             ∑Wi                 W1 + W2 + W3 + ... + Wn
                                                                                                 i

      ‫א‬ γ  t ‫א‬، Ai  A ‫א‬‫א‬
                                                W‫א‬‫א‬‫א‬‫א‬

                          ∑ xi ( Ai tγ )                tγ ∑ xi ( Ai tγ )          ∑ xi ( Ai tγ )            x1 A1 + x2 A2 + x3 A3 + ... + x3 A3
             xc =          i
                                                =           i
                                                                              =     i
                                                                                                         =
                           ∑ ( Ai tγ )                              tγA                          A                A1 + A2 + A3 + ... + An
                               i

                          ∑ yi ( Ai tγ )                tγ ∑ yi ( Ai tγ )          ∑ yi ( Ai tγ )             y1 A1 + y 2 A2 + y3 A3 + ... + y3 A3
             YG =          i
                                                    =           i
                                                                               =        i
                                                                                                         =
                               ∑ ( Ai tγ )                          tγA                              A              A1 + A2 + A3 + ... + An
                                i

                                                                              - ١٠٠ -
          ‫א‬‫א‬                                          ١٧١                                             ‫א‬
     ‫א‬‫א‬                                                                            



                 ‫א‬‫א‬‫א‬ xc , yc ‫א‬ A ‫א‬‫א‬‫א‬
                               K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
    ‫א‬                         ‫א‬                                                          A            xc             yc
    
                                     y                                                                                 
                                             xc                                                                        
                                                                                                                       
     ١                                                      a                    a2              
                                                                                                               a
                                                                                                                       
                                                                                                                               a
                                                                                                               2               2
                               yc
                                                                             x
                                                         a

                                  y                                                                                    
                                               xc
                                                                                                                       
                                                                                    ab              
                                                                                                               a
                                                                                                                       
                                                                                                                               b
                                                                                                               2               2
     ٢                                                        b
                              yc
                                                                             x
                                                     a

                                     y                                                                                 
                                                  xc                                                                   
     ٣                                                                  b       
                                                                                                 ab     J             
                                                                                                                               b
                                                                                                  2                            3
                              yc                                                 x                                         
                                 y                                                                                     
                                             xc                                                                        
     ٤      ‫א‬‫א‬                                                                                                 
                                                                        b                            a + b cosθ           b sin θ
                              yc
                                           θ
                                                                                         ab sin θ                
                                                                                 x                         2                  2
                                                         a




                                                               - ١٠١ -
        ‫א‬‫א‬                                                        ١٧١                                                ‫א‬
    ‫א‬‫א‬                                                                                            



                                    y         xc                                                                                         
                                                                                                                                         
                                                                                                                                         
    ٥         ‫א‬                                                            d
                                                                                                 π a2          a=
                                                                                                                               d
                                                                                                                                    a=
                                                                                                                                                      d
                             yc                                  a                                                             2                      2
                                                                                            x


                                    y
                                                    xc
                                                                                                                                           
    ٦  ‫א‬                                                                                   π a2            a               
                                                                                                                                                 4a
                                                                                                         2                                       3π
                              yc                                     a                                                                       
                                                                                        x

                                    y        xc                                                                                          
                                                                                                                                         
    ٧                                                                                              π a2       
                                                                                                                           4a
                                                                                                                                     
                                                                                                                                                 4a
                                                                                                         4                 3π                    3π
           ‫א‬         yc
                                                                          x
                                          90
                                                     a

                                y                                                                                                        
                                                                                                                                         
    ٨  ‫א‬                                                                                      a 2θ       2a sin θ         4a sin 2 θ
                                               θ
                                                                                                                                                         2
                                                                               yc
                                                                                                         2           3θ
                                                                                    x                                                            3θ
                                                         a
                                               xc

                                              xc                                                                                         
                                                                                                                                         
    ٩                                                                      b               π ab                   a                     b
                              yc                                                                                                        
                                                                                            x            4                 2                     2

                                                             a




                                                                         - ١٠٢ -
           ‫א‬‫א‬                                    ١٧١                                             ‫א‬
   ‫א‬‫א‬                                                                        



                                                                       ‫א‬‫א‬‫א‬K٢
‫א‬‫א‬‫א‬‫א‬‫א‬     
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
                                                                                                                   ‫א‬
                                                                                             W‫א‬
           y
                                                                    ∑ xi Ai             x1 A1 + x2 A2 + x3 A3 + ... + x3 A3
                                                            xc =     i
                                                                                   =
                                                                    ∑ Ai                     A1 + A2 + A3 + ... + An
                                                                        i

                                                                    ∑ yi Ai             y1 A1 + y 2 A2 + y3 A3 + ... + y3 A3
   yc                                                       yc =     i
                                                                                    =
                                                                     ∑ Ai                     A1 + A2 + A3 + ... + An
                                                                         i
                                                  x
                                                           ‫א‬‫א‬‫א‬ A1 , A2 , A3 ,...., An 
                      xc
                                                                                  x1 , x2 , x3 ,...., xn ،‫א‬‫א‬
                             K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫ א‬y1 , y 2 , y3 ,...., y n 
               ‫א‬‫א‬‫א‬
       y           xc                            y           xc                                           ‫א‬‫א‬

                                1                            1                     2
                                          yc
                    3       2                                                3                                                
  yc

                                          x                                                    x




                                                          - ١٠٣ -
             ‫א‬‫א‬                                              ١٧١                                    ‫א‬
          ‫א‬‫א‬                                                                     



           ‫א‬‫א‬‫א‬K‫א‬‫א‬‫א‬‫א‬‫א‬W١
                                                                                                                      K‫א‬
                                                                y



                                              175mm                                                                               
                                                                                                                                  
                                                                                                  ⎛
                                                                                                  ⎜ 40 +
                                                                                                         175 ⎞
                                                                                                             ⎟m                   
                                                                                                  ⎝       2 ⎠
                                                                                             yc                                   
                                       40mm
                                                                                      20mm                 x
                                                                                                                                  
                                                             190mm                                                                
                                                                                                                         W‫א‬
                                                            ‫א‬‫א‬‫א‬‫א‬‫א‬
                                                                                                   xc = 0 ‫א‬‫א‬‫א‬
                                                             yc ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
                           ∑ yi Ai            y1 A1 + y 2 A2 (40 + 175 2)(40)(175) + 20(190)40
                 yc =         i
                                         =                  =                                  = 71.5 mm
                               ∑ Ai              A1 + A2             40(175) + 190(40)
                                   i

      y                                                    xc = 0, yc = 71.5 mm                 
                                                                                                  ‫א‬‫א‬‫א‬
            1in
                                                      ‫א‬W٢
                          0.625in
            1                                                           ‫א‬‫א‬‫א‬‫א‬‫א‬                          E١
                                                        ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬                           E٢
3in
                                                                                   K‫א‬‫א‬‫א‬
                                                1.5in
                           2
                                                 x                                                                       W‫א‬
          0.8in                                              ‫א‬‫א‬‫א‬‫א‬E١
                       3.5in
                                                          ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬




                                                                      - ١٠٤ -
            ‫א‬‫א‬                                               ١٧١                                        ‫א‬
  ‫א‬‫א‬                                                                                 



                       ∑ xi Ai             x1 A1 + x2 A2 0.5(1)(3) + [1 + 0.333(2.5)](0.5)(2.5)(3)
                xc =       i
                                       =                =                                          = 1.24 in
                           ∑ Ai               A1 + A2                1(3) + 0.5(2.5)(3)
                              i
                                                                                                                                         
                       ∑ yi Ai             y1 A1 + y 2 A2 1.5(1)(3) + 1(0.5)(2.5)(3)
                yc =       i
                                       =                 =                           = 1.22 in
                           ∑ Ai               A1 + A2         1(3) + 0.5(2.5)(3)
                               i

             ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬E٢
             W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
                           ∑ xi Ai         0.5(1)(3) + [1 + 0.333(2.5)](0.5)(2.5)(3) − 0.8[π (0.625) 2 / 4]
                 xc =          i
                                       =                                                                    = 1.26 in
                               ∑ Ai                       1(3) + 0.5(2.5)(3) − π (0.625) 2 / 4
                                  i
                                                                                                                                          
                           ∑ yi Ai         1.5(1)(3) + 1(0.5)(2.5)(3) − 1.5[π (0.625) / 4]       2
                 yc =          i
                                       =                                                   = 1.21in
                               ∑ Ai              1(3) + 0.5(2.5)(3) − π (0.625) 2 / 4
                                   i

‫א‬‫א‬‫א‬W
                                                                                                          ‫א‬
                                W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W٣
            y                                                                    y                                                    
                     3in                     3.5in                                     3in                    3.5in




                                                            2in                       1      2                              2in




 4in
                                                            2in                                           3                 2in
                                                                         4in


                                                                                                     4
                                                            2.5in                                                           2.5in
1.5in                                                                    1.5in
                                                                     x                                                                x


                  2.5in                    4in                                       2.5in               4in




                                                                    - ١٠٥ -
        ‫א‬‫א‬                                                                      ١٧١                                                  ‫א‬
    ‫א‬‫א‬                                                                                                           



                                                                                                                                                         W‫א‬
‫א‬‫א‬
                                                                                ‫א‬‫א‬
 ‫ א‬Ai                                         xi                                                xi Ai      yi                                    yi Ai

١                      2.5(6.5) = 16.3 0.5(2.5) = 1.25                                                20.5       0.5(6.5) = 3.25                            53.0

٢                      0.5(2) = 1                   2.5 + 0.5(0.5) = 2.75                             2.75       4.5 + 0.5(2) = 5.5 5.5

٣                      4(2) = 8                     2.5 + 0.5(4) = 4.5                                36.0       2.5 + 0.5(2) = 3.5                         28

٤                      0.5(2.5)(4) = 5 2.5 + 0.333(4) = 3.83 19.2                                                0.667(2.5) = 1.67 8.35
        A = ∑ Ai = 30.3                                                   ∑ xi Ai = 78.4                         ∑ yi Ai = 94.9
                  i                                                        i                                      i
                                                           ∑ xi Ai
                                                               78.4
                                                                                                                        ∑y A    i       i
                                                                                                                                                94.9
                                                    xc =    i
                                                             =      = 2.59 in                                    yc   = i
                                                                                                                                            =        = 3.13 in
                                                         ∑ Ai 30.3
                                                                i
                                                                                                                        ∑A  i
                                                                                                                                    i           30.3



                                                                                                        ‫א‬‫א‬W٤
        y
                      3in                   3.5in                                                        ‫א‬‫א‬‫א‬K‫א‬
                                                                                                                                             K‫א‬
                                         3                                                                                                                    W‫א‬
                                           in                                  2in
                                         4
                       5                                                                         ‫א‬
                                                8                   1in
                                                                               2in             ‫א‬
4in                                 7
                                        1                                                             ‫א‬
                        6                                                                        ‫א‬‫א‬K‫א‬
                                                                               2.5in
1.5in                                                                                                   ‫א‬‫א‬‫א‬‫א‬
                                                                                        x
                                                                                                      ٨٥‫א‬
            1in             2.5in
                                                     1in
              2.5in                     4in
                                                                                                  ‫א‬‫א‬‫א‬‫א‬
                                                                                                                                                                 
                                                                                                                                                                   




                                                                                       - ١٠٦ -
           ‫א‬‫א‬                                      ١٧١                                            ‫א‬
      ‫א‬‫א‬                                                                       




‫ א‬Ai                               xi                                 xi Ai           yi                          yi Ai

      ١          2.5(6.5) = 16.3        0.5(2.5) = 1.25                    20.5            0.5(6.5) = 3.25               53.0

      ٢          0.5(2) = 1             2.5 + 0.5(0.5) = 2.75              2.75            4.5 + 0.5(2) = 5.5          5.5
      ٣          4(2) = 8               2.5 + 0.5(4) = 4.5                 36.0            2.5 + 0.5(2) = 3.5            28

  ٤              0.5(2.5)(4) = 5        2.5 + 0.333(4) = 3.83 19.2                         0.667(2.5) = 1.67          8.35
 5   − π (0.75) 2                     1                                   − 0.44 1.5 + 4 = 5.5                        − 2.42
                   = −0.44
           4
 6   − 0.44                           1                        − 0.44 1.5                                             − 0.6 6
 7   − 0.44                           1 + 2 .5 = 3 .5          − 1.54 2.5                                               − 1.10
 8   − 0.44                           1 + 2 .5 + 2 = 5 .5      − 2.42 2.5 + 1 = 3.5                                   − 1.54
    A = ∑ Ai = 28.5                              ∑ xi Ai = 73.5        ∑ yi Ai = 89.1
                    i                               i                                       i

                                              ∑ xi Ai
                                                   73.5
                                                                                                  ∑ yi Ai       89.1
                                        xc = i   =      = 2.58 in                          yc =    i
                                                                                                            =        = 3.13 in
                                             ∑ Ai 28.5                                             ∑ Ai         28.5
                                                i                                                      i

                                        li                                                ‫א‬‫א‬K٣
                                                                       l ‫א‬‫א‬
                                                                 ‫א‬‫א‬‫א‬K‫א‬
                                                                       K
 yi
           ′
          yc                                                  ‫א‬‫א‬‫א‬
                                                                  ‫א‬K‫א‬
                                                        x       li 
                                 ′
                                xc                            ‫א‬‫א‬‫א‬K‫א‬
                                xi                              xi , yi ‫א‬‫א‬
                                              W‫א‬ xc , yc ‫א‬‫א‬‫א‬
                   ∑ xi lix l + x2l 2 + x3l3 + ... + x3l3
                                                                            ∑ y i li       y1l1 + y 2 l2 + y3l3 + ... + y3l3
               xc =     i
                         = 11                             , yc =             i
                                                                                       =
                    ∑ li      l1 + l2 + l3 + ... + ln                        ∑ li               l1 + l 2 + l3 + ... + ln
                            i                                                    i




                                                            - ١٠٧ -
         ‫א‬‫א‬                                       ١٧١                                             ‫א‬
    ‫א‬‫א‬                                                                         



                                                                                  W
          ‫א‬K‫א‬‫א‬‫א‬
        ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬،‫א‬
                             K‫א‬‫א‬‫א‬‫א‬
                ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬                                     ‫א‬                                        A
                                                                                                       xc                 yc

                                    y                                                                              
                                              xc                                                                   
                                                                                                                  
 ١                                                      a                             a2
                                                                                                       a                  a
                                                                                                       2                  2
                              yc
                                                                         x
                                                     a

                                    y                                                                              
                                                xc
                                                                                                                   
                                                                                                    a                  b
                                                                                 2( a + b )            2                  2
    ٢                                                      b
                             yc
                                                                         x
                                                     a

                                y                                                                                  
                                           xc
                                                                                                                   
    ٣                                                            b                          a(a + 2b + c)        b(b + c)
                                                                                 a+b+c           2(a + b + c)      2( a + b + c )
                                                              yc             x




                                                           - ١٠٨ -
        ‫א‬‫א‬                                               ١٧١                                             ‫א‬
    ‫א‬‫א‬                                                                                



                                          y    xc

                
                
                                                                                             2π a               a                a
 ٤                  ‫א‬                                             d

                                   yc                     a

                                                                                x

                                          y
                                                 xc
    ٥        ‫א‬
                                                                                             πa                 a                2a
                                                                                                                                 π
                                   yc                         a
                                                                            x

                                      y                                                                                   
                                                                                                                          
    ٦        ‫א‬                                                                                      a sin θ      2a sin 2 θ 2
                                                                                             aθ
                                                 θ                    yc                                        θ
                                                                            x                                                    θ
                                                      a
                                                 xc


                                                                                                                                       
                                                                         ‫א‬‫א‬K٤
                                             W‫א‬‫א‬
               ∑ xi l i
                   x l + x2l 2 + x3l3 + ... + x3l3
                                                                                    ∑ y i li          y1l1 + y 2l2 + y3l3 + ... + y3l3
       xc =     i
                  = 11                             , yc =                            i
                                                                                                  =
             ∑ li      l1 + l2 + l3 + ... + ln                                       ∑ li                  l1 + l2 + l3 + ... + ln
                     i                                                                   i

                          x1 , x2 , x3 ,...., xn ،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ l1 , l2 , l3 ,...., ln 
                                             ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫ א‬y1 , y2 , y3 ,...., yn 




                                                                  - ١٠٩ -
           ‫א‬‫א‬                                             ١٧١                                       ‫א‬
        ‫א‬‫א‬                                                                       



  ‫א‬‫א‬K‫א‬‫א‬W٥
                                                                                                              K‫א‬‫א‬‫א‬
          y                      400mm                                       y                       400mm
                                                                                        200mm
                                                                                                              1
                                                                                                          •
                                                                                       xc

                                                                  300mm
300mm                                                                            •
                                                                             2        150mm          yc

                                                             x                                                           x
                                                                                             •   3
                                                                                     100mm
                     200mm                                                              200mm

                                                                                                                         W‫א‬
                   ،‫א‬٣٢١‫א‬‫א‬
                                                                                      ‫א‬‫א‬‫א‬
                                        ∑ xi l i       400(200) + 300(0) + 200(100)
                                 xc =    i
                                                   =                                = 111mm
                                         ∑ li                400 + 300 + 200
                                            i

                                        ∑ y i li       400(300) + 300(150) + 200(0)
                                 yc =    i
                                                   =                                = 183 mm
                                         ∑ li                400 + 300 + 200
                                             i




                                                                   - ١١٠ -
          ‫א‬‫א‬                                  ١٧١                                      ‫א‬
  ‫א‬‫א‬                                                               



                                                              ‫א‬‫א‬K٥
     ، xy ‫א‬‫א‬ A ‫א‬‫א‬
     ‫א‬‫א‬‫א‬K yi  xi ‫א‬ Ai ‫א‬
                                 ‫א‬‫א‬‫א‬‫א‬‫א‬
      y
                           xi
                                                     Ai

                                                                     W A ‫א‬K
                                                                     I x = ∑ yi Ai ,         I y = ∑ xi Ai
                                                                                 2                          2

                                                                            i                         i

 b                                                                    A ‫א‬ I x 
                            ri                       yi
                                  A
                                                               A ‫א‬ I y  x ‫א‬
                                                                                                           y ‫א‬
                                                          x



                                                                                              
                      K‫א‬ yi 2  xi 2 ، A ‫א‬   E١
     ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬                                E٢
                                                                          mm 4 , cm 4 , m 4 , ... ‫ﺑ‬
‫א‬‫א‬‫א‬                                          E٣
      ‫א‬‫א‬‫א‬‫א‬‫ א‬
                                                     K‫א‬‫א‬‫א‬‫א‬
                                  K‫א‬‫א‬‫א‬                         E٤

                                                                                                                    
                                                                                                                    
                                                                                                                    



                                                      - ١١١ -
   ‫א‬‫א‬                                                        ١٧١                     ‫א‬
‫א‬‫א‬                                                                



                                    ‫א‬‫א‬
                                                                                               
‫א‬                             ‫א‬                         I x0       I y0   Ix       Iy
                             y
                                   y0



                                                    x0                             a4     a4      a3         a3
                         a
   ١                                                                               12     12      3          3
                                                                 x
                                       a

                                  y0



                                                        x0
                         b
                                                                                   ab 3   ba 3   ab 3       ab 3
   ٢                                                         x                     12     12      3          3
                                   a

                y                               a
                                                3
                                       y0


                                                             b
                                                                                   ab 3   ba 3   ab 3       ba 3
                                                     x0                            36     36     12          4
   ٣
                                                                         x
                                   a                b
                                                    3
                         y
                                   y0



   4                                                x0           d                 πa 2   πa 4   5πa 4      5πa 4
                yc                          a                                       4      4       4          4
                                                                     x



                                                             - ١١٢ -
     ‫א‬‫א‬                                             ١٧١                                              ‫א‬
‫א‬‫א‬                                                                             



                           y
                                y0

     5                                             x0
                                                                            ⎛π   8 ⎞             πa 4        πa 4        5πa 4
                                              a                           a4⎜ −    ⎟
                                                            x               ⎝ 8 9π ⎠              8             8          8
                                              4a
                                              3π

                           y
                                y0


     6                                             x0
                                                        b                    πab 3               πba 3      5πab 3       5πba 3
                                                            x                 64                  64          64           64
                                     a


                                                                                   ‫א‬‫א‬K٦
 y                    xi                                                 ‫א‬‫א‬
                                                                                            W‫א‬‫א‬
                                         li
                                                                   I x = ∑ y i 2 li ,              I y = ∑ xi l i
                                                                                                                    2

                                                                              i                             i

                                                                                       ‫א‬ li 
                                              yi
                           ri

                                                                ‫א‬‫א‬
                                                    x
                                                                                    ‫ﺑ‬‫א‬
                y                                                                                      mm 3 , cm 3 , m 3 , ...
                           a                       ‫א‬‫א‬‫א‬‫א‬‫א‬W٦
                                l                                                                 y  x ‫א‬
                                2                                     l3
                                                         Ix =           ,        I y = a 2l                                    
                                              x                       12

                                l
                                2
                                                                                                                                


                                                            - ١١٣ -
       ‫א‬‫א‬                                                      ١٧١                                           ‫א‬
    ‫א‬‫א‬                                                                                 



                                                                   ‫א‬‫א‬‫א‬‫א‬‫א‬
     ‫א‬                                         ‫א‬                            I x0                I y0       Ix                   Iy

                   y
                                  y0


      1                      2a
                                                   x0                   a3 2
                                                                        2π
                                                                            (
                                                                           π −8        )       πa 3
                                                                                                2
                                                                                                                 πa 3
                                                                                                                  2
                                                                                                                                     3πa 3
                                                                                                                                       2
                                            a
                             π                                x


                         a sin θ
                   y         θ         y0
                                                         2θ
                                                                                                a3
                                                2a sin                                     I x = (2θ − sin 2θ )
                                                          2                                     4
      2                                            θ
                                                                                                a3
                                                         x0
                                                                                           I y = (2θ + sin 2θ )
                                                                                                4
                             θ                                x
                                   a

                                                                                     ‫א‬‫א‬‫א‬K٧
y                                                                      ‫א‬‫א‬‫א‬
                 xi
                                                          Ai                                      W‫א‬
                                                                                   I x + I y = ∑ ( xi 2 + yi 2 ) Ai           
                                                                                                    i

                                                                                      ri = xi 2 + yi 2 ‫א‬
                                                                                                2


                                                                                 J = ∑ ri 2 Ai = I x + I y                    
                   ri                                     yi                               i

                         A                                             ‫א‬ J ‫א‬‫א‬
                                                                                   K‫א‬‫א‬‫א‬ xy
                                                                  x                                                                     




                                                                        - ١١٤ -
   ‫א‬‫א‬                                       ١٧١                                         ‫א‬
‫א‬‫א‬                                                                



                                                                       ‫א‬‫א‬‫א‬K٨
                                     W‫א‬‫א‬‫א‬
                                                    J = ∑ ( xi 2 + yi 2 ) li = ∑ ri 2 li = I x + I y 
                                                                  i                       i

                                                                                                                   
                         y
                                         ky
                                                                       A
                                                                                                                   
                                                                                                                   
                                    xc
                                                           •                                                       
                                                                                                                   
                                                                                                                   
                                                                                kx
                                                                           yc                                      
                                                                                                                   
                                                                                     x                             
                                                                  ‫א‬K٩
   ‫א‬‫א‬ A ‫א‬‫א‬
       y  x ‫א‬‫א‬‫א‬ k y  k x ‫א‬‫א‬
                                                ‫א‬‫א‬‫א‬
                                                                      Ix                 Iy
                                                          kx =          ,      ky =          
                                                                      A                  A
             W k p ‫א‬‫א‬‫א‬‫א‬
                                                                        J
                                                          kp =
                                                                        A
                                                                                 k p 2 = k x 2 + k y 2 
      W‫א‬‫א‬‫א‬‫א‬ k p  k y ، k x ‫א‬W٧
                    y                                                                                        W‫א‬
                                                   ab 3                ba 3
                                             Ix =      ,         Iy =      ,        A = ab                
                                                   12                  12
                               x                                                                             
        b

                                                  - ١١٥ -

                     a
         ‫א‬‫א‬                                               ١٧١                                       ‫א‬
    ‫א‬‫א‬                                                                        



                                                                      Ix    ab 3   b
                                                        kx =             =       =   ,
                                                                      A    12ab 2 3
                                                
                                                                      Iy          ba 3   a
                                                        ky =               =           =   ,
                                                                      A          12ab 2 3
                                                                                 b2 + a2        b2 + a2
                                           k p = kx + k y
                                                    2            2         2
                                                                               =         ⇒ kp =
                                                                                   12            2 3
                                                                      ‫א‬،‫א‬،‫א‬‫א‬‫א‬K١٠
y
                      x                                                        K A ‫א‬‫א‬
                                          dA
                             y0
                                                                                              x ‫א‬‫א‬
                dx
                                  •                                      I x = I x0 + A d y 2                      
                                           x0
                                                                                              y ‫א‬‫א‬
                                                                        I y = I oy + A d x 2                        

                 d           A
                                                        y                               x = d x + x0  y = d y + y0 
                                           dy
                                                                               ‫א‬‫א‬
                                                                           ‫א‬‫א‬‫א‬
                                                             x
                                                                       ‫א‬‫א‬
         K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
                           x ‫א‬‫א‬‫א‬‫א‬‫א‬ W٨
     y                                                                                                                         
                                                                                                                        W‫א‬
                                                             W‫א‬‫א‬‫א‬‫א‬
                                                                                                      2
                                                                                         ab 3     ⎛b⎞   ab 3
    b                                                I x = I xo + A d y           2
                                                                                       =      + ab⎜ ⎟ =                
                                                                                         12       ⎝2⎠    3
                                  b
                                                                                                                               
                                  2
                                      x
                                                                                                                               
                  a
                                                                                                                               
     ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
                                                                                                                     K‫א‬

                                                                 - ١١٦ -
     ‫א‬‫א‬                                                   ١٧١                                     ‫א‬
  ‫א‬‫א‬                                                                        



                                                   ‫א‬‫א‬‫א‬K١١
،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
             K‫א‬‫א‬‫א‬‫א‬
‫א‬‫א‬ x0 ‫א‬،‫א‬‫א‬‫א‬W٩
                               K‫א‬‫א‬‫א‬ x′  x ‫א‬

                                                                                          40mm
                40mm
                             x′                                                                        x′


                                                                                              1
 175mm                                                                          175mm                 143mm


                                                             x0                                             x0
                                                                                              2       32mm
                                                          yc =72mm
                                              40mm                                            3                           40mm
                                                                                                                                 x
                                                                  x
              190mm                                                                      190mm

                                                                        
                                                                                                                      W‫א‬
                                                 ‫א‬‫א‬‫א‬‫א‬‫א‬
                               ‫א‬ x0 ‫א‬١‫א‬
                                                          ab 3 40(143) 3
                                                             =I x0 ,1 =                 
                                                           3         3
                                      ‫א‬ x0 ‫א‬٢‫א‬
                                                                            ab 3 40(32) 3
                                                             I x0 , 2 =        =                                  
                                                                             3      3
                             x0 ‫א‬٣‫א‬‫א‬‫א‬‫א‬
                                                                    190(40) 3
                                   I x0 , 3 = I xc , 3   + Ad y =
                                                                  2
                                                                              + 190(40)(32 + 20) 2       
                                                                       12
                                                             W‫א‬ I x0 ‫א‬


                                                              - ١١٧ -
           ‫א‬‫א‬                                              ١٧١                                          ‫א‬
       ‫א‬‫א‬                                                                         



                                         40(143) 3 40(32) 3 ⎛ 190( 40) 3                      ⎞
I x0 = I x0 ,1 + I x0 , 2 + I x0 , 3 =            +        +⎜
                                                            ⎜ 12         + 190(40)(32 + 20) 2 ⎟ = 6.10 × 10 7 mm 4
                                                                                              ⎟
                                             3        3     ⎝                                 ⎠
                                                                                                  x ‫א‬ I x ‫א‬J
        ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
                                                                                                            W‫א‬‫א‬
          I x = I xo + A d y 2 = 6.10 × 10 7 + [40(175) + 190(40)](72) 2 = 1.3 × 108 mm 4                                               
                                                                                                 x′ ‫א‬ I x′ ‫א‬J
     ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
                                                                                            W‫א‬‫א‬‫א‬
                          I x′ == 6.10 × 107 + [40(175) + 190(40)](143) 2 = 3.60 × 108 mm 4                                             
                                                                                    
     ‫א‬‫א‬‫א‬‫א‬
                 y0                               ‫א‬
                                               ‫א‬‫א‬‫א‬‫א‬‫א‬
                                                                                                                           K‫א‬
                                    x0
                                                       ‫א‬‫א‬‫א‬ I x0 ‫א‬W١٠
                                                                                            ‫א‬‫א‬‫א‬
                                                                                                                               W‫א‬
                  I x0
                                                 y0                                 y0
                                                                                                                   y0
                                                                                                 1
                                                                                                                  2
                                                                  x0                                  x0   _                          x0
                                                                       =


                                                 I x0                              Ix0,1                               I x0 , 2
                                     
                                                                                                  W‫א‬‫א‬

                                                                                           (               )
                                                                   3         3
                                                                a1b1  ab      1
                                 I x0 = I x0 ,1 − I x0 , 2 =        − 2 2 =    a1b1 − a2 b2 
                                                                                    3       3
                                                                                                                                         
                                                                 12    12    12

                                                                   - ١١٨ -
            ‫א‬‫א‬                                                           ١٧١                                              ‫א‬
     ‫א‬‫א‬                                                                                             



                     y                                                                           ‫א‬ I x0 ‫א‬W١١
                               1in                2.5in
                                                                                     ‫א‬‫א‬‫א‬‫א‬
                                           y0     0.625in
                               1
                                                                                                                                              W‫א‬
3in                                  3
                                                                                          ‫א‬‫א‬‫א‬‫א‬
            1.5in
                                                                     x0                                                          ‫א‬‫א‬
                                                  2              1.22in
                                                                          x                               W x ‫א‬١‫א‬‫א‬
                                                                                                  1(3) 3
                           0.8in
                                                                                     I x ,1 =                                   
                               1.24in                                                               3
                                             3.5in
                                                                                                     W x ‫א‬٢‫א‬‫א‬
                                                                                  2.5(3) 3
                                                                     I x , 2 =                               
                                                                                    12
                                             ‫א‬‫א‬‫א‬‫א‬W x ‫א‬٣‫א‬‫א‬
                                                            π (0.625) 4           π (0.625) 2
                                              I x ,3 =                     +                       (1.5) 2                                   
                                                                64                        4
                                                                                     x ‫א‬
                                            1(3) 3 2.5(3) 3 ⎛ π (0.625) 4 π (0.625) 2            ⎞
                          Ix =                   +         −⎜⎜            +             (1.5) 2 ⎟ = 13.9 in 4 
                                                                                                 ⎟                       
                                              3       12      ⎝    64              4             ⎠
                                                                        π (6.25) 2
                                              A = 1(3) + 0.5(2.5)(3) −            = 6.44 in 2        W‫א‬‫א‬
                                                                            4
                                                                                                W‫א‬‫א‬‫א‬‫א‬
                                          I x0 = I x − A d y 2 = 13.9 − 6.44(1.22) 2 = 4.31in 4                                                 
             y                                                                     ‫א‬‫א‬‫א‬W
                          3in                     3.5in
                                                                                          

                               1 2                                2in
                                                                                                                        ‫א‬
    1.6in                 5                                                                    x ‫א‬ I x ‫א‬W١٢
                                                      3           2in                                              ‫א‬‫א‬
                         1in
    3.5in
                         0.5in
                                                                                                                                              W‫א‬
                                             4
                                                                  2.5in
                                                                              x       ‫א‬‫א‬‫א‬‫א‬
                                                                                                  W‫א‬‫א‬‫א‬‫א‬
                         2.5in                        4in

                                                                                  - ١١٩ -
      ‫א‬‫א‬                               ١٧١                                    ‫א‬
  ‫א‬‫א‬                                                      



                                                                                                         
                                                                                     2
‫א‬                 I oi            d1                      Ai           Ai d i               Ii
                                       _              2.5(6.5) = 16.3         _          2.5(6.5) 2
  ١                    J                                                                            = 228.9
                                                                                             3
  ٢            0.5(2) 3           2 .5 + 2 + 1 = 5 .5    0.5(2) = 1   30.25                    30.6
                        = 0.33
                  12
           4(2) 3
                                2.5 + 1 = 3.5         4( 2) = 8       98        100.7
  ٣                = 2.67
             12
  ٤            4(2.5) 3         0.667(2.5) = 1.67 0.5(4)2.5 = 5       13.94     15.7
                        = 1.74
                  36
             1(1.6) 3           3.5 + 0.8 = 4.3       − 1(1.6) = −1.6 − 29.58 − 29.9
  ٥        −          = −0.34
                12
                   x ‫א‬          A = ∑ Ai = 28.7in 2              I I = 346in 4
                           _
                                               i

                                                                    II   346
                                                        kI =           =      = 3.47in
                                                                    A    28.7
                                                                                                           
                                                                                                           




                                              - ١٢٠ -
                    ‫א‬‫א‬                                                           ١٧١                                                  ‫א‬
          ‫א‬‫א‬                                                                                                       



                                                           ‫א‬‫א‬‫א‬‫א‬K١٢
                    ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
                           K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬K‫א‬

                                               ‫א‬‫א‬
                                                                                     y                
                                                                                                      

                                                                                                   
                                                                               x                          x




                                                                                     y
                                                in2 ‫א‬
    in‫א‬‫א‬




                                                                in‫א‬
                             ‫א‬




                                                                                    in‫א‬




                                                                                                                          4               4
                                                                                                              I xx : in       I yy : in       k x : in     k y : in



     36 × 12                             194              194        57.11               36.48                   12.11        12.103             355          2.49
     24 × 12                             194              120        53.29               24.31                   12.08         3.635             254          2.68
      14 × 8                             194               53        15.59               13.94                    8.06           542             57.5         1.92
           1                             194               20         5.88                8.14                    5.27          69.2              8.5         1.20
    36 × 5
           4

                                                                                                                                                                    
                                                                                                                                                                    
                                                                                                                                                                    
                                                                                                                                                                    
                                                                                                                                                                    
                                                                                                                                                                    
                                                                                                                                                                    

                                                                                      - ١٢١ -
        ‫א‬‫א‬                                    ١٧١                                        ‫א‬
  ‫א‬‫א‬                                                                 



                                                     
                          ‫א‬‫א‬‫א‬ xc , yc ‫א‬‫א‬‫א‬ J ١
 x ‫א‬‫א‬‫א‬‫א‬‫א‬ J ٢
                                                                                                        y ‫א‬

                                                                         y
           y                                                                  25mm
               25mm                  25mm




                                                                 125mm                  90mm
  125mm


                                                                                       25mm
                                                                                                             x
                              25mm
                                                x                                                        125mm

                         115mm                                                                 25mm




    y
                                                                  y
               a

                          a
  2a                                 l                  850mm


                                                                             50 o                     70 o
                                            x                                                                         x
                     2a
                                                                                     1950mm




                                                       - ١٢٢ -
       ‫א‬‫א‬                                          ١٧١                                          ‫א‬
    ‫א‬‫א‬                                                                    


                                                                                      y
                                                                                 15mm            50mm
           y
                 10in           12in

                                                                                                                    30mm

                                                                                                                              x
                                                 18in
                                                                                                                    10mm
    12in
                                                      x                                                 15mm
                                                                                                                    30mm

                              28in


                                                                                          50mm



           y   3in
                                                                          y
                                                                                      600mm

                     90                                                               100mm                               x


                                  12in                                                                    100mm
                                                                  200mm
                                                                                                                    800mm

                                                                  200mm                                   550mm


                                  4in                             200mm
                                         x
                                                                              200mm
                     6in


y
                      150mm




                                                50mm
               30o         25mm    30o
                                                  x
    30mm                                 30mm




                                                            - ١٢٣ -
                                       ‫א‬‫א‬‫א‬
                                    ã¹]<gè…‚jÖ]æ<ËÖ]<Üé×Ãj×Ö<íÚ^ÃÖ]<퉉ö¹]
                                        ‫א‬‫א‬‫א‬‫א‬




                   
                   ‫א‬
 ‫א‬




٨
     ‫א‬‫א‬                             ١٧١                          ‫א‬
    ‫א‬                                              



‫א‬‫א‬‫א‬‫א‬‫א‬W‫א‬‫א‬
                                                                               KKK‫א‬
                                                                                               
                               W‫א‬‫א‬‫א‬W‫א‬‫א‬
                                                                           ‫א‬     •

                                                                     ‫א‬         •

                                                               ‫א‬‫א‬         •

                            ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬     •

                        K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬     •

                                                                                               
                                                            KW‫א‬‫א‬
                                           




                                               -١٢٤-
       ‫א‬‫א‬                                     ١٧١                                       ‫א‬
      ‫א‬                                                                   



                                              ‫א‬
                                                                                                                     
،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬K‫א‬‫א‬‫א‬‫א‬
‫א‬‫א‬K‫א‬‫א‬
K١٦٥٤ Blaise Pascal  Pierre de Fermat ‫א‬‫א‬
‫א‬،١٦٦٢ John Graunt ‫א‬
                                                                       K‫א‬
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
                                                                                                              K
                                                                                                     W‫א‬K١
‫א‬،‫א‬W١
                                                                                         K‫א‬
  
‫א‬‫א‬
                                                                                                  K
                                                                        WW١
                                            A = {0,1,2,3,4,5}
                                        B = {1,3,5,7,9}
                                           C = {−6,−3,0,3,6,9,12,15}
                                                                     W‫א‬
                                 A = { ٥٠ }
                                    B = { ٩١ }
                         15 C = {  − 6 ٣‫א‬ }
                                         
‫א‬ S W٢
                                K card ( S ) W S ،‫א‬K
                                                          W١‫א‬‫א‬‫א‬W٢
                                card ( A) = 6, card ( B) = 5, card (C ) = 8

                                                        -١٢٥-
      ‫א‬‫א‬                                        ١٧١                                      ‫א‬
     ‫א‬                                                                      



  A ‫א‬ S  A W٣
                                            KK A ⊄ S W A ⊆ S WK S
                                                                                  W‫א‬‫א‬W٣
              A = {1,2,3}, B = {0,1,2,3,4,8}, C = {−1,0,2,3,5,8}, D = {−1,2,5}
 1  C ‫א‬ B ‫א‬ A ‫א‬
                                                                                               K C 
                                                           K C ‫א‬ D ‫א‬
                         K card (φ ) = 0 W،‫א‬ φ ‫א‬‫א‬W٤
  (a, b) ‫א‬‫א‬ B  A  A × B ‫א‬‫א‬‫א‬W٥
 K‫א‬‫א‬‫א‬K B  b  A  a
                                                                                  W‫א‬‫א‬W٤
                                   A = {0,1,2,5}, B = {5,9}, C = {0,1}
                                                                                             W‫א‬
                                                                            A × B × C E٢                 A × B E١
                                                                                                               W‫א‬
           1) A × B = {(0,5), (0,9), (1,5), (1,9), (2,5), (2,9), (5,5), (5,9)}
        2) A × B × C = {(0,5,0), (0,9,0), (1,5,0), (1,9,0), (2,5,0), (2,9,0), (5,5,0), (5,9,0),
                                   (0,5,1), (0,9,1), (1,5,1), (1,9,1), (2,5,1), (2,9,1), (5,5,1), (5,9,1)}
                                                         W B  A ‫א‬‫א‬W١
                                     card ( A × B) = card ( A) × card ( B)
                                                            K‫א‬‫א‬
                                                                                          W‫א‬‫א‬W٥
                       card ( A × B ) = card ( A) × card ( B) = 4 × 2 = 8
               
                       card ( A × B × C ) = card ( A) × card ( B ) × card (C ) = 4 × 2 × 2 = 16
                                                                                                                       
                                                                                                     W‫א‬K٢
                                KW٦

                                                          -١٢٦-
        ‫א‬‫א‬                                                 ١٧١                                                 ‫א‬
       ‫א‬                                                                                         



                                                                                                                  WW٦
                                                                   ‫؟‬E١
                              K1,2,3,4,5,6 ‫א‬E٢
                                              ‫؟‬‫א‬‫א‬‫א‬‫א‬‫א‬E٣
                           KE‫א‬‫א‬F‫א‬‫א‬‫א‬E٤
                                                               K‫א‬‫א‬E٥
            K S K‫א‬‫א‬ W٧
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
                                                                                                                                    K‫א‬
                                                                       K٦‫א‬‫א‬W٧
                                                                                                                                      W‫א‬
 D ‫א‬‫א‬K‫א‬،‫א‬W‫א‬‫א‬E١
                                                                                 
                                                                  K S = {N , D} WKK N ‫א‬‫א‬
                                              K S = {1,2,3,4,5,6} W‫א‬‫א‬‫א‬‫א‬‫א‬E٢
                                                                      
، L W‫א‬W‫א‬‫א‬‫א‬‫א‬‫ א‬E٣
                                        
             K S = {L, D,U } WKU W، D W
      
 H W‫א‬، T W‫א‬‫א‬‫ א‬E٤
                                          
                                                                                                            K S = {H ,T } WK
                                             
‫א‬K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬E٥
                                                     K S = {HH , HT , TH ,TT } = {H ,T } × {H , T } W٤‫א‬
  
W‫א‬‫א‬‫א‬‫א‬
                                                                                              KK S = {{H , H },{H , T },{T ,T }}
                                                                                         K‫א‬‫א‬
      K‫א‬‫א‬‫א‬‫א‬K‫א‬W٨
        1,2,3,4,5,6 ‫א‬W٨
                                                                                   W‫א‬‫א‬
                                                                          ، A = {1,3,5} WW A 
                                                                     -١٢٧-
      ‫א‬‫א‬                                    ١٧١                                     ‫א‬
     ‫א‬                                                                 



                                                          ، B = {2,4,6} WW B
                                                        K C = {5,6} W٤W C
                        
‫א‬K٢١W ‫א‬٤١٤W٩
                                                                                                       K‫א‬
                                                            KE١
‫א‬W B ،‫א‬W A W‫א‬‫א‬‫א‬‫א‬E٢
        K‫א‬‫א‬‫א‬W D ،‫א‬W C ،‫א‬
                                                                                                          W‫א‬
                       1) S = {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}           card ( S ) = 6
                  2) A = {{3,4}}, B = {{1,3},{1,4},{2,3},{2,4}}, C = {{1,2}},
                          D = {{1,3},{1,4},{2,3},{2,4},{1,2}}
                                                 W‫א‬‫א‬‫א‬‫א‬W٩
                    K B  A ‫א‬‫א‬‫א‬ A U B ‫א‬E١
                   K‫א‬ B  A ‫א‬‫א‬‫א‬ A I B E٢
                        K A ‫א‬‫א‬‫א‬‫א‬ A E3
                                                   K A I B = A U B  A U B = A I B W
      1,2,3,4,5,6 ‫א‬W١٠
  KW C ،٦W B ،٣W A W‫א‬‫א‬‫א‬
                                                                          KE١
                                                               K C  B  A W‫א‬‫א‬‫א‬‫א‬E٢
 A U C  A U B  A I B I C  B I C  A I C  A I B W ‫א‬‫א‬‫א‬‫א‬E٣
                                                                  
                                                                                     K A U B U C  B U C
                                             K A I B  A U B  B  A W‫א‬‫א‬‫א‬‫א‬E٤
                                                                   K٤‫א‬‫א‬‫א‬E٥




                                                      -١٢٨-
      ‫א‬‫א‬                                    ١٧١                                   ‫א‬
     ‫א‬                                                              



                                                                                                       W‫א‬
       1) S = {1,2,3,4,5,6}    card ( S ) = 6
       2) A = {4,5,6}, B = {1,2,3,4,5}, C = {2,4,6},
      3) A I B = {4,5}, A I C = {4,6}, B I C = {2,4}, A I B I C = {4}
          A U B = {1,2,3,4,5,6} = S , A U C = {2,4,5,6}, B U C = S , A U B U C = S
         4) A = {1,2,3}, B = {6},            A U B = {} = φ ,     A I B = {} = φ
W A U B ،٦‫א‬٥W B ،٤W A E٥
                              
                               K٥٤W A I B ،٦٣
                                                                                         W‫א‬K٣
‫א‬‫א‬ S W١٠
                                                                                                      K‫א‬
                                            card ( A)
                             K P ( A) =              W A ‫א‬،‫א‬
                                            card ( S )
                                                 K‫א‬‫א‬W١١
                                                                     
‫א‬‫א‬ S = {H , T } W‫א‬‫א‬
                                                                    W‫א‬‫א‬K
                                                   card ({H }) 1
                                    P({H }) =                 = = 50%
                                                    card ( S )  2
‫א‬‫א‬‫א‬K  P({H })  P( H ) 
                                                   
                                                                                                      K‫א‬
                                                                           W‫א‬‫א‬
                                                       card ({T }) 1
                                P(T ) = P({T }) =                 = = 50%
                                                        card ( S )  2
                                                     KK‫א‬W١٢
                                                                        
 S = {HH , HT , TH , TT } = {H , T } × {H , T } W‫א‬‫א‬
                        W‫א‬‫א‬K‫א‬‫א‬
                                                  card ({HH }) 1
                                   P( HH ) =                  = = 25%
                                                    card ( S )  4
                                                             W‫א‬‫א‬‫א‬

                                                       -١٢٩-
      ‫א‬‫א‬                                   ١٧١                                    ‫א‬
     ‫א‬                                                              


                                                  card ({HT , TH }) 2
                            P({HT ,TH }) =                        = = 50%
                                                      card ( S )    4
‫א‬ S ′ = {{H , H }, {H , T }, {T , T }} W‫א‬‫א‬‫א‬
                                                                     
{H , H } ‫א‬‫א‬‫א‬ {H , T } ‫א‬
                                                                                          K‫א‬
                                  K C  B  A ‫א‬‫א‬‫א‬‫א‬K١٠‫א‬W١٣
                                                                                                        W‫א‬
                       S = {1,2,3,4,5,6}     card ( S ) = 6
                    A = {4,5,6}, B = {1,2,3,4,5}, C = {2,4,6}
                                 3                5                 3
                       P( A) =     = 50%, P( B ) = ≅ 83.33%, P(C ) = = 50%
                                 6                6                 6
           W‫א‬‫א‬‫א‬‫א‬W١١
     K١٠‫א‬E١
       K٪١٠٠١E‫א‬‫א‬F‫א‬‫א‬‫א‬E٢
                                                                                      
‫א‬ ٠‫א‬‫א‬‫א‬،‫א‬
                                                                         K‫א‬‫א‬‫א‬‫א‬
       K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
     W‫א‬‫א‬١٤٥‫א‬‫א‬W١٤
    ١٤         ١٣  ١٢  ١١               ١٠        ٩         ٨         ٧       ٦      ٥      
                                                                                                      ‫א‬‫א‬
   ٠{٠٣       ٠{١٣      ٠{١٤      ٠{٠٦     ٠{٢٣      ٠{١١        ٠{٠٨      ٠{١٧     ٠{٠٠     ٠{٠٥      ‫א‬
                                                                                 W‫א‬‫א‬
                                                               K‫א‬١١E١
                                                           K‫א‬‫א‬٩E٢
                                                                                 K١٣٨E٣
                                                                                  K‫א‬٤E٤
                                                                              K٢٠E٥



                                                      -١٣٠-
       ‫א‬‫א‬                                      ١٧١                                       ‫א‬
      ‫א‬                                                                     



                                                                                                               W‫א‬
 D  C  B  A K S = {5,6,7,8,9,10,11,12,13,14} W‫א‬‫א‬
                                                                         
                                                          K‫א‬٥١‫א‬‫א‬‫א‬‫א‬ E 
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
                                                               K١١٠
                             1) A = {5,6,7,8,9,10}
                             P( A) = P(5) + P (6) + P(7) + P(8) + P (9) + P (10)
                                   = 0.05 + 0.00 + 0.17 + 0.08 + 0.11 + 0.23 = 64%
                             2) B = {10,11,12,13,14}
                             P( A) = P(10) + P (11) + P(12) + P(13) + P(14)
                                  = 0.23 + 0.06 + 0.14 + 0.13 + 0.03 = 59%
                             3) C = {8,9,10,11,12,13}
                             P( A) = P(8) + P(9) + P (10) + P(11) + P (12) + P (13)
                                   = 0.08 + 0.11 + 0.23 + 0.06 + 0.14 + 0.13 = 75%
                             4) D = {} = φ ⇒ P ( D) = 0
                             5) E = {} = φ ⇒ P( E ) = 0
                                                                                                                       
                                                                                               W‫א‬‫א‬K٤
F A ‫א‬‫א‬ B ‫א‬ B A ‫א‬‫א‬W١٢
                                                   K‫א‬‫א‬‫א‬KE P( A) ≠ 0
            K P(B A) W‫א‬‫א‬ S A W‫א‬‫א‬
                                                       K‫א‬W١٥
                             K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
                                                                                                               W‫א‬
                                                                                         W‫א‬
                              
‫א‬‫א‬K‫א‬‫א‬‫א‬W B ،‫א‬‫א‬‫א‬W A
                                                                                                               KB A
 S A = {HH , HT } W S = {HH , HT , TH , TT } W‫א‬‫א‬
                                                             
                                                                                              W B A = {HT } 
                                                         -١٣١-
      ‫א‬‫א‬                                    ١٧١                                     ‫א‬
     ‫א‬                                                                 


                                                  card ( B A)        1
                                   P( B A) =                    =     = 50%
                                                  card ( S A)        2
                                                                   
٣‫א‬‫א‬٥‫א‬‫א‬W١٦
                                                                      W‫א‬‫א‬K‫א‬‫א‬
                                                                                      K‫א‬‫א‬‫א‬E١
                                                                                      K‫א‬‫א‬‫א‬E٢
                                                           K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬E٣
                                                              K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬E٤
                                                                                                          W‫א‬
                                                                             W‫א‬‫א‬
                        K‫א‬‫א‬‫א‬W C ،‫א‬‫א‬‫א‬W B ،‫א‬‫א‬‫א‬W A
 C A  B  A W‫א‬٤١‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
                                                                                                        K C B 
 R1 , R2 , R3 , R4 , R5  S = {R1 , R2 , R3 , R4 , R5 , B1 , B2 , B3 } W‫א‬‫א‬E١
                                                                               
                                    WK‫א‬‫א‬‫א‬ B1 , B2 , B3 ‫א‬‫א‬‫א‬
                                                                       5
                        A = {R1 , R2 , R3 , R4 , R5 }, P( A) = = 62.5% ≠ 0
                                                                       8
                                                                       W،‫א‬‫א‬E٢
                                                             3
                             B = {B1 , B2 , B3 }, P( B) = = 37.5% ≠ 0
                                                             8
                                                                   
‫א‬‫א‬‫א‬‫א‬ S A = {R2 , R3 , R4 , R5 , B1 , B2 , B3 } W‫א‬‫א‬E٣
                                  W C A = {B1 , B2 , B3 } WK
                                                      3
                                         K P(C A) = ≅ 42.86%
                                                      7
                                                                     
‫א‬‫א‬‫א‬‫א‬ S B = {R1 , R2 , R3 , R4 , R5 , B2 , B3 } W‫א‬‫א‬E٤
                                     W C B = {B2 , B3 } WK
                                                         2
                                         K P(C B) = ≅ 28.57%
                                                         7
                             W‫א‬‫א‬‫א‬ B  A  S W٢

                                                      -١٣٢-
      ‫א‬‫א‬                                  ١٧١                                ‫א‬
     ‫א‬                                                          


                               0) P( S ) = 1,       P (φ ) = 0
                               1) P ( A) = 1 − P ( A),       P( A) = 1 − P ( A)
                             2) P( A U B) = P( A) + P( B) − P( A I B)
                                                P( B I A)
                               3) P ( B A) =                     ( P ( A) ≠ 0)
                                                  P( A)
                                                    K‫א‬‫א‬W١٧
                                                                              W‫א‬‫א‬
                                                                              K‫א‬‫א‬E١
                                                                     K‫א‬‫א‬‫א‬E٢
                                                                                                   W‫א‬
                                                                       W‫א‬‫א‬
                  S = {HHHH , HHHT , HHTH , HTHH , THHH , HHTT ,L, TTTT }
               K = {H , T } × {H , T } × {H ,T } × {H , T }
                  card ( S ) = 2 × 2 × 2 × 2 = 16
                                K‫א‬ B  A ٢١‫א‬‫א‬‫א‬‫א‬
                                                         1
                               1) A = {TTTT } ⇒ P( A) =    = 6.25%
                                                        16
                  W،‫א‬‫א‬W B W B ‫א‬E٢
                                                                        1
                        B = A ⇒ P( B) = P( A) = 1 − P( A) = 1 −          = 93.75%
                                                                       16
                                                           K‫א‬‫א‬E١‫א‬‫א‬
‫א‬‫א‬ 70% ‫א‬‫א‬‫א‬ W١٨
‫א‬‫א‬ 73% ‫א‬‫א‬ 85% 
                                                                                               ‫؟‬‫א‬
                                                                                                   W‫א‬
        K‫א‬K‫א‬‫א‬
‫א‬‫א‬W B ‫א‬‫א‬W A W‫א‬‫א‬‫א‬
                                                                                  
                                W‫א‬ A U B ‫א‬‫א‬
                           P( A) = 0.7, P( B) = 0.85, P( A I B) = 0.73
                                                 W‫א‬‫א‬E٢‫א‬‫א‬
                                                     -١٣٣-
      ‫א‬‫א‬                              ١٧١                              ‫א‬
     ‫א‬                                                    



                P( A U B) = P( A) + P( B) − P( A I B) = 0.7 + 0.85 − 0.73 = 82%
                                   K‫א‬‫א‬E٣‫א‬‫א‬١٥‫א‬W١٩
                                                                                             W‫א‬
                                       K S = {HH , HT ,TH , TT } W‫א‬‫א‬
                                                                          W‫א‬
                             
‫א‬‫א‬K‫א‬‫א‬‫א‬W B ،‫א‬‫א‬‫א‬W A
                                                                                    WK B A
                           A = {HH , HT }, B = {HT , TT },      A I B = {HT }
                                                1
                                    P( A I B) 4 2
                           P( B A) =          =   = = 50%
                                       P ( A)   1 4
                                                2
،‫א‬‫א‬‫א‬‫א‬‫א‬ B  A W١٣
                                                                                            W
                                    P( A I B) = P( A) × P( B)
                                               K‫א‬
                              W‫א‬‫א‬K B  A W٣
                                     P( A I B) = P( A) × P( B)       W‫א‬‫א‬
                                   P( A I B) = P( A) × P( B A)       W‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
                                                                             K‫؟‬
  
‫א‬‫א‬‫א‬‫א‬‫א‬W٢٠
                                                                       
‫א‬‫א‬‫א‬K٠{٢‫א‬‫א‬‫א‬‫א‬‫א‬٠{١
                                                                               
                                                                            K‫א‬‫א‬‫א‬
                                                                                             W‫א‬
        K‫א‬K‫א‬‫א‬
                                     
‫א‬‫א‬‫א‬W B ‫א‬‫א‬W A W‫א‬
                                                                        
                                                          W‫א‬ A I B ‫א‬‫א‬

                                                 -١٣٤-
         ‫א‬‫א‬                               ١٧١                               ‫א‬
        ‫א‬                                                      



                                    P( A) = 0.1, P( B A) = 0.2
                                  W‫א‬‫א‬‫א‬
                             P( A I B) = P( A) × P( B A) = 0.1 × 0.2 = 2%
                                 W‫א‬‫א‬‫א‬‫א‬W٢١
                                                 
‫א‬W C ،‫א‬‫א‬‫א‬W B ،‫א‬‫א‬‫א‬W A
                                             K‫א‬‫א‬‫א‬‫א‬‫א‬K‫א‬‫א‬
                                                                                                  W‫א‬
                                          
 C  A ‫א‬K‫א‬ B  A ‫א‬‫א‬‫א‬
                                                                 K‫א‬ C  B 
                                                                                W‫א‬‫א‬
           S = {HHH , HHT , HTH , THH , HTT , THT , TTH , TTT } = {H , T } × {H , T } × {H , T }
    
           card ( S ) = 2 × 2 × 2 = 8
                                                              W‫א‬‫א‬‫א‬‫א‬
                                                                4
                            A = {HHH , HHT , HTH , HTT }, P ( A) == 50%
                                                                8
                                                                4
                           B = {HHH , HHT ,THH , THT }, P( B) = = 50%
                                                                8
                                                      2
                            C = {HHT , THH }, P (C ) = = 25%
                                                      8
                                                                                W B  A ‫א‬E١
                                                       2
         A I B = {HHH , HHT } ⇒ P( A I B) = = 25% = 50% × 50% = P( A) × P( B)
                                                       8
                                                                                                     
                                                                                  K B  A ‫א‬
                                                                                W C  A ‫א‬E٢
                                                 1
             A I C = {HHT } ⇒ P( A I C ) = = 12.5% = 50% × 25% = P( A) × P(C )
                                                 8
                                                                                                     
                                                                                  K C  A ‫א‬
                                                                                W C  B ‫א‬E١
                                                2
  B I C = {HHT , THH } ⇒ P( B I C ) = = 25% ≠ 12.5% = 50% × 25% = P( B) × P(C )
                                                8
                                                                                                     
                                                                                  K C  B ‫א‬
                                                     -١٣٥-
      ‫א‬‫א‬                                ١٧١                                   ‫א‬
     ‫א‬                                                           



                        K٣١٠‫א‬W٢٢
                                          W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
                                  K‫א‬E٢                  ،‫א‬‫א‬E١
                                                                                                    W‫א‬
                                                                                   W‫א‬‫א‬‫א‬
                                 
‫א‬‫א‬K‫א‬‫א‬‫א‬W B ،‫א‬‫א‬‫א‬W A 
                                                                                       AI B
                                      KW B  A ‫א‬‫א‬‫א‬E١
                                                               7 7
                            P( A I B) = P( A) × P( B) =        × = 49%
                                                              10 10
                                  KW B  A ‫א‬‫א‬‫א‬E٢
                                                               7 6
                          P( A I B) = P( A) × P( B A) =        × ≅ 46.67%
                                                              10 9
                                                                                                            




                                                   -١٣٦-
     ‫א‬‫א‬                               ١٧١                             ‫א‬
    ‫א‬                                                    



                                              
                                                                                                    
                  KK{1,2,3,4,5,6} ‫א‬W١
                                                                       KE١
                                                                  W‫א‬E٢
                       KW C ،W B ،W A 
                                                           KE٢‫א‬‫א‬‫א‬E٣
  W‫א‬K S = {1,2,3,4} W‫א‬‫א‬W٢
                                    1            1           1         1
                          1) P (1) = ,   P ( 2) = ,   P(3) = , P(4) =
                                    2            3           4         5
                                    1             1             1        1
                          2) P (1) = ,   P ( 2) = ,   P (3) = − , P(4) =
                                    2            4             4        2
                                    1            1            1        1
                          3) P (1) = ,   P (2) = ,    P(3) = , P (4) =
                                    2            4            8        8
                                     1            1           1
                          4) P (1) = ,   P ( 2) = ,   P (3) = , P(4) = 0
                                     2            4           4
      ‫؟‬‫א‬٣‫א‬‫א‬‫א‬‫א‬W٣
٣‫א‬٤‫א‬‫א‬W٤
                                                                                ‫؟‬٥‫א‬
                              K٥٠١٥٠W٥
                                                                     W‫א‬‫א‬
                                                                               K٧‫א‬E١
                                                                       K١١E٢
                                                                       K١٩E٣
                           K١٠٠٥١٥٠W٦
                                                                     W‫א‬‫א‬
                                                                              K٥٣‫א‬E١
                                                                       K٦١E٢
                                                                       K٧٩E٣
                                                 -١٣٧-
      ‫א‬‫א‬                              ١٧١                                ‫א‬
     ‫א‬                                                      



                        W‫א‬‫א‬KW٧
                                                                         ‫؟‬‫א‬‫א‬E١
                                                                 ‫؟‬‫א‬‫א‬‫א‬E٢
                                                      ‫؟‬‫א‬‫א‬‫א‬‫א‬E٣
 KK٦١‫א‬ W٨
                                              W‫א‬‫א‬‫א‬
                                                                                K٦E١
                                                                                  K١‫א‬E٢
                                                                    KK٤E٣
                                                                                                       
                                                                                    W‫א‬K٥
‫א‬‫א‬ W١٤
                                                                            KK
           K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
                                                              
‫א‬‫א‬‫א‬W٢٣
      K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
                           K‫א‬‫א‬‫א‬‫א‬‫א‬
                                                       
‫א‬‫א‬‫א‬W٢٤
  K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
                          K‫א‬‫א‬‫א‬‫א‬‫א‬
                                      
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
                                                                    K‫א‬
                                            W٤٥‫א‬W٢٥
                       2, 1, 3, 5, 2, 2, 2, 1, 4, 2, 6, 2, 4, 3, 1,
                   2, 4, 3, 1, 4, 4, 2, 4, 4, 2, 2, 3, 1, 4, 2,
                       3, 1, 5, 2, 4, 1, 3, 2, 4, 4, 2, 5, 1, 3, 4
     K‫א‬‫א‬‫א‬‫א‬‫א‬W١٥

                                                 -١٣٨-
     ‫א‬‫א‬                                ١٧١                                ‫א‬
    ‫א‬                                                        



                                                                         K‫א‬‫א‬W٢٦
                                                                            ‫؟‬‫א‬E١
                                                                    K‫א‬‫א‬E٢
                                                                                                W‫א‬
                                                                                 K٤٥‫א‬E١
                                                               W‫א‬‫א‬‫א‬E٢


                                                                                                        
                      ‫א‬‫א‬‫א‬‫ א‬ ‫א‬‫א‬                ‫א‬
                           ٨                 ٨                   ١
                           ٢٢               ١٤                   ٢
                           ٢٩                ٧                   ٣
                           ٤١               ١٢                   ٤
                           ٤٤                ٣                   ٥
                           ٤٥                ١                   ٦
                                            ٤٥                 ‫א‬
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
                                     W‫א‬‫א‬‫א‬
                               14

                               12

                               10

                               8

                               6

                               4

                               2


                                   1    2   3        4     5       6

            W‫א‬‫א‬ {x1 , x2 ,L, xn } W‫א‬‫א‬W١٦
                                                                            W‫א‬
                                            x1 + x2 + L + xn
                                     x=
                                                    n
                                                -١٣٩-
             ‫א‬‫א‬                                               ١٧١                                             ‫א‬
            ‫א‬                                                                                    



                                                                                                                   W‫א‬

                                       s   2
                                                =
                                                  (x − x ) + (x − x )
                                                         1
                                                             2                             2
                                                                                   + L + x − xn
                                                                                   2                  (      )2


                                                                               n −1
                                                                                                       W‫א‬‫א‬‫א‬
                                        s = s2
     W‫א‬E‫א‬F‫א‬‫א‬‫א‬‫א‬
                                                      med = x n+1                            W‫א‬ n ‫א‬
                                                                          2
                                                         xn + xn
                                                                          +1
                                           med =            2        2
                                                                                              W‫א‬ n ‫א‬
                                                                 2
               K mode WK‫א‬‫א‬‫א‬‫א‬‫א‬
                                                              K‫א‬‫א‬
                                       K‫א‬‫א‬‫א‬‫א‬‫א‬
    ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
                                                                                                                   K‫א‬
                                                              K‫א‬‫א‬‫א‬‫א‬W٢٧
                                                                                                                                    W‫א‬
    ‫א‬‫א‬E١
                                    W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
                                   (8 × 1) + (14 × 2) + (7 × 3) + (12 × 4) + (3 × 5) + (1 × 6)
                         x=                                                                   = 2.8
                                                               45
                                                                                                   W‫א‬‫א‬E٢
           8(2.8 − 1) + 14(2.8 − 2 ) + 7(2.8 − 3) + 12(2.8 − 4 ) + 3(2.8 − 5) + 1(2.8 − 6 )
                               2                 2                    2                               2             2              2
       2
       s =                                                                                  ≅ 1.75
                                                45 − 1
                                                                                                      W‫א‬‫א‬‫א‬E٣
                                                   s = 1.75 ≅ 1.32
                                                                                 
                               W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ n = 45 E٤
                                             med = x n+1 = x 45+1 = x23 = 3
                                                                  2                    2

                                                                                                W‫א‬‫א‬‫א‬‫א‬‫א‬E٤
                                                                      -١٤٠-
         ‫א‬‫א‬                               ١٧١                                  ‫א‬
     ‫א‬                                                            



                                                mode = 2
                                                                                       W
 WE CASIO fx − 82TL WF‫א‬‫א‬‫א‬‫א‬
                                            KE MODE SD F‫א‬‫א‬‫א‬‫א‬W‫א‬‫א‬
                                         KE SHIFT Scl = F‫א‬‫א‬‫א‬W‫א‬‫א‬
                                       KE DT ‫א‬‫א‬F‫א‬W‫א‬‫א‬
F‫א‬E SHIFT xσ n −1 F‫א‬‫א‬‫א‬E SHIFT x F‫א‬W‫א‬‫א‬‫א‬
                                                                                KE‫א‬‫א‬‫א‬
           WEF‫א‬‫א‬W٢٨
                  0.80 0.81 0.81 0.82 0.81 0.82 0.80 0.82 0.81 0.81
                              K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
                                                                                                     W‫א‬
                                                     W‫א‬‫א‬
                           ‫א‬‫א‬‫א‬‫א‬             ‫א‬‫א‬             EF‫א‬‫א‬
                                 ٢                        ٢                    ٠{٨٠
                                 ٧                        ٥                    ٠{٨١
                                ١٠                        ٣                    ٠{٨٢
                                                                 W‫א‬‫א‬‫א‬‫א‬
               (2 × 0.80) + (5 × 0.81) + (3 × 0.82)
            x=                                      = 0.811 cm
                                10
             2  2(0.811 − 0.80) 2 + 5(0.811 − 0.81) 2 + 3(0.811 − 0.82) 2
            s =                                                           ≅ 0.000 054 cm 2
                                          10 − 1
           s = 0.000054 ≅ 0.007348 cm
                  1⎛            ⎞ 1⎛             ⎞ 1           1
            med = ⎜ x n + x n ⎟ = ⎜ x10 + x10 ⎟ = ( x5 + x6 ) = (0.81 + 0.81) = 0.81 cm
                  2⎜ 2
                    ⎝
                             +1 ⎟
                            2 ⎠
                                  2⎜ 2
                                   ⎝        2
                                              +1 ⎟
                                                 ⎠ 2           2
            mode = 0.81 cm
                                                                                                             



                                                     -١٤١-
  ‫א‬‫א‬                                   ١٧١                             ‫א‬
 ‫א‬                                                        



                                             
         W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W١
                                    4 6 6 7 9 10
                                           W‫א‬٢٠‫א‬W٢
   ٢٠ ١٩           ١٨    ١٧   ١٦ ١٥        ١٤     ١٣ ١٢ ١٠         ٩     ‫א‬‫א‬
    ٤       ٦      ٢     ٧     ١    ٢      ٧      ٢       ١    ٢   ١               ‫א‬
                                                                           K‫א‬
                    W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W٣
                                         3 4 10 4 4
                                              K‫א‬١٠‫א‬
 K105 110 115 109 110 111  W‫א‬‫א‬‫א‬‫א‬‫א‬W٤
                                                                                 K‫א‬
                                                K‫א‬‫א‬‫א‬‫א‬‫א‬ W٥




                                                  -١٤٢-
        ‫א‬‫א‬                                 ١٧١                               ‫א‬
                                                                     



                                            ‫א‬‫א‬

                                                        
،،‫א‬،‫א‬‫א‬،E١
                                                                                K١٩٨٣J ١٤٠٣
،‫א‬،‫א‬‫א‬‫א‬،E٢
                                                                                  K١٩٨١،
3) Alfred Aho, John Hopcroft and Jeffrey Ullman, Design and Analysis of Computer
Algorithms, Addison Wesley, Reading, England, 1974.
4) Gwyn Davies and Gordon Hick, Mathematics for scientific and technical students,
Addison Wesley Longman, Harlow, England, 1998.
5) C. L. Lui, Elements of Discrete Mathematics, McGraw-Hill, New York, 1977.
6) Alexander Schrijver, Theory of Linear and Integer Programming, John Wiley & Sons,
Chichester, England, 1986.
7) Seymour Lipschutz and Marc Lipson, Discrete Mathematics, McGraw-Hill, New York,
1997.
8) Peter Tebbutt, Basic Mathematics, John Wiley & Sons, Chichester, England, 1998.




                                               - ١٤٣ -
‫א‬              ١٧١                                          ‫א‬
                                                      



                    ‫א‬

                      
 ٢                                            ‫א‬‫א‬W‫א‬‫א‬
 ٢                                           ‫א‬‫א‬W‫א‬‫ א‬
 ٢                                      ‫א‬‫א‬K١
 ٢                 ‫א‬‫א‬‫א‬‫א‬K٢
 ٢                             ‫א‬‫א‬
 ٣                                   ‫א‬‫א‬
 ٣        
 ٤                                  ‫א‬‫א‬
 ٤                                                          
 ٥                                      ‫א‬‫א‬K٣ 
 ٥                          ‫א‬‫א‬‫א‬
 ٥             ax 2 + bx + c ‫א‬
 ٧                                  
 ٧                         
 ٨                           ‫א‬‫א‬ 
 ٩                                                            
 ١٠                                           ‫א‬‫א‬K٤
 ١٠                                ‫א‬‫א‬‫א‬ 
 ١٢                                                           
 ١٤                                               ‫א‬W‫א‬‫ א‬
 ١٤                                             ‫א‬K١
 ١٤                                       ‫א‬‫א‬K٢ 
 ١٦                                                           
 ١٧                            ‫א‬‫א‬‫א‬K٣ 
‫א‬          ١٧١                                              ‫א‬
                                                     



 ١٧                                         ‫א‬
 ١٧                                ‫א‬‫א‬
 ١٨                                 ‫א‬
 ١٩                                           ‫א‬
 ٢٠                                                           
 ٢٠                              ‫א‬‫א‬W‫א‬‫א‬
 ٢٠                                               ‫א‬‫א‬K١
 ٢٤                                                ‫א‬‫א‬‫א‬K٢
 ٢٨                                             ‫א‬‫א‬‫א‬K٣
 ٣٠                                             ‫א‬‫א‬‫א‬K٤ 
 ٣٣                                        ‫א‬‫א‬‫א‬K٥ 
 ٣٧                                                         
 ٣٩                                               ‫א‬W‫א‬‫א‬
 ٣٩                        ‫א‬‫א‬‫א‬K1
 ٤٠                                             ‫א‬K٢
 ٤١                                ‫א‬‫א‬‫א‬K٣
 ٤٤                                                         
 ٤٥                            ‫א‬‫א‬‫א‬‫א‬‫א‬K٤
 ٤٦                                                         
  ٤٧                    ‫א‬ ‫א‬‫א‬‫א‬‫א‬K٥
  ٤٧                       ‫א‬‫א‬‫א‬‫א‬‫א‬
  ٤٧                 ‫א‬‫א‬‫א‬‫א‬‫א‬
  ٤٨                                                         
 ٤٩        f ( x) ‫א‬‫א‬‫א‬‫א‬‫א‬K٦
 ٥٢                                                         
  ٥٤                                   ‫א‬‫א‬W‫א‬‫א‬‫א‬
  ٥٤                                ‫א‬‫א‬K١
‫א‬        ١٧١                                            ‫א‬
                                                 



  ٥٥                                    ‫א‬K٢
  ٥٦                                ‫א‬‫א‬K٣ 
  ٥٧                                                     
  ٥٧                                    ‫א‬‫א‬K٤
  ٥٩                                ‫א‬‫א‬K٥
  ٦٠              ‫א‬‫א‬‫א‬‫א‬‫א‬K٦
  ٦١             ‫א‬‫א‬‫א‬K٧
  ٦٢                                                    
  ٦٤                            ‫א‬W‫א‬‫א‬
  ٦٤                                          ‫א‬‫א‬K١
 ٦٧                                                    
 ٦٨                                    ‫א‬K٢
 ٦٩                   ‫א‬‫א‬‫א‬
 ٧٠                          ‫א‬‫א‬‫א‬‫א‬ 
 ٧١                                                    
  ٧٢                                      ‫א‬K٣
 ٧٤                                                    
 ٧٥                      ‫א‬‫א‬K٤
 ٧٦                      ‫א‬‫א‬
 ٧٨                                                    
 ٨٠                 ‫א‬‫א‬‫א‬‫א‬W‫א‬‫א‬
 ٨٠                                       ‫א‬‫א‬K١
 ٨٠                                 ‫א‬‫א‬
 ٨٤                                               ‫א‬
 ٨٥                                               ‫א‬‫א‬
 ٨٦                                                     
 ٨٨                                       ‫א‬‫א‬‫א‬K٢ 
‫א‬                  ١٧١                                       ‫א‬
                                                       



 ٨٨                                        ‫א‬‫א‬
 ٨٨                                                    ‫א‬
 ٨٩                                                 ‫א‬‫א‬
 ٨٩                                                    ‫א‬
 ٩٠                                                       ‫א‬
 ٩١                                                     ‫א‬
 ٩٢                                                            
 ٩٣                                            ‫א‬‫א‬‫א‬K٣
 ٩٣                             ‫א‬‫א‬‫א‬
 ٩٤                                W‫א‬‫א‬‫א‬
 ٩٥                                            ‫א‬‫א‬
 ١٠٠                                   ‫א‬‫א‬W‫א‬‫א‬
 ١٠٠                               ‫א‬‫א‬K١
 ١٠٣                        ‫א‬‫א‬‫א‬K٢
 ١٠٧                            ‫א‬‫א‬KK٣
 ١٠٩                    ‫א‬‫א‬K٤
 ١١١                            ‫א‬‫א‬K٥
 ١١٣                           ‫א‬‫א‬K٦
 ١١٤                   ‫א‬‫א‬‫א‬K٧
 ١١٥                    ‫א‬‫א‬‫א‬K٨
 ١١٥                                          ‫א‬K٩ 
 ١١٦      ‫א‬،‫א‬،‫א‬‫א‬‫א‬K١٠
 ١١٧               ‫א‬‫א‬‫א‬K١١
 ١٢١       ‫א‬‫א‬‫א‬‫א‬K١٢
 ١٢٢                                                          
 ١٢٥                                         ‫א‬W‫א‬‫א‬
 ١٢٥                                                   ‫א‬K١
‫א‬        ١٧١                               ‫א‬
                                     



 ١٢٦                               ‫א‬K٢
 ١٢٩                          ‫א‬K٣
 ١٣١                         ‫א‬‫א‬K٤
 ١٣٧                                        
 ١٣٨                             ‫א‬K٥
 ١٤٢                                        
 ١٤٣                                        ‫א‬‫א‬ 
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               ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

             ‫א‬E‫א‬F‫א‬‫א‬

GOTEVOT appreciates the financial support provided by BAE SYSTEMS

								
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