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					                               FINALTERM EXAMINATION

                                       Spring 2010

                                  MTH301- Calculus II

                                                                      Time: 90 min

                                                                            Marks: 60

Student Info

    Student ID:

    Center:

    Exam Date:




For Teacher's Use Only

 Q No.         1     2    3       4         5        6    7    8    Total

Marks

 Q No.         9     10   11      12        13       14   15   16

Marks

 Q No.        17     18   19      20        21       22   23   24

Marks

 Q No.        25     26   27      28        29       30   31   32

Marks

 Q No.        33     34   35      36        37       38   39

Marks
       Question No: 1       ( Marks: 1 ) - Please choose one




Q 1.   Intersection of two straight lines is --------------




           ►Surface



           ►Curve



a.         ►Plane



b.         ►Point




       Question No: 2       ( Marks: 1 ) - Please choose one




       Plane is a --------------- surface.



           ►One-dimensional



           ►Two-dimensional
    ►Three-dimensional



    ►Dimensionless




Question No: 3       ( Marks: 1 ) - Please choose one




Let w = f(x, y, z) and x = g(r, s), y = h(r, s), z = t(r, s) then by chain rule

w
   
r




     w x w y w z
                
    ► x r y r z r



       w x w y w z
                 
       r r r r r r
    ►



       w x x w y y w z z
                       
       x r s y r s z r s
    ►
       w r w r w r
                 
       r x r y r z
    ►




Question No: 4         ( Marks: 1 ) - Please choose one




What are the parametric equations that correspond to the following vector equation?

               ^                 ^
r (t )  sin 2 t i  (1  cos 2t ) j




       x  sin 2 t ,       y  1  cos 2t , z  0
    ►



       y  sin 2 t ,       x  1  cos 2t , z  0
    ►



       x  sin 2 t ,       y  1  cos 2t , z  1
    ►



       x  sin 2 t ,       y  cos 2t   , z 1
    ►




Question No: 5         ( Marks: 1 ) - Please choose one
What are the parametric equations that correspond to the following vector equation?

                 ^        ^          ^
r (t )   2t  1 i  3 t j  sin 3t k




       z  2t  1     ,       x  3 t    ,   y  sin 3t
    ►



       y  2t  1     ,       x  3 t    ,   z  sin 3t
    ►



       x  2t  1     ,       z  3 t    ,   y  sin 3t
    ►



       x  2t  1     ,       y  3 t    ,   z  sin 3t
    ►




Question No: 6        ( Marks: 1 ) - Please choose one




What is the derivative of following vector-valued function?


r (t )  (cos5t , tan t , 6sin t )
      
                 sin 5t                   
      r (t )          , sec t , 6 cos t 
                 5                        
    ►



                    sin 5t
      r (t )  (            , sec t , 6cos t )
                       5
    ►



      
      r (t )  (5sin 5t , sec2 t , 6cos t )
    ►



      
      r (t )  (sin 5t , sec2 t ,  6cos t )
    ►




Question No: 7             ( Marks: 1 ) - Please choose one




What is the derivative of following vector-valued function?


                        3
r (t )   t 4 , t  1 , 2 
                       t 




      
                             1   6 
      r (t )   4t 3 ,         , 3 
                            t 1 t 
    ►
      
                          1     6
      r (t )   4t 3 ,       , 3
                        2 t 1 t 
    ►



      
                          1    6 
      r (t )   4t 4 ,       , 3 
                        2 t 1 t 
    ►



      
                          1    6 
      r (t )   4t 3 ,       , 3 
                        2 t 1 t 
    ►




Question No: 8      ( Marks: 1 ) - Please choose one




The following differential is exact

dz  ( x 2 y  y ) dx  x dy




    ►True



    ►False
Question No: 9     ( Marks: 1 ) - Please choose one




Which one of the following is correct Wallis Sine formula when
                                                                       n
                                                                           is even and
                                                                                         n  2?



      
      2
                           n  1  n  3  n  5                 5 3 1
      
      0
          sin n x dx 
                         2     n      n  2  n  4
                                                         
                                                                      6 4 2
   ►



      
      2
                          n  1  n  3  n  5        6   4 2
      
      0
          sin n x dx 
                             n  n  2  n  4                  7   5 3
   ►



      
      2
                               n   n  2  n  4        6   4 2
      
      0
          sin n x dx 
                         2    n  1  n  3  n  5              5   3 1
   ►



      
      2
                            n   n  2  n  4        6    4 2
      
      0
          sin n x dx 
                          n  1  n  3  n  5              5    3 1
   ►




Question No: 10      ( Marks: 1 ) - Please choose one
Match the following equation in polar co-ordinates with its graph.

r cos   a
where a is an arbitrary cons tan t




   ►




   ►




   ►




   ►
Question No: 11        ( Marks: 1 ) - Please choose one




                                                                                       (r ,  )
If the equation of a curve, in polar co-ordinates, remains unchanged after replacing              by
(r ,    )
               then the curve is said to be symmetric about which of the following?



    ►Initial line



    ►y-axis



    ►Pole




Question No: 12        ( Marks: 1 ) - Please choose one




                                                                                       (r ,  )
If the equation of a curve, in polar co-ordinates, remains unchanged after replacing              by
(r ,  )
            then the curve is said to be symmetric about which of the following?



    ►Initial line



    ►y-axis
   ►Pole




Question No: 13      ( Marks: 1 ) - Please choose one




                                                                          x
                                                           f ( x)  sin
                                                                          3
What is the amplitude of a periodic function defined by                       ?



   ►0



   ►1



      1
      3
   ►



   ►Does not exist




Question No: 14      ( Marks: 1 ) - Please choose one




                                                        f ( x)  4cos3x
What is the period of a periodic function defined by                      ?
      
      4
   ►



      
      3
   ►



      2
       3
   ►



     
   ►




Question No: 15     ( Marks: 1 ) - Please choose one




Match the following periodic function with its graph.

         3       0 x4
f ( x)  
         0       4 x6




   ►
   ►




   ►




   ►




Question No: 16    ( Marks: 1 ) - Please choose one




What is the period of periodic function whose graph is as below?




   ►2
   ►5



   ►6



   ►8




Question No: 17    ( Marks: 1 ) - Please choose one




What is the period of periodic function whose graph is as below?




   ►0



   ►4



   ►6



   ►8
Question No: 18        ( Marks: 1 ) - Please choose one




Let L denotes the Laplace Transform.

                                                             F (t ) 
                                                       lim          
                                                       t 0
                                                             t 
  L{F (t )}  f ( s)           s
If                     where       is a constant.and                     exists then which of the following equation
holds?



       F (t ) 
     L          f (s  a)
       t 
   ►



       F (t ) 
     L          f (s  a)
       t 
   ►



                   
       F (t ) 
     L           f (s) ds
       t  s
   ►



       F (t )     d
     L           { f ( s )}
       t          ds
   ►




Question No: 19        ( Marks: 1 ) - Please choose one
                                                                                   f ( s)
Which of the following is Laplace inverse transform of the function                         defined by
             3    2
f ( s)         
            s2   s
                           ?




        3te2t  2
      ►



        3 e 2t  2t
      ►



        3 e 2t  2
      ►



      ►None of these.




Question No: 20                ( Marks: 1 ) - Please choose one




      ( x1 , y1 , z1 )         ( x2 , y2 , z2 )
Let                      and                      be any two points in three dimensional space. What does the formula
  ( x2  x1 )2  ( y2  y1 )2  ( z2  z1 )2
                                                     calculates?



      ►Distance between these two points
   ►Midpoint of the line joining these two points



   ►Ratio between these two points




Question No: 21      ( Marks: 1 ) - Please choose one




                     P( x, y)         Q( x, y)
 Let the functions              and              are finite and continuous inside and at the boundary of a

                                         P dx  Q dy 
 closed curve C in the xy-plane. If                       is an exact differential then



  C
     P dx  Q dy  



      ►Zero




   ►One



   ►Infinite




Question No: 22      ( Marks: 1 ) - Please choose one
What is Laplace transform of the function F (t ) if F (t )  t ?



                1
      L t 
    ►           s



                1
      L t 
    ►           s2



      L t  e s
    ►



      L t  s
    ►




Question No: 23          ( Marks: 1 ) - Please choose one




                            L{e5t }
  What is the value of                if L denotes laplace transform?



                       1
      L{e5t } 
    ►                s 5



                      s
      L{e5t } 
    ►               s  25
                     2
                     5
       L{e5t } 
   ►               s  25
                   2




                   5!
       L{e5t } 
   ►               s6




Question No: 24         ( Marks: 1 ) - Please choose one




                               
                               C
                                   (3x  2 y) dx  (2 x  y) dy

 Evaluate the line integral                                       where C is the line segment from (0, 0) to (0,
 2).




   ►1



   ►0



   ►2



   ►-2
Question No: 25     ( Marks: 1 ) - Please choose one




                               
                               C
                                   (2 x  y) dx  ( x 2  y) dy

  Evaluate the line integral                                      where C is the line segment from (0, 0) to (2,
  0).




    ►0



    ►-4



    ►4



    ► not exist
     Do




Question No: 26     ( Marks: 1 ) - Please choose one




Which of the following are direction ratios for the line joining the points (1, 3, 5) and (2,  1, 4) ?




    ► 2 and 9
     3,




    ► -4 and -1
     1,
     ► -3 and 20
      2,




     ►0.5, -3 and 5/4




Question No: 27         ( Marks: 1 ) - Please choose one




If R  {( x, y ) / 0  x  2 and 1  y  4}, then


R
     (6 x 2  4 xy 3 )dA 




       4   2

        
       1   0
               (6 x 2  4 xy 3 )dydx

     ►



       2   4

        
       0   1
               (6 x 2  4 xy 3 )dxdy

     ►



       4   2

        
       1   0
               (6 x 2  4 xy 3 )dxdy

     ►
      4   1

       
      2   0
              (6 x 2  4 xy 3 )dxdy

   ►




Question No: 28        ( Marks: 1 ) - Please choose one




Which of the following is true for a periodic function whose graph is as below?




   ►Even function



   ►Odd function



   ►Neither even nor odd function




Question No: 29        ( Marks: 1 ) - Please choose one
Which of the following is true for a function whose graph is given above




    ► odd function
     An




    ► even function
     An




    ►Neither even nor odd




Question No: 30      ( Marks: 1 ) - Please choose one




At each point of domain, the function ----------------
   ► defined
    Is




   ► continuous
    Is




   ► infinite
    Is




   ► a limit
    Has




Question No: 31      ( Marks: 2 )




 Determine whether the following differential is exact or not.

  dz  4 x3 y 3 dx  3x 4 y 2 dy


 Solution:


dz  4 x3 y 3 dx  3x 4 y 2 dy
p
     12 x3 y 2
y
Q
     12 x3 y 2
X
p Q
    
y X
yes
Question No: 32        ( Marks: 2 )



    Evaluate

      

      
     
          sin nx dx



    where n is an integer other than zero.



Solution:








      sin nx dx

                  
   cos nx 

  n     
   cos n cos n 
           
      n       n   
 1
 ( cos n  cos n )
 n
0



Question No: 33        ( Marks: 2 )




                                          F (t ) if F (t )  e
                                                               3t

    Find Laplce transform of the function



Solution:
                              
         L(e )   e3t  e  st
                 3t

                              0
             
           e  ( s 3)t .dt
             0

             e  ( s 3)t
         {                }lim 0  
            ( s  3)
            1            1
                    ( ( s 3)t )
           s 3 e
            1
                    (0  1)
           s 3
             1
                   ......Ans
           s 3




Question No: 34                   ( Marks: 3 )




                                                 a0
Determine the Fourier co-efficient                    of the periodic function defined below:

f ( x)  2 x  1                         0 x2

Solution:

         1       

          
a            f ( x)dx
             

 f ( x)  (2 x  1)
(0, 2)
     2
  (2 x  1)dx
   0
                      2
  x2  x
                        0

6
Question No: 35       ( Marks: 3 )




 Determine whether the following differential is exact or not.

  dz  (3x 2e2 y  2 y 2e3 x ) dx  (2 x3e2 y  2 ye3 x ) dy


Solution:



dz = Pdx + Qdy



Therefore,
                                                               P   Q
For dz to be an exact differential it must satisfy                =
                                                               y   x
                                P   Q
But this test fails becuase        
                                y   x
Not Exact




Question No: 36       ( Marks: 3 )



                                                


                                                  sin       x  sin 5 x  dx
                                                2
                                                          3

                                                0

      Use Wallis sine formula to evaluate
Solution:



    

   0
     2
         sin 3 xdx
  n 1
      .
    n
  3 1

    3
  2

  3
    

   0
     2
         sin 5 xdx
  n 1 n  3
     .
    n n2
  5 1 5  3
     .
    5 52
  4 2
 .
  5 3



  sin           x  sin 5 x  dx
2
              3

0

        2 4 2
         .
        3 5 3




Question No: 37               ( Marks: 5 )




Evaluate the following line integral which is independent of path.

(3,2)

    
(0,0)
           (2 xe y ) dx  ( x 2e y ) dy
Solution:




   z
p     2e y                             2e dx
                                             y

   x
   z
Q     x 2e y                           x e dy
                                            2 y

   y
        (3,2)
z             2 xe y  x 2 ye y
        (0,0)

z  6e 2  18e 2
z  24e 2



Question No: 38              ( Marks: 5 )




                                                  bn                             f (t )
Determine the Fourier coefficients                     for a periodic function            of period 2 defined by

         4 (1 + t)                           -1 < t < 0
f (t )  
         0                                       0<t<1


Solution:



         1      
bn 
              f ( x) sin nxdx
                

    1    1

    
            4(1  t ) sin nxdx
        1


  1  4(1  t ) cos nx 
                                    1

                      
             n         1
  4(1  t )
            cos n(1)  cos n(1) 
    n
  4(1  t )
            (cos n  cos n)
    n
Question No: 39        ( Marks: 5 )




                                                             

 Determine whether the following vector field F is conservative or not.

                            ^             ^            ^
  F ( x, y, z )  (4 x  z ) i  (3 y  z ) j  ( y  x) k

…………………………………..

				
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