Class XII Mathematics 2012 Question Paper (Set 1) by gyanguru

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									I Series : SMAil I                                                                                                           m.r.
                                                                                                                                                 65/1/1
                                                                                                                             Code No.
-m-r.r.                                                                                                  W&wff         m-     CfiT~-~               ~ ~- ~
Roll No.                                                                                                 lR~~1
                                                                                                          Candidates must write the Code on
                                                                                                          the title page of the answer-book.


     •    ~~Cfl{Bfq)~~-11;f -q:~~8                                                  t   I
     •    ~11;f -q:OO~citam-~~m-~CfiT~~-~~~-~lR~                                                                                            I

     •    ~~Cfl{Bfq)~~ -11;f-q:                              29   ~t            I

     •    ~~cnr~fM&"'11                            wm~~,                       ~cnr~~W-                                        I
     •    ~      ~-11;f   CfiT~         ~ ~      15 fl:Rc: Cf)f~                    R<:rr IT<:{T     ~ I ~-11;f Cf)ffcrrRur ~                    -q: 10.15     ~
                 ~
          fcf;<:rr          I   10.15 ~        ~ 10.30 ~ ~~                             ~      ~-11;f        t$T 31tt~ ~                  ~~          ~~ -
          ~lR~~~~1

     •    Please check that this question paper contains 8 printed pages.
     •    Code number given on the right hand side of the question paper should be written on the
          title page of the answer-book by the candidate.
     •    Please check that this question paper contains 29 questions.
     •    Please write down the Serial Number of the question before attempting it.
.•        15 minutes time has been allotted to read this question paper. The question paper will be
          distributed at 10.15 a.m. From 10.15 a.m. to 10.30 a.m., the students will read the
          question paper only and will not write any answer on the answer script during this period.

                                                                          1"ffUto
                                                            MATHEMATICS·
~             "fW{ : 3 "f!TR.]                                                                                                          [~3fcn:                 100
Time allowed : 3 hours                  J                                                                                      [ Maximum marks: 100

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65/111                                                                     1                                                                                [P.T.D.
    General Instructions:
         (i)   All questions are compulsory.
         (ii)  The question paper consists of 29 questions divided into three Sections A, Band C.
               Section A comprises of 10 questions of one mark each, Section B comprises of 12
               questions of four marks each and Section C comprises of 7 questions of six marks
               each.
         (iii) All questions in Section A are to be answered in one word, one sentence or as per
               the exact requirement of the question.
         (iv) There is no overall choice. However, internal choice has been provided in 4
               questions of four marks each and 2 questions of six marks each. You 'have to attempt
               only one of the alternatives in all such questions.
         (v)   Use of calculators is not permitted.



                                                                           ~-3l
                                                                       SECTION-A

             ~~            1 ~ 10 "ffCf)~~CflT                        1 .aicf;~ I
             Question numbers 1 to 10 carry 1 mark each.


~                   ~Wr         o);~          3FfIIil2, -1, -2        t, <it ~             ~        "bJ;;'"Q 'I'IT~?                                                         1

             If a line has direction ratios 2, -1, -2, then what are its direction cosines?


             ~             1\       1\        1\        ~   1\   1\        1\          n+""         ~             ~    ....,4.                            ~
                    a = Ai + j + 4k CflT b = 2i + 6j + 3k lR" ~~'i 4

/
             "II~                                                                                   ~"fll~        s,    \II      'A' CflTl1H ~            "P1I"l~        I   1

             Find 'A' when the projection ofa                         =   Ai + j + 4k on                     b = 2i + 6j + 3k is 4 'units.

                                                                                                                                      n               ~

/
             ~~             1\           1\    1\   ~       1\   1\        1\          ~       1\            1\         1\

             ~I<O;"11   a = i - 2j + k, b = -2i + 4j + 5k fMT c = i - 6j -7k CflT '4 111fli;1 ~                                                       "P1I"l~        I       1
                                                            ~    1\       1\     I\~                    1\        1\             1\           41\         1\        1\

             Find the sum of the vectors a = i - 2j + k, b = -2i + 4j + 5k and c = i - 6j - 7k.




I'~~~:[~dx                                                                                                                                                                   1



                                f~                                                                                                                              "
                                                                                                                                          "    t ••



             Evaluate:                   dx
                                2


    65/1/1                                                                             2
 /.            llR"ffif ~          f   (1 - x) --[x dx.                                                                1


               Evaluate     f   (1 - x) --[x dx-.




                            5 3 8
               ~L\=         2 0 1            t or ~       aZ3 CfiTQ'(lRfUI':f)~
                                                                 3                 :                                   1
~                           1 2 3

                          5 3 8
               If L\ =    2 0 1 , write the minor of the element aZ3'
                          1 2 3




/_
      a-:" ~
       ..
                   (25 3) ( -21 -3) = (-4 6) t, orxCfiTllR~
                        7         4     -9                x                            I                               1


                    2 3) ( -21 -3) = (-4 x6),write the value of x.
               If ( 5 7         4     -9




/-                                       [     cos O   sin e ]    . [sin e -cos e ]
 __ -.-        ~~:cose                                         +sm8                                                    1
                                                .
                                             -sm e     cos e           cos e sin e

                                        [      cos e   sin e ]    .    [sin e     -cos e ]
               Simplify:        cos e           . e            +sme
                                             -sm       cose               cose     sin e




                                                                                                                       1


               Write the principal value of cos-{i)             - 2 sin-{-i).




               l1RTN#ri             (a, b)    E   N*"~~~~*,a*b=~.~.                          (a, b)rnr~t         I
~.
               5*7"ffif~1                                                                                              1

               Let * be a 'binary' operation on N given by a * b = LCM (a, b) for all a, b            E    N. Find
               5   * 7.

      65/1/1                                                          3                                          [P.T.O.
                                                                        ~-~
                                                                  SECTION-B

            w.1~            11 ~ 22 0Cfl~w.14                          ~CfiTt                      I

            Question numbers 11 to 22 carry 4 marks each,

     1',    ~(cosx)Y=(cosy)Xt                           'ffi~~~                            I                                                                                    4

                                    ~
      /'                                                                       . Qy                         sin2(a + y) ~
            ~siny=xsin(a+y)t                             'ffi~~~                           d.x=                sin a    t' I


            If(cosx)Y = (cos y)X, find~,
                                    OR
                                                      sin2(a + y)  Qy
            If sin y = x sinea + y), prove that d.x =    sin a

     12/    ~     cx:lfcffi
                          ~             ~         fuqq;r fcmFiT ~ ~                        fcf>~             ~ ~    ~    m 3lR         q;1l'lIf4Cf)(11            80% ~
 c7 ~m?                                                                                                                                                                     .   4
              How many times must a man toss a fair coin, so that the probability of having at least
              one head is more than 80% ?

     ~ /'   ~       (1, 2, - 4) ~ ~.~                        qrffi ~ ~                     CfiT~               ~    cnm       ~Ji"ICf)<OI ~          ~              \ilT GT
~'          ~:m x -     8 = y + 19 = z - 10 ~ x - 15 = y - 29 = z - 5 "tR"
                                                                         <1k\q(1 I                                                     m                                        4
                      3      -16        7        3        8      -5
              Find the Vector and Cartesian equations of the line passing through the point
                      4) an d perpen dilCU1ar t 0 th e tw 0 lines x - 8 -_ Y + 19 -_ z ~ 10 and
              (1 , 2 ,-
                                                             .      3       -16         7
            x - 15 _ Y - 29 _ z - 5
            . 3 -       8 - -5 '

            ~      -+ -+            -+ -t:+-. ,+,+ ~      .>I- ~  -+                       -+
            <.il". a, b    ~                                       I
                                    c (11'1 ~'t1 't11"'~1 t' ICfl a 1=         5, I b 1= 12 ~                      I -+ 1=
                                                                                                                     c
                                                                                                                                ~
                                                                                                                             13 t' ~
                                                                                                                                           -+
                                                                                                                                           a+b+
                                                                                                                                                     -+       -+
                                                                                                                                                              C    =
                                                                                                                                                                        -+ ~
                                                                                                                                                                        0 t',
            ~-+       -+       -+       -+   -+    -+                  ~
            (11 a   ' b + b ' c + c ' a CfiTl1R ~                      '"Tl1I"l~       I                                                                                        4
                 -+ -+ -+                                                     -+                       -+               -+,                     -+        4        -+     _+
              If a, b, c are three vectors such that I a 1= 5, I b 1= 12 and I c I = 13, and a + b + c =                                                                  0,
                                             -+    -+   -+   -+   -+     -+
              find the value of a ' b + b ' c + c ' a,


     i
     ,
              f.l",:f •• ct -
              2x2~-        2xy+ y2 = 0,
                                             wil""., 'liT 0<'1
                                                             ~                     :                                                                                            4



              Solve the following differential equation:
              2x2~-        2xy + y2 = o.


     65/111                                                                                    4
              r"18f<:1fuI('1~                            ~
                                            ~41Cf)"'Ot'Cf)T                rn mo ~              :                                                   4

           ~ = 1 + X2 + y2 + x2y2,                       m t fct;"y = 1 ~         x=0t              t

              Find the particular solution of the following differential equation;
           ~ = 1 + x2 + y2 + x2y2, given that y = 1 when x = O.




                                                                                                                                                    4



              llRmo~:                     f(1-X)~1         +x2) dx

              Evaluate :         f    sin x sin 2x sin 3x dx
                                 OR

              Evaluate : f(1-X)~1                 + x2) dx



/-            ern;- y = x3   -       llx .+ 5 'q-{   ~     ~         mo ~       fi:m   'q-{ ~           WT 'Cf)T
                                                                                                               ~41Cf)"'Ot Y =x   - 11 t   I         4
                                            ~
              ~'Cf)Tw:Wr~-V49.5                ~f~Cf)l (approximate) llRmo~
                                           'Cf)T                                     I
              Find the point on the curve y = x3 - llx + 5 at which the equation of tangent is y = x-II.
                                            OR
              Using differentials, find the approximate value of-V49.5.

)Y~y=(tan-lx)2t,                                m~fct;"                                                                                            4
                         d2y                             . dv
              (x2 +   If ctx2 + 2x(x2 + 1) ~=                        2.
              Ify = (tan-1x)2, show that
                      _d2v              dv
              (x2 + 1)d;'2 + 2x(x2 + 1)~ = 2.


 ~\4t",fOtCflI          t-~                   w:Wr~~~fct;"
                                           'Cf)T                                                                                                   4
                b+c    q+r     y+z        [a                              p X]
                c+a    r+p     z+x     =2, b                              q Y
                a+b p+q x+y                  c                            r z
              Using properties of determinants,                           prove that
                b
                c+a
                   +c  q+ y+
                       r+p     z+x
                                      r=2 b
                                                 z              [a        p X]
                                                                          q Y
                a+b p+q 'x+y                 c                            r z

     65/111                                                                       5                                                           [P.T.O.
r    r
           . .
                  ~~fq;tan-I(

                  !   .,--".'.
                                             .
                                                 ~
                                                      cos.x )=~-!'x
                                                     1 + smx      4      2
                                                                                  E (-~,
                                                                                         2 2
                                                                                               ~).                                             4




                 ~~                    sin-{1
                                             8
                                                 i) + sin-{~)    = cos-{~~).


                  Prove that tan-{1 :o:i~ ;) = ~ -~,                     X E (-~,        ~).


                                                 OR
                                   8
                  Prove that sin-{1 7) + sin-r(~) = cos-{~~).


/.                >:iRT A ~ IR - {3} _                B ~ IR - {I} t I 'Ii\'R f: A ...• B ;;IT f{x) ~ (~             =;) rm 'If\'!Tffir t
                  ~ ~              ~         I~        fq; fl~tctt ~                  t
                                                                             3l1'C0lqCf) I 3m: II ~          ~         I                       4
                  Let A = IR- {3} and B = IR- {1}. Consider the function f : A ~. B defined by

                                 (X-2)
                  f(x)= x _ 3 . Show that fis one-one and onto and hence find r'.


                                                                         ~-lf

                                                                  SECTION-C

                  w.l~                 23 ~ 29 (lCfi~w.l~6.atcnt                     I

                      uestion numbers 23 to 29 carry 6 marks each.

                  ~     A(3, -1, 2), B(5, 2, 4) ~                     C(-1, -1,6) IDU           Rmfuf ~          cnr *l4\Cf)(OI ~
                                                                                                                             ~             I
                  W~ctft~P(6,5,9)~~~~                                            I                                                             6
                  Find the equation of the plane determined by the points A(3, -1,2), B(5, 2, 4) and
                  C(-1, -1,6) and hence find the distance between the plane and the point P(6, 5, 9).

                  ~~              tfc!;:~ qi\lfq~M1~~~              II ~ 60% gl;jlqRl II ~ t~                   40% gl;jlqlRll ~       ~
                  t    I~              Cflt~ -qfturm ~              t
                                                               "Cf>«f fq; gl;jlqRl II (i\~ql<'l30%            31tt gl;jlql*l II ~ ~    ~
                  20%~~~"1l                                A   ~Wf<rr I Cflt~,a:ffill qi\lfqfJl(1~~~~"Cf>T                   ~I~~I TfIT
                  ll<IT ~          ~     ~   ll<IT fq; ~    A~        fl:R;rr t   IW ~          ctft~            t
                                                                                                        l'il~c.;ctl fq; eft ~ gl;jlqRl II
                  ~        GffiYlT   t?                                                                                    -                   6
                  Of the students in a college, it is known that 60% reside in hostel and 40% are day
                  scholars (not residing in hostel). Previous year results report that 30% of all students
                  who reside in hostel attain 'A' grade and 20% of day scholars attain 'A' grade in
                  their annual examination. At the end of the year.tone student is' chosen at random
                  from the college and he has an 'A' grade, what is the probability that the student is a
                                                                                          1
                  hostlier?
                                                                                                                                      ".


         65/1/1                                                                      6
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          ~
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                                        afu" ~

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                                                          f
                                                       Cf)f.:rqlur~



                                                                       ~m
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                                                                                                           3 ~~
                                                                                                 17.50om~'tRm~~
                                                                                            12 ~~~"ffi~mr~-q~
                                                                                                                        A 'tR"1 tkJ 31tt
                                                                                                                            A 'tRom 1

                                                                                                                                  .'
                                                                                                                                       7;
                                                                                                                                           I




          ~*"~~3~lf~(1           ~~fcf;-~ffi'qm?
                 cnT~~~~~mqjCbl~~ffl~
          3Q'tlCR1                                                                                            I"                               6
          A manufacturer produces nuts and bolts. It takes 1 hour of work on machine A and
          3 hours on machine B to produce a package of nuts. It takes 3 hours on machine l¥.
          and 1 hour on machine B to produce a package of bolts. He earns a profit 'of ~ 17.50
          per package on nuts and ~ 7 per package of bolts. How many packages of each
          should be produced each day so as to maximize his profits if he operates his "
          machines for at the most 12 hours a day? Form the above as a linear programming
          problem and solve it graphically.

               :/                      7tl4

                                       f
          I'


  2j.     m-~fcf;-                            (~tanx    +~cotx)dx=~.                ~                                                          6
./'
                                        o

           3

          f         (2x2 + 5x)dx Cf)f1H
                                    l               <WIT Cbl-mm*"~           -q ~       ~       1
           1
                                7tl4

          Prove that            f      (~tanx+~cotx)dx=~.~
                                 o
                                               OR
                            3

          Evaluate         f     (2x2 + 5x)dx as a limit of a sum.
                            1


  27.';;;'1'''''''       flllif~         m~~aif                    3x- 2y + 1 ~ 0, 2x + 3y- 21 ~ 0<1"1I x- 5y + 9 ~ o"ll
          M-~CfiT~~~'1                                                                                                                         6
          Using the method of integration, find the area of the region bounded by the lines
          3x - 2y + 1 = 0, 2x + 3y - 21 = 0 and x - 5y + 9 = O.


~1\~Q'fcf;-~~~~~,                                              ~~                  ~Wntom              ~          ~t,          Cbl~
          ~~*"a(ffi*"~t                                       I'      ,.'                                                              "       6
                                                                                                                                       )

           Show that the height of a closed right circular cylinder of given surface and
           maximum volume, is equal to the diameter of its base.

  65/111            -                                                          7                                                       [P.T.O.
/
          '. /-.   ..~         'liT   m "::"lf<'1iil1"
                                          H              ~   «<ll<WI ~    'lit m ~     :
        ,d;:~~::"
          .                   x - y + 2z-7
                              3x + 4y- 5z =-5
                              2x-y+3z=12                                                          61



                                                                                                       -
                         Using matrices, solve the following system of linear equations:
                              x-y+ 2z=7
                              3x + 4y- 5z =-5
                              2x-y+ 3z= 12
                                                             , OR
                         Using elementary operations, find the inverse of the following matrix:
                                -1
                                      123
                                          1 2   J
                               [
                                      3   1 1




l




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             65/1/1                                                 8
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