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                                          Guess Paper – 2009
                                               Class - XII
                                          Subject – Mathematics

        neral
     General Instructions:
                Instructions:
     The question paper consists of three sections A, B and C.
      (i) Question numbers 1 to 10 in section A are of 1 mark each.
      (ii) Question numbers 11 to 22 in section B are of 4 marks each.
      (iii) Question numbers 23 to 29 in section C are of 6 marks each.
      (iv) All questions are compulsory.
      (v) Internal choices have been provided in some questions. You have to
            attempt only one of the choices in such questions.
      (vi) Use of calculators is not permitted. However, you may ask for logarithmic
            and statistical tables, if required.

                                                         SECTION – A

1.       Show that * : R × R → R given by (a, b) → a + 4b2 is a binary operation.

2.       Find the value of cos (sec–1 x + cosec–1 x), | x | ≥ 1
                2 3          1 0 
3.       If A =      and I = 0 1  Find x and y Such that A = xA + yI.
                                                                2

                1 2               
                                      3 x     3 2
4.       Find values of x for which         =
                                      x 1     4 1
                           cos  sin           sin   cos 
5.       Simplify : cos                + sin cos  sin  
                           sin  cos                       
                         2
                     sec (log x )
6.       Evaluate :       x
                                   dx .
                   1
                           2x 
7.                     
         Evaluate sin1 
                   0
                                 2
                          1  x 
                                   dx

8.       Find a - b if two vectors a and b                          are such that a  2 ; b  3 and a .b  4
9.       Find the direction cosines of x-axis.
                                                      
10.      If a  2 i  2 j  3k, b  - i  2 j  k and c  3 i  j ,                   are       such        that          a+    b   is
         perpendicular to c , then find the value of .




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                                                         SECTION-B

11.   Let A = R – {3} and B = R – {1}. Consider the function f : A → B defined by
              x 2
      f(x)        . Show that f is bijective.
              x 3
12.   Using properties of determinants, prove that :
                1 a2          ab         ac
                  ab         1 b   2
                                          bc         
                                                    1  a 2  b2  c 2     
                                               2
                   ac          bc       1 c

                          1 x  1 x   1                 1
      Prove that : tan1                            1
                          1  x  1  x   4  2 cos x
13.                                                      ,      x 1
                                                            2

                                        OR

                            x 1       1  2x  1       1  33 
      Solve for x : tan 1         tan             tan       .
                            x  1          2x  1           36 

                                  1: x  3
                               
14.   If the function f (x )  ax  b;3  x  5 is continuous at x = 3 and x = 5, then find
                                   7; x  5
                               
      the value of a & b
               log x             d 2 y 2 log x  3
15.   If y          , Show that      
                 x               dx 2      x3

                                        OR
                                                          d2 y    dy
      If y = (sin-1x)2 , show that (1  x 2 )                2
                                                               x    2  0
                                                          dx      dx
                                                                                                         
16.   Find the intervals in which the function f(x) = sin 3x; 0  x 
                                                                                                         2
      (i) is increasing (ii) is decreasing

                                             1 
17.   Evaluate :           loglog x              dx
                            
                                          log x 2 
                                                     
                                    OR


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                     3

                      x                
                            2
      Evaluate :                 5x  1 dx as a limit of a sum
                     1

                                                         dy 
18.   Solve the following differential equation : sin1     xy
                                                         dx 
19.   Find the particular solution of the differential equation 1  x 2                                           dx  xy  x
                                                                                                                    dy            2
                                                                                                                                      ,
      given that y(0) = 2.

20.   If with reference to the right handed system of mutually perpendicular unit
                                                 
      vectors i , j and k ,   3 i  j,   2 i  j  3k then express  in the form of
       = 1 +  2 , where 1 is parallel to  and  2 is perpendicular to  .

21.   Find the equation of the plane that contains the point (1, – 1, 2) and is
      perpendicular to each of the planes 2x + 3y – 2z = 5 and x + 2y – 3z = 8.

22.   A football match may be either won, drawn or lost by the host country’s
      team, So there are three ways of forecasting the result of any one match, one
      correct and two incorrect. Find the probability of forecasting at least three
      correct results for four matches.


                                                        SECTION-C

                         1 2  3
23.   Find A , where A  2 3
               1
                                   2  . Hence, Solve the system of linear
                                      
                         3  3  4
                                     
      equations : x + 2y - 3z = -4,     2x + 3y + 2z = 2, and     3x - 3y - 4z = 11
                                                              OR
                                                                             1 1      1
      Using elementary row transformations, find inverse of the matrix      1 2  3
                                                                                         
                                                                             2  1 3 
                                                                                        
24.   Show that the height of the cylinder of maximum volume that can be
                                            2r
      inscribed in a sphere of radius r is       .
                                             3
                                            OR
      A given quantity of metal is to be cast into a solid half circular cylinder (i.e.,
      with rectangular base and semicircular ends). Show that in order that the
      total surface area may be minimum, the ratio of the length of the cylinder to
      the diameter of its circular ends is  : (  + 2 )

25.   Evaluate:          cot x dx dx.


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26.      Sketch the region enclosed between circles x2 + y2 = 1 and x2 + (y – 1)2 = 1.
         Also, find the area of region using integration.
                                              OR
         Sketch the region common to the circle x 2 + y2 = 16 and the parabola x2 = 6y.
         Also find the area of the region using integration

27.      Find the foot of the perpendicular from P(1, 2, 3) on the line
          x 6 y 7 z 7
                            . Also, obtain the equation of the plane containing the
            3      2     2
         line and the point (1, 2, 3).

28.      Every gram of wheat provides 0.1 gm of proteins and 0.25 gm of
         carbohydrates. The corresponding values for rice are 0.05 gm and 0.5 gm
         respectively. Wheat costs Rs. 4 per kg and rice Rs. 6 per kg. The minimum
         daily requirements of proteins and carbohydrates for an average child are 50
         gms and 200 gms respectively. In what quantities should wheat and rice be
         mixed in the daily diet to provide minimum daily requirements of proteins
         and carbohydrates at minimum cost. Frame an L.P.P. and solve it
         graphically.

      29. A candidate has to reach the examination centre in time. Probability of him
                                                                          3 1          3
          going by bus or scooter or by other means of transport are        ,    and
                                                                         10 10         5
                                                                1     1
          respectively. The probability that he will be late is   and    respectively, if
                                                                4     3
          he travels by bus or scooter. But he reaches in time if he uses any other
          mode of transport. He reached late at the centre. Find the probability that he
          travelled by bus.


Paper Submitted By:
DILIP BISWAS
Email Id : - dilipbiswas2006@gmail.com
Telephone No. 09214927974




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