# CBSE Board Math Sample Papers for Class 12

CBSE MATHEMATICS SAMPLE PAPER 2013 - I I I
CLASS - XII
SECTION A

I    Write the  principal  value of   cos  -2.

2.  write  the angle of  the principle  ,"",cl(x)  defined on the domain R-( -1,  1). 3  4  2x  4

3.  Find x i f   -5  2  =  -5  3·

4.If A is  a square matrix of  order 3 such that  ladj AI  = 64. Find  IA'I 1 r A is a square matrix satisfying N= 1, then what is the inverse of  A? dy If f(x) =  sinxo, find  dx

5.What is the degree of  the  following differential equation? y d'y +(d Y) '  =  x(d3y) ' dx2dx  dx' ?

6.  If ii and  [;  represent the two adjacent sides of  a parallelogram, then write the area of  parallelogram
in  terms of  ii and b.

7.  Find the angle between two vectors  ii and  [;  i f  lii =3,  1[;1  = 4 and  Iii x [;1  = 6

8.  Find  the  direction  cosines of  a line, passing through  origin and lying in  the  first  octant, making
equal angles with the three coordinate axes.

SECTION B

9 .  Show  that the relation R  the s e t   A = {x;  X E  Z, 0 s:;  x s:;  12}  gIven  by R =  {(a, b)  : la-bl is  divisible  by 4}  is an equivalence relation. Find the set of  all elements related to I.

10.  Solve for  x:   2 tan -I (sin x)  = tan -I (2s e c  x), O<x< ;

11.  I f  none of a, b and c is zero, using properties of  determinants.

12.Find all the points of  dicontinuity of  the function f(x) = (x' )  on [1 , 2), where [.J  denotes the greatest
integer function.

13.Differentiate  sin"  (2x. J l -x' )  w.r.t. cos"
(I -x'  )I + x' f

14.Evaluate :   dx cos(x-a) cos (x-b)

15.Evaluate :  f x(logx)"  dx

16.  Evaluate : f : I dx x -

17.  Using properties of  definite integrals, evaluate.
f'  xdx4-cos 'x

18.  The dot products of  a vector with the vectors  i-3k, i-2k and i+j+4k are 0, 5 and  8 respectively. Find
the vector.

19.  Find the equation of  plane passing through the point ( I ,  2,  1) and perpendicular to the line joining
the points (1, 4, 2) and (2, 3,  5). Also, find the perpendicular distance of  the plane from the ori gin.

20.  A biased die is twice as likely to  show an even number as  an odd number. The die is rolled three
times. I f  occurance of  an even number is considered a success, then write the probability distribution
of  number of  successes. Also find the mean number of  successes.

SECTIONC

21.  Using matrices, solve the following system of  equations:

y ; t, z ; t
xyz   xyz   xyz

22.  Show that the  volume  of  the  greatest cylinder which can be  inscribed in  a  cone  of  height  h and
semivertical angle  a , is  ~ rr  h'  tan' a

OR

Show that the normal at any point e to the curve x = a cos e +a e  sin e and y = a sin e -a e  cos e is at
a constant distance from the origin.

23.  Find the area of  the region: {(x,y) : 0 ~ y ~ x', 0 ~ y ~ x+2; 0 ~ x ~ 3}

24.  Fi nd  the particular solution of  the differential equation (xdy-ydx)y· sin (~) = (ydx+xdy)x cos ~ ,given that y=rr when x=3.

25.  Find the equation of the  plane passing through the point ( I , 1,  I) and containing the line i' = (-3i+J+5k) + A(3 i-J +5k). Also, show that the plane contains the line

26.  A  company sells two  different products A and  B. The two  products  are  produced in a  common production process which has a total capacity of  500 man hours. It takes 5 hours to produce a unit of  A and  3 hours to  produce a unit ofB. The demand in the market shows that the maximum number of  units  of A that can be sold is 70 and that ofB is  125 . Profit on each unit of  A is Rs.  20 and on B is Rs.  15.  How many units of  A and B should be produced to maximise the profit. Form an L.P.P. and  solve it graphically.

27.  Two bags A and B contain 4 white and 3 black balls  and 2 white and 2 black ball s respectively. From bag A.  two balls are drawn at random and then transferred to bag B. A ball is then drawn from  bag B and is found to be a black ball. What is the probability that the transferred balls were  I white and I  black?

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