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) : labl 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(xa) cos (xb)
15.Evaluate : f x(logx)" dx
16. Evaluate : f : I dx x 
17. Using properties of definite integrals, evaluate.
f' xdx4cos 'x
18. The dot products of a vector with the vectors i3k, i2k 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 (xdyydx)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 iJ +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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