CBSE Board Class 12 Math Previous Year Question Papers 2008

Previous Paper
Class – XII
Subject – MATHEMATICS

Time : 3 hrs                            Max. Marks : 100
General Instructions:

SECTION – A

1. Show that the binary operation  defined by a*b = ab + 1 on Q is commutative.

2.  Solve : tan-12x + tan-13x = π/4.

3. Find a matrix X such that B – 2A + X = O, where A =.

4. If A is a square matrix of order 3 such that  = 64, find.

5. Construct a 2  2 matrix whose elements  are given by:   =.

6.  The Cartesian equation of a line AB is. Find the direction cosines of a line parallel to AB.

7. If, find a unit vector parallel to the vector

8. Given that A =  and B =  find AB. Hence using this product solve the system of equations : x – y + z = 4, x – 2y – 2z = 9, 2x + y + 3z = 1
OR
  Using elementary row transformation, find the inverse of the matrix.

9. Show that a right circular cylinder, which is open at the top and has a given surface area, will have the greatest volume if its height is equal to the radius of its base.
OR
Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is 8/27 of the volume of the sphere.

10. Evaluate:  dx.

11. Using the method of integration, find the area of the region bounded by the lines
2x + y = 4, 3x – 2y = 6 and x – 3y + 5 = 0.
OR
Make a rough sketch of the region given below and find the area using the method of integration :

12. Find the image of the point (1, 2, 3) in the plane x + 2y + 4z = 38. Also find the perpendicular distance from the point to the plane.
OR
A line makes angles α, β, γ and δ with the diagonals of a cube, prove that cos2α + cos2β + cos2γ + cos2δ = 4/3

13. An aeroplane can carry a maximum of 200 passengers. A profit of Rs. 400 is made on each first class ticket and a profit of Rs. 300 is made on each economy class ticket. The airline reserves at least 20 seats for first class. However, at least 4 times as many passengers prefer to travel by economy class to by the first class. Determine how many of each type ticket must be sold in order to maximize the profit for the airline. What is the maximum profit? Frame an L.P.P and solve it graphically.

14. In a bolt factory, machines A, B and C, manufacture respectively 25%, 35%, 40% of the total bolts. Of their output 5%, 4% & 2% respectively are defective bolts. A bolt is drawn at random and is found to be defective. Find the probability that it is manufactured by machine B.

Q.15Find the slope of the tangent to the curve  3 ,8 2 2 5
2 2 x = t + t − y = t − t − at the point.(2, -1)   

Q.16 .If  a i j k b i j k ˆ ˆ , ˆ 2 ˆ = 2ˆ + 2 ˆ + 3 = − + + r r and  c i j = 3ˆ + ˆ r are such that  a br r  + λ is perpendicular to  c r , then find the value of  λ .

Q.17If  f : R→ R be defined as f(x) =,93 x + 7then find  ( ).

Q.18  Find values of k if area of triangle is 4 square units and vertices are
(k,0),(4,0),(0,2).    Ans k=0,8            

Q.19  The number of all possible matrices of order  3× 3 with each entry 0 or 1 .

Q.20  Write the total  number of binary operation on a set consisting of n
element .

Q.21 If the points (1, 1, p) and (-3, 0, 1) be equidistant from the plane
( ) 13 ,0ˆ r 3. i ˆ + 4 ˆ j −12k + = r then find the value of p. 

Q.22 A toy manufacturers produce two types of dolls ; a basic version doll  A and deluxe version doll B. Each doll of type B takes twice as long to produce as one doll of type A . The company have time to make a maximum of 2000 , dolls of type A per day , the supply of plastic is sufficient to produce 1500 dolls per day and each type requires equal amount of it .The deluxe version  i.e. type  B requires a fancy dress of which there are only 600 per day available  . If the company makes profitof  3 and  5 per doll respectively on doll A and B , how many of each should be produced weekly in order to maximize the profit ? Solve it by graphical method