CBSE Board Class 12 Math Sample Papers 2010

Sample Paper - 2010
Subject – Mathematics
Class – XII
General Instructions
                                                  SECTION A:
1. If f: R  be given by  f ( x) =  then find

2. Simplify :  a+b=10;a+2b=4
                     
3. . Write the points where the function is differentiable.

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

5. Find a unit vector perpendicular to each of the vectors
    

SECTION B:

6.  Let f: N   be a function defined  as  f ( x ) = 4x2 + 12x + 15 .Show that f: N where S is the range of  f, is invertible .Find the inverse of f .

7. If x , y, z are different and 

8. If the function  is continuous at x = 3 and x = 5, then find the value of a & b.
                
9. Evaluate:   dx  and hence find  :   dx , where n is positive integer.

10. Solve:   ( 1  +  ex/y ) dx  +  ex/y ( 1 – x/y ) dy  =  0

11. Find the intervals in which the function f(x) = sin x - cos x is increasing or decreasing on [0,]. 

12.Prove that the curves y2 = 4ax and x y = c2 cut at right angles if c4 = 32 a4 

13.Perpendicular to the sum of the other two find 

14.If with reference to the right handed system of mutually perpendicular unit vectors, and ,  then express in the form of = + , whereis parallel to  and  is perpendicular to . (hots)

15. Find the shortest distance between the lines whose vector equations are
            
16.Find the image of the point P( 6,5,9) on the plane determined by the points
       A ( 3,-1,2 ) , B ( 5,2,4 ) and C ( -1,-1,6 )

17. The probability of a shooter hitting a target is 3/4. How many minimum number of times must he/ she fire so that the probability of hitting the target at least once is more than 0.99?

SECTION C:

18. Using elementary transformation find the inverse of the matrix

19. Prove that of all the triangles inscribed in a circle ,the equilateral triangle triangle has the maximum area.   

20.A water tank has the shape of an inverted right circular cone with its axis vertical and  vertex  lowermost .Its semi vertical angle is tan-1 ( 0.5) .Water is poured in to it at a  constant rate of 5 cubic  meter per hour .Find the rate at which the level of the water is rising at an  instant when the depth of  the water tank  is 4m.

21. Suppose a girl throws a die. If she gets a 5 or 6, she tosses a coin three times and notes the number of heads. If she gets 1, 2,3 or 4 she tosses a coin once and notes whether a head or tail is obtained. If she obtained exactly one head, what is the probability that she threw 1, 2,3 or 4 with the die.

22. A line makes angles  with the diagonals of a cube , prove that

23. A manufacturer of patent medicines is preparing a production plan on medicines A and B. There is sufficient raw material available to make 20,000 bottles of A and 40,000 bottles of B,  but there are only 45,000 bottles into which either of medicines can be put. Further, it takes 3 hours to prepare enough material to fill 1,000 bottles of A and it takes one hour to prepare enough material to fill 1,000 bottles of B and there are 66 hours available for this operation. The profit is Rs.8 per bottle for A and Rs. 7 per Bottle for B. How  should the manufacturer schedule his production in order to maximize his profit.