Math sample paper for exam 2013

Sample Paper – 2014
Class – XII
 Subject – Mathematics

SECTION A ( 1X10=1M)

(Questions 1 to 10 carry 1 mark each)

1.       Let * be the binary operation on N given by a *b = HCF of a and b. Find 20*16

2.       What is Sin^-1(Sin 7π/6) ?  

3.       Find x and y if =

4.       If A is a square matrix of order 3 and  = 64 then find .

5.       Find

6.       Find the adj A of  .

7.      

8.       Find the value of  α so that   = αi + 2j + k is perpendicular to  = 4i – 9j +  2k

9.       Find the unit vector in the direction of if

10.    Find k if the lines and are perpendicular.

SECTION B(Q. 11 to 22 carry 4 marks each)

11.   Show that the relation R on NXN defined by ( a,b) R (c,d) a+d= b+c is an equivalence relation.     (or)

 Let f : R R                be a function defined by f(x) = 4 + 3x . Show that f is invertible and find the inverse of f.

12.   Prove that   tan^-1 (     -   )/+) = π/4 – ½ Cos^-1x                    .              

13.               Using properties of determinants Prove that     = 4.              

14.         Test the continuity of the following function at x = 0 ,

                If x =  a ( t + Sint ) , y = a ( 1 – Cost ) , show that y’’ = 1/a, at t=   ( or ) If  x^p y ^q = ,Prove that y’ = y/x.

15.   Find the intervals where the function f (x) =2x³ – 9x² + 12x + 30 is a) increasing b) decreasing.

16.   Evaluate: 

 (or)

  Evaluate as sum of limits

17.   Solve the differential equation  x²y’ = x²-2 +xy

 ( or)

  Form the differential equation representing the family of ellipses having foci on x-axis and centre at the origin.

Solve the differential equation   Cos²x y’ + y = tanx.

                                                                                                                                                                                                                          i.       

18.   Three vectors  ,    satisfying  the condition   ++  = 0 . Evaluate the quantity  +    +   if  = 1 ,  = 4 = 2.

19.   Find the shortest distance  between the lines  =   I + j + K ( 2i – j + k ) and  = ( 2i + j  - k ) + p( 3i -5j+2k).

20.   In a factory which manufactures bolts, machine A, B and C respectively 25%, 35% and 40% of the bolts, Of their output s 5,4,and 2 percent are respectively defective bolts. A nolt is drawn random from the product and is found to be defective. What is the probability that it is manufactured from machine A?                                                                                                                              SECTION C ( Each question carries 6 marks)

21.   Find the inverse of  using elementary transformation.  ( or)  if  A =  find A^-1 and hence solve the equations  2x+3y+z= 11, -3x+2y+z=4,                                                                                              5x-4y-2z = -9

24 .Find the maximum  area of the isosceles triangle inscribed in an ellipse  x²/a² + y²/b² = 1, whose vertex lies along the major axis. (or)   Show that the maximum value of the cylinder which can be inscribed in a sphere of radius 5 is 500π cm³.

25.Prove that

26. Make a rough sketch of the region given below and find its area using integration.  { (x,y) : 0≤y≤2x+3,}.

27. Find the foot of the perpendicular and the perpendicular distance of the point (3,2,1) from the plane   2x-y+z +1=0. Find the image of the point in the plane.

28. From a lot of 30 bulbs which includes 6 defective, a sample of 4 bulbs is drawn at random with replacement. Find the mean and variance of the number of defective bulbs.

29. A furniture firm manufactures chairs and tables each requiring the use of three machines A,B and C . Production of the chair requires 2 hrs on machine A, 1 hr on machine B, and  1 hr on machine C.Each table requires  1 hr on machine A, 1 hr on machine B and  3 hrs on machine C. The profit obtained   by selling one chair is Rs. 30 while by selling one table Rs. 60. The total time available per week on machine A is 70 hrs,  machine B 40 hrs, and on machine  C 90 hrs. How many chairs and tables should be made per week so as to maximize profit? Formulate the problem as LPP and solve it graphically.