CBSE Board Sample Paper – 2011
Class – XII
Subject - MATHEMATICS
M.M : 100 TIME: 3 HRS
GENERAL INSTRUCTION:
(a) All questions are compulsory.
(b) This question paper consists of 29 questions divided into three section A, B, and C. Section A comprises of 10 question of 1 mark each, section B comprises of 12 questions of 4 marks each and section C comprises of 7 questions of 6 marks each.
(c) All questions in Section A are to be answered in one word, one sentence or as per the exact requirement of the question.
(d) There is no overall choice. However, internal choice has been provided in 03 questions of four marks each and 02 questions of six marks each. You have to attempt only one of the alternatives in all such questions.
(e) Use of calculators is not permitted. You may ask for logarithmic tables, if required.
SECTION – A
1. If B is skew symmetric matrix. Write whether the(ABA’) is symmetric or skew symmetric.
2. Show that the function f : N → N given by F(x) = 3x . Show that f is not an onto.
3. A matrix A of order 3 x 3 has determinant 5. What is the value of |3A|.
4. Let * be a binary operator of Ngiven by a*b = L.C.M. of (a,b) a,b N. Find (16*20)*40.
5. What is the pricipal values of cos-1(cos2π/3) + sin-1(sin2π/3).
6. Find the value of cos (sec–1 x + cosec–1 x), | x | ≥ 1
7. Evaluate ∫ x √ (x+2)dx
8. Evaluate : ∫ [{sec2(logx)}/x]dx .
9. 
10. Write the value of 
SECTION – B
1. Prove that the relation R on Z set of all integers defined by (x, y) R x – y is divisible by n is an equivalence relation on Z.
2. Two cards are drawn simultaneously from a well shuffled pack of 52 cards. Determine the probability distribution of the number of aces.
3. Show that the curves x = y2 and xy = k2 cut orthogonally, if 8k2 = 1.
4. If sin y = x sin (a + y), prove that dy/dx = {sin2(a+y)}/sina .
5. Solve for x : tan-1{(x-1)/(x+2)} + tan-1{(x+1)/(x+2)} = π/4.
6. Using properties of determinants, prove that :
7. Evaluate : 
8. If x = a[cost + log|tan(t/2)|] and then y = a sint find dy/dx at t = π/4.
9. Solve the differential equation: (1+x2) dy/dx + y = tan-1x.
10. If the function 
is continuous at x = 3 and x = 5, then find the value of a & b.
11. If y = e(acos-1x), -1≤x≤1 show that (1 – x2) d2y/dx2 – x dy/dx – a2y = 0.
12. If y = (tan-1x)2, show that (x2 + 1)2 y2 + 2x (x2 + 1)y1 = 2.
SECTION – C
1. Show that the height of a cylinder, which is open at the top, having a given surface and greatest volume, is equal to the radius of its base .
2. Find the equation of the plane passing through the point ( -1,-1, 2 ) and perpendicular to each of the following planes : 2x + 3y -3z = 2 and 5x -4y +z = 6.
3. Find the area of region bounded between two circle x2 + y2 = 4 and (x-2)2 + y2 = 4. Using Integration.
4. Show that the volume of the greatest right circular cylinder that can be inscribed in a cone of height ‘h’ and semivertical angle α is (4/27)πh3tan2α. .
5. Find the distance of the point (-2,3,-4) from a line (x+2)/3 = (2y+3)/4 = (3z+4)/5 measured parallel to plane 4x + 12y – 3z + 1 = 0.
6. 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 on each second class ticket. The air lines reserves at least 20 seats for first class. However, at least four times as many passengers prefer to travel in second class than by the first class ticket. Determine how many tickets of each class must be sold to maximize profit for airlines. What is the maximum profit? Solve it as Graphically ?
7. Every gram of wheat provides 0.1 gm of proteins and 0.25 gm of carbohydrates. The corresponding values for rice are 0.05 gm and 0.5 gm respectively. Wheat costs Rs. 4 per kg and rice Rs. 6 per kg. The minimum daily requirements of proteins and carbohydrates for an average child are 50 gms and 200 gms respectively. In what quantities should wheat and rice be mixed in the daily diet to provide minimum daily requirements of proteins and carbohydrates at minimum cost. Frame an L.P.P. and solve it graphically.