CBSE Board Sample Paper – 2011
Class – 12th
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. Write the identity element for the binary operation * defined on the set R of real numbers by the rule a * b = 3ab/8, for all a, b R.
2. The side of a square is increasing at the rate of 0.2 cm/sec. Find the rate of increase of the perimeter.
3. Find ∫[(1+cotx)/(x+log sinx)]dx.
4. :R – {-3/5} R be a function defined or f(x) = 2x/(5x+3) find -1; range of - {-3/5}
5. Find tan-1 [ 2 cos(2 sin-1 1/2 )].
6. If f (x) = ex sin x in [ 0, π ], then find c in Rolle’s theorem .where 0<c<π.
7. What is the principal value of sin-1(sin 5π/6) + cos-1(cos π/6) ?
8. The side of a square is increasing at the rate of 0.2 cm/sec. Find the rate of increase of the perimeter.
9. A fair die is rolled. Let the events A= {1, 3, 5}, B = {2, 3}, C ={2, 3, 4, 5}. Find P (A|B).
10. The slope at any point of a curve y = f(x) is given by dy/dx = 3x2 and it passes through (–1, 1). Find the equation of the curve.
SECTION – B
1. Find the equation of the normals to the curve y = x3 + 2x + 6, which are parallel to the line x + 14y + 4 = 0.
2. Show that the lines r1 = î + ĵ – k + λ(3î - ĵ) and r2 = 4î – k + μ(2î + 3k)
intersect. Find their point of intersection .
3. Find the intervals in which f(x) = sin x – cos x, where 0 < x < 2 is increasing or decreasing.
4. If x = 2 cos θ – cos 2θ, y = 2 sin θ – sin 2θ, find d2y/dx2 at θ = π/2.
5. Ten eggs are drawn successively with replacement from a lot containing 10% defective eggs . Find the probability that there is alteast one defective eggs.
6. Prove that 9π/8 – 9/4 sin-1.1/3 = 9/4 sin-1 (2√2)/3.
7. Find the image of the point ( 1,6,3) in the line x/1 = (y-1)/2 = ( z-2)/3.
8. Solve the differential equation : ( 1+ y2 ) dx = (tan-1y – x) dy ; given y (0) = 0.
9. Prove that: sin-1(12/13) . cos-1(4/5) . tan-1(63/16) = π.
10. Find the angle between the line (x-2)/3 = (y+1)/-1=(z-3)/2 and the plane 3x+4y+z+5=0.
11. Solve the Differential equation :
(y/x)cos(y/x)dx – {(y/x)sin(y/x) + cos(y/x)}dy = 0
12. Find the equation of the plane containing the line of intersection of the plane x+2y+3z-4=0 and 2x+y-z+5=0 and which is perpendicular to the plane 5x+3y+6z+8=0.
SECTION – C
1. Using Matrices solve the following linear equation-
2x-y+z=3, -x+2y-z=-4, x-y+2z=1
OR
Show that the semivertical angle of the right circular cone of given total surface area and maximum volume is sin-1(1/3).
2. Using the method of integration, find the area of the smaller region bounded by ellipse : x2/16 + y2/9 =1 and line x/4 + y/3 = 1.
OR
Evaluate: ∫(sinx/sin4x)dx
3. Using differentials,evaluate [17/81](1/4) approximately.
OR
Find the intervals in which the following function are increasing or decreasing: f(x) = (x-1)(x+2)2.
4. An aeroplane can carry a maximum of 200 passengers. A profit of Rs. 400 is made on each 1st is class ticket and a profit of Rs. 300 is made on each economy class ticket. The airline reserves at least 20 seats for Ist class. However, at least four times as many passengers prefer to travel by economy class to first class. Determine how many tickets of each type should be sold in order to maximise the profit.
5. Find the distance of the point (1,-2,3) from the plane x - y + z = 5 measured parallel to the line (x+1)/2 = (y+3)/3 = (1-z)/6.
6. 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.
7. If cos-1a + cos-1b + cos-1c = π ,prove that a2 + b2 + c2 + 2abc =1.
OR
Show that the volume of the greatest cylinder which can be inscribed in a cone of height ‘h’ and semi vertical angle α is (4/27) πh3 tan2α.