CBSE Board Question paper 2009
Class XII (Foreign)
Subject: Mathematics
Time allowed : 3 hours Maximum marks: 100
General Instructions:
(i) All questions are compulsory.
(ii) The question paper consists of 29 questions divided into three Sections A, B and C. Section A comprises of 10 questions of one marks each, Section B comprises of 12 questions of four marks each and Section C comprises of 7 questions of six marks each.
(iii) All questions in Section A are to be answered in one word, one sentence or as per the exact requirement of the question.
(iv) There is no overall choice. However, internal choice has been provided in 4 questions of four marks each and 2 questions of six marks each. You have to attempt only one of the alternatives in all such questions.
(v) Use of calculators is not permitted.
SECTION-A
Questions number 1 to 10 carry 1 mark each.
1. Evaluate:
∫ [1/(x + x logx)] dx
2. Evaluate:
01∫[1/√(2x + 3)] dx
3. If the binary opration *, defined on Q, is defined as a * b = 2a +b – ab, for all a, b ε Q, find the value of 3 * 4.
4.
5. Find a unit vector in the direction of a = 2i – 3j + 6k.
6. Find the direction cosines of the line passing through the following points: (-2, 4, -5), (1, 2, 3)
7.
8.
9.
10. Write the principal value of tan-1[tan(3π/4)].
SECTION-B
Questions number 11 to 22 carry 4 marks each.
11. Evalute:
∫ [cosx/{(2 + sinx)(3 + 4sinx)}] dx
OR
∫ x2.cos-1x dx
12. Show that the relation R in the set of real numbers, defined as R = {(a, b) : a ≤ b2} is neither reflexive, nor symmetric, nor transitive.
13. If log (x2 + y2) = 2 tan-1(y/x), then show that dy/dx = (x+y)/(x-y).
OR
If x = a (cos t + t sin t) and y = a (sin t – t cos t), then find d2y/dx2.
14. Find the equation of the tangent to the curve y = √(4x - 2) which is parallel to the line 4x – 2y + 5 =0.
OR
Using differentials, find the approximate value of f(2.01), where f(x) = 4x3 +5x2 + 2.
15. Prove the following:
tan-1(1/4) + tan-1(2/9) = ½ cos-1 (3/5).
OR
Solve the following for x:
cos-1[(x2-1)/(x2+1)] + tan-1(2x/(x2 - 1)) = 2π/3
16. Find the angle between the line (x+1)/2 = (3y+5)/9 = (3 – z)/-6 and the plane 10x + 2y – 11z =3.
17. Solve the following differential equation:
(x3 + y3) dy – x2y dx = 0
18. Find the particular solution of the differential equation dy/dx + y cot x = 4x cosec x, (x ≠ 0), given that y = 0 when x = π/2.
19.
20. The probability that A hits a target is 1/3 and the probability that B hits it is 2/5. If each one of A and B shoots at the target, what is the probability that
(i) The target is hit?
(ii) Exactly one of them hits the target?
21. Find dy/dx, if yx + xy = ab, where a, b are constants.
22.
SECTION-C
Questions number 23 to 29 carry six marks each.
23. One kind of cake requires 200g of flour and 25g of fat, and another kind of cake requires 100g of flour and 50gof fat. Find the maximum number of cakes which can be made from 5 kg of flour and 1 kg of fat assuming that there is no shortage of the other ingredients used in making the cakes. Formulate the above as a linear programming problem and solve graphically.
24. Using integration, find the area of the region:
{(x, y) : 9x2 + y2 ≤ 36 and 3x + y ≥ 6}
25. Show that the lines (x+3)/-3 = (y - 1)/1 = (z - 5)/5; (x +1)/-1 = (y - 2)/2 = (z - 5)/5 are coplanar. Also find the equation of the plane containing the lines.
26. Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is 2R/√3. Also find the maximum volume.
OR
Show that the total surface area of a closed cuboids with square base and given volume, is minimum, when it is a cube.
27. Using matrices, solve the following system of linear equestions:
3x – 2y + 3z = 8
2x + y – z = 1
4x – 3y + 2z = 4
28. Evakuate:
∫[x4/{(x-1)(x2+1)}] dx
OR
14∫[|x - 1| + |x - 2| + |x - 4|]dx
29. Two cards are drawn simultaneously (or successively without replacement) from a well shuffled pack of 52 cards. Find the mean and variance of the number of red cards.
CBSE Board Question Papers Class 12th 2009