CBSE Question Paper 2009 for Class 12 - Mathematics

by saumil shrivastava

General Instructions for CBSE Question Paper :

(i) All the question are compulsory.

(ii) The question paper consists of 29 question divided into three Sections A, B and C. Section A comprises of 10 questions of one marks each, Section B comprises of 12 questions for four marks each and Section C comprises of 7 question of six marks each.

(iii) All question 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 question of four marks each and 2 questions of six marks each. You have to attempt only one of the alternative in all such questions.

(v) Use of calculator is not permitted.

SECTION - A

1. Find the projections of $\overset{\rightarrow}{a}\ on\ \overset{\rightarrow}{b} \ if$ $\overset{\rightarrow}{a}, \overset{\rightarrow}{b} = 8\ and\ b =$ $\overset{\wedge}{2i} + \overset{\wedge}{6j} + \overset{\wedge}{3k}.$

2. Write a unit vector in the direction of $\overset{\rightarrow}{a} = \overset{\wedge}{2i} - \overset{\wedge}{6j} + \overset{\wedge}{3k}.$

3. Write the value of p for which $\overset{\rightarrow}{a} = \overset{\wedge}{3i} + \overset{\wedge}{2j} + \overset{\wedge}{9k}$ and $\overset{\rightarrow}{b} = \overset{\wedge}{i} + \overset{\wedge}{pj} + \overset{\wedge}{3k}$ are parallel vectors.

4. If matrix A = (123), write AA', where A' is the transpose of matrix A.

5. Write the value of the determinant $\left |\overset{2}{\underset{6x}{5}}\ \overset{3}{\underset{9x}{6}}\ \overset{4}{\underset{12x}{8}}\right |$

6. Using the principal value, evaluate the following:

$\sin^{-1} \Bigg(\sin \cfrac{3\pi}{5}\Bigg)$

7. Evaluate : $\int \cfrac{\sec^2x}{3 + \tan\ x}dx.$

8. If $\int\limits_{0}^{1} (3x^2 + 2x + k) dx = 0,$ find the value of k.

9. If the binary operation on the set of integers Z, is defined by a * b = a + 3b^2, then find the value of 2 * 4

10. If A is an invertible matrix of order 3 and | A | = 5, then find |adj. A |.

SECTION - B

11. If $\overset{\rightarrow}{a} \times \overset{\rightarrow}{b} = \overset{\rightarrow}{c} \times \overset{\rightarrow}{d}$ $\overset{\rightarrow}{a} \times \overset{\rightarrow}{c} \times \overset{\rightarrow}{d},$ show that $\overset{\rightarrow}{a} - \overset{\rightarrow}{d}$ is parallel to $\overset{\rightarrow}{b} - \overset{\rightarrow}{c}$ where $\overset{\rightarrow}{a} \not = \overset{\rightarrow}{d}$ and $\overset{\rightarrow}{b} \not = \overset{\rightarrow}{c}$

12. Prove that : $\sin^{-1} \Bigg(\cfrac{4}{5}\Bigg) + \sin^{-1}\Bigg(\cfrac{5}{13}\Bigg) +$ $\sin^{-1}\Bigg(\cfrac{16}{65}\Bigg) = \cfrac{\pi}{2}$

Solve for $x : \tan^{-1} 3x + \tan^{-1} 2x = \cfrac{\pi}{4}$

13. Find the value of $\delta$ so that the lines

$\cfrac{1-x}{3} = \cfrac{7y-14}{2\delta} = \cfrac{5z-10}{11}$ $and\ \cfrac{7-7x}{3\delta} = \cfrac{y-5}{1} = \cfrac{6-z}{5}$

14. Solve the following differential equations:

$\cfrac{dy}{dx} + y = \cos \ x - \sin \ x.$

15. Find the particular solution, satisfying the given condition, for the following differential equation:

$\cfrac{dy}{dx} - \cfrac{y}{x} + cosec \Bigg(\cfrac{y}{x}\Bigg) = 0; y = 0\ when \ x = 1$

16. By using properties of determinants, prove the following:

$\left | \overset{x+4}{\underset{2x}{2x}}\ \overset{2x}{\underset{2x}{x+4}}\ \overset{2x}{\underset{x+4}{2x}}\right |$ = (5x+4)(4-x)^2.

17. A die through again and again until three sixes are obtained. find the the probability of obtaining the third six in the sixth throw of the die.

18. Differentiate the following function w.r.t.x :

$X^{\sin\ x} + (\sin\ X)^{\cos\ x}$

19. Evaluate : $\int \cfrac{e^x}{\sqrt{5} - 4\ e^x - e^{2x}} \ dx.$

Evaluate : $\int \cfrac{(x-4)e^x}{(x-2)^3}\ dx.$

20. Prove that the relation R in the set A = {1, 2, 3, 4, 5} given by R = { (a, b) : |a - b| is even }, is an equivalence relation.

21. Find $\cfrac{dy}{dx}\ if \ (x^2 + y^2)^2 = xy.$

If $y = 3 \cos (\log \ x) + 4 \sin (\log \ x),$ then show that $X^2 . \cfrac{d^2y}{dx^2} + \cfrac{dy}{dx} + y = 0$

22. Find the equation of tangents to the curve $y = \sqrt{3x-2}$ which is parallel to the line 4x - 2y + 5 = 0

Find the interval in which the function f given by $f(x) = x^3 + \cfrac{1}{x^3}, x \not = 0$ (i) Increasing (ii) Decreasing

SECTION - C

23. Find the volume of the largest cylinder that can be inscribed in a sphere of radius r.

A tank with rectangular base and rectangular sides, Open at the top is to be constructed so that its depth is 2 cm and volume is 8 m^3. If building of tank costs Rs. 70 per sq. metre for the basis and Rs. 45 per sq. metre for sides, what is the cost of least expensive tank?

24. A diet is contain at least 80 units of Vitamin A and 100 units of minerals. Two foods $F_1\ and F_2$ are available. Food F_1 costs Rs. 4 per unit and F_2 costs Rs. 6 per unit. one food F_1 contains 3 units of Vitamin A and 4 units of minerals. One unit of food F_2 contains 6 units of Vitamin A and 3 units of minerals. Formulate this as a linear programming problem and find graphically the minimum cost of diet that consists of mixture of these two foods and also meets the minimal nutritional requirements.

25. Three bags contains balls as shown in the table below:

A bag is chosen at random and two balls are drawn from it. They happen to be white and red. what is the probability that they came from III bags?

26. Using matrices, solve the following system of equations:

27. Evaluate : $\int\limits_{0}^{\pi} \cfrac{e^{\cos\ x}}{e^{\cos \ x + e^{- \cos\ x}}}\ dx.$

Evaluate : $\int\limits_{0}^{\pi/2} (2 \log \sin \ x - \log \sin 2x) dx$

28. Using the method of integration, find the area of the region bounded by the lines 2x + y = 4, 3x - 2y = 6 $and \ x - 3y + 5 = 0$

29. Find the equation of the plane passing through the point (-1, 3, 2) and perpendicular to each of the planes $x + 2y + 3z = 5 \ and \ 3x + 3y + z = 0.$