CBSE Class 12 math paper-2024 set-2 series QSS4R/4

Here we provide CBSE Class 12 math paper-2024 set-2 series QSS4R/4 for practice and student get more marks after understanding the question type in board exam. Case study question are in the paper of 12 marks.

MATHEMATICS

Cbse Class 12 math paper-2023 set-1 series-EF1GH/C

CBSE Class 12 math paper-2023 set-2 series-EF1GH/4 Download

CBSE Class 12 math paper-2024 set-2 series QSS4R/4. Download

Time allowed : 3 hours Maximum Marks: 80

General instruction:

Read the following instructions very carefully and strictly follow them:

(i) This question paper cantains 38 questions. All are compulsory.

(ii) This question paper is divided into five section – A, B, C, D and E

(iii) In Section A, question no. 1 to 18 are multiple choice questions(MCQ) and question number 19 and 20 are Assertion-Reason based question of 1 mark each.

(iv) In Section B, Question no. 21 to 25 are very short answer (VSA) type question, carrying 2 marks each.

(v) In Section C, Question no. 26 to 31 are short answer (SA) type question, carrying 3 marks each.

(vi) In Section D, Question no. 32 to 35 are long answer (LA) type question carrying 5 marks each.

(vii) In Section E, Question no. 36 to 38 are case study based question carrying 4 marks each.

(viii) There is no overall choice. However, an internal choice has been provided in 2 questions in Section B, 3 question in Section C, 2 question in Section D and 2 question in Section E.

(ix) Use of calculators is not allowed.

SECTION A

This section consists of 20 multiple choice question (MCQs) of 1 marks each. 20×1 = 20

1. The lines $\dfrac{1 - x}{2} = \dfrac{y - 1}{3} = \dfrac{z}{1}$ and $\dfrac{2x-3}{2p} = \dfrac{y}{-1} = \dfrac{z-4}{7}$ are perpendicular to each other for p equal to

(i) -1/2 (ii) 1/2

(iii) 2 (iv) 3

2. The maximum value of Z = 4x + y for a L.P.P. whose feasible region is given below is :

(i) 50 (ii) 110

(iii) 120 (iv) 170

3. The probability distribution of a random variable X is:

Where k is some unknown constant.

The probability that the randomvariable X takes the value 2 is :

(a) 1/5 (b) 2/5

4. If $A = [a_{ij}] = \begin{bmatrix} 2 & -1 & 5 \\ 1 & 3 & 2 \\ 5 & 0 & 4 \end{bmatrix}$ and $c_{ij}$ is the cofactor of element $a_{ij}$ , then the value of $a_{21}.c_{11} + a_{22}.c_{12} + a_{23}.c_{13}$ is :

(a) -57 (b) 0

5. If $\begin{bmatrix} 1 & 3 \\ 3 & 4\end{bmatrix}$ and A² – kA – 5I = O, then the valueof k is :

(a) 3 (b) 5

6. If $e^{x^2y} = c$ , then $\dfrac{dy}{dx}$ is :

(a) $\dfrac{xe^{x^2y}}{2y}$

(b) $\dfrac{-2y}{x}$

(d) $\dfrac{x}{2y}$

7. The value of constant c that makes the function f defined by

$f(x) =\begin{cases} x^2 - c^2, & \text{if} x<4 \\cx + 20, & \text{if} x geq 4 \end{cases}$

continuous for all real number is :

(a) -2 (b) -1

8. $\displaystyle \int |x| dx$ is :

(a) -2 (b) -1

9. The number of arbitrary constants in the particular solution of the differential equation

$\log (\dfrac{dy}{dx}) = 3x + 4y$ ; y(0) = 0 is/are

(a) 2 (b) 1

10. If $\begin{bmatrix} a & c & 0 \\ b & d & 0 \\ 0 & 0 & 5 \end{bmatrix}$ is a scalar matrix, then the value of a + 2b + 3c + 4d is:

(a) 0 (b) 5

11. If $A = \begin{bmatrix} 2 & 1 \\ -4 & -2 \end{bmatrix}$ , then the value of I – A + A² – A³ + . . . is :

(a) $A = \begin{bmatrix} -1 & -1 \\ 4 & 3 \end{bmatrix}$

(b) $A = \begin{bmatrix} 3 & 1 \\ -4 & -1 \end{bmatrix}$

(d) $A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$

12. Given that $A^{-1} = \dfrac{1}{7}\begin{bmatrix} 2 & 1 \\ -3 & 2 \end{bmatrix}$

(a) $7\begin{bmatrix} 2 & -1 \\ 3 & 2 \end{bmatrix}$

(b) $\begin{bmatrix} 2 & -1 \\ 3 & 2 \end{bmatrix}$

(d) $\dfrac{1}{49}\begin{bmatrix} 2 & -1 \\ 3 & 2 \end{bmatrix}$

13. The integrating factor of the differential equation $(x + 2y^2)\dfrac{dy}{dx} = y$ (y>0) is :

(a) 1/x (b) x

14. A vector perpendicular to the line $\vec{r} = \hat{i} + \hat{j} - \hat{k} + \lamda(3\hat{i} - \hat{j})$ is:

(a) $5\hat{i} + \hat{j} + 6\hat{k}$

(b) $\hat{i} + 3\hat{j} + 5\hat{k}$

(d) $9\hat{i} - 3\hat{j}$

15. The vectors $\vec{a} = 2 \hat{i} - \hat{j} + \hat{k}, \vec{b} = \hat{i} - 3\hat{j} - 5\hat{k}$ and $\vec{c} = -3\hat{i} + 4\hat{j} + 4\hat{k}$ represnts the sides of

(a) An equilateral triangle

(b) An obtuse-angled triangle

(d) A right-angle triangle

16. Let $\vec{a}$ be any vector such that $|\vec{a}| = a$ . the value of

$|\vec{a} \times \hat{i}|^2 + |\vec{a} \times{a} \times \hat{j}|^2 +|\vec{a} \times \hat{k}|^2$ is :

(a) a² (b) 2a²

17. If $\vec{a}$ and $\vec{b}$ are two vectors such that $|\vec{a}| = 1,|\vec{b}| = 2$ and $\vec{a}.\vec{b} = \sqrt{3}$ , then the angle between $2\vec{a}$ and $-\vec{b}$ is :

(a) π/6 (b) π/3

18. The function f(x) = kx – sin x is strictly increasing for

(a) k > 1 (b) k < 1

Questions number 19 and 20 are Assertion and Reason based questions carrying 1 mark each. Two statements are given, one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer from the codes (a), (b), (c) and (d as given below.

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).

(b) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assértion (A).

(c) Assertion (A) is true and Reason (R) is false.

(d) Assertion (A) is false and Reason (R) is true.

19. Assertion (A) : The corner points of the boundedfeasible region of a L.P.P. are shown below. The maximum value of Z = x + 2y occurs at infinites points.

Reason (R): The optimal solution of a L.P.P. having bounded feasible region must occur at corner points.

20. Assertion (A): The relation R = {(x, y) : (x + y) is prime number and x, y ∈ N} is not a reflexive relation.

Reason (R) : The number ‘2n’ is composite for all natural number n.

SECTION B

Case Study Question

In this section there are 5 very short answer type question of 2 marks each.

21. The volume of a cube is increasing at the rate of 6 cm³/s. How fast is the surface area of cube increasing, when the length of an edge is 8 cm ?

22. (a) Express $\tan^{-1}(\dfrac{\cos x}{1 - \sin x})$ , where $\dfrac{-\pi}{2} < x < \dfrac{\pi}{2}$ in the simplest form.

(b) Find the principal value of $\tan^{-1}(1) + \cos^{-1}(\frac{-1}{2}) + \sin^{-1}(\frac{-1}{\sqrt{2}})$ .

23. Show that $f(x) = \dfrac{4\sin x}{2 + \cos x} - x$ is an increasing function of x in [0, π/2].

24. If $y = \cos^3(\sec^2 2t)$ , find $\dfrac{dy}{dx}$ .

(b) If $x^y = e^{x-y}$ , prove that $\dfrac{dy}{dx} = \dfrac{\log x}{(1 + \log x)^2}$ .

25. Evaluate : $\displaystyle \int_{-1/2}^{1/2} \cos x.\log({\dfrac{1 + x}{1 - x}) dx$

SECTION C

In this section there are 6 short answer type questions of 3 marks each.

26. Given that $x^y + y^x = a^b$ , where a and b are positive constants, find $\dfrac{dy}{dx}$ .

27. (a) Find the particular solution of the differential equation $\dfrac{dy}{dx} = y \cot 2x$ , given that y(π/4) = 2.

(b) Find the particular solution of the differential equation $(xe^{y/x} + y)dx = x dy$ , given that y = 1 when x = 1.

28. (a) Find : $\displaystyle \int \dfrac{2x + 3}{x^2(x + 3)} dx$ .

29. (a) A card from a well shuffled deck of 52 playing cards is lost . From the remaining cards of the pack, a card is drawn at random and is found to be a King. Find the probability of the lost card being a King.

(b) A biased die is twice as likely to show an even number as an odd number. If such a die is thrown twice, find the probability distribution of the number of sixes. Also, find the mean of the distribution.

30. Solve the following L.P.P. graphically :

Maximise Z = x + 3y

subject to the constraints :

x + 2y ≤ 200

x + y ≤ 150

y ≤ 75

x, y ≥ 0

31. (a) Evaluate : $\displaystyle \int_0^{\pi/4} \dfrac{x dx}{1 + \cos 2x + \sin 2x}$

(b) Find : $\displaystyle \int e^x[\dfrac{1}{(1+x^2)^{3/2}}+\dfrac{x}{\sqrt{1+x^2}}]dx$

SECTION D

In this section there are 4 long answer type questions of 5 marks each.

32. (a) Let A = R – {5} and B = R – {1}. Consider the function f : A → B, defined by $f(x) = \dfrac{x - 3}{x - 5}$ . Show that f is one-one and onto.

(b) Check whether the relation S in the set of real numbers R defined by S = {(a,b): where a – b + √2 is an irrational number} is reflexve, symmetric or transitive.

33. (a) Find the distance between the line $\dfrac{x}{2} = \dfrac{2y - 6}{4} = \dfrac{1 - z}{-1}$ and another line parallel to it passing through the point (4, 0, -5).

(b) If the lines $\dfrac{x - 1}{-3} = \dfrac{y - 2}{2k} = \dfrac{z-3}{2}$ and $\dfrac{x - 1}{3k} = \dfrac{y -1}{1} = \dfrac{z - 6}{-7}$ are perpendicular to each other, find the value of k and hence write the vector equation of a line perpendicular to these two lines and passing through the point (3, -4, 7).

34. Use the product of matrices $\begin{bmatrix} 1 & 2 & -3 \\ 3 & 2 & -2 \\ 2 & -1 & 1 \end{bmatrix}.\begin{bmatrix} 0 & 1 & 2 \\ -7 & 7 & -7 \\ -7 & 5 & -4 \end{bmatrix}$ to solve the following system of equations :

x + 2y – 3z = 6

3x + 2y – 2z = 3

2x – y + z = 2

35. Sketch the graph of y = x|x| and hence find the area bounded by this curve X-axis and the ordinates x = -2 and x = 2, using integration.

(b) Using integration, find the area bounded by the ellipse 9x² + 25y² = 225, the lines x = -2, x = 2 and the x-axis.

SECTION E

In this section, there are 3 case study based questions of 4 marks each.

36.

An instructor at the astronomical centre shows three among the brightest stars in a particular constellation. Assume that the telescope is located at O(0, 0, 0) and the three stars have their locations at the points D, A and V having position vectors $2\hat{i} + 3\hat{j} + 4\hat{k}, 7\hat{i} + 5\hat{j} + 8\hat{k}$ and $-3\hat{i} + 7\hat{j} + 11\hat{k}$ respectively. [CBSE 2024]

An instructor at the astronomical centre shows three among

Based on the above information, answer the following question:

(i) How far is the star V from star A ? 1

(ii) Find a unit vector in the direction of $\vec{DA}$ . 1

(iii) Find the measure of ∠VDA . 2

(iii) What is the projection of vector $\vec{DV}$ on vector $\vec{DA}$ ? 2

37. Rohit, Jaspreet and Alia appeared for an interview for three vacancies in the same post. The probability of Rohit’s selection is 1/5, Jaspreet;s selection is 1/3 and Alia’s selection is 1/4. The event of selection is independent of each other. [CBSE 2024]

Based on the above information, answer the following question:

(i) What is the probability that at least one of them is selected ? 1

(ii) Find $P(G/\overline{H})$ where G is the event of Jaspreet’s selection and $\overline{H}$ denotes the event that Rohit is not selected. 1

(iii) Find the probability that exactly one of them is selected. 2

(iii) Find the probability that exactly two of them are selected. 2

38.

A Store has been selling calculators at Rs 350 each. A market survey indicates that a reduction in price (p) of calculator increases the number of units (x) sold. The relation between the price and quantity sold is given by the demand function $p = 450 - \dfrac{1}{2} x$ . [CBSE 2024]

A Store has been selling calculators at Rs 350 each

Based on the above information, answer the following question :

(i) Determine the number of units (x) that should be to maximise the revenue R(x) = x p(x). Also, verify the result. 2

(ii) What rebate in price of calculator should the store give to maximise the revenue ? 2

According to the CBSE syllabus