Roll No.

Q.P. Code 65/1/1

Mathematics

CBSE Class XII Board Examination 2024

Time allowed: 3 hoursMaximum Marks: 80

General Instructions:

The paper has 38 questions in five sections (A to E); all of them must be answered.
Section A: Q1–18 are multiple choice and Q19–20 are Assertion-Reason, 1 mark each.
Section B: Q21–25, very short answer (VSA), 2 marks each.
Section C: Q26–31, short answer (SA), 3 marks each.
Section D: Q32–35, long answer (LA), 5 marks each.
Section E: Q36–38, case-study based, 4 marks each.
There is no overall choice, but internal choices (OR) are given in 2 questions of Section B, 3 of Section C, 2 of Section D and 2 of Section E. Calculators are not allowed.

SECTION A

Q1–20 carry 1 mark each (Q1–18 MCQ; Q19–20 Assertion-Reason).

The function \( f : \mathbb{R}_+ \to \mathbb{R} \) (where \( \mathbb{R}_+ \) is the set of all non-negative real numbers) is given by \( f(x) = 4x + 3 \). Then f is:

(A) one-one but not onto
(B) onto but not one-one
(C) both one-one and onto
(D) neither one-one nor onto

A matrix has 36 elements. How many different orders can it have?

(A) 13
(B) 3
(C) 5
(D) 9

Consider \( f(x) = \begin{cases} x^2 + 3, & x \ne 0 \\ 1, & x = 0 \end{cases} \). Which statement is true?

(A) f(x) is continuous and differentiable for all x ∈ ℝ
(B) f(x) is continuous for all x ∈ ℝ
(C) f(x) is continuous and differentiable for all x ∈ ℝ − {0}
(D) f(x) is discontinuous at infinitely many points

f(x) is continuous on [a, b] and differentiable on (a, b). f(x) is strictly increasing on (a, b) if:

(A) f′(x) < 0 for all x ∈ (a, b)
(B) f′(x) > 0 for all x ∈ (a, b)
(C) f′(x) = 0 for all x ∈ (a, b)
(D) f(x) > 0 for all x ∈ (a, b)

Given \( \begin{bmatrix} x+y & 2 \\ 5 & xy \end{bmatrix} = \begin{bmatrix} 6 & 2 \\ 5 & 8 \end{bmatrix} \), find \( \dfrac{24}{x} + \dfrac{24}{y} \).

(A) 7
(B) 6
(C) 8
(D) 18

\( \int_a^b f(x)\,dx \) equals:

(A) \( \int_a^b f(a - x)\,dx \)
(B) \( \int_a^b f(a + b - x)\,dx \)
(C) \( \int_a^b f(x - (a + b))\,dx \)
(D) \( \int_a^b f((a - x) + (b - x))\,dx \)

θ is the angle between two unit vectors \( \hat{a} \) and \( \hat{b} \), and sin θ = 3/5. Then \( \hat{a} \cdot \hat{b} \) equals:

(A) ± 3/5
(B) ± 3/4
(C) ± 4/5
(D) ± 4/3

Find the integrating factor of \( (1 - x^2)\dfrac{dy}{dx} + xy = ax \), where −1 < x < 1.

(A) \( \dfrac{1}{x^2 - 1} \)
(B) \( \dfrac{1}{\sqrt{x^2 - 1}} \)
(C) \( \dfrac{1}{1 - x^2} \)
(D) \( \dfrac{1}{\sqrt{1 - x^2}} \)

A line has direction cosines √3 k, √3 k, √3 k. Find k.

(A) ± 1
(B) ± √3
(C) ± 3
(D) ± 1/3

10.

A linear programming problem is about optimising (maximising or minimising) a:

(A) logarithmic function
(B) linear function
(C) quadratic function
(D) exponential function

11.

If P(A | B) = P(A′ | B), which statement is true?

(A) P(A) = P(A′)
(B) P(A) = 2 P(B)
(C) P(A ∩ B) = ½ P(B)
(D) P(A ∩ B) = 2 P(B)

12.

Evaluate \( \begin{vmatrix} x+1 & x-1 \\ x^2+x+1 & x^2-x+1 \end{vmatrix} \).

(A) 2x³
(B) 2
(C) 0
(D) 2x³ − 2

13.

Find the derivative of sin(x²) with respect to x at \( x = \sqrt{\pi} \).

(A) 1
(B) −1
(C) \( -2\sqrt{\pi} \)
(D) \( 2\sqrt{\pi} \)

14.

Find the order and degree of \( \left[1 + \left(\dfrac{dy}{dx}\right)^2\right]^3 = \dfrac{d^2y}{dx^2} \).

(A) 1, 2
(B) 2, 3
(C) 2, 1
(D) 2, 6

15.

Find the vector whose initial point is B(3, −4, 7) and terminal point is A(2, −3, 5).

(A) \( \hat i - \hat j + 2\hat k \)
(B) \( \hat i + \hat j + 2\hat k \)
(C) \( -\hat i - \hat j - 2\hat k \)
(D) \( -\hat i + \hat j - 2\hat k \)

16.

The distance of the point P(a, b, c) from the y-axis is:

(A) b
(B) b²
(C) \( \sqrt{a^2 + c^2} \)
(D) a² + c²

17.

How many corner points does the feasible region of x ≥ 0, y ≥ 0, x + y ≥ 4 have?

(A) 0
(B) 1
(C) 2
(D) 3

18.

A and B are non-zero square matrices of the same order with (A + B)² = A² + B². Then:

(A) AB = O
(B) AB = −BA
(C) BA = O
(D) AB = BA

19.

Assertion (A): For \( A = \begin{bmatrix} 1 & \cos\theta & 1 \\ -\cos\theta & 1 & \cos\theta \\ -1 & -\cos\theta & 1 \end{bmatrix} \), θ ∈ [0, 2π], we have |A| ∈ [2, 4].
Reason (R): cos θ ∈ [−1, 1] for all θ ∈ [0, 2π].

(A) Both A and R are true, and R is the correct explanation of A.
(B) Both A and R are true, but R is not the correct explanation of A.
(C) A is true, but R is false.
(D) A is false, but R is true.

20.

Assertion (A): No line in space can be perpendicular to all three of the x, y and z axes at the same time.
Reason (R): If a line makes angles α, β, γ with the positive x, y, z axes, then cos²α + cos²β + cos²γ = 1.

(A) Both A and R are true, and R is the correct explanation of A.
(B) Both A and R are true, but R is not the correct explanation of A.
(C) A is true, but R is false.
(D) A is false, but R is true.

SECTION B

Q21–25 are very short answer (VSA) questions, 2 marks each.

21.

Is the function f(x) = x²|x| differentiable at x = 0? Justify.

Given \( y = \sqrt{\tan\sqrt{x}} \), prove that \( \sqrt{x}\,\dfrac{dy}{dx} = \dfrac{1 + y^4}{4y} \).

22.

Show that f(x) = 4x³ − 18x² + 27x − 7 has no maximum or minimum value.

23.

Find \( \int x\sqrt{1 + 2x}\,dx \).

Evaluate \( \displaystyle\int_0^{\pi^2/4} \dfrac{\sin\sqrt x}{\sqrt x}\,dx \).

24.

\( \vec a \) and \( \vec b \) are non-zero vectors with \( (\vec a + \vec b) \perp \vec a \) and \( (2\vec a + \vec b) \perp \vec b \). Prove that \( |\vec b| = \sqrt2\,|\vec a| \).

25.

ABCD is a parallelogram (figure: AB is the top side, DC the bottom side, and the diagonal DB is drawn from D to B). Given \( \vec{AB} = 2\hat i - 4\hat j + 5\hat k \) and \( \vec{DB} = 3\hat i - 6\hat j + 2\hat k \), find \( \vec{AD} \) and then the area of the parallelogram.

SECTION C

Q26–31 are short answer (SA) questions, 3 marks each.

26.

On A = {1, 2, 3, 4, 5}, the relation R = {(x, y) : |x² − y²| < 8}. Check whether R is reflexive, symmetric and transitive.

f : ℝ → ℝ is given by f(x) = ax + b with f(1) = 1 and f(2) = 3. Find f(x) and check whether f is one-one and onto.

27.

Given \( \sqrt{1 - x^2} + \sqrt{1 - y^2} = a(x - y) \), prove that \( \dfrac{dy}{dx} = \sqrt{\dfrac{1 - y^2}{1 - x^2}} \).

Find dy/dx if y = (tan x)^x.

28.

Find \( \displaystyle\int \dfrac{x^2}{(x^2 + 4)(x^2 + 9)}\,dx \).

Evaluate \( \displaystyle\int_1^3 \big(|x - 1| + |x - 2| + |x - 3|\big)\,dx \).

29.

Find the particular solution of \( x^2\dfrac{dy}{dx} - xy = x^2\cos^2\left(\dfrac{y}{2x}\right) \), given y = π/2 when x = 1.

30.

Solve graphically: maximise Z = 500x + 300y subject to x + 2y ≤ 12, 2x + y ≤ 12, 4x + 5y ≥ 20, x ≥ 0, y ≥ 0.

31.

E and F are independent events with P(Ē) = 0.6 and P(E ∪ F) = 0.6. Find P(F) and P(Ē ∪ F̄).

SECTION D

Q32–35 are long answer (LA) questions, 5 marks each.

32.

For \( A = \begin{bmatrix} 1 & -2 & 0 \\ 2 & -1 & -1 \\ 0 & -2 & 1 \end{bmatrix} \), find A⁻¹ and use it to solve x − 2y = 10, 2x − y − z = 8, −2y + z = 7.

Given \( A = \begin{bmatrix} -1 & a & 2 \\ 1 & 2 & x \\ 3 & 1 & 1 \end{bmatrix} \) and \( A^{-1} = \begin{bmatrix} 1 & -1 & 1 \\ -8 & 7 & -5 \\ b & y & 3 \end{bmatrix} \), find (a + x) − (b + y).

33.

Evaluate \( \displaystyle\int_0^{\pi/4} \dfrac{\sin x + \cos x}{9 + 16\sin 2x}\,dx \).

Evaluate \( \displaystyle\int_0^{\pi/2} \sin 2x\,\tan^{-1}(\sin x)\,dx \).

34.

Using integration, find the area of the part of the ellipse \( \dfrac{x^2}{16} + \dfrac{y^2}{4} = 1 \) that lies between the lines x = −2 and x = 2.

35.

The image of a point P(x, y, z) in the line \( \dfrac x1 = \dfrac{y-1}{2} = \dfrac{z-2}{3} \) is P′(1, 0, 7). Find P.

SECTION E

Q36–38 are case-study based questions, 4 marks each.

36.

Case Study 1: Traffic police use speed cameras that can catch a speeding vehicle from up to 300 m away. One camera is fixed on a pole 5 m high, and a car C moves away from the pole at 20 m/s. When the car is x m from the foot of the pole, the angle of elevation of the camera from the car is θ.

(i) Express θ in terms of the height of the camera and x. (1)
(ii) Find dθ/dx. (1)
(iii)(a) Find the rate of change of the angle of elevation with respect to time when the car is 50 m from the pole. (2)

(iii)(b)

(iii)(b) For another car 50 m from the foot of the pole, the angle of elevation is changing at the rate of 3/101 rad/s (in magnitude). Find the speed of this car. (2)

37.

Case Study 2: Climate change has made air turbulence more common. An airplane meets severe, moderate or light turbulence with equal probability, and the chance of arriving late is 55%, 37% and 17% respectively for these three.

(i) Find the probability that an airplane reaches its destination late. (2)
(ii) Given that the airplane reached late, find the probability that this was due to moderate turbulence. (2)

38.

Case Study 3: A function has an inverse only when it is one-one and onto. Sine on ℝ is neither (its graph is the familiar wave between −1 and 1), so sine is restricted to an interval A, mapped onto [−1, 1], so that sin⁻¹ : [−1, 1] → A exists.

(i) Give an example of such an interval A other than the principal value branch. (1)
(ii) Using the principal value branch, find \( \sin^{-1}\left(-\tfrac12\right) - \sin^{-1}(1) \). (1)
(iii)(a) Draw the graph of sin⁻¹x on [−1, 1] for its principal value branch. (2)

(iii)(b)

(iii)(b) Find the domain and range of f(x) = 2 sin⁻¹(1 − x). (2)

— END —

Questions restated from the CBSE Class XII Board Examination 2024 (set 65/1/1) for practice. Reading passages and map items are described, not reproduced.