Roll No.

Q.P. Code 65/1/1

Mathematics

CBSE Class XII Board Examination 2026

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 questions, 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 some questions in Sections B to E offer an internal choice (OR). Calculators are not allowed.

SECTION A

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

Given \(2\cos^{-1}x = y\), the values that y can take lie in

(A) \(0 \le y \le \pi\)
(B) \(-\pi \le y \le \pi\)
(C) \(0 \le y \le 2\pi\)
(D) \(-\pi \le y \le 0\)

Which one of these can NOT be the order of a row matrix?

(A) 2 × 1
(B) 1 × 2
(C) 1 × 1
(D) 1 × n

For matrices A and B of suitable orders, which of these properties hold?
(i) (A + B)′ = A′ + B′
(ii) (A − B)′ = B′ − A′
(iii) (AB)′ = A′B′
(iv) (kAB)′ = kB′A′, k a scalar

(A) (i) only
(B) (i), (ii) and (iii)
(C) (i) and (ii)
(D) (i) and (iv)

Let \(\Delta_1 = \begin{vmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{vmatrix}\) and \(\Delta_2 = \begin{vmatrix} 0 & 2 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 6 \end{vmatrix}\). Then

(A) \(\Delta_1 = 2\Delta_2\)
(B) \(\Delta_2 = -2\Delta_1\)
(C) \(\Delta_1 = \Delta_2\)
(D) \(\Delta_2 = -\Delta_1\)

One value of x satisfying \(\begin{vmatrix} \cos x & \sin x \\ -\cos x & \sin x \end{vmatrix} = 1\) is

(A) 0
(B) \(\frac{\pi}{4}\)
(C) \(\frac{\pi}{3}\)
(D) \(\frac{\pi}{2}\)

A and B are skew-symmetric matrices of the same order. Which of the following is also skew-symmetric?

(A) AB
(B) AB + BA
(C) \((A + B)^2\)
(D) A − B

On the interval [0, 3], the smallest value of \(f(x) = x^3 - 12x\) is

(A) −16
(B) −9
(C) 0
(D) 16

If \(\int \frac{3ax}{b^2 + c^2x^2}\,dx = A\log|b^2 + c^2x^2| + K\), then A equals

(A) 3a
(B) \(\frac{3a}{2b^2}\)
(C) \(\frac{3a}{b^2c^2}\)
(D) \(\frac{3a}{2c^2}\)

Evaluate \(\displaystyle\int_{-1}^{1} \frac{x^3}{x^2 + 2|x| + 1}\,dx\).

(A) 0
(B) log 2
(C) 2 log 2
(D) \(\frac{1}{2}\log 2\)

10.

Find the area enclosed by the curve y = x|x|, the x-axis and the lines x = −1 and x = 1.

(A) 0
(B) \(\frac{1}{3}\)
(C) \(\frac{2}{3}\)
(D) \(\frac{4}{3}\)

11.

For the differential equation \(R\frac{dx}{dy} + Px = Q\), where P, Q, R are functions of y only, the integrating factor is

(A) \(e^{\int \frac{P}{Q}dy}\)
(B) \(e^{\int P\,dy}\)
(C) \(e^{\int \frac{P}{R}dy}\)
(D) \(e^{\int \frac{P}{R}dx}\)

12.

Give the order and the degree (in that order) of the differential equation \(\frac{d}{dx}(e^{y}) = 0\).

(A) 0, 1
(B) 1, 1
(C) 2, 1
(D) 1, not defined

13.

For what value of p are the vectors \(\hat i + 2\hat j + 3\hat k\) and \(2\hat i - p\hat j + \hat k\) perpendicular?

(A) 0
(B) 1
(C) \(\frac{5}{2}\)
(D) \(-\frac{5}{2}\)

14.

Points with position vectors \(-\hat i - \hat j + 2\hat k\), \(2\hat i + m\hat j + 5\hat k\) and \(3\hat i + 11\hat j + 6\hat k\) lie on one line. Find m.

(A) 8
(B) −8
(C) 2
(D) \(\frac{5}{2}\)

15.

(A) \(6\sqrt3\)
(B) \(8\sqrt3\)
(C) \(12\sqrt3\)
(D) \(3\sqrt{12}\)

16.

Find the length of the perpendicular from the point (2, 5, 7) to the line \(\frac{x}{1} = \frac{y}{0} = \frac{z}{0}\).

(A) 2
(B) 5
(C) \(\sqrt{74}\)
(D) \(\sqrt{78}\)

17.

An LPP has objective function Z = 5x + 7y. Its feasible region (shown in a figure) is the quadrilateral with corners O(0, 0), (0, 2), (3, 4) and (7, 0). Find (maximum value of Z) − (minimum value of Z).

(A) 8
(B) 29
(C) 35
(D) 43

18.

What is the degree of the objective function of a linear programming problem?

(A) 0
(B) 1
(C) 2
(D) Any natural number

19.

Assertion (A): When a fair die is thrown, the probability of getting a prime number, given that the number shown is odd, is \(\frac{2}{3}\).
Reason (R): For any two events A and B, \(P(A|B) = \frac{P(A \cup B)}{P(B)}\).

(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): The lines x = py + q, z = ry + s and x = p′y + q′, z = r′y + s′ are perpendicular when pp′ + rr′ = 1.
Reason (R): The lines \(\vec r = \vec a_1 + \lambda \vec b_1\) and \(\vec r = \vec a_2 + \mu \vec b_2\) are perpendicular if \(\vec b_1 \cdot \vec b_2 = 0\).

(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 of 2 marks each.

21.

A function is defined by \(f(x) = \frac{|x-3|}{2(x-3)}\) for x < 3 and \(f(x) = \frac{x-6}{6}\) for x ≥ 3. Is f continuous at x = 3?

Given \(\sqrt3\,(x^2 + y^2) = 4xy\), find \(\frac{dy}{dx}\) at the point \(\left(\frac12, \frac{\sqrt3}{2}\right)\).

22.

A room-freshener bottle is an inverted cone with semi-vertical angle \(\frac{\pi}{6}\). The volume of perfume in it falls steadily at 1 mm³/min. How fast is the perfume level falling when the level (height of liquid) is 10 mm?

23.

P and Q are the points (1, 3, 2) and (−1, 0, 8). Find the vector of magnitude 14 along \(\overrightarrow{QP}\).

24.

The vectors \(\vec a = 3\hat i - 2\hat j + 2\hat k\) and \(\vec b = \hat i + 2\hat k\) are adjacent sides of a parallelogram. Find vectors along its diagonals and their lengths.

25.

Write in simplest form: \(\tan^{-1}\left(\frac{\cos 2x - \sin 2x}{\cos 2x + \sin 2x}\right)\), where \(0 < x < \frac{\pi}{4}\).

Find the value of \(\tan\left(\sin^{-1}1 - \cos^{-1}\left(-\frac12\right)\right)\).

SECTION C

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

26.

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

27.

Find \(\displaystyle\int \sqrt{\frac{x+2}{x-2}}\,dx\).

Find \(\displaystyle\int \frac{x^2}{(x^2+9)(x^2+16)}\,dx\).

28.

Let \(I_1 = \displaystyle\int_{-\pi/4}^{\pi/4} \frac{dx}{1 + \cos 2x}\) and \(I_2 = \displaystyle\int_{-1/2}^{1/2} |x|\,dx\). Show that \(I_1 - 4I_2 = 0\).

29.

Solve (general solution): \(x^2\frac{dy}{dx} = x^2 + xy + y^2\).

Find the particular solution of \(xy\frac{dy}{dx} = (x+2)(y+2)\) given that y(1) = −1.

30.

Solve graphically: minimise Z = 13x − 15y subject to x + y ≤ 7, 2x − 3y + 6 ≥ 0, x ≥ 0, y ≥ 0.

31.

Bag I has 3 red and 4 white balls; bag II has 8 red and 6 white balls. A die is rolled: if it shows a number less than 3, a ball is drawn from bag I, otherwise from bag II. Find the probability that the ball drawn is red.

For two events X and Y, the probability that at least one of them happens is a, and the probability that exactly one of them happens is b. Prove that P(X′) + P(Y′) = 2 − 2a + b.

SECTION D

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

32.

On the set Z of integers, a relation is defined by R = {(x, y) : |x − y| is divisible by a fixed prime p}. Determine whether R is an equivalence relation.

A function \(f: \mathbb{R} - \{\frac35\} \to \mathbb{R} - \{\frac35\}\) is given by \(f(x) = \frac{3x+2}{5x-3}\). Prove that f is both one-one and onto.

33.

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

The matrix \(\begin{bmatrix} 3 & -1 & \sin 3x \\ -7 & 4 & \cos 2x \\ -11 & 7 & 2 \end{bmatrix}\) is singular. Find all x in \(\left[0, \frac{\pi}{2}\right]\).

34.

Given x = cos t and y = cos mt, prove that \((1 - x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx} + m^2y = 0\).

35.

Are the lines \(\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}\) and \(\frac{x-4}{5} = \frac{y-1}{2} = z\) parallel? If they are, find the distance between them; if not, find where they meet (if they intersect).

SECTION E

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

36.

An online delivery company has 5000 subscribers, each paying ₹300 a year for unlimited free deliveries. It plans to raise the annual fee; for every ₹1 increase, 10 subscribers are expected to leave. Suppose the fee is raised by ₹x.
(i) How many subscribers leave after a rise of ₹x? (1 mark)
(ii) Write the total revenue R(x) after the rise as a function of x. (1 mark)
(iii)(a) Find the value of x for which R(x) is maximum. (2 marks)

(iii)(b)

Find the sub-intervals of (0, 5000) on which R(x) is increasing and on which it is decreasing.

37.

An online jackpot has one first prize of ₹3,00,000, two second prizes of ₹2,00,000 each and three third prizes of ₹50,000 each. In all 1,00,000 tickets were sold at ₹100 each. Rohan buys one ticket.
(i) What amounts can he possibly win? (1 mark)
(ii)(a) Find the probability that he wins at least ₹2,00,000. (2 marks)
(iii) Rohan also buys a ticket in another jackpot with a single prize of ₹5,00,000, where the chance of winning is 1 in 1,00,000. Find the probability that he wins on exactly one of his two tickets. (1 mark)

(ii)(b)

Find the probability that he does not win any amount.

38.

A roundabout on a busy road has its outer boundary along the circle C₁: x² + y² = 64; a circular pond with a fountain at its centre is C₂: x² + y² = 4.
(i) Draw C₁ and C₂ in one diagram. (1 mark)
(ii) Write y as a function of x for C₁ and for C₂. (1 mark)
(iii)(a) Using integration, find the area covered by the roundabout (region inside C₁). (2 marks)

(iii)(b)

Using integration, find the area of the circular pond.

— END —

Questions restated from the CBSE Class XII Board Examination 2026 (set 65/1/1) for practice. Reading passages and map items are described, not reproduced.