CBSE Class 12 Mathematics 2026 question paper (65/1)

Maximum marks 80 · Time 3 hours · 3 sets

Try each question first, then open its model answer. Where the paper offers a choice, both questions are shown with OR between them.

Section A

1 mark each

Q.11 mark $2 \cos^{-1} x = y$
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Q.21 mark $Which of the following cannot be the order of a row-matrix ?$
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Q.31 mark $(A + B)' = A' + B'$
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Q.41 mark $\Delta_1 = \begin{vmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{vmatrix}$
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Q.51 mark $x$
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Q.61 mark $If A and B are skew symmetric matrices of same order, then which of the following matrices is also skew symmetric ?$
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Q.71 mark $f(x) = x^3 - 12x$
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Q.81 mark $\int \frac{3ax}{b^2 + c^2x^2}\,dx = A \log \left| b^2 + c^2x^2 \right| + K$
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Q.91 mark $\int_{-1}^{1} \frac{x^3}{x^2 + 2\left| x \right| + 1}\,dx$
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Q.101 mark $y = x\left| x \right|$
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Q.111 mark $R \frac{dx}{dy} + Px = Q$
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Q.121 mark $\frac{d}{dx}(e^y) = 0$
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Q.131 mark $\hat{i} + 2\hat{j} + 3\hat{k}$
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Q.141 mark $-\hat{i} - \hat{j} + 2\hat{k}$
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Q.151 mark $\left| \vec{a} \right| = 8$
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Q.161 mark $\frac{x}{1} = \frac{y}{0} = \frac{z}{0}$
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Q.171 mark $Z = 5x + 7y$
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Q.181 mark $The degree of an objective function of a linear programming problem is$
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Direction : Questions number 19 and 20 are Assertion and Reason based questions carrying 1 mark each. Two statements are given, one labelled Assertion (A) and other labelled Reason (R). Select the correct answer from the codes (A), (B), (C) and (D) as given below.
Q.191 mark $\frac{2}{3}$
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Q.201 mark $x = py + q$
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Section B

2 marks each

Q.21 (a)2 marks $f(x) = \begin{cases} \frac{\left| x - 3 \right|}{2(x - 3)}, & x < 3 \\ \frac{x - 6}{6}, & x \geq 3 \end{cases}$
OR
Q.21 (b)2 marks $\sqrt{3}(x^2 + y^2) = 4xy$
Q.222 marks $^3$
Q.232 marks $\overrightarrow{QP}$
Q.242 marks $\vec{a} = 3\hat{i} - 2\hat{j} + 2\hat{k}$
Q.25 (a)2 marks $\tan^{-1} \left( \frac{\cos 2x - \sin 2x}{\cos 2x + \sin 2x} \right)$
OR
Q.25 (b)2 marks $\tan \left( \sin^{-1} 1 - \cos^{-1} \left( -\frac{1}{2} \right) \right)$

Section C

3 marks each

Q.263 marks $\int_{0}^{1} x \tan^{-1} x \, dx$
Q.27 (a)3 marks $\int \sqrt{\frac{x + 2}{x - 2}} \, dx$
OR
Q.27 (b)3 marks $\int \frac{x^2}{(x^2 + 9) \, (x^2 + 16)} \, dx$
Q.283 marks $I_1 = \int_{-\pi/4}^{\pi/4} \frac{dx}{1 + \cos 2x}$
Q.29 (a)3 marks $x^2 \frac{dy}{dx} = x^2 + xy + y^2$
OR
Q.29 (b)3 marks $xy \frac{dy}{dx} = (x + 2) \, (y + 2)$
Q.303 marks $Z = 13x - 15y$
Q.31 (a)3 marks $Out of two bags, bag I contains 3 red and 4 white balls and bag II contains 8 red and 6 white balls. A die is thrown. If it shows a number less than 3 then a ball is drawn at random from bag I, otherwise a ball is drawn at random from bag II. Find the probability that the ball drawn from one of the bags is a red ball.$
OR
Q.31 (b)3 marks $P(X') + P(Y') = 2 - 2a + b$

Section D

5 marks each

Q.32 (a)5 marks $R = \{(x, y) : \left| x - y \right| \text{ is divisible by a prime number 'p'}, x, y \in Z\}$
OR
Q.32 (b)5 marks $f : R - \left\{ \frac{3}{5} \right\} \longrightarrow R - \left\{ \frac{3}{5} \right\}$
Q.33 (a)5 marks $A = \begin{bmatrix} 0 & 2 & 1 \\ -2 & -1 & -2 \\ 1 & -1 & 0 \end{bmatrix}$
OR
Q.33 (b)5 marks $\begin{bmatrix} 3 & -1 & \sin 3x \\ -7 & 4 & \cos 2x \\ -11 & 7 & 2 \end{bmatrix}$
Q.345 marks $x = \cos t$
Q.355 marks $\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4}$

Section E

An online delivery company in a city has 5000 subscribers and collects annual subscription fees of ₹ 300 per subscriber for unlimited free deliveries. The company wishes to increase the annual subscription fee. It is predicted that, for every increase of ₹ 1, ten subscribers will discontinue. Assume that the company increased the annual fee by ₹ $x$ . Based on the given information, answer the following questions :
Q.36 (i)1 mark $x$
Q.36 (ii)1 mark $R(x)$
Q.36 (iii) (a)2 marks $x$
OR
Q.36 (iii) (b)2 marks $R(x)$
In an online jackpot, there is one first prize of ₹ 3,00,000, two second prizes of ₹ 2,00,000 each and three third prizes of ₹ 50,000 each. A total of 1,00,000 jackpot tickets each costing ₹ 100 were sold there by raising a fund of ₹ 1,00,00,000. Rohan bought one ticket. Based on given information, answer the following questions :
Q.37 (i)1 mark $What are the possible amounts, the person can win ?$
Q.37 (ii) (a)2 marks $What is the probability that the person wins atleast ₹ 2,00,000 ?$
OR
Q.37 (ii) (b)2 marks $What is the probability that the person does not win any amount ?$
Q.37 (iii)1 mark $In another jackpot, Rohan also bought a ticket having a prize money of ₹ 5,00,000. The chances of winning the jackpot are 1 in 1,00,000. Find the probability that on exactly one of tickets he wins the jackpot.$
Roundabouts are often made on busy roads to ease the traffic and avoid red lights. One such round-about is made such that equation representing its boundary is given by $C_1$ ; $x^2 + y^2 = 64$ . There is a circular pond with a fountain in the middle of the roundabout whose equation is given by $C_2 : x^2 + y^2 = 4$ . Based on the given information, answer the following questions :
Q.38 (i)1 mark $C_1$
Q.38 (ii)1 mark $x$
Q.38 (iii) (a)2 marks $Using integration find the area of region covered by the roundabout.$
OR
Q.38 (iii) (b)2 marks $Using integration, find the area of region covered by circular pond.$

Section A

1 mark each

Q.11 mark $\int \frac{3ax}{b^2 + c^2x^2}\,dx = A \log \left| b^2 + c^2x^2 \right| + K$
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Q.21 mark $\int_{-1}^{1} \frac{x^3}{x^2 + 2\left| x \right| + 1}\,dx$
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Q.31 mark $y = x\left| x \right|$
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Q.41 mark $R \frac{dx}{dy} + Px = Q$
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Q.51 mark $\frac{d}{dx} (\sin y) = y^2$
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Q.61 mark $\hat{i} + 2\hat{j} + 3\hat{k}$
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Q.71 mark $-\hat{i} - \hat{j} + 2\hat{k}$
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Q.81 mark $\left| \vec{a} \right| = 8$
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Q.91 mark $\frac{x}{0} = \frac{y}{1} = \frac{z}{0}$
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Q.101 mark $Z = 5x + 7y$
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Q.111 mark $The degree of an objective function of a linear programming problem is$
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Q.121 mark $\tan^{-1} x = 3y$
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Q.131 mark $Which of the following cannot be an order of a column-matrix ?$
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Q.141 mark $(A + B)' = A' + B'$
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Q.151 mark $\Delta_1 = \begin{vmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{vmatrix}$
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Q.161 mark $x$
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Q.171 mark $If A and B are symmetric matrices of same order, then which of the following matrices is a skew-symmetric matrix ?$
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Q.181 mark $f(x) = x^2 + 1$
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Direction : Questions number 19 and 20 are Assertion and Reason based questions carrying 1 mark each. Two statements are given, one labelled Assertion (A) and other labelled Reason (R). Select the correct answer from the codes (A), (B), (C) and (D) as given below.
Q.191 mark $x = py + q$
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Q.201 mark $\frac{2}{3}$
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Section B

2 marks each

Q.212 marks $\vec{a}$
Q.222 marks $\vec{a} = 3\hat{i} - 2\hat{j} + 2\hat{k}$
Q.23 (a)2 marks $\tan^{-1} \left( \frac{\cos 2x - \sin 2x}{\cos 2x + \sin 2x} \right)$
OR
Q.23 (b)2 marks $\tan \left( \sin^{-1} 1 - \cos^{-1} \left( -\frac{1}{2} \right) \right)$
Q.24 (a)2 marks $f(x) = \begin{cases} \frac{\left| x - 3 \right|}{2(x - 3)}, & x < 3 \\ \frac{x - 6}{6}, & x \geq 3 \end{cases}$
OR
Q.24 (b)2 marks $\sqrt{3}(x^2 + y^2) = 4xy$
Q.25 (a)2 marks $\cot^{-1} \sqrt{\frac{1 + \cos 2x}{1 - \cos 2x}}$
OR
Q.25 (b)2 marks $\sin \left( \tan^{-1} \left( -\sqrt{3} \right) - \sec^{-1} 2 \right)$

Section C

3 marks each

Q.263 marks $I_1 = \int_{-\pi/4}^{\pi/4} \frac{dx}{1 + \cos 2x}$
Q.27 (a)3 marks $y^2\,dx + (x^2 - xy + y^2)\,dy = 0$
OR
Q.27 (b)3 marks $\frac{dy}{dx} = y \tan x$
Q.283 marks $Z = 13x - 15y$
Q.29 (a)3 marks $Out of two bags, bag I contains 3 red and 4 white balls and bag II contains 8 red and 6 white balls. A die is thrown. If it shows a number less than 3 then a ball is drawn at random from bag I, otherwise a ball is drawn at random from bag II. Find the probability that the ball drawn from one of the bags is a red ball.$
OR
Q.29 (b)3 marks $P(X') + P(Y') = 2 - 2a + b$
Q.303 marks $\int_{0}^{1} x \sin^{-1} x \, dx$
Q.31 (a)3 marks $\int \sqrt{\frac{x + 2}{x - 2}} \, dx$
OR
Q.31 (b)3 marks $\int \frac{x^2}{(x^2 + 9) \, (x^2 + 16)} \, dx$

Section D

5 marks each

Q.325 marks $x = 3 \sin t - \sin 3t$
Q.335 marks $Prove that the line through points A(0, -1, -1) and B(4, 5, 1) intersects the line through points C(3, 9, 4) and D(-4, 4, 4). Hence, write the equation of line passing through the point of intersection of lines AB and CD as well as origin.$
Q.34 (a)5 marks $R = \{(x, y) : \left| x - y \right| \text{ is divisible by a prime number 'p'}, x, y \in Z\}$
OR
Q.34 (b)5 marks $f : R - \left\{ \frac{3}{5} \right\} \longrightarrow R - \left\{ \frac{3}{5} \right\}$
Q.35 (a)5 marks $A = \begin{bmatrix} 0 & 2 & 1 \\ -2 & -1 & -2 \\ 1 & -1 & 0 \end{bmatrix}$
OR
Q.35 (b)5 marks $\begin{bmatrix} 3 & -1 & \sin 3x \\ -7 & 4 & \cos 2x \\ -11 & 7 & 2 \end{bmatrix}$

Section E

In an online jackpot, there is one first prize of ₹ 3,00,000, two second prizes of ₹ 2,00,000 each and three third prizes of ₹ 50,000 each. A total of 1,00,000 jackpot tickets each costing ₹ 100 were sold there by raising a fund of ₹ 1,00,00,000. Rohan bought one ticket. Based on given information, answer the following questions :
Q.36 (i)1 mark $What are the possible amounts, the person can win ?$
Q.36 (ii) (a)2 marks $What is the probability that the person wins atleast ₹ 2,00,000 ?$
OR
Q.36 (ii) (b)2 marks $What is the probability that the person does not win any amount ?$
Q.36 (iii)1 mark $In another jackpot, Rohan also bought a ticket having a prize money of ₹ 5,00,000. The chances of winning the jackpot are 1 in 1,00,000. Find the probability that on exactly one of tickets he wins the jackpot.$
Roundabouts are often made on busy roads to ease the traffic and avoid red lights. One such round-about is made such that equation representing its boundary is given by $C_1$ ; $x^2 + y^2 = 64$ . There is a circular pond with a fountain in the middle of the roundabout whose equation is given by $C_2 : x^2 + y^2 = 4$ . Based on the given information, answer the following questions :
Q.37 (i)1 mark $C_1$
Q.37 (ii)1 mark $x$
Q.37 (iii) (a)2 marks $Using integration find the area of region covered by the roundabout.$
OR
Q.37 (iii) (b)2 marks $Using integration, find the area of region covered by circular pond.$
An online delivery company in a city has 5000 subscribers and collects annual subscription fees of ₹ 300 per subscriber for unlimited free deliveries. The company wishes to increase the annual subscription fee. It is predicted that, for every increase of ₹ 1, ten subscribers will discontinue. Assume that the company increased the annual fee by ₹ $x$ . Based on the given information, answer the following questions :
Q.38 (i)1 mark $x$
Q.38 (ii)1 mark $R(x)$
Q.38 (iii) (a)2 marks $x$
OR
Q.38 (iii) (b)2 marks $R(x)$

Section A

1 mark each

Q.11 mark $\left| \vec{a} \right| = 8$
Tap an option to check your answer.
Q.21 mark $\frac{x}{0} = \frac{y}{0} = \frac{z}{1}$
Tap an option to check your answer.
Q.31 mark $Z = 5x + 7y$
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Q.41 mark $The degree of an objective function of a linear programming problem is$
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Q.51 mark $\sin^{-1} x + \pi = y$
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Q.61 mark $A = [a_{ij}]_{3 \times 3}$
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Q.71 mark $(A + B)' = A' + B'$
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Q.81 mark $\Delta_1 = \begin{vmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{vmatrix}$
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Q.91 mark $x$
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Q.101 mark $(AB - BA)$
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Q.111 mark $f(x) = e^{-x}$
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Q.121 mark $\int \frac{3ax}{b^2 + c^2x^2}\,dx = A \log \left| b^2 + c^2x^2 \right| + K$
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Q.131 mark $\int_{-1}^{1} \frac{x^3}{x^2 + 2\left| x \right| + 1}\,dx$
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Q.141 mark $y = x\left| x \right|$
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Q.151 mark $R \frac{dx}{dy} + Px = Q$
Tap an option to check your answer.
Q.161 mark $\frac{d}{dx}(y')^3 + (y')^3 = 1$
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Q.171 mark $\hat{i} + 2\hat{j} + 3\hat{k}$
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Q.181 mark $-\hat{i} - \hat{j} + 2\hat{k}$
Tap an option to check your answer.
Direction : Questions number 19 and 20 are Assertion and Reason based questions carrying 1 mark each. Two statements are given, one labelled Assertion (A) and other labelled Reason (R). Select the correct answer from the codes (A), (B), (C) and (D) as given below.
Q.191 mark $\frac{2}{3}$
Tap an option to check your answer.
Q.201 mark $x = py + q$
Tap an option to check your answer.

Section B

2 marks each

Q.21 (a)2 marks $\tan^{-1} \left( \frac{\cos 2x - \sin 2x}{\cos 2x + \sin 2x} \right)$
OR
Q.21 (b)2 marks $\tan \left( \sin^{-1} 1 - \cos^{-1} \left( -\frac{1}{2} \right) \right)$
Q.22 (a)2 marks $f(x) = \begin{cases} \frac{\left| x - 3 \right|}{2(x - 3)}, & x < 3 \\ \frac{x - 6}{6}, & x \geq 3 \end{cases}$
OR
Q.22 (b)2 marks $\sqrt{3}(x^2 + y^2) = 4xy$
Q.23 (a)2 marks $\sin^{-1} \sqrt{\frac{1 + \cos 2x}{2}}$
OR
Q.23 (b)2 marks $\cos[\sin^{-1}(-1) - \tan^{-1}(-\sqrt{3})]$
Q.242 marks $\Delta ABC$
Q.252 marks $\vec{a} = 3\hat{i} - 2\hat{j} + 2\hat{k}$

Section C

3 marks each

Q.263 marks $\int_{0}^{1} \log(1 + x^2)\,dx$
Q.27 (a)3 marks $Out of two bags, bag I contains 3 red and 4 white balls and bag II contains 8 red and 6 white balls. A die is thrown. If it shows a number less than 3 then a ball is drawn at random from bag I, otherwise a ball is drawn at random from bag II. Find the probability that the ball drawn from one of the bags is a red ball.$
OR
Q.27 (b)3 marks $P(X') + P(Y') = 2 - 2a + b$
Q.28 (a)3 marks $\int \sqrt{\frac{x + 2}{x - 2}} \, dx$
OR
Q.28 (b)3 marks $\int \frac{x^2}{(x^2 + 9) \, (x^2 + 16)} \, dx$
Q.293 marks $I_1 = \int_{-\pi/4}^{\pi/4} \frac{dx}{1 + \cos 2x}$
Q.30 (a)3 marks $(y^2 - x^2)dx = 2xy \, dy$
OR
Q.30 (b)3 marks $(1 + e^{2x})dy + (1 + y^2)e^x\,dx = 0$
Q.313 marks $Z = 13x - 15y$

Section D

5 marks each

Q.325 marks $Show that line AB passing through points A(0, 4, 1), B(2, 3, -1) and the line CD passing through points C(4, 5, 0), D(2, 6, 2) are parallel. Also, find distance between them.$
Q.33 (a)5 marks $R = \{(x, y) : \left| x - y \right| \text{ is divisible by a prime number 'p'}, x, y \in Z\}$
OR
Q.33 (b)5 marks $f : R - \left\{ \frac{3}{5} \right\} \longrightarrow R - \left\{ \frac{3}{5} \right\}$
Q.34 (a)5 marks $A = \begin{bmatrix} 0 & 2 & 1 \\ -2 & -1 & -2 \\ 1 & -1 & 0 \end{bmatrix}$
OR
Q.34 (b)5 marks $\begin{bmatrix} 3 & -1 & \sin 3x \\ -7 & 4 & \cos 2x \\ -11 & 7 & 2 \end{bmatrix}$
Q.355 marks $x = a(\sin t - t \cos t)$

Section E

Roundabouts are often made on busy roads to ease the traffic and avoid red lights. One such round-about is made such that equation representing its boundary is given by $C_1$ ; $x^2 + y^2 = 64$ . There is a circular pond with a fountain in the middle of the roundabout whose equation is given by $C_2 : x^2 + y^2 = 4$ . Based on the given information, answer the following questions :
Q.36 (i)1 mark $C_1$
Q.36 (ii)1 mark $x$
Q.36 (iii) (a)2 marks $Using integration find the area of region covered by the roundabout.$
OR
Q.36 (iii) (b)2 marks $Using integration, find the area of region covered by circular pond.$
An online delivery company in a city has 5000 subscribers and collects annual subscription fees of ₹ 300 per subscriber for unlimited free deliveries. The company wishes to increase the annual subscription fee. It is predicted that, for every increase of ₹ 1, ten subscribers will discontinue. Assume that the company increased the annual fee by ₹ $x$ . Based on the given information, answer the following questions :
Q.37 (i)1 mark $x$
Q.37 (ii)1 mark $R(x)$
Q.37 (iii) (a)2 marks $x$
OR
Q.37 (iii) (b)2 marks $R(x)$
In an online jackpot, there is one first prize of ₹ 3,00,000, two second prizes of ₹ 2,00,000 each and three third prizes of ₹ 50,000 each. A total of 1,00,000 jackpot tickets each costing ₹ 100 were sold there by raising a fund of ₹ 1,00,00,000. Rohan bought one ticket. Based on given information, answer the following questions :
Q.38 (i)1 mark $What are the possible amounts, the person can win ?$
Q.38 (ii) (a)2 marks $What is the probability that the person wins atleast ₹ 2,00,000 ?$
OR
Q.38 (ii) (b)2 marks $What is the probability that the person does not win any amount ?$
Q.38 (iii)1 mark $In another jackpot, Rohan also bought a ticket having a prize money of ₹ 5,00,000. The chances of winning the jackpot are 1 in 1,00,000. Find the probability that on exactly one of tickets he wins the jackpot.$