CBSE Class 12 Mathematics 2025 question paper (65/2)

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 $\vec{a}$
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Q.21 mark $f(x) = x^2 - 4x + 6$
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Q.31 mark $f(2a - x) = f(x)$
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Q.41 mark $A = \begin{bmatrix} 1 & 12 & 4y \\ 6x & 5 & 2x \\ 8x & 4 & 6 \end{bmatrix}$
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Q.51 mark $y = \sin^{-1} x$
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Q.61 mark $\dfrac{3\pi}{4}, \dfrac{\pi}{3}$
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Q.71 mark $P(E) > 0$
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Q.81 mark $Which of the following can be both a symmetric and skew-symmetric matrix ?$
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Q.91 mark $3\hat{i} + \hat{j} + 2\hat{k}$
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Q.101 mark $4\,AB + 3(AB + BA) - 4\,BA$
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Q.111 mark $100\,\pi$
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Q.121 mark $\vec{p}$
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Q.131 mark $x = 1 + 5\mu$
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Q.141 mark $If A denotes the set of continuous functions and B denotes set of differentiable functions, then which of the following depicts the correct relation between set A and B ?$
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Q.151 mark $y = x^2$
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Q.161 mark $Z = 5x + 7y$
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Q.171 mark $A^2 - B^2 = (A - B)(A + B)$
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Q.181 mark $\dfrac{d}{dx}\left(\dfrac{dy}{dx}\right)^3 = 0$
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ASSERTION – REASON BASED QUESTIONS Direction : Question number 19 and 20 are Assertion (A) and Reason (R) based questions. Two statements are given, one labelled Assertion (A) and other labelled Reason (R). Select the correct answer from the options (A), (B), (C) and (D) as given below :
Q.191 mark $A = \text{diag}\,[\,3\ \ 5\ \ 2\,]$
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Q.201 mark $Assertion (A) : Every point of the feasible region of a Linear Programming Problem is an optimal solution. Reason (R) : The optimal solution for a Linear Programming Problem exists only at one or more corner point(s) of the feasible region.$
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Section B

2 marks each

Q.21 (a)2 marks $\vec{a}$
OR
Q.21 (b)2 marks $\vec{\alpha}$
Q.222 marks $\displaystyle\int_0^{\pi/4} \sqrt{1 + \sin 2x}\;dx$
Q.232 marks $f(x) = \sin x - ax + b$
Q.242 marks $\vec{a}$
Q.25 (a)2 marks $x = e^{\frac{x}{y}}$
OR
Q.25 (b)2 marks $f(x) = \begin{cases} 2x - 3, & -3 \le x \le -2 \\ x + 1, & -2 < x \le 0 \end{cases}$

Section C

3 marks each

Q.26 (a)3 marks $2(y + 3) - xy \dfrac{dy}{dx} = 0$
OR
Q.26 (b)3 marks $(1 + x^2)\dfrac{dy}{dx} + 2xy = 4x^2$
Q.273 marks $\in$
Q.283 marks $Z = x - 5y$
Q.29 (a)3 marks $y = \log\left(\sqrt{x} + \dfrac{1}{\sqrt{x}}\right)^2$
OR
Q.29 (b)3 marks $x\sqrt{1 + y} + y\sqrt{1 + x} = 0$
Q.30 (a)3 marks $P(2) = \dfrac{3}{10}$
OR
Q.30 (b)3 marks $A = \{(x, y) : x + y = 9\}$
Q.313 marks $\displaystyle\int \dfrac{1}{x} \sqrt{\dfrac{x + a}{x - a}}\;dx$

Section D

5 marks each

Q.325 marks $y = 5x + 2$
Q.335 marks $\displaystyle\int \dfrac{x^2 + x + 1}{(x + 2)(x^2 + 1)}\;dx$
Q.34 (a)5 marks $\dfrac{x + 1}{2} = \dfrac{y - 1}{1} = \dfrac{z - 9}{-3}$
OR
Q.34 (b)5 marks $l : \vec{r} = 4\hat{i} + 2\hat{j} + 2\hat{k} + \lambda\,(\hat{i} - \hat{j} - \hat{k})$
Q.35 (a)5 marks $A = \begin{bmatrix} -4 & 4 & 4 \\ -7 & 1 & 3 \\ 5 & -3 & -1 \end{bmatrix}$
OR
Q.35 (b)5 marks $A = \begin{bmatrix} 1 & 2 & 0 \\ -2 & -1 & -2 \\ 0 & -1 & 1 \end{bmatrix}$

Section E

A school is organizing a debate competition with participants as speakers $S = \{S_1, S_2, S_3, S_4\}$ and these are judged by judges $J = \{J_1, J_2, J_3\}$ . Each speaker can be assigned one judge. Let R be a relation from set S to J defined as $R = \{(x, y) : \text{speaker } x \text{ is judged by judge } y,\ x \in S,\ y \in J\}$ . Based on the above, answer the following :
Q.36 (i)1 mark $How many relations can be there from S to J ?$
Q.36 (ii)1 mark $f = \{(S_1, J_1), (S_2, J_2), (S_3, J_2), (S_4, J_3)\}$
Q.36 (iii) (a)2 marks $How many one-one functions can be there from set S to set J ?$
OR
Q.36 (iii) (b)2 marks $R_1 = \{(S_1, S_2), \{S_2, S_4)\}$
Three persons viz. Amber, Bonzi and Comet are manufacturing cars which run on petrol and on battery as well. Their production share in the market is 60%, 30% and 10% respectively. Of their respective production capacities, 20%, 10% and 5% cars respectively are electric (or battery operated). Based on the above, answer the following :
Q.37 (i) (a)2 marks $What is the probability that a randomly selected car is an electric car ?$
OR
Q.37 (i) (b)2 marks $What is the probability that a randomly selected car is a petrol car ?$
Q.37 (ii)1 mark $A car is selected at random and is found to be electric. What is the probability that it was manufactured by Comet ?$
Q.37 (iii)1 mark $A car is selected at random and is found to be electric. What is the probability that it was manufactured by Amber or Bonzi ?$
A small town is analyzing the pattern of a new street light installation. The lights are set up in such a way that the intensity of light at any point $x$ metres from the start of the street can be modelled by $f(x) = e^x \sin x$ , where $x$ is in metres. Based on the above, answer the following :
Q.38 (i)2 marks $f(x)$
Q.38 (ii)2 marks $x \in [0, \pi]$

Section A

1 mark each

Q.11 mark $x$
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Q.21 mark $\dfrac{3\pi}{4}, \dfrac{\pi}{3}$
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Q.31 mark $\displaystyle\int \dfrac{dx}{\sin^2 x \cos^2 x}$
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Q.41 mark $\det(P) = \alpha$
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Q.51 mark $\sin^{-1}\left(\cos \dfrac{43\pi}{5}\right)$
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Q.61 mark $If A denotes the set of continuous functions and B denotes set of differentiable functions, then which of the following depicts the correct relation between set A and B ?$
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Q.71 mark $y = x^2$
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Q.81 mark $4\,AB + 3(AB + BA) - 4\,BA$
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Q.91 mark $\dfrac{d}{dx}\left(\dfrac{dy}{dx}\right)^3 = 0$
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Q.101 mark $f(x) = x^2 - 4x + 6$
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Q.111 mark $In the following probability distribution, the value of p is : X 0123 P(X) p p 0.3 2p$
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Q.121 mark $\overrightarrow{PQ} \times \overrightarrow{PR} = 4\hat{i} + 8\hat{j} - 8\hat{k}$
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Q.131 mark $P(E) > 0$
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Q.141 mark $Which of the following can be both a symmetric and skew-symmetric matrix ?$
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Q.151 mark $3\hat{i} + \hat{j} + 2\hat{k}$
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Q.161 mark $A^2 - B^2 = (A - B)(A + B)$
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Q.171 mark $x = 1 + 5\mu$
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Q.181 mark $Z = 5x + 7y$
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ASSERTION – REASON BASED QUESTIONS Direction : Question number 19 and 20 are Assertion (A) and Reason (R) based questions. Two statements are given, one labelled Assertion (A) and other labelled Reason (R). Select the correct answer from the options (A), (B), (C) and (D) as given below :
Q.191 mark $A = \text{diag}\,[\,3\ \ 5\ \ 2\,]$
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Q.201 mark $Assertion (A) : Every point of the feasible region of a Linear Programming Problem is an optimal solution. Reason (R) : The optimal solution for a Linear Programming Problem exists only at one or more corner point(s) of the feasible region.$
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X	0	1	2	3
P(X)	p	p	0.3	2p

Section B

2 marks each

Q.212 marks $f(x) = \sin x - ax + b$
Q.222 marks $\displaystyle\int_0^{\pi} \dfrac{\sin 2px}{\sin x}\;dx$
Q.23 (a)2 marks $x = e^{\frac{x}{y}}$
OR
Q.23 (b)2 marks $f(x) = \begin{cases} 2x - 3, & -3 \le x \le -2 \\ x + 1, & -2 < x \le 0 \end{cases}$
Q.242 marks $\vec{p} = 2\hat{i} - 3\hat{j} - \hat{k}$
Q.25 (a)2 marks $\vec{a}$
OR
Q.25 (b)2 marks $\vec{\alpha}$

Section C

3 marks each

Q.26 (a)3 marks $y = \log\left(\sqrt{x} + \dfrac{1}{\sqrt{x}}\right)^2$
OR
Q.26 (b)3 marks $x\sqrt{1 + y} + y\sqrt{1 + x} = 0$
Q.273 marks $f : N \rightarrow N$
Q.283 marks $The feasible region along with corner points for a linear programming problem are shown in the graph. Write all the constraints for the given linear programing problem.$
Q.29 (a)3 marks $2(y + 3) - xy \dfrac{dy}{dx} = 0$
OR
Q.29 (b)3 marks $(1 + x^2)\dfrac{dy}{dx} + 2xy = 4x^2$
Q.30 (a)3 marks $P(2) = \dfrac{3}{10}$
OR
Q.30 (b)3 marks $A = \{(x, y) : x + y = 9\}$
Q.313 marks $f(a - x) = f(x)$

Section D

5 marks each

Q.32 (a)5 marks $\dfrac{x + 1}{2} = \dfrac{y - 1}{1} = \dfrac{z - 9}{-3}$
OR
Q.32 (b)5 marks $l : \vec{r} = 4\hat{i} + 2\hat{j} + 2\hat{k} + \lambda\,(\hat{i} - \hat{j} - \hat{k})$
Q.335 marks $\displaystyle\int \dfrac{5x}{(x + 1)(x^2 + 9)}\;dx$
Q.34 (a)5 marks $A = \begin{bmatrix} -4 & 4 & 4 \\ -7 & 1 & 3 \\ 5 & -3 & -1 \end{bmatrix}$
OR
Q.34 (b)5 marks $A = \begin{bmatrix} 1 & 2 & 0 \\ -2 & -1 & -2 \\ 0 & -1 & 1 \end{bmatrix}$
Q.355 marks $y = 5x + 2$

Section E

Three persons viz. Amber, Bonzi and Comet are manufacturing cars which run on petrol and on battery as well. Their production share in the market is 60%, 30% and 10% respectively. Of their respective production capacities, 20%, 10% and 5% cars respectively are electric (or battery operated). Based on the above, answer the following :
Q.36 (i) (a)2 marks $What is the probability that a randomly selected car is an electric car ?$
OR
Q.36 (i) (b)2 marks $What is the probability that a randomly selected car is a petrol car ?$
Q.36 (ii)1 mark $A car is selected at random and is found to be electric. What is the probability that it was manufactured by Comet ?$
Q.36 (iii)1 mark $A car is selected at random and is found to be electric. What is the probability that it was manufactured by Amber or Bonzi ?$
A small town is analyzing the pattern of a new street light installation. The lights are set up in such a way that the intensity of light at any point $x$ metres from the start of the street can be modelled by $f(x) = e^x \sin x$ , where $x$ is in metres. Based on the above, answer the following :
Q.37 (i)2 marks $f(x)$
Q.37 (ii)2 marks $x \in [0, \pi]$
A school is organizing a debate competition with participants as speakers $S = \{S_1, S_2, S_3, S_4\}$ and these are judged by judges $J = \{J_1, J_2, J_3\}$ . Each speaker can be assigned one judge. Let R be a relation from set S to J defined as $R = \{(x, y) : \text{speaker } x \text{ is judged by judge } y,\ x \in S,\ y \in J\}$ . Based on the above, answer the following :
Q.38 (i)1 mark $How many relations can be there from S to J ?$
Q.38 (ii)1 mark $f = \{(S_1, J_1), (S_2, J_2), (S_3, J_2), (S_4, J_3)\}$
Q.38 (iii) (a)2 marks $How many one-one functions can be there from set S to set J ?$
OR
Q.38 (iii) (b)2 marks $R_1 = \{(S_1, S_2), \{S_2, S_4)\}$

Section A

1 mark each

Q.11 mark $\vec{p}$
Tap an option to check your answer.
Q.21 mark $Which of the following can be both a symmetric and skew-symmetric matrix ?$
Tap an option to check your answer.
Q.31 mark $\displaystyle\int_0^{a} x\,dx \le \dfrac{a}{2} + 6$
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Q.41 mark $(AB^T - BA^T)$
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Q.51 mark $\cos\left(\dfrac{\pi}{6} + \cot^{-1}(-\sqrt{3})\right)$
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Q.61 mark $\dfrac{d}{dx}\left(\dfrac{dy}{dx}\right)^3 = 0$
Tap an option to check your answer.
Q.71 mark $f(x) = x^2 - 4x + 6$
Tap an option to check your answer.
Q.81 mark $x = 1 + 5\mu$
Tap an option to check your answer.
Q.91 mark $y = x^2$
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Q.101 mark $P(E) > 0$
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Q.111 mark $-4$
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Q.121 mark $\vec{a} = \alpha\hat{i} + \hat{j} + 4\hat{k}$
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Q.131 mark $3\hat{i} + \hat{j} + 2\hat{k}$
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Q.141 mark $\dfrac{3\pi}{4}, \dfrac{\pi}{3}$
Tap an option to check your answer.
Q.151 mark $Z = 5x + 7y$
Tap an option to check your answer.
Q.161 mark $If A denotes the set of continuous functions and B denotes set of differentiable functions, then which of the following depicts the correct relation between set A and B ?$
Tap an option to check your answer.
Q.171 mark $4\,AB + 3(AB + BA) - 4\,BA$
Tap an option to check your answer.
Q.181 mark $A^2 - B^2 = (A - B)(A + B)$
Tap an option to check your answer.
ASSERTION – REASON BASED QUESTIONS Direction : Question number 19 and 20 are Assertion (A) and Reason (R) based questions. Two statements are given, one labelled Assertion (A) and other labelled Reason (R). Select the correct answer from the options (A), (B), (C) and (D) as given below :
Q.191 mark $Assertion (A) : Every point of the feasible region of a Linear Programming Problem is an optimal solution. Reason (R) : The optimal solution for a Linear Programming Problem exists only at one or more corner point(s) of the feasible region.$
Tap an option to check your answer.
Q.201 mark $A = \text{diag}\,[\,3\ \ 5\ \ 2\,]$
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X	$-4$	$-3$	$-2$	$-1$	$0$
P(X)	$0.1$	$0.2$	$0.3$	$0.2$	$0.2$

Section B

2 marks each

Q.212 marks $f(x) = \sin x - ax + b$
Q.222 marks $\displaystyle\int 2x^3\,e^{x^2}\,dx$
Q.23 (a)2 marks $x = e^{\frac{x}{y}}$
OR
Q.23 (b)2 marks $f(x) = \begin{cases} 2x - 3, & -3 \le x \le -2 \\ x + 1, & -2 < x \le 0 \end{cases}$
Q.242 marks $|\vec{a}| = 2$
Q.25 (a)2 marks $\vec{a}$
OR
Q.25 (b)2 marks $\vec{\alpha}$

Section C

3 marks each

Q.26 (a)3 marks $y = \log\left(\sqrt{x} + \dfrac{1}{\sqrt{x}}\right)^2$
OR
Q.26 (b)3 marks $x\sqrt{1 + y} + y\sqrt{1 + x} = 0$
Q.273 marks $\mathbb{R}$
Q.283 marks $Z = 2x + y$
Q.29 (a)3 marks $P(2) = \dfrac{3}{10}$
OR
Q.29 (b)3 marks $A = \{(x, y) : x + y = 9\}$
Q.30 (a)3 marks $2(y + 3) - xy \dfrac{dy}{dx} = 0$
OR
Q.30 (b)3 marks $(1 + x^2)\dfrac{dy}{dx} + 2xy = 4x^2$
Q.313 marks $\displaystyle\int_a^{b} x^3\,dx = 0$

Section D

5 marks each

Q.32 (a)5 marks $\dfrac{x + 1}{2} = \dfrac{y - 1}{1} = \dfrac{z - 9}{-3}$
OR
Q.32 (b)5 marks $l : \vec{r} = 4\hat{i} + 2\hat{j} + 2\hat{k} + \lambda\,(\hat{i} - \hat{j} - \hat{k})$
Q.335 marks $\displaystyle\int \left(\sqrt{\tan x} + \sqrt{\cot x}\right) dx$
Q.345 marks $y = 5x + 2$
Q.35 (a)5 marks $A = \begin{bmatrix} -4 & 4 & 4 \\ -7 & 1 & 3 \\ 5 & -3 & -1 \end{bmatrix}$
OR
Q.35 (b)5 marks $A = \begin{bmatrix} 1 & 2 & 0 \\ -2 & -1 & -2 \\ 0 & -1 & 1 \end{bmatrix}$

Section E

A school is organizing a debate competition with participants as speakers $S = \{S_1, S_2, S_3, S_4\}$ and these are judged by judges $J = \{J_1, J_2, J_3\}$ . Each speaker can be assigned one judge. Let R be a relation from set S to J defined as $R = \{(x, y) : \text{speaker } x \text{ is judged by judge } y,\ x \in S,\ y \in J\}$ . Based on the above, answer the following :
Q.36 (i)1 mark $How many relations can be there from S to J ?$
Q.36 (ii)1 mark $f = \{(S_1, J_1), (S_2, J_2), (S_3, J_2), (S_4, J_3)\}$
Q.36 (iii) (a)2 marks $How many one-one functions can be there from set S to set J ?$
OR
Q.36 (iii) (b)2 marks $R_1 = \{(S_1, S_2), \{S_2, S_4)\}$
Three persons viz. Amber, Bonzi and Comet are manufacturing cars which run on petrol and on battery as well. Their production share in the market is 60%, 30% and 10% respectively. Of their respective production capacities, 20%, 10% and 5% cars respectively are electric (or battery operated). Based on the above, answer the following :
Q.37 (i) (a)2 marks $What is the probability that a randomly selected car is an electric car ?$
OR
Q.37 (i) (b)2 marks $What is the probability that a randomly selected car is a petrol car ?$
Q.37 (ii)1 mark $A car is selected at random and is found to be electric. What is the probability that it was manufactured by Comet ?$
Q.37 (iii)1 mark $A car is selected at random and is found to be electric. What is the probability that it was manufactured by Amber or Bonzi ?$
A small town is analyzing the pattern of a new street light installation. The lights are set up in such a way that the intensity of light at any point $x$ metres from the start of the street can be modelled by $f(x) = e^x \sin x$ , where $x$ is in metres. Based on the above, answer the following :
Q.38 (i)2 marks $f(x)$
Q.38 (ii)2 marks $x \in [0, \pi]$