CBSE Class 12 Mathematics 2025 question paper (65/4)

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 $\sin^{-1}\left(\sin\left(-\frac{10\pi}{3}\right)\right)$
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Q.21 mark $A^2 + B^2$
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Q.31 mark $f(x) = x^3 + 5x + 1$
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Q.41 mark $y = \sin^{-1} x$
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Q.51 mark $\lambda$
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Q.61 mark $(3, 6, -1)$
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Q.71 mark $If M and N are square matrices of order 3 such that det (M) = m and MN = mI, then det (N) is equal to :$
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Q.81 mark $f(x) = \begin{cases} 3x - 2, & 0 < x \le 1 \\ 2x^2 + ax, & 1 < x < 2 \end{cases}$
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Q.91 mark $f : N \to W$
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Q.101 mark $\begin{bmatrix} 0 & 1 & -2 \\ -1 & 0 & -7 \\ 2 & 7 & 0 \end{bmatrix}$
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Q.111 mark $\Delta$
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Q.121 mark $f(x) = \begin{cases} x, & \text{if } x \le 1 \\ 5, & \text{if } x > 1 \end{cases}$
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Q.131 mark $f(x) = 2x + \cos x$
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Q.141 mark $\int \frac{\cos 2x - \cos 2\alpha}{\cos x - \cos \alpha}\, dx$
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Q.151 mark $\int\limits_0^1 \frac{dx}{e^x + e^{-x}}$
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Q.161 mark $\left(\frac{d^2y}{dx^2}\right)^2 + \left(\frac{dy}{dx}\right)^2 = x \sin\left(\frac{dy}{dx}\right)$
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Q.171 mark $y = \sqrt{x}$
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Q.181 mark $The corner points of the feasible region of a Linear Programming Problem are (0, 2), (3, 0), (6, 0), (6, 8) and (0, 5). If Z = ax + by; (a, b > 0) be the objective function, and maximum value of Z is obtained at (0, 2) and (3, 0), then the relation between a and b is :$
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Questions number 19 and 20 are Assertion and Reason based questions. 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.
Q.191 mark $P(A \cap B) = 0$
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Q.201 mark $Assertion (A) : In a Linear Programming Problem, if the feasible region is empty, then the Linear Programming Problem has no solution. Reason (R) : A feasible region is defined as the region that satisfies all the constraints.$
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Section B

2 marks each

Q.212 marks $\det (A) = 3$
Q.22 (a)2 marks $f(x) = 2x^2 - ax + 3$
OR
Q.22 (b)2 marks $f(x) = x + \frac{1}{x}$
Q.23 (a)2 marks $\sin^{-1}\left(\frac{x}{\sqrt{1 + x^2}}\right)$
OR
Q.23 (b)2 marks $\sin^{-1}\sqrt{x - 1}$
Q.242 marks $\frac{x^2}{9} + \frac{y^2}{4} = 1$
Q.252 marks $y = 5x - 2x^3$

Section C

3 marks each

Q.26 (a)3 marks $f : R^+ \to R$
OR
Q.26 (b)3 marks $A = \{1, 2, 3\}$
Q.27 (a)3 marks $f(x) = \begin{cases} \frac{x^2 - 2x - 3}{x + 1}, & x \ne -1 \\ k, & x = -1 \end{cases}$
OR
Q.27 (b)3 marks $f(x) = x|x|$
Q.283 marks $\int\limits_{\pi/2}^{\pi} e^x \left(\frac{1 - \sin x}{1 - \cos x}\right) dx$
Q.29 (a)3 marks $Find the probability distribution of the number of boys in families having three children, assuming equal probability for a boy and a girl.$
OR
Q.29 (b)3 marks $A coin is tossed twice. Let X be a random variable defined as number of heads minus number of tails. Obtain the probability distribution of X and also find its mean.$
Q.303 marks $(-1, -5, -10)$
Q.313 marks $Z = 100x + 50y$

Section D

5 marks each

Q.325 marks $3 \times 3$
Q.335 marks $y = 4x - \frac{1}{2}x^2$
Q.34 (a)5 marks $\int \frac{\cos x}{(4 + \sin^2 x)(5 - 4\cos^2 x)}\, dx$
OR
Q.34 (b)5 marks $\int\limits_0^{\pi} \frac{dx}{a^2 \cos^2 x + b^2 \sin^2 x}$
Q.35 (a)5 marks $\vec{a}$
OR
Q.35 (b)5 marks $(1, 2, -4)$

Section E

Some students are having a misconception while comparing decimals. For example, a student may mention that $78{\cdot}56 > 78{\cdot}9$ as 7856 > 789. In order to assess this concept, a decimal comparison test was administered to the students of class VI through the following question : In the recently held Sports Day in the school, 5 students participated in a javelin throw competition. The distances to which they have thrown the javelin are shown below in the table :
Name of student Distance of javelin (in meters)
Ajay $47{\cdot}7$
Bijoy $47{\cdot}07$
Kartik $43{\cdot}09$
Dinesh $43{\cdot}9$
Devesh $45{\cdot}2$
The students were asked to identify who has thrown the javelin the farthest. Based on the test attempted by the students, the teacher concludes that 40% of the students have the misconception in the concept of decimal comparison and the rest do not have the misconception. 80% of the students having misconception answered Bijoy as the correct answer in the paper. 90% of the students who are identified with not having misconception, did not answer Bijoy as their answer. On the basis of the above information, answer the following questions :
Q.36 (i)1 mark $What is the probability of a student not having misconception but still answers Bijoy in the test ?$
Q.36 (ii)1 mark $What is the probability that a randomly selected student answers Bijoy as his answer in the test ?$
Q.36 (iii) (a)2 marks $What is the probability that a student who answered as Bijoy is having misconception ?$
OR
Q.36 (iii) (b)2 marks $What is the probability that a student who answered as Bijoy is amongst students who do not have the misconception ?$
An engineer is designing a new metro rail network in a city. Initially, two metro lines, Line A and Line B, each consisting of multiple stations are designed. The track for Line A is represented by $l_1 : \frac{x - 2}{3} = \frac{y + 1}{-2} = \frac{z - 3}{4}$ , while the track for Line B is represented by $l_2 : \frac{x - 1}{2} = \frac{y - 3}{1} = \frac{z + 2}{-3}$ . Based on the above information, answer the following questions :
Q.37 (i)1 mark $Find whether the two metro tracks are parallel.$
Q.37 (ii)1 mark $(l_1)$
Q.37 (iii) (a)2 marks $To connect the stations, a pedestrian pathway perpendicular to the two metro lines is to be constructed which passes through point (3, 2, 1). Determine the equation of the pedestrian walkway.$
OR
Q.37 (iii) (b)2 marks $Find the shortest distance between Line A and Line B.$
During a heavy gaming session, the temperature of a student's laptop processor increases significantly. After the session, the processor begins to cool down, and the rate of cooling is proportional to the difference between the processor's temperature and the room temperature $(25^\circ\text{C})$ . Initially the processor's temperature is $85^\circ\text{C}$ . The rate of cooling is defined by the equation $\frac{d}{dt}(T(t)) = -k(T(t) - 25)$ , where T(t) represents the temperature of the processor at time t (in minutes) and k is a constant. Based on the above information, answer the following questions :
Q.38 (i)2 marks $T(0) = 85^\circ\text{C}$
Q.38 (ii)2 marks $40^\circ\text{C}$

Name of student	Distance of javelin (in meters)
Ajay	$47{\cdot}7$
Bijoy	$47{\cdot}07$
Kartik	$43{\cdot}09$
Dinesh	$43{\cdot}9$
Devesh	$45{\cdot}2$

Section A

1 mark each

Q.11 mark $\cos^{-1} x$
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Q.21 mark $\lambda$
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Q.31 mark $A^2 + B^2$
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Q.41 mark $f(x) = \begin{cases} \frac{1 - \sin^3 x}{3\cos^2 x}, & \text{for } x \ne \frac{\pi}{2} \\ k, & \text{for } x = \frac{\pi}{2} \end{cases}$
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Q.51 mark $f(x) = x^3 + 5x + 1$
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Q.61 mark $\lambda, \lambda, \lambda$
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Q.71 mark $\begin{vmatrix} -1 & 2 & 4 \\ 1 & x & 1 \\ 0 & 3 & 3x \end{vmatrix} = -57$
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Q.81 mark $\begin{bmatrix} 0 & 1 & -2 \\ -1 & 0 & -7 \\ 2 & 7 & 0 \end{bmatrix}$
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Q.91 mark $f(x) = \begin{cases} 3x - 2, & 0 < x \le 1 \\ 2x^2 + ax, & 1 < x < 2 \end{cases}$
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Q.101 mark $f : N \to W$
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Q.111 mark $f(x) = 2x + \cos x$
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Q.121 mark $\Delta$
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Q.131 mark $f(x) = \begin{cases} x, & \text{if } x \le 1 \\ 5, & \text{if } x > 1 \end{cases}$
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Q.141 mark $\int e^x (\cos x - \sin x)\, dx$
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Q.151 mark $y = \sqrt{x}$
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Q.161 mark $\frac{dy}{dx} + y \tan x - \sec x = 0$
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Q.171 mark $The corner points of the feasible region of a Linear Programming Problem are (0, 2), (3, 0), (6, 0), (6, 8) and (0, 5). If Z = ax + by; (a, b > 0) be the objective function, and maximum value of Z is obtained at (0, 2) and (3, 0), then the relation between a and b is :$
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Q.181 mark $\int\limits_0^1 \frac{dx}{e^x + e^{-x}}$
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Questions number 19 and 20 are Assertion and Reason based questions. 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.
Q.191 mark $P(A \cap B) = 0$
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Q.201 mark $Assertion (A) : In a Linear Programming Problem, if the feasible region is empty, then the Linear Programming Problem has no solution. Reason (R) : A feasible region is defined as the region that satisfies all the constraints.$
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Section B

2 marks each

Q.212 marks $5x - ky = 2$
Q.22 (a)2 marks $\sin^{-1}\left(\frac{x}{\sqrt{1 + x^2}}\right)$
OR
Q.22 (b)2 marks $\sin^{-1}\sqrt{x - 1}$
Q.232 marks $\frac{x^2}{9} + \frac{y^2}{4} = 1$
Q.24 (a)2 marks $f(x) = 2x^2 - ax + 3$
OR
Q.24 (b)2 marks $f(x) = x + \frac{1}{x}$
Q.252 marks $f(x) = \frac{8}{3}x^3 - 12x^2 + 18x + 5$

Section C

3 marks each

Q.26 (a)3 marks $Find the probability distribution of the number of boys in families having three children, assuming equal probability for a boy and a girl.$
OR
Q.26 (b)3 marks $A coin is tossed twice. Let X be a random variable defined as number of heads minus number of tails. Obtain the probability distribution of X and also find its mean.$
Q.27 (a)3 marks $f : R^+ \to R$
OR
Q.27 (b)3 marks $A = \{1, 2, 3\}$
Q.283 marks $\int \frac{\cos x\, dx}{1 + \cos x + \sin x}$
Q.29 (a)3 marks $Consider the experiment of tossing a coin. If the coin shows head, toss it again; but if it shows a tail, then throw a die. Find the conditional probability of the event A : 'the die shows a number greater than 3' given that B : 'there is at least one tail'.$
OR
Q.29 (b)3 marks $2\lambda$
Q.303 marks $(-1, -5, -10)$
Q.313 marks $Z = 3x + 5y$

X	1	2	3	$2\lambda$	$3\lambda$	$4\lambda$
P(X)	$\frac{11}{30}$	$\frac{1}{15}$	$\frac{1}{10}$	$\frac{3}{10}$	$\frac{1}{15}$	$\frac{1}{10}$

Section D

5 marks each

Q.325 marks $y = 4x - \frac{1}{2}x^2$
Q.335 marks $3 \times 3$
Q.34 (a)5 marks $\int\limits_0^{\pi/4} \frac{\sin x \cos x}{\cos^4 x + \sin^4 x}\, dx$
OR
Q.34 (b)5 marks $\int \frac{\sqrt{x^2 + 1}\left[\log(x^2 + 1) - 2\log x\right]}{x^2}\, dx$
Q.35 (a)5 marks $\vec{a}$
OR
Q.35 (b)5 marks $(1, 2, -4)$

Section E

Some students are having a misconception while comparing decimals. For example, a student may mention that $78{\cdot}56 > 78{\cdot}9$ as 7856 > 789. In order to assess this concept, a decimal comparison test was administered to the students of class VI through the following question : In the recently held Sports Day in the school, 5 students participated in a javelin throw competition. The distances to which they have thrown the javelin are shown below in the table :
Name of student Distance of javelin (in meters)
Ajay $47{\cdot}7$
Bijoy $47{\cdot}07$
Kartik $43{\cdot}09$
Dinesh $43{\cdot}9$
Devesh $45{\cdot}2$
The students were asked to identify who has thrown the javelin the farthest. Based on the test attempted by the students, the teacher concludes that 40% of the students have the misconception in the concept of decimal comparison and the rest do not have the misconception. 80% of the students having misconception answered Bijoy as the correct answer in the paper. 90% of the students who are identified with not having misconception, did not answer Bijoy as their answer. On the basis of the above information, answer the following questions :
Q.36 (i)1 mark $What is the probability of a student not having misconception but still answers Bijoy in the test ?$
Q.36 (ii)1 mark $What is the probability that a randomly selected student answers Bijoy as his answer in the test ?$
Q.36 (iii) (a)2 marks $What is the probability that a student who answered as Bijoy is having misconception ?$
OR
Q.36 (iii) (b)2 marks $What is the probability that a student who answered as Bijoy is amongst students who do not have the misconception ?$
An engineer is designing a new metro rail network in a city. Initially, two metro lines, Line A and Line B, each consisting of multiple stations are designed. The track for Line A is represented by $l_1 : \frac{x - 2}{3} = \frac{y + 1}{-2} = \frac{z - 3}{4}$ , while the track for Line B is represented by $l_2 : \frac{x - 1}{2} = \frac{y - 3}{1} = \frac{z + 2}{-3}$ . Based on the above information, answer the following questions :
Q.37 (i)1 mark $Find whether the two metro tracks are parallel.$
Q.37 (ii)1 mark $(l_1)$
Q.37 (iii) (a)2 marks $To connect the stations, a pedestrian pathway perpendicular to the two metro lines is to be constructed which passes through point (3, 2, 1). Determine the equation of the pedestrian walkway.$
OR
Q.37 (iii) (b)2 marks $Find the shortest distance between Line A and Line B.$
During a heavy gaming session, the temperature of a student's laptop processor increases significantly. After the session, the processor begins to cool down, and the rate of cooling is proportional to the difference between the processor's temperature and the room temperature $(25^\circ\text{C})$ . Initially the processor's temperature is $85^\circ\text{C}$ . The rate of cooling is defined by the equation $\frac{d}{dt}(T(t)) = -k(T(t) - 25)$ , where T(t) represents the temperature of the processor at time t (in minutes) and k is a constant. Based on the above information, answer the following questions :
Q.38 (i)2 marks $T(0) = 85^\circ\text{C}$
Q.38 (ii)2 marks $40^\circ\text{C}$

Name of student	Distance of javelin (in meters)
Ajay	$47{\cdot}7$
Bijoy	$47{\cdot}07$
Kartik	$43{\cdot}09$
Dinesh	$43{\cdot}9$
Devesh	$45{\cdot}2$

Section A

1 mark each

Q.11 mark $\sin^{-1}(2x^2 - 3)$
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Q.21 mark $\begin{bmatrix} 0 & 1 & -2 \\ -1 & 0 & -7 \\ 2 & 7 & 0 \end{bmatrix}$
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Q.31 mark $f(x) = \begin{cases} 3x - 2, & 0 < x \le 1 \\ 2x^2 + ax, & 1 < x < 2 \end{cases}$
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Q.41 mark $y = \log_{2x}(\sqrt{2x})$
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Q.51 mark $f : N \to W$
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Q.61 mark $(-2, 3, 5)$
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Q.71 mark $3 \times 3$
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Q.81 mark $f(x) = x^3 + 5x + 1$
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Q.91 mark $\lambda$
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Q.101 mark $A^2 + B^2$
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Q.111 mark $y = \sqrt{x}$
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Q.121 mark $\int\limits_0^1 \frac{dx}{e^x + e^{-x}}$
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Q.131 mark $The corner points of the feasible region of a Linear Programming Problem are (0, 2), (3, 0), (6, 0), (6, 8) and (0, 5). If Z = ax + by; (a, b > 0) be the objective function, and maximum value of Z is obtained at (0, 2) and (3, 0), then the relation between a and b is :$
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Q.141 mark $\int e^{-3\log x}\, dx = f(x) + C$
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Q.151 mark $f(x) = \begin{cases} x, & \text{if } x \le 1 \\ 5, & \text{if } x > 1 \end{cases}$
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Q.161 mark $\frac{dy}{dx} = \frac{-x}{y}$
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Q.171 mark $\Delta$
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Q.181 mark $f(x) = 2x + \cos x$
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Questions number 19 and 20 are Assertion and Reason based questions. 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.
Q.191 mark $P(A \cap B) = 0$
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Q.201 mark $Assertion (A) : In a Linear Programming Problem, if the feasible region is empty, then the Linear Programming Problem has no solution. Reason (R) : A feasible region is defined as the region that satisfies all the constraints.$
Tap an option to check your answer.

Section B

2 marks each

Q.212 marks $A = \begin{bmatrix} 2 & -2 \\ -2 & 2 \end{bmatrix}$
Q.22 (a)2 marks $\sin^{-1}\left(\frac{x}{\sqrt{1 + x^2}}\right)$
OR
Q.22 (b)2 marks $\sin^{-1}\sqrt{x - 1}$
Q.232 marks $\frac{x^2}{9} + \frac{y^2}{4} = 1$
Q.24 (a)2 marks $f(x) = 2x^2 - ax + 3$
OR
Q.24 (b)2 marks $f(x) = x + \frac{1}{x}$
Q.252 marks $5\ \text{cm}^3/\text{s}$

Section C

3 marks each

Q.263 marks $\int \frac{\sqrt{x}}{1 + \sqrt{x^{3/2}}}\, dx$
Q.273 marks $(-1, -5, -10)$
Q.28 (a)3 marks $f : R^+ \to R$
OR
Q.28 (b)3 marks $A = \{1, 2, 3\}$
Q.29 (a)3 marks $\frac{p}{3}$
OR
Q.29 (b)3 marks $In a city, a survey was conducted among residents about their preferred mode of commuting. It was found that 50% people preferred using public transport, 35% preferred using a bicycle and 20% use both public transport and a bicycle. If a person is selected at random, find the probability that : (i) The person uses only public transport. (ii) The person uses a bicycle, given that they also use the public transport. (iii) The person uses neither public transport nor a bicycle.$
Q.30 (a)3 marks $f(x) = \begin{cases} \frac{x^2 - 2x - 3}{x + 1}, & x \ne -1 \\ k, & x = -1 \end{cases}$
OR
Q.30 (b)3 marks $f(x) = x|x|$
Q.313 marks $For the given graph of a Linear Programming Problem, write all the constraints satisfying the given feasible region.$

X	0	1	2	3
P(X)	p	$\frac{p}{3}$	$\frac{p}{6}$	$\frac{p}{12}$

Section D

5 marks each

Q.325 marks $y = 4x - \frac{1}{2}x^2$
Q.33 (a)5 marks $\vec{a}$
OR
Q.33 (b)5 marks $(1, 2, -4)$
Q.34 (a)5 marks $\int\limits_0^{3/2} |x \cos \pi x|\, dx$
OR
Q.34 (b)5 marks $\int \frac{dx}{\sin x + \sin 2x}$
Q.355 marks $3 \times 3$

Section E

Some students are having a misconception while comparing decimals. For example, a student may mention that $78{\cdot}56 > 78{\cdot}9$ as 7856 > 789. In order to assess this concept, a decimal comparison test was administered to the students of class VI through the following question : In the recently held Sports Day in the school, 5 students participated in a javelin throw competition. The distances to which they have thrown the javelin are shown below in the table :
Name of student Distance of javelin (in meters)
Ajay $47{\cdot}7$
Bijoy $47{\cdot}07$
Kartik $43{\cdot}09$
Dinesh $43{\cdot}9$
Devesh $45{\cdot}2$
The students were asked to identify who has thrown the javelin the farthest. Based on the test attempted by the students, the teacher concludes that 40% of the students have the misconception in the concept of decimal comparison and the rest do not have the misconception. 80% of the students having misconception answered Bijoy as the correct answer in the paper. 90% of the students who are identified with not having misconception, did not answer Bijoy as their answer. On the basis of the above information, answer the following questions :
Q.36 (i)1 mark $What is the probability of a student not having misconception but still answers Bijoy in the test ?$
Q.36 (ii)1 mark $What is the probability that a randomly selected student answers Bijoy as his answer in the test ?$
Q.36 (iii) (a)2 marks $What is the probability that a student who answered as Bijoy is having misconception ?$
OR
Q.36 (iii) (b)2 marks $What is the probability that a student who answered as Bijoy is amongst students who do not have the misconception ?$
An engineer is designing a new metro rail network in a city. Initially, two metro lines, Line A and Line B, each consisting of multiple stations are designed. The track for Line A is represented by $l_1 : \frac{x - 2}{3} = \frac{y + 1}{-2} = \frac{z - 3}{4}$ , while the track for Line B is represented by $l_2 : \frac{x - 1}{2} = \frac{y - 3}{1} = \frac{z + 2}{-3}$ . Based on the above information, answer the following questions :
Q.37 (i)1 mark $Find whether the two metro tracks are parallel.$
Q.37 (ii)1 mark $(l_1)$
Q.37 (iii) (a)2 marks $To connect the stations, a pedestrian pathway perpendicular to the two metro lines is to be constructed which passes through point (3, 2, 1). Determine the equation of the pedestrian walkway.$
OR
Q.37 (iii) (b)2 marks $Find the shortest distance between Line A and Line B.$
During a heavy gaming session, the temperature of a student's laptop processor increases significantly. After the session, the processor begins to cool down, and the rate of cooling is proportional to the difference between the processor's temperature and the room temperature $(25^\circ\text{C})$ . Initially the processor's temperature is $85^\circ\text{C}$ . The rate of cooling is defined by the equation $\frac{d}{dt}(T(t)) = -k(T(t) - 25)$ , where T(t) represents the temperature of the processor at time t (in minutes) and k is a constant. Based on the above information, answer the following questions :
Q.38 (i)2 marks $T(0) = 85^\circ\text{C}$
Q.38 (ii)2 marks $40^\circ\text{C}$