CBSE Class 12 Mathematics 2025 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 $A = \begin{bmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$
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Q.21 mark $\vec{a} = 3\hat{i} + 2\hat{j} - \hat{k}$
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Q.31 mark $\int_{-1}^{1} \frac{|x|}{x}\,dx$
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Q.41 mark $x$
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Q.51 mark $f(x) = |x| + |x - 1|$
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Q.61 mark $If A is a square matrix of order 2 such that det (A) = 4, then det (4 adj A) is equal to :$
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Q.71 mark $P(E) = \frac{2}{3}$
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Q.81 mark $f(x) = x^3 - 3x + 2$
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Q.91 mark $A = \begin{bmatrix} 1 & -2 & -1 \\ 0 & 4 & -1 \\ -3 & 2 & 1 \end{bmatrix}$
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Q.101 mark $\int \frac{2^{\frac{1}{x}}}{x^2}\,dx = k \cdot 2^{\frac{1}{x}} + C$
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Q.111 mark $\vec{a} + \vec{b} + \vec{c} = \vec{0}$
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Q.121 mark $(x + 2y^3)\frac{dy}{dx} = 2y$
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Q.131 mark $A = \begin{bmatrix} 7 & 0 & x \\ 0 & 7 & 0 \\ 0 & 0 & y \end{bmatrix}$
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Q.141 mark $Z = 18x + 9y$
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Q.151 mark $If A and B are invertible matrices, then which of the following is not correct ?$
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Q.161 mark $Z = ax + by$
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Q.171 mark $y^2 = x$
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Q.181 mark $The graph of a trigonometric function is as shown. Which of the following will represent graph of its inverse ?$
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Direction : Question numbers 19 and 20 are Assertion (A) and Reason (R) 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 options (A), (B), (C) and (D) as given below.
Q.191 mark $f : Z \to Z$
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Q.201 mark $f(x) = \begin{cases} 3x - 8, & x \leq 5 \\ 2k, & x > 5 \end{cases}$
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Section B

2 marks each

Q.21 (a)2 marks $2^{\cos^2 x}$
OR
Q.21 (b)2 marks $\tan^{-1}(x^2 + y^2) = a^2$
Q.222 marks $\tan^{-1}\left[2\sin\left(2\cos^{-1}\frac{\sqrt{3}}{2}\right)\right]$
Q.232 marks $\vec{a} = 2\hat{i} - \hat{j} + \hat{k}$
Q.242 marks $f(x) = 5x^{\frac{3}{2}} - 3x^{\frac{5}{2}}$
Q.25 (a)2 marks $\vec{a} = 3\hat{i} + \hat{j} + 2\hat{k}$
OR
Q.25 (b)2 marks $\overrightarrow{AB}$

Section C

3 marks each

Q.263 marks $The side of an equilateral triangle is increasing at the rate of 3 cm/s. At what rate its area increasing when the side of the triangle is 15 cm ?$
Q.273 marks $Z = x + 2y$
Q.28 (a)3 marks $\int \frac{x + \sin x}{1 + \cos x}\,dx$
OR
Q.28 (b)3 marks $\int_{0}^{\frac{\pi}{4}} \frac{dx}{\cos^3 x \sqrt{2 \sin 2x}}$
Q.29 (a)3 marks $\vec{r} = (1 - \lambda)\hat{i} + (\lambda - 2)\hat{j} + (3 - 2\lambda)\hat{k}$
OR
Q.29 (b)3 marks $\vec{B} = 2\hat{i} + 8\hat{j}$
Q.30 (a)3 marks $The probability distribution for the number of students being absent in a class on a Saturday is as follows : X 0245 P(X) p 2p 3p p Where X is the number of students absent. (i) Calculate p. (ii) Calculate the mean of the number of absent students on Saturday.$
OR
Q.30 (b)3 marks $For the vacancy advertised in the newspaper, 3000 candidates submitted their applications. From the data it was revealed that two third of the total applicants were females and other were males. The selection for the job was done through a written test. The performance of the applicants indicates that the probability of a male getting a distinction in written test is 0.4 and that a female getting a distinction is 0.35. Find the probability that the candidate chosen at random will have a distinction in the written test.$
Q.313 marks $y = |x + 3|$

X	0	2	4	5
P(X)	p	2p	3p	p

Section D

5 marks each

Q.32 (a)5 marks $\sqrt{1 - x^2} + \sqrt{1 - y^2} = a(x - y)$
OR
Q.32 (b)5 marks $x = a\left(\cos\theta + \log\tan\frac{\theta}{2}\right)$
Q.335 marks $f(x) = 2x^3 - 15x^2 + 36x + 1$
Q.34 (a)5 marks $\frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3}$
OR
Q.34 (b)5 marks $\frac{x + 5}{1} = \frac{y + 3}{4} = \frac{z - 6}{-9}$
Q.355 marks $A school wants to allocate students into three clubs : Sports, Music and Drama, under following conditions : The number of students in Sports club should be equal to the sum of the number of students in Music and Drama club. The number of students in Music club should be 20 more than half the number of students in Sports club. The total number of students to be allocated in all three clubs are 180. Find the number of students allocated to different clubs, using matrix method.$

Section E

A technical company is designing a rectangular solar panel installation on a roof using 300 metres of boundary material. The design includes a partition running parallel to one of the sides dividing the area (roof) into two sections. Let the length of the side perpendicular to the partition be $x$ metres and with parallel to the partition be y metres. Based on this information, answer the following questions :
Q.36 (i)1 mark $x$
Q.36 (ii)1 mark $x$
Q.36 (iii) (a)2 marks $Find the critical points of the area function. Use second derivative test to determine critical points at the maximum area. Also, find the maximum area.$
OR
Q.36 (iii) (b)2 marks $Using first derivative test, calculate the maximum area the company can enclose with the 300 metres of boundary material, considering the parallel partition.$
A class-room teacher is keen to assess the learning of her students the concept of "relations" taught to them. She writes the following five relations each defined on the set A = {1, 2, 3} : $R_1 = \{(2, 3), (3, 2)\}$ $R_2 = \{(1, 2), (1, 3), (3, 2)\}$ $R_3 = \{(1, 2), (2, 1), (1, 1)\}$ $R_4 = \{(1, 1), (1, 2), (3, 3), (2, 2)\}$ $R_5 = \{(1, 1), (1, 2), (3, 3), (2, 2), (2, 1), (2, 3), (3, 2)\}$ The students are asked to answer the following questions about the above relations :
Q.37 (i)1 mark $Identify the relation which is reflexive, transitive but not symmetric.$
Q.37 (ii)1 mark $Identify the relation which is reflexive and symmetric but not transitive.$
Q.37 (iii) (a)2 marks $Identify the relations which are symmetric but neither reflexive nor transitive.$
OR
Q.37 (iii) (b)2 marks $R_2$
A bank offers loan to its customers on different types of interest namely, fixed rate, floating rate and variable rate. From the past data with the bank, it is known that a customer avails loan on fixed rate, floating rate or variable rate with probabilities 10%, 20% and 70% respectively. A customer after availing loan can pay the loan or default on loan repayment. The bank data suggests that the probability that a person defaults on loan after availing it at fixed rate, floating rate and variable rate is 5%, 3% and 1% respectively. Based on the above information, answer the following :
Q.38 (i)2 marks $What is the probability that a customer after availing the loan will default on the loan repayment ?$
Q.38 (ii)2 marks $A customer after availing the loan, defaults on loan repayment. What is the probability that he availed the loan at a variable rate of interest ?$

Section A

1 mark each

Q.11 mark $P(E) = \frac{2}{3}$
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Q.21 mark $\vec{\alpha} = \hat{i} - 4\hat{j} + 9\hat{k}$
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Q.31 mark $\int \frac{1 - 2\sin x}{\cos^2 x}\,dx$
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Q.41 mark $\vec{a} + \vec{b} + \vec{c} = \vec{0}$
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Q.51 mark $The graph of a trigonometric function is as shown. Which of the following will represent graph of its inverse ?$
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Q.61 mark $\det(9\,A^{-1})$
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Q.71 mark $f(x) = |x| + |x - 1|$
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Q.81 mark $x$
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Q.91 mark $m \times n$
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Q.101 mark $Z = ax + by$
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Q.111 mark $A = \begin{bmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$
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Q.121 mark $\frac{dy}{dx} + y = \frac{1 + y}{x}$
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Q.131 mark $A = [a_{ij}]$
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Q.141 mark $f(x) = x^3 - 3x + 2$
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Q.151 mark $\int \frac{2^{\frac{1}{x}}}{x^2}\,dx = k \cdot 2^{\frac{1}{x}} + C$
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Q.161 mark $If A and B are invertible matrices, then which of the following is not correct ?$
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Q.171 mark $Z = 18x + 9y$
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Q.181 mark $y^2 = x$
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Direction : Question numbers 19 and 20 are Assertion (A) and Reason (R) 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 options (A), (B), (C) and (D) as given below.
Q.191 mark $f : Z \to Z$
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Q.201 mark $f(x) = \begin{cases} 3x - 8, & x \leq 5 \\ 2k, & x > 5 \end{cases}$
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Section B

2 marks each

Q.21 (a)2 marks $\vec{a} = 3\hat{i} + \hat{j} + 2\hat{k}$
OR
Q.21 (b)2 marks $\overrightarrow{AB}$
Q.222 marks $f(x) = x^2 - 2ax + b$
Q.23 (a)2 marks $2^{\cos^2 x}$
OR
Q.23 (b)2 marks $\tan^{-1}(x^2 + y^2) = a^2$
Q.242 marks $\sin^{-1}\left(\sin \frac{3\pi}{5}\right)$
Q.252 marks $\vec{a} = 2\hat{i} - \hat{j} + \hat{k}$

Section C

3 marks each

Q.26 (a)3 marks $\vec{r} = (1 - \lambda)\hat{i} + (\lambda - 2)\hat{j} + (3 - 2\lambda)\hat{k}$
OR
Q.26 (b)3 marks $\vec{B} = 2\hat{i} + 8\hat{j}$
Q.273 marks $Z = 20x + 30y$
Q.283 marks $^2$
Q.29 (a)3 marks $The probability distribution for the number of students being absent in a class on a Saturday is as follows : X 0245 P(X) p 2p 3p p Where X is the number of students absent. (i) Calculate p. (ii) Calculate the mean of the number of absent students on Saturday.$
OR
Q.29 (b)3 marks $For the vacancy advertised in the newspaper, 3000 candidates submitted their applications. From the data it was revealed that two third of the total applicants were females and other were males. The selection for the job was done through a written test. The performance of the applicants indicates that the probability of a male getting a distinction in written test is 0.4 and that a female getting a distinction is 0.35. Find the probability that the candidate chosen at random will have a distinction in the written test.$
Q.30 (a)3 marks $\int \frac{\cos 2x}{(\sin x + \cos x)^2}\,dx$
OR
Q.30 (b)3 marks $\int_{0}^{\frac{\pi}{2}} \frac{5\sin x + 3\cos x}{\sin x + \cos x}\,dx$
Q.313 marks $y = |x + 3|$

X	0	2	4	5
P(X)	p	2p	3p	p

Section D

5 marks each

Q.32 (a)5 marks $\tan^{-1} \frac{\sqrt{1 - x^2}}{x}$
OR
Q.32 (b)5 marks $\frac{dy}{dx}$
Q.335 marks $f(x) = 2x^3 - 15x^2 + 36x + 1$
Q.345 marks $A school wants to allocate students into three clubs : Sports, Music and Drama, under following conditions : The number of students in Sports club should be equal to the sum of the number of students in Music and Drama club. The number of students in Music club should be 20 more than half the number of students in Sports club. The total number of students to be allocated in all three clubs are 180. Find the number of students allocated to different clubs, using matrix method.$
Q.35 (a)5 marks $\frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3}$
OR
Q.35 (b)5 marks $\frac{x + 5}{1} = \frac{y + 3}{4} = \frac{z - 6}{-9}$

Section E

A bank offers loan to its customers on different types of interest namely, fixed rate, floating rate and variable rate. From the past data with the bank, it is known that a customer avails loan on fixed rate, floating rate or variable rate with probabilities 10%, 20% and 70% respectively. A customer after availing loan can pay the loan or default on loan repayment. The bank data suggests that the probability that a person defaults on loan after availing it at fixed rate, floating rate and variable rate is 5%, 3% and 1% respectively. Based on the above information, answer the following :
Q.36 (i)2 marks $What is the probability that a customer after availing the loan will default on the loan repayment ?$
Q.36 (ii)2 marks $A customer after availing the loan, defaults on loan repayment. What is the probability that he availed the loan at a variable rate of interest ?$
A technical company is designing a rectangular solar panel installation on a roof using 300 metres of boundary material. The design includes a partition running parallel to one of the sides dividing the area (roof) into two sections. Let the length of the side perpendicular to the partition be $x$ metres and with parallel to the partition be y metres. Based on this information, answer the following questions :
Q.37 (i)1 mark $x$
Q.37 (ii)1 mark $x$
Q.37 (iii) (a)2 marks $Find the critical points of the area function. Use second derivative test to determine critical points at the maximum area. Also, find the maximum area.$
OR
Q.37 (iii) (b)2 marks $Using first derivative test, calculate the maximum area the company can enclose with the 300 metres of boundary material, considering the parallel partition.$
A class-room teacher is keen to assess the learning of her students the concept of "relations" taught to them. She writes the following five relations each defined on the set A = {1, 2, 3} : $R_1 = \{(2, 3), (3, 2)\}$ $R_2 = \{(1, 2), (1, 3), (3, 2)\}$ $R_3 = \{(1, 2), (2, 1), (1, 1)\}$ $R_4 = \{(1, 1), (1, 2), (3, 3), (2, 2)\}$ $R_5 = \{(1, 1), (1, 2), (3, 3), (2, 2), (2, 1), (2, 3), (3, 2)\}$ The students are asked to answer the following questions about the above relations :
Q.38 (i)1 mark $Identify the relation which is reflexive, transitive but not symmetric.$
Q.38 (ii)1 mark $Identify the relation which is reflexive and symmetric but not transitive.$
Q.38 (iii) (a)2 marks $Identify the relations which are symmetric but neither reflexive nor transitive.$
OR
Q.38 (iii) (b)2 marks $R_2$

Section A

1 mark each

Q.11 mark $Z = ax + by$
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Q.21 mark $\hat{i} - \hat{j}$
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Q.31 mark $\int_{0}^{1} \frac{e^x}{1 + x}\,dx = \alpha$
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Q.41 mark $\int \frac{2^{\frac{1}{x}}}{x^2}\,dx = k \cdot 2^{\frac{1}{x}} + C$
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Q.51 mark $A = \begin{bmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$
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Q.61 mark $\begin{bmatrix} x + y & 3y \\ 3x & x + 3 \end{bmatrix} = \begin{bmatrix} 9 & 4x + y \\ x + 6 & y \end{bmatrix}$
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Q.71 mark $P(M \cap N) = 0.5$
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Q.81 mark $x$
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Q.91 mark $\vec{a} + \vec{b} + \vec{c} = \vec{0}$
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Q.101 mark $f(x) = |x| + |x - 1|$
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Q.111 mark $A system of linear equations is represented as AX = B, where A is coefficient matrix, X is variable matrix and B is the constant matrix. Then the system of equations is$
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Q.121 mark $f(x) = x^3 - 3x + 2$
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Q.131 mark $\left[1 + \left(\frac{dy}{dx}\right)^2\right]^5 = \frac{d^2y}{dx^2}$
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Q.141 mark $The graph of a trigonometric function is as shown. Which of the following will represent graph of its inverse ?$
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Q.151 mark $Z = 18x + 9y$
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Q.161 mark $A = \begin{bmatrix} 1 & -2 & -1 \\ 0 & 4 & -1 \\ -3 & 2 & 1 \end{bmatrix}$
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Q.171 mark $If A and B are invertible matrices, then which of the following is not correct ?$
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Q.181 mark $y^2 = x$
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Direction : Question numbers 19 and 20 are Assertion (A) and Reason (R) 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 options (A), (B), (C) and (D) as given below.
Q.191 mark $f(x) = \begin{cases} 3x - 8, & x \leq 5 \\ 2k, & x > 5 \end{cases}$
Tap an option to check your answer.
Q.201 mark $f : Z \to Z$
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Section B

2 marks each

Q.212 marks $\vec{a} = 2\hat{i} - \hat{j} + \hat{k}$
Q.222 marks $f(x) = \sqrt{3} \sin x - \cos x - 2ax + b$
Q.23 (a)2 marks $\vec{a} = 3\hat{i} + \hat{j} + 2\hat{k}$
OR
Q.23 (b)2 marks $\overrightarrow{AB}$
Q.242 marks $x$
Q.25 (a)2 marks $2^{\cos^2 x}$
OR
Q.25 (b)2 marks $\tan^{-1}(x^2 + y^2) = a^2$

Section C

3 marks each

Q.263 marks $Z = 8x + 9y$
Q.27 (a)3 marks $\int \frac{2x - 1}{(x - 1)(x + 2)(x - 3)}\,dx$
OR
Q.27 (b)3 marks $\int_{0}^{5} (|x - 1| + |x - 2| + |x - 5|)\,dx$
Q.283 marks $A spherical medicine ball when dropped in water dissolves in such a way that the rate of decrease of volume at any instant is proportional to its surface area. Calculate the rate of decrease of its radius.$
Q.293 marks $y = |x + 3|$
Q.30 (a)3 marks $\vec{r} = (1 - \lambda)\hat{i} + (\lambda - 2)\hat{j} + (3 - 2\lambda)\hat{k}$
OR
Q.30 (b)3 marks $\vec{B} = 2\hat{i} + 8\hat{j}$
Q.31 (a)3 marks $The probability distribution for the number of students being absent in a class on a Saturday is as follows : X 0245 P(X) p 2p 3p p Where X is the number of students absent. (i) Calculate p. (ii) Calculate the mean of the number of absent students on Saturday.$
OR
Q.31 (b)3 marks $For the vacancy advertised in the newspaper, 3000 candidates submitted their applications. From the data it was revealed that two third of the total applicants were females and other were males. The selection for the job was done through a written test. The performance of the applicants indicates that the probability of a male getting a distinction in written test is 0.4 and that a female getting a distinction is 0.35. Find the probability that the candidate chosen at random will have a distinction in the written test.$

X	0	2	4	5
P(X)	p	2p	3p	p

Section D

5 marks each

Q.325 marks $A school wants to allocate students into three clubs : Sports, Music and Drama, under following conditions : The number of students in Sports club should be equal to the sum of the number of students in Music and Drama club. The number of students in Music club should be 20 more than half the number of students in Sports club. The total number of students to be allocated in all three clubs are 180. Find the number of students allocated to different clubs, using matrix method.$
Q.335 marks $\int \sin^{-1} \sqrt{\frac{x}{a + x}}\,dx$
Q.34 (a)5 marks $\sqrt{1 - x^2} + \sqrt{1 - y^2} = a(x - y)$
OR
Q.34 (b)5 marks $x = a\left(\cos\theta + \log\tan\frac{\theta}{2}\right)$
Q.35 (a)5 marks $\frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3}$
OR
Q.35 (b)5 marks $\frac{x + 5}{1} = \frac{y + 3}{4} = \frac{z - 6}{-9}$

Section E

A class-room teacher is keen to assess the learning of her students the concept of "relations" taught to them. She writes the following five relations each defined on the set A = {1, 2, 3} : $R_1 = \{(2, 3), (3, 2)\}$ $R_2 = \{(1, 2), (1, 3), (3, 2)\}$ $R_3 = \{(1, 2), (2, 1), (1, 1)\}$ $R_4 = \{(1, 1), (1, 2), (3, 3), (2, 2)\}$ $R_5 = \{(1, 1), (1, 2), (3, 3), (2, 2), (2, 1), (2, 3), (3, 2)\}$ The students are asked to answer the following questions about the above relations :
Q.36 (i)1 mark $Identify the relation which is reflexive, transitive but not symmetric.$
Q.36 (ii)1 mark $Identify the relation which is reflexive and symmetric but not transitive.$
Q.36 (iii) (a)2 marks $Identify the relations which are symmetric but neither reflexive nor transitive.$
OR
Q.36 (iii) (b)2 marks $R_2$
A bank offers loan to its customers on different types of interest namely, fixed rate, floating rate and variable rate. From the past data with the bank, it is known that a customer avails loan on fixed rate, floating rate or variable rate with probabilities 10%, 20% and 70% respectively. A customer after availing loan can pay the loan or default on loan repayment. The bank data suggests that the probability that a person defaults on loan after availing it at fixed rate, floating rate and variable rate is 5%, 3% and 1% respectively. Based on the above information, answer the following :
Q.37 (i)2 marks $What is the probability that a customer after availing the loan will default on the loan repayment ?$
Q.37 (ii)2 marks $A customer after availing the loan, defaults on loan repayment. What is the probability that he availed the loan at a variable rate of interest ?$
A technical company is designing a rectangular solar panel installation on a roof using 300 metres of boundary material. The design includes a partition running parallel to one of the sides dividing the area (roof) into two sections. Let the length of the side perpendicular to the partition be $x$ metres and with parallel to the partition be y metres. Based on this information, answer the following questions :
Q.38 (i)1 mark $x$
Q.38 (ii)1 mark $x$
Q.38 (iii) (a)2 marks $Find the critical points of the area function. Use second derivative test to determine critical points at the maximum area. Also, find the maximum area.$
OR
Q.38 (iii) (b)2 marks $Using first derivative test, calculate the maximum area the company can enclose with the 300 metres of boundary material, considering the parallel partition.$