CBSE Class 12 Mathematics 2025 question paper (65/6)

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 $AB'$
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Q.21 mark $A = \begin{bmatrix} -1 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \end{bmatrix}$
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Q.31 mark $The following graph is a combination of :$
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Q.41 mark $Sum of two skew-symmetric matrices of same order is always a/an :$
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Q.51 mark $\left[ \sec^{-1}(-\sqrt{2}) - \tan^{-1}\left(\dfrac{1}{\sqrt{3}}\right) \right]$
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Q.61 mark $f(x) = \begin{cases} \dfrac{\log(1 + ax) + \log(1 - bx)}{x}, & \text{for } x \ne 0 \\ k, & \text{for } x = 0 \end{cases}$
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Q.71 mark $\tan^{-1}(x^2 - y^2) = a$
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Q.81 mark $y = a\cos(\log x) + b\sin(\log x)$
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Q.91 mark $f(x) = |x|$
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Q.101 mark $f'(x) = 3(x^2 + 2x) - \dfrac{4}{x^3} + 5$
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Q.111 mark $\displaystyle\int \dfrac{x + 5}{(x + 6)^2}\, e^x\, dx$
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Q.121 mark $-\dfrac{d^4y}{dx^4} + 2e^{dy/dx} + y^2 = 0$
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Q.131 mark $\log\left(\dfrac{dy}{dx}\right) = 3x + 4y$
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Q.141 mark $Z = x + 2y$
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Q.151 mark $Z = 2x + 5y$
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Q.161 mark $|\vec{a}| = 5$
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Q.171 mark $y^2 = x$
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Q.181 mark $A box has 4 green, 8 blue and 3 red pens. A student picks up a pen at random, checks its colour and replaces it in the box. He repeats this process 3 times. The probability that at least one pen picked was red is :$
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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 $|\vec{a} \times \vec{b}|^2 + |\vec{a} \cdot \vec{b}|^2 = 256$
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Q.201 mark $f(x) = e^x$
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Section B

2 marks each

Q.212 marks $f(x) = \sin^{-1}(-x^2)$
Q.22 (a)2 marks $\sqrt{e^{\sqrt{2x}}}$
OR
Q.22 (b)2 marks $(x)^y = (y)^x$
Q.232 marks $f(x) = \dfrac{x - 4}{x + 1}$
Q.24 (a)2 marks $\vec{a}$
OR
Q.24 (b)2 marks $\vec{r}$
Q.252 marks $\vec{r} = (\hat{i} + \hat{j} - \hat{k}) + \lambda(3\hat{i} - \hat{j})$

Section C

3 marks each

Q.263 marks $A = \begin{bmatrix} 1 \\ 4 \\ -2 \end{bmatrix}$
Q.27 (a)3 marks $y = \sin^{-1}(3x - 4x^3)$
OR
Q.27 (b)3 marks $y = \cos^{-1}\left(\dfrac{1 - x^2}{1 + x^2}\right)$
Q.28 (a)3 marks $2x + y = 41$
OR
Q.28 (b)3 marks $f : N \to N$
Q.293 marks $Z = (x + 4y)$
Q.30 (a)3 marks $P(2,\ 4,\ -1)$
OR
Q.30 (b)3 marks $3\hat{i} - \hat{j} - 2\hat{k}$
Q.313 marks $0{\cdot}03$

Section D

5 marks each

Q.32 (a)5 marks $\displaystyle\int \dfrac{x^2 + 1}{(x - 1)^2 (x + 3)}\, dx$
OR
Q.32 (b)5 marks $\displaystyle\int_{0}^{\pi/2} \dfrac{x}{\sin x + \cos x}\, dx$
Q.335 marks $y = 2 + |x + 1|$
Q.34 (a)5 marks $x^2 y\, dx - (x^3 + y^3)\, dy = 0$
OR
Q.34 (b)5 marks $(1 + x^2)\dfrac{dy}{dx} + 2xy - 4x^2 = 0$
Q.355 marks $\dfrac{x}{1} = \dfrac{1 - y}{-2} = \dfrac{2z - 4}{6}$

Section E

Three students, Neha, Rani and Sam go to a market to purchase stationery items. Neha buys 4 pens, 3 notepads and 2 erasers and pays ₹ 60. Rani buys 2 pens, 4 notepads and 6 erasers for ₹ 90. Sam pays ₹ 70 for 6 pens, 2 notepads and 3 erasers. Based upon the above information, answer the following questions :
Q.36 (i)1 mark $AX = B$
Q.36 (ii)1 mark $|A|$
Q.36 (iii) (a)2 marks $A^{-1}$
OR
Q.36 (iii) (b)2 marks $A^2 - 8I$
A ladder of fixed length ‘h’ is to be placed along the wall such that it is free to move along the height of the wall. Based upon the above information, answer the following questions :
Q.37 (i)1 mark $Express the distance (y) between the wall and foot of the ladder in terms of ‘h’ and height (x) on the wall at a certain instant. Also, write an expression in terms of h and x for the area (A) of the right triangle, as seen from the side by an observer.$
Q.37 (ii)1 mark $Find the derivative of the area (A) with respect to the height on the wall (x), and find its critical point.$
Q.37 (iii) (a)2 marks $Show that the area (A) of the right triangle is maximum at the critical point.$
OR
Q.37 (iii) (b)2 marks $If the foot of the ladder whose length is 5 m, is being pulled towards the wall such that the rate of decrease of distance (y) is 2 m/s, then at what rate is the height on the wall (x) increasing, when the foot of the ladder is 3 m away from the wall ?$
A shop selling electronic items sells smartphones of only three reputed companies A, B and C because chances of their manufacturing a defective smartphone are only 5%, 4% and 2% respectively. In his inventory he has 25% smartphones from company A, 35% smartphones from company B and 40% smartphones from company C. A person buys a smartphone from this shop.
Q.38 (i)2 marks $Find the probability that it was defective.$
Q.38 (ii)2 marks $What is the probability that this defective smartphone was manufactured by company B ?$

Section A

1 mark each

Q.11 mark $Sum of two skew-symmetric matrices of same order is always a/an :$
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Q.21 mark $A = \begin{bmatrix} 0 & -3 & 8 \\ 3 & 0 & 5 \\ -8 & -5 & 0 \end{bmatrix}$
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Q.31 mark $The graph shown below depicts :$
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Q.41 mark $AB'$
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Q.51 mark $f(x) = \begin{cases} \dfrac{\log(1 + ax) + \log(1 - bx)}{x}, & \text{for } x \ne 0 \\ k, & \text{for } x = 0 \end{cases}$
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Q.61 mark $y = a\cos(\log x) + b\sin(\log x)$
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Q.71 mark $\left[ \sec^{-1}(-\sqrt{2}) - \tan^{-1}\left(\dfrac{1}{\sqrt{3}}\right) \right]$
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Q.81 mark $\tan^{-1}(x^2 - y^2) = a$
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Q.91 mark $f(x) = x^2$
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Q.101 mark $\displaystyle\int \dfrac{x + 5}{(x + 6)^2}\, e^x\, dx$
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Q.111 mark $f'(x) = 3(x^2 + 2x) - \dfrac{4}{x^3} + 5$
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Q.121 mark $\dfrac{d^2y}{dx^2} + 4\left(\dfrac{dy}{dx}\right) = x \log\left(\dfrac{d^2y}{dx^2}\right)$
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Q.131 mark $Z = x + 2y$
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Q.141 mark $y^2 = x$
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Q.151 mark $|\vec{a}| = 5$
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Q.161 mark $\log\left(\dfrac{dy}{dx}\right) = 3x + 4y$
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Q.171 mark $Z = 2x + 5y$
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Q.181 mark $Chances that three persons A, B, and C go to the market are 30%, 60% and 50% respectively. The probability that at least one will go to the market is :$
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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 $|\vec{a} \times \vec{b}|^2 + |\vec{a} \cdot \vec{b}|^2 = 256$
Tap an option to check your answer.
Q.201 mark $f(x) = e^x$
Tap an option to check your answer.

Section B

2 marks each

Q.21 (a)2 marks $\sqrt{e^{\sqrt{2x}}}$
OR
Q.21 (b)2 marks $(x)^y = (y)^x$
Q.22 (a)2 marks $\vec{a}$
OR
Q.22 (b)2 marks $\vec{r}$
Q.232 marks $f(x) = \dfrac{2}{x} - 5$
Q.242 marks $f(x) = \sin^{-1}(-x^2)$
Q.252 marks $\lambda$

Section C

3 marks each

Q.263 marks $A = \begin{bmatrix} 1 & -1 & 0 \end{bmatrix}$
Q.273 marks $Z = (x + 4y)$
Q.28 (a)3 marks $P(2,\ 4,\ -1)$
OR
Q.28 (b)3 marks $3\hat{i} - \hat{j} - 2\hat{k}$
Q.29 (a)3 marks $y = \sin^{-1}(3x - 4x^3)$
OR
Q.29 (b)3 marks $y = \cos^{-1}\left(\dfrac{1 - x^2}{1 + x^2}\right)$
Q.30 (a)3 marks $2x + y = 41$
OR
Q.30 (b)3 marks $f : N \to N$
Q.313 marks $A coin is biased so that the head is 3 times as likely to occur as tail. If the coin is tossed three times, find the probability distribution of number of tails. Hence, find the mean of the distribution.$

Section D

5 marks each

Q.32 (a)5 marks $x^2 y\, dx - (x^3 + y^3)\, dy = 0$
OR
Q.32 (b)5 marks $(1 + x^2)\dfrac{dy}{dx} + 2xy - 4x^2 = 0$
Q.335 marks $y = -x^2$
Q.34 (a)5 marks $\displaystyle\int \dfrac{x^2 + 1}{(x - 1)^2 (x + 3)}\, dx$
OR
Q.34 (b)5 marks $\displaystyle\int_{0}^{\pi/2} \dfrac{x}{\sin x + \cos x}\, dx$
Q.355 marks $(2, -1, 5)$

Section E

A shop selling electronic items sells smartphones of only three reputed companies A, B and C because chances of their manufacturing a defective smartphone are only 5%, 4% and 2% respectively. In his inventory he has 25% smartphones from company A, 35% smartphones from company B and 40% smartphones from company C. A person buys a smartphone from this shop.
Q.36 (i)2 marks $Find the probability that it was defective.$
Q.36 (ii)2 marks $What is the probability that this defective smartphone was manufactured by company B ?$
Three students, Neha, Rani and Sam go to a market to purchase stationery items. Neha buys 4 pens, 3 notepads and 2 erasers and pays ₹ 60. Rani buys 2 pens, 4 notepads and 6 erasers for ₹ 90. Sam pays ₹ 70 for 6 pens, 2 notepads and 3 erasers. Based upon the above information, answer the following questions :
Q.37 (i)1 mark $AX = B$
Q.37 (ii)1 mark $|A|$
Q.37 (iii) (a)2 marks $A^{-1}$
OR
Q.37 (iii) (b)2 marks $A^2 - 8I$
A ladder of fixed length ‘h’ is to be placed along the wall such that it is free to move along the height of the wall. Based upon the above information, answer the following questions :
Q.38 (i)1 mark $Express the distance (y) between the wall and foot of the ladder in terms of ‘h’ and height (x) on the wall at a certain instant. Also, write an expression in terms of h and x for the area (A) of the right triangle, as seen from the side by an observer.$
Q.38 (ii)1 mark $Find the derivative of the area (A) with respect to the height on the wall (x), and find its critical point.$
Q.38 (iii) (a)2 marks $Show that the area (A) of the right triangle is maximum at the critical point.$
OR
Q.38 (iii) (b)2 marks $If the foot of the ladder whose length is 5 m, is being pulled towards the wall such that the rate of decrease of distance (y) is 2 m/s, then at what rate is the height on the wall (x) increasing, when the foot of the ladder is 3 m away from the wall ?$

Section A

1 mark each

Q.11 mark $\tan^{-1}(x^2 - y^2) = a$
Tap an option to check your answer.
Q.21 mark $A = \begin{bmatrix} 0 & 0 & -5 \\ 0 & 3 & 0 \\ 4{\cdot}3 & 0 & 0 \end{bmatrix}$
Tap an option to check your answer.
Q.31 mark $The graph shown below depicts :$
Tap an option to check your answer.
Q.41 mark $\left[ \sec^{-1}(-\sqrt{2}) - \tan^{-1}\left(\dfrac{1}{\sqrt{3}}\right) \right]$
Tap an option to check your answer.
Q.51 mark $AB'$
Tap an option to check your answer.
Q.61 mark $Sum of two skew-symmetric matrices of same order is always a/an :$
Tap an option to check your answer.
Q.71 mark $y = a\cos(\log x) + b\sin(\log x)$
Tap an option to check your answer.
Q.81 mark $f(x) = \begin{cases} \dfrac{\log(1 + ax) + \log(1 - bx)}{x}, & \text{for } x \ne 0 \\ k, & \text{for } x = 0 \end{cases}$
Tap an option to check your answer.
Q.91 mark $f(x) = x^x$
Tap an option to check your answer.
Q.101 mark $\log\left(\dfrac{dy}{dx}\right) = 3x + 4y$
Tap an option to check your answer.
Q.111 mark $Z = x + 2y$
Tap an option to check your answer.
Q.121 mark $\left[\dfrac{d^2y}{dx^2} - 1\right]^2 = \dfrac{dy}{dx}$
Tap an option to check your answer.
Q.131 mark $f'(x) = 3(x^2 + 2x) - \dfrac{4}{x^3} + 5$
Tap an option to check your answer.
Q.141 mark $Z = 2x + 5y$
Tap an option to check your answer.
Q.151 mark $y^2 = x$
Tap an option to check your answer.
Q.161 mark $\displaystyle\int \dfrac{x + 5}{(x + 6)^2}\, e^x\, dx$
Tap an option to check your answer.
Q.171 mark $|\vec{a}| = 5$
Tap an option to check your answer.
Q.181 mark $0{\cdot}10$
Tap an option to check your answer.
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 $|\vec{a} \times \vec{b}|^2 + |\vec{a} \cdot \vec{b}|^2 = 256$
Tap an option to check your answer.
Q.201 mark $f(x) = e^x$
Tap an option to check your answer.

Section B

2 marks each

Q.21 (a)2 marks $\vec{a}$
OR
Q.21 (b)2 marks $\vec{r}$
Q.222 marks $f(x) = \sin^{-1}(-x^2)$
Q.232 marks $f(x) = x + \dfrac{1}{x}$
Q.24 (a)2 marks $\sqrt{e^{\sqrt{2x}}}$
OR
Q.24 (b)2 marks $(x)^y = (y)^x$
Q.252 marks $l_1 : \dfrac{x - 5}{2} = \dfrac{y + 3}{1} = \dfrac{z - 1}{-3}$

Section C

3 marks each

Q.263 marks $\begin{bmatrix} 1 & x & 1 \end{bmatrix} \begin{bmatrix} 1 & 3 & 2 \\ 2 & 5 & 1 \\ 15 & 3 & 2 \end{bmatrix} \begin{bmatrix} 1 \\ 2 \\ x \end{bmatrix} = O.$
Q.27 (a)3 marks $P(2,\ 4,\ -1)$
OR
Q.27 (b)3 marks $3\hat{i} - \hat{j} - 2\hat{k}$
Q.283 marks $Z = (x + 4y)$
Q.29 (a)3 marks $2x + y = 41$
OR
Q.29 (b)3 marks $f : N \to N$
Q.30 (a)3 marks $y = \sin^{-1}(3x - 4x^3)$
OR
Q.30 (b)3 marks $y = \cos^{-1}\left(\dfrac{1 - x^2}{1 + x^2}\right)$
Q.313 marks $Bag I contains 4 white and 5 black balls. Bag II contains 6 white and 7 black balls. A ball drawn randomly by from bag I is transferred to bag II and then a ball is drawn randomly from bag II. Find the probability that the ball drawn is white.$

Section D

5 marks each

Q.32 (a)5 marks $x^2 y\, dx - (x^3 + y^3)\, dy = 0$
OR
Q.32 (b)5 marks $(1 + x^2)\dfrac{dy}{dx} + 2xy - 4x^2 = 0$
Q.335 marks $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$
Q.34 (a)5 marks $\displaystyle\int \dfrac{x^2 + 1}{(x - 1)^2 (x + 3)}\, dx$
OR
Q.34 (b)5 marks $\displaystyle\int_{0}^{\pi/2} \dfrac{x}{\sin x + \cos x}\, dx$
Q.355 marks $A(0,\ -1,\ -1)$

Section E

A ladder of fixed length ‘h’ is to be placed along the wall such that it is free to move along the height of the wall. Based upon the above information, answer the following questions :
Q.36 (i)1 mark $Express the distance (y) between the wall and foot of the ladder in terms of ‘h’ and height (x) on the wall at a certain instant. Also, write an expression in terms of h and x for the area (A) of the right triangle, as seen from the side by an observer.$
Q.36 (ii)1 mark $Find the derivative of the area (A) with respect to the height on the wall (x), and find its critical point.$
Q.36 (iii) (a)2 marks $Show that the area (A) of the right triangle is maximum at the critical point.$
OR
Q.36 (iii) (b)2 marks $If the foot of the ladder whose length is 5 m, is being pulled towards the wall such that the rate of decrease of distance (y) is 2 m/s, then at what rate is the height on the wall (x) increasing, when the foot of the ladder is 3 m away from the wall ?$
A shop selling electronic items sells smartphones of only three reputed companies A, B and C because chances of their manufacturing a defective smartphone are only 5%, 4% and 2% respectively. In his inventory he has 25% smartphones from company A, 35% smartphones from company B and 40% smartphones from company C. A person buys a smartphone from this shop.
Q.37 (i)2 marks $Find the probability that it was defective.$
Q.37 (ii)2 marks $What is the probability that this defective smartphone was manufactured by company B ?$
Three students, Neha, Rani and Sam go to a market to purchase stationery items. Neha buys 4 pens, 3 notepads and 2 erasers and pays ₹ 60. Rani buys 2 pens, 4 notepads and 6 erasers for ₹ 90. Sam pays ₹ 70 for 6 pens, 2 notepads and 3 erasers. Based upon the above information, answer the following questions :
Q.38 (i)1 mark $AX = B$
Q.38 (ii)1 mark $|A|$
Q.38 (iii) (a)2 marks $A^{-1}$
OR
Q.38 (iii) (b)2 marks $A^2 - 8I$