CBSE Class 12 Mathematics 2025 question paper (65/7)

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 $The given graph illustrates :$
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Q.21 mark $f(x) = \cos^{-1} x + \sin x$
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Q.31 mark $3 \times 3$
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Q.41 mark $A = \begin{bmatrix} \sqrt{3} & 0 & 0 \\ 0 & \sqrt{2} & 0 \\ 0 & 0 & \sqrt{5} \end{bmatrix}$
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Q.51 mark $|A| = 3$
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Q.61 mark $|A| = 5$
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Q.71 mark $\begin{bmatrix} 2x - 1 & 3x \\ 0 & y^2 - 1 \end{bmatrix} = \begin{bmatrix} x + 3 & 12 \\ 0 & 35 \end{bmatrix}$
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Q.81 mark $f(x) = \begin{cases} 1, & \text{if } x \le 3 \\ ax + b, & \text{if } 3 < x < 5 \\ 7, & \text{if } 5 \le x \end{cases}$
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Q.91 mark $f(x) = -2x^8$
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Q.101 mark $\dfrac{5}{2}(3x + 1)$
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Q.111 mark $f : R \to R$
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Q.121 mark $\displaystyle\int \dfrac{e^{9\log x} - e^{8\log x}}{e^{6\log x} - e^{5\log x}}\, dx$
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Q.131 mark $For a function f(x), which of the following holds true ?$
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Q.141 mark $\displaystyle\int \dfrac{e^x}{\sqrt{4 - e^{2x}}}\, dx$
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Q.151 mark $3\hat{i} + 15\hat{j} + 6\hat{k}$
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Q.161 mark $|\vec{a} + \vec{b}| = |\vec{a} - \vec{b}|$
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Q.171 mark $P(A) = \dfrac{1}{7}$
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Q.181 mark $A coin is tossed and a card is selected at random from a well shuffled pack of 52 playing cards. The probability of getting head on the coin and a face card from the pack is :$
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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 $f(x) = \begin{cases} x \sin \dfrac{1}{x}, & x \ne 0 \\ 0, & x = 0 \end{cases}$
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Q.201 mark $\sec^{-1}\left(\dfrac{\sqrt{3}}{2}\right)$
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Section B

2 marks each

Q.212 marks $f : A \to B$
Q.222 marks $A = \begin{bmatrix} 2 & 3 \\ -1 & 2 \end{bmatrix}$
Q.23 (a)2 marks $\left(\dfrac{5^x}{x^5}\right)$
OR
Q.23 (b)2 marks $-2x^2 - 5xy + y^3 = 76$
Q.242 marks $Z = 5x + 4y$
Q.25 (a)2 marks $10 identical blocks are marked with '0' on two of them, '1' on three of them, '2' on four of them and '3' on one of them and put in a box. If X denotes the number written on the block, then write the probability distribution of X and calculate its mean.$
OR
Q.25 (b)2 marks $In a village of 8000 people, 3000 go out of the village to work and 4000 are women. It is noted that 30% of women go out of the village to work. What is the probability that a randomly chosen individual is either a woman or a person working outside the village ?$

Section C

3 marks each

Q.26 (a)3 marks $f : R \to R$
OR
Q.26 (b)3 marks $R = \{(x, y) : xy \text{ is a square of a natural number},\ x,\ y \in N\}$
Q.27 (a)3 marks $2x + 5y - 1 = 0$
OR
Q.27 (b)3 marks $A shopkeeper sells 50 Chemistry, 60 Physics and 35 Maths books on day I and sells 40 Chemistry, 45 Physics and 50 Maths books on day II. If the selling price for each such subject book is ₹ 150 (Chemistry), ₹ 175 (Physics) and ₹ 180 (Maths), then find his total sale in two days, using matrix method. If cost price of all the books together is ₹ 35,000, what profit did he earn after the sale of two days ?$
Q.283 marks $y = \sqrt{\log\left\{\sin\left(\dfrac{x^3}{3} - 1\right)\right\}}$
Q.293 marks $Amongst all pairs of positive integers with product as 289, find which of the two numbers add up to the least.$
Q.303 marks $Z = 18x + 10y$
Q.31 (a)3 marks $\vec{a} = \hat{i} - \hat{j} + 2\hat{k}$
OR
Q.31 (b)3 marks $\vec{r} = (2\hat{i} - \hat{j} + 3\hat{k}) + \lambda(\hat{i} - 2\hat{j} + 3\hat{k})$

Section D

5 marks each

Q.32 (a)5 marks $\displaystyle\int \dfrac{x^2 + 1}{(x^2 + 2)(2x^2 + 1)}\, dx$
OR
Q.32 (b)5 marks $\displaystyle\int_{0}^{\pi} \dfrac{x \tan x}{\sec x + \tan x}\, dx$
Q.335 marks $\dfrac{\pi}{4}$
Q.345 marks $\dfrac{dy}{dx} = \cos x - 2y$
Q.35 (a)5 marks $\dfrac{2x + 4}{6} = \dfrac{y + 1}{2} = \dfrac{-2z + 6}{-4}$
OR
Q.35 (b)5 marks $(-1, 5, 2)$

Section E

Three friends A, B and C move out from the same location O at the same time in three different directions to reach their destinations. They move out on straight paths and decide that A and B after reaching their destinations will meet up with C at his predecided destination, following straight paths from A to C and B to C in such a way that $\overrightarrow{OA} = \vec{a}$ , $\overrightarrow{OB} = \vec{b}$ and $\overrightarrow{OC} = 5\vec{a} - 2\vec{b}$ respectively. Based upon the above information, answer the following questions :
Q.36 (i)1 mark $Complete the given figure to explain their entire movement plan along the respective vectors.$
Q.36 (ii)1 mark $\overrightarrow{AC}$
Q.36 (iii) (a)2 marks $\vec{a} \cdot \vec{b} = 1$
OR
Q.36 (iii) (b)2 marks $\vec{a} = 2\hat{i} - \hat{j} + 4\hat{k}$
Camphor is a waxy, colourless solid with strong aroma that evaporates through the process of sublimation, if left in the open at room temperature. (Cylindrical-shaped Camphor tablets) A cylindrical camphor tablet whose height is equal to its radius (r) evaporates when exposed to air such that the rate of reduction of its volume is proportional to its total surface area. Thus, $\dfrac{dV}{dt} = kS$ is the differential equation, where V is the volume, S is the surface area and t is the time in hours. Based upon the above information, answer the following questions :
Q.37 (i)1 mark $Write the order and degree of the given differential equation.$
Q.37 (ii)1 mark $V = \pi r^3$
Q.37 (iii) (a)2 marks $r = 3$
OR
Q.37 (iii) (b)2 marks $r = 1$
Based upon the results of regular medical check-ups in a hospital, it was found that out of 1000 people, 700 were very healthy, 200 maintained average health and 100 had a poor health record. Let $A_1$ : People with good health, $A_2$ : People with average health, and $A_3$ : People with poor health. During a pandemic, the data expressed that the chances of people contracting the disease from category $A_1$ , $A_2$ and $A_3$ are 25%, 35% and 50%, respectively. Based upon the above information, answer the following questions :
Q.38 (i)2 marks $A person was tested randomly. What is the probability that he/she has contracted the disease ?$
Q.38 (ii)2 marks $A_2$

Section A

1 mark each

Q.11 mark $Study the given graph. It illustrates :$
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Q.21 mark $\begin{bmatrix} 2x - 1 & 3x \\ 0 & y^2 - 1 \end{bmatrix} = \begin{bmatrix} x + 3 & 12 \\ 0 & 35 \end{bmatrix}$
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Q.31 mark $|A| = 5$
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Q.41 mark $|A| = 3$
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Q.51 mark $A = \begin{bmatrix} \sqrt{3} & 0 & 0 \\ 0 & \sqrt{2} & 0 \\ 0 & 0 & \sqrt{5} \end{bmatrix}$
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Q.61 mark $3 \times 3$
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Q.71 mark $f(x) = \cos^{-1} x + \sin x$
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Q.81 mark $f(x) = \begin{cases} \dfrac{\sin^2 ax}{x^2}, & \text{if } x \ne 0 \\ 1, & \text{if } x = 0 \end{cases}$
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Q.91 mark $f : R \to R$
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Q.101 mark $a\,R\,b$
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Q.111 mark $f(x) = -2x^8$
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Q.121 mark $For a function f(x), which of the following holds true ?$
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Q.131 mark $\displaystyle\int \dfrac{e^{9\log x} - e^{8\log x}}{e^{6\log x} - e^{5\log x}}\, dx$
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Q.141 mark $\displaystyle\int \dfrac{a^x}{\sqrt{1 - a^{2x}}}\, dx$
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Q.151 mark $A coin is tossed and a card is selected at random from a well shuffled pack of 52 playing cards. The probability of getting head on the coin and a face card from the pack is :$
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Q.161 mark $3\hat{i} + 15\hat{j} + 6\hat{k}$
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Q.171 mark $P(A) = \dfrac{1}{5}$
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Q.181 mark $|\vec{a} + \vec{b}| = |\vec{a} - \vec{b}|$
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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 $f(x) = \begin{cases} x \sin \dfrac{1}{x}, & x \ne 0 \\ 0, & x = 0 \end{cases}$
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Q.201 mark $\sec^{-1}\left(\dfrac{\sqrt{3}}{2}\right)$
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Section B

2 marks each

Q.21 (a)2 marks $10 identical blocks are marked with '0' on two of them, '1' on three of them, '2' on four of them and '3' on one of them and put in a box. If X denotes the number written on the block, then write the probability distribution of X and calculate its mean.$
OR
Q.21 (b)2 marks $In a village of 8000 people, 3000 go out of the village to work and 4000 are women. It is noted that 30% of women go out of the village to work. What is the probability that a randomly chosen individual is either a woman or a person working outside the village ?$
Q.222 marks $\begin{bmatrix} 3 & -1 \\ 0 & 1 \\ 2 & -3 \end{bmatrix} A = \begin{bmatrix} 2 \\ -5 \\ -17 \end{bmatrix}$
Q.232 marks $f : A \to B$
Q.242 marks $Z = 14x - 10y$
Q.25 (a)2 marks $\left(\dfrac{5^x}{x^5}\right)$
OR
Q.25 (b)2 marks $-2x^2 - 5xy + y^3 = 76$

Section C

3 marks each

Q.26 (a)3 marks $2x + 5y - 1 = 0$
OR
Q.26 (b)3 marks $A shopkeeper sells 50 Chemistry, 60 Physics and 35 Maths books on day I and sells 40 Chemistry, 45 Physics and 50 Maths books on day II. If the selling price for each such subject book is ₹ 150 (Chemistry), ₹ 175 (Physics) and ₹ 180 (Maths), then find his total sale in two days, using matrix method. If cost price of all the books together is ₹ 35,000, what profit did he earn after the sale of two days ?$
Q.27 (a)3 marks $f : R \to R$
OR
Q.27 (b)3 marks $R = \{(x, y) : xy \text{ is a square of a natural number},\ x,\ y \in N\}$
Q.283 marks $\tan^{-1} (\sec x + \tan x)$
Q.293 marks $Find dimensions of a rectangle of perimeter 12 cm which will generate maximum volume when swept along a circular rotation keeping the shorter side fixed as the axis.$
Q.30 (a)3 marks $\vec{a} = \hat{i} - \hat{j} + 2\hat{k}$
OR
Q.30 (b)3 marks $\vec{r} = (2\hat{i} - \hat{j} + 3\hat{k}) + \lambda(\hat{i} - 2\hat{j} + 3\hat{k})$
Q.313 marks $Z = 18x + 10y$

Section D

5 marks each

Q.32 (a)5 marks $\displaystyle\int \dfrac{x}{(x - 1)(x^2 + 4)}\, dx$
OR
Q.32 (b)5 marks $\displaystyle\int_{0}^{\pi} \dfrac{x \sin x}{1 + \cos^2 x}\, dx$
Q.33 (a)5 marks $\dfrac{2x + 4}{6} = \dfrac{y + 1}{2} = \dfrac{-2z + 6}{-4}$
OR
Q.33 (b)5 marks $(-1, 5, 2)$
Q.345 marks $(x^2 + y^2)\, dx + xy\, dy = 0$
Q.355 marks $\dfrac{\pi}{4}$

Section E

Based upon the results of regular medical check-ups in a hospital, it was found that out of 1000 people, 700 were very healthy, 200 maintained average health and 100 had a poor health record. Let $A_1$ : People with good health, $A_2$ : People with average health, and $A_3$ : People with poor health. During a pandemic, the data expressed that the chances of people contracting the disease from category $A_1$ , $A_2$ and $A_3$ are 25%, 35% and 50%, respectively. Based upon the above information, answer the following questions :
Q.36 (i)2 marks $A person was tested randomly. What is the probability that he/she has contracted the disease ?$
Q.36 (ii)2 marks $A_2$
Three friends A, B and C move out from the same location O at the same time in three different directions to reach their destinations. They move out on straight paths and decide that A and B after reaching their destinations will meet up with C at his predecided destination, following straight paths from A to C and B to C in such a way that $\overrightarrow{OA} = \vec{a}$ , $\overrightarrow{OB} = \vec{b}$ and $\overrightarrow{OC} = 5\vec{a} - 2\vec{b}$ respectively. Based upon the above information, answer the following questions :
Q.37 (i)1 mark $Complete the given figure to explain their entire movement plan along the respective vectors.$
Q.37 (ii)1 mark $\overrightarrow{AC}$
Q.37 (iii) (a)2 marks $\vec{a} \cdot \vec{b} = 1$
OR
Q.37 (iii) (b)2 marks $\vec{a} = 2\hat{i} - \hat{j} + 4\hat{k}$
Camphor is a waxy, colourless solid with strong aroma that evaporates through the process of sublimation, if left in the open at room temperature. (Cylindrical-shaped Camphor tablets) A cylindrical camphor tablet whose height is equal to its radius (r) evaporates when exposed to air such that the rate of reduction of its volume is proportional to its total surface area. Thus, $\dfrac{dV}{dt} = kS$ is the differential equation, where V is the volume, S is the surface area and t is the time in hours. Based upon the above information, answer the following questions :
Q.38 (i)1 mark $Write the order and degree of the given differential equation.$
Q.38 (ii)1 mark $V = \pi r^3$
Q.38 (iii) (a)2 marks $r = 3$
OR
Q.38 (iii) (b)2 marks $r = 1$

Section A

1 mark each

Q.11 mark $The given graph illustrates :$
Tap an option to check your answer.
Q.21 mark $|A| = 5$
Tap an option to check your answer.
Q.31 mark $|A| = 3$
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Q.41 mark $3 \times 3$
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Q.51 mark $f(x) = \cos^{-1} x + \sin x$
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Q.61 mark $A = \begin{bmatrix} \sqrt{3} & 0 & 0 \\ 0 & \sqrt{2} & 0 \\ 0 & 0 & \sqrt{5} \end{bmatrix}$
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Q.71 mark $f(x) = -2x^8$
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Q.81 mark $f(x) = \begin{cases} 3ax - b, & x > 1 \\ 11, & x = 1 \\ -5ax - 2b, & x < 1 \end{cases}$
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Q.91 mark $\begin{bmatrix} 2x - 1 & 3x \\ 0 & y^2 - 1 \end{bmatrix} = \begin{bmatrix} x + 3 & 12 \\ 0 & 35 \end{bmatrix}$
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Q.101 mark $Edge of a variable cube increases at the rate of 5 cm/s. The rate at which the surface area of the cube increases when the edge is 2 cm long is :$
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Q.111 mark $\displaystyle\int \dfrac{e^{9\log x} - e^{8\log x}}{e^{6\log x} - e^{5\log x}}\, dx$
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Q.121 mark $f : R \to R$
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Q.131 mark $3\hat{i} + 15\hat{j} + 6\hat{k}$
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Q.141 mark $\displaystyle\int \dfrac{e^{-x}}{16 + 9e^{-2x}}\, dx$
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Q.151 mark $|\vec{a} + \vec{b}| = |\vec{a} - \vec{b}|$
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Q.161 mark $A coin is tossed and a card is selected at random from a well shuffled pack of 52 playing cards. The probability of getting head on the coin and a face card from the pack is :$
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Q.171 mark $P(B) = \dfrac{1}{5}$
Tap an option to check your answer.
Q.181 mark $For a function f(x), which of the following holds true ?$
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 $f(x) = \begin{cases} x \sin \dfrac{1}{x}, & x \ne 0 \\ 0, & x = 0 \end{cases}$
Tap an option to check your answer.
Q.201 mark $\sec^{-1}\left(\dfrac{\sqrt{3}}{2}\right)$
Tap an option to check your answer.

Section B

2 marks each

Q.21 (a)2 marks $\left(\dfrac{5^x}{x^5}\right)$
OR
Q.21 (b)2 marks $-2x^2 - 5xy + y^3 = 76$
Q.222 marks $A = \begin{bmatrix} 1 & 0 \\ -1 & 5 \end{bmatrix}$
Q.23 (a)2 marks $10 identical blocks are marked with '0' on two of them, '1' on three of them, '2' on four of them and '3' on one of them and put in a box. If X denotes the number written on the block, then write the probability distribution of X and calculate its mean.$
OR
Q.23 (b)2 marks $In a village of 8000 people, 3000 go out of the village to work and 4000 are women. It is noted that 30% of women go out of the village to work. What is the probability that a randomly chosen individual is either a woman or a person working outside the village ?$
Q.242 marks $Z = 5x + 3y$
Q.252 marks $f : A \to B$

Section C

3 marks each

Q.263 marks $Z = 18x + 10y$
Q.27 (a)3 marks $\vec{a} = \hat{i} - \hat{j} + 2\hat{k}$
OR
Q.27 (b)3 marks $\vec{r} = (2\hat{i} - \hat{j} + 3\hat{k}) + \lambda(\hat{i} - 2\hat{j} + 3\hat{k})$
Q.283 marks $\log (x^x + \mathrm{cosec}^2 x)$
Q.293 marks $Show that of all the rectangles with a fixed perimeter, the square has the greatest area.$
Q.30 (a)3 marks $f : R \to R$
OR
Q.30 (b)3 marks $R = \{(x, y) : xy \text{ is a square of a natural number},\ x,\ y \in N\}$
Q.31 (a)3 marks $2x + 5y - 1 = 0$
OR
Q.31 (b)3 marks $A shopkeeper sells 50 Chemistry, 60 Physics and 35 Maths books on day I and sells 40 Chemistry, 45 Physics and 50 Maths books on day II. If the selling price for each such subject book is ₹ 150 (Chemistry), ₹ 175 (Physics) and ₹ 180 (Maths), then find his total sale in two days, using matrix method. If cost price of all the books together is ₹ 35,000, what profit did he earn after the sale of two days ?$

Section D

5 marks each

Q.32 (a)5 marks $\displaystyle\int \dfrac{3x + 1}{(x - 2)^2 (x + 2)}\, dx$
OR
Q.32 (b)5 marks $\displaystyle\int_{0}^{\pi/2} \dfrac{x}{\cos x + \sin x}\, dx$
Q.33 (a)5 marks $\dfrac{2x + 4}{6} = \dfrac{y + 1}{2} = \dfrac{-2z + 6}{-4}$
OR
Q.33 (b)5 marks $(-1, 5, 2)$
Q.345 marks $(x - \sin y)\, dy + (\tan y)\, dx = 0$
Q.355 marks $\dfrac{\pi}{4}$

Section E

Camphor is a waxy, colourless solid with strong aroma that evaporates through the process of sublimation, if left in the open at room temperature. (Cylindrical-shaped Camphor tablets) A cylindrical camphor tablet whose height is equal to its radius (r) evaporates when exposed to air such that the rate of reduction of its volume is proportional to its total surface area. Thus, $\dfrac{dV}{dt} = kS$ is the differential equation, where V is the volume, S is the surface area and t is the time in hours. Based upon the above information, answer the following questions :
Q.36 (i)1 mark $Write the order and degree of the given differential equation.$
Q.36 (ii)1 mark $V = \pi r^3$
Q.36 (iii) (a)2 marks $r = 3$
OR
Q.36 (iii) (b)2 marks $r = 1$
Based upon the results of regular medical check-ups in a hospital, it was found that out of 1000 people, 700 were very healthy, 200 maintained average health and 100 had a poor health record. Let $A_1$ : People with good health, $A_2$ : People with average health, and $A_3$ : People with poor health. During a pandemic, the data expressed that the chances of people contracting the disease from category $A_1$ , $A_2$ and $A_3$ are 25%, 35% and 50%, respectively. Based upon the above information, answer the following questions :
Q.37 (i)2 marks $A person was tested randomly. What is the probability that he/she has contracted the disease ?$
Q.37 (ii)2 marks $A_2$
Three friends A, B and C move out from the same location O at the same time in three different directions to reach their destinations. They move out on straight paths and decide that A and B after reaching their destinations will meet up with C at his predecided destination, following straight paths from A to C and B to C in such a way that $\overrightarrow{OA} = \vec{a}$ , $\overrightarrow{OB} = \vec{b}$ and $\overrightarrow{OC} = 5\vec{a} - 2\vec{b}$ respectively. Based upon the above information, answer the following questions :
Q.38 (i)1 mark $Complete the given figure to explain their entire movement plan along the respective vectors.$
Q.38 (ii)1 mark $\overrightarrow{AC}$
Q.38 (iii) (a)2 marks $\vec{a} \cdot \vec{b} = 1$
OR
Q.38 (iii) (b)2 marks $\vec{a} = 2\hat{i} - \hat{j} + 4\hat{k}$