CBSE Class 12 Mathematics 2026 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 $The following graph represents :$
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Q.21 mark $A = [a_{ij}]$
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Q.31 mark $\sec^{-1}\left(\sqrt{2}\right) + 2\operatorname{cosec}^{-1}\left(-\sqrt{2}\right)$
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Q.41 mark $If points (2, 3), (0, 4) and (p, 2) are collinear, then the value of p is :$
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Q.51 mark $e^{e^x}$
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Q.61 mark $The surface area of a sphere when its volume changes at the same rate as its radius is :$
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Q.71 mark $f(x) = \begin{cases} \dfrac{\sin x}{x} + \cos x, & x \neq 0 \\ k, & x = 0 \end{cases}$
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Q.81 mark $f(x) = [x]$
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Q.91 mark $\displaystyle\int \dfrac{dx}{\sqrt{25 - 16x^2}}$
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Q.101 mark $\displaystyle\int_0^1 \dfrac{dx}{e^x + e^{-x}} = \tan^{-1} e + k$
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Q.111 mark $y = x$
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Q.121 mark $1 + \left(\dfrac{dy}{dx}\right)^3 = \lambda \left(\dfrac{d^3y}{dx^3}\right)^2$
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Q.131 mark $Which of the following is not a Linear Differential Equation ?$
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Q.141 mark $3x + y \geq 3$
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Q.151 mark $Z = px + qy$
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Q.161 mark $(3\hat{i} - 2\hat{j} + 5\hat{k}) \times (4\hat{i} + p\hat{j} + q\hat{k}) = \vec{0}$
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Q.171 mark $-1$
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Q.181 mark $A box contains 4 red, 5 blue and 1 green marble. A child randomly takes out a marble from the box, notes down the colour and puts it back in the box. If the activity is repeated 3 times, what is the probability that at least one marble is red ?$
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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 $Assertion (A) : If A and B are two square matrices such that AB and BA are defined, then it is not necessary that AB = BA. Reason (R) : Product of two diagonal matrices of same order is commutative.$
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Q.201 mark $f : N \to N$
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Section B

2 marks each

Q.212 marks $\tan^{-1}\left(-\dfrac{1}{\sqrt{3}}\right) + \cot^{-1}\left(\dfrac{1}{\sqrt{3}}\right) + \tan^{-1}\left(\sin\left(-\dfrac{\pi}{2}\right)\right) + \tan^{-1}\left(\tan\dfrac{2\pi}{3}\right)$
Q.22 (a)2 marks $f(x) = \begin{cases} \dfrac{\cos x}{-x + \dfrac{\pi}{2}}, & x \neq \dfrac{\pi}{2} \\[2ex] 1, & x = \dfrac{\pi}{2} \end{cases}$
OR
Q.22 (b)2 marks $f(x) = \begin{cases} x - 1, & x < 2 \\ 2x - 3, & x \geq 2 \end{cases}$
Q.232 marks $f(x) = x^x$
Q.242 marks $3\hat{i} + \hat{j}$
Q.25 (a)2 marks $4\hat{i} - \hat{j} + 3\hat{k}$
OR
Q.25 (b)2 marks $\vec{a}$

Section C

3 marks each

Q.26 (a)3 marks $A = \mathbb{R} - \{3\}$
OR
Q.26 (b)3 marks $\mathbb{Z}$
Q.273 marks $A = \begin{bmatrix} 3 & 2 & 0 \\ 1 & 4 & 0 \\ 0 & 0 & 5 \end{bmatrix}$
Q.28 (a)3 marks $xy = e^{x - y}$
OR
Q.28 (b)3 marks $\tan^{-1}\left(\dfrac{\sqrt{1 + x^2} - \sqrt{1 - x^2}}{\sqrt{1 + x^2} + \sqrt{1 - x^2}}\right)$
Q.293 marks $Z = 200x + 120y$
Q.30 (a)3 marks $\dfrac{x - 2}{3} = \dfrac{1 - y}{2} = \dfrac{z - 3}{2}$
OR
Q.30 (b)3 marks $\vec{r} = (4 + \lambda)\hat{i} + (2\lambda - 1)\hat{j} - 3\lambda\hat{k}$
Q.313 marks $\dfrac{1}{3}$

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 $\displaystyle\int_0^1 \dfrac{x \tan^{-1} x}{\left(1 + x^2\right)^{3/2}}\, dx$
Q.335 marks $y = |x - 6|$
Q.34 (a)5 marks $y e^y\, dx = \left(y^3 + 2x e^y\right) dy$
OR
Q.34 (b)5 marks $\left(x^3 - 3xy^2\right) dx = \left(y^3 - 3x^2 y\right) dy$
Q.355 marks $\dfrac{x - 1}{2} = \dfrac{y - 2}{3} = \dfrac{z - 3}{4}$

Section E

At a birthday party, children are being served orange juice in conical cups, as shown in the figure. Each cup is 15 cm deep and has a radius 5 cm. The juice is being poured into this cup at a rate of $0{\cdot}1$ cm $^3$ /s. On the basis of the above information, answer the following questions :
Q.36 (i)1 mark $\alpha$
Q.36 (ii)1 mark $At what rate is the juice level in the cup rising when the juice is 6 cm deep ?$
Q.36 (iii) (a)2 marks $When the juice is 6 cm deep, then find at what rate is the upper surface area of juice increasing ?$
OR
Q.36 (iii) (b)2 marks $When the juice is 6 cm deep, then find the rate at which the wetted surface area of the cup is increasing.$
A carpenter needs to design a wooden box in the shape of a cuboid such that the sum of its length and breadth is 3 cm more than its height. Twice of its length, thrice of its breadth and its height add up to 10 cm. Its breadth added to 7 times its height is 1 cm less than 3 times its length. On the basis of the above information, answer the following questions :
Q.37 (i)1 mark $AX = B$
Q.37 (ii)1 mark $A^{-1}$
Q.37 (iii) (a)2 marks $A^{-1}$
OR
Q.37 (iii) (b)2 marks $A^2 + 7\,I$
An NGO organises a charity event in which they decide to distribute woollen caps to protect children from winter. The caps to be distributed are in three separate boxes, Box I has 30 red caps, Box II has 20 red and 10 green caps, and Box III has 30 green caps. The probability that a Box i is selected and a cap picked out is $\dfrac{i}{6}$ , where i = 1, 2, 3. Based on the above information, answer the following questions : A person selects a cap.
Q.38 (i)2 marks $What is the probability that he selects a red cap ?$
Q.38 (ii)2 marks $If he selects a green cap, what is the probability that the cap has come from Box II ?$

Section A

1 mark each

Q.11 mark $The following graph represents :$
Tap an option to check your answer.
Q.21 mark $A = [a_{ij}]$
Tap an option to check your answer.
Q.31 mark $If points (2, 3), (0, 4) and (p, 2) are collinear, then the value of p is :$
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Q.41 mark $e^{e^x}$
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Q.51 mark $\sec^{-1}\left(\sqrt{2}\right) + 2\operatorname{cosec}^{-1}\left(-\sqrt{2}\right)$
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Q.61 mark $S = 72t + 3t^2 - t^3$
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Q.71 mark $\displaystyle\int \dfrac{dx}{\sqrt{25 - 16x^2}}$
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Q.81 mark $\displaystyle\int_0^1 \dfrac{dx}{e^x + e^{-x}} = \tan^{-1} e + k$
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Q.91 mark $f(x) = \begin{cases} \dfrac{\sin x}{x} + \cos x, & x \neq 0 \\ k, & x = 0 \end{cases}$
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Q.101 mark $y = x$
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Q.111 mark $f(x) = [x]$
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Q.121 mark $4\left[\dfrac{d^2y}{dx^2}\right]^2 + 3\left[1 + \left(\dfrac{dy}{dx}\right)^2 - y\right] = 0$
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Q.131 mark $(3\hat{i} - 2\hat{j} + 5\hat{k}) \times (4\hat{i} + p\hat{j} + q\hat{k}) = \vec{0}$
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Q.141 mark $Z = px + qy$
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Q.151 mark $-1$
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Q.161 mark $3x + y \geq 3$
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Q.171 mark $Which of the following is not a Linear Differential Equation ?$
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Q.181 mark $The probability that it will rain tomorrow in cities A, B and C is 60%, 70% and 80% respectively. The probability that it will rain tomorrow in at least one of the cities is :$
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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 $Assertion (A) : If A and B are two square matrices such that AB and BA are defined, then it is not necessary that AB = BA. Reason (R) : Product of two diagonal matrices of same order is commutative.$
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Q.201 mark $f : N \to N$
Tap an option to check your answer.

Section B

2 marks each

Q.21 (a)2 marks $4\hat{i} - \hat{j} + 3\hat{k}$
OR
Q.21 (b)2 marks $\vec{a}$
Q.222 marks $\tan^{-1}\left(-\dfrac{1}{\sqrt{3}}\right) + \cot^{-1}\left(\dfrac{1}{\sqrt{3}}\right) + \tan^{-1}\left(\sin\left(-\dfrac{\pi}{2}\right)\right) + \tan^{-1}\left(\tan\dfrac{2\pi}{3}\right)$
Q.232 marks $f(x) = \dfrac{x}{\log x}$
Q.242 marks $(2\hat{i} + 3\hat{j} + \hat{k})$
Q.25 (a)2 marks $f(x) = \begin{cases} \dfrac{\cos x}{-x + \dfrac{\pi}{2}}, & x \neq \dfrac{\pi}{2} \\[2ex] 1, & x = \dfrac{\pi}{2} \end{cases}$
OR
Q.25 (b)2 marks $f(x) = \begin{cases} x - 1, & x < 2 \\ 2x - 3, & x \geq 2 \end{cases}$

Section C

3 marks each

Q.263 marks $\dfrac{1}{3}$
Q.273 marks $A = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix}$
Q.28 (a)3 marks $A = \mathbb{R} - \{3\}$
OR
Q.28 (b)3 marks $\mathbb{Z}$
Q.293 marks $Z = 20x + 10y$
Q.30 (a)3 marks $xy = e^{x - y}$
OR
Q.30 (b)3 marks $\tan^{-1}\left(\dfrac{\sqrt{1 + x^2} - \sqrt{1 - x^2}}{\sqrt{1 + x^2} + \sqrt{1 - x^2}}\right)$
Q.31 (a)3 marks $\dfrac{x - 2}{3} = \dfrac{1 - y}{2} = \dfrac{z - 3}{2}$
OR
Q.31 (b)3 marks $\vec{r} = (4 + \lambda)\hat{i} + (2\lambda - 1)\hat{j} - 3\lambda\hat{k}$

Section D

5 marks each

Q.32 (a)5 marks $y e^y\, dx = \left(y^3 + 2x e^y\right) dy$
OR
Q.32 (b)5 marks $\left(x^3 - 3xy^2\right) dx = \left(y^3 - 3x^2 y\right) dy$
Q.335 marks $y = x|x|$
Q.34 (a)5 marks $\displaystyle\int \dfrac{x}{(x - 1)(x^2 + 4)}\, dx$
OR
Q.34 (b)5 marks $\displaystyle\int_0^1 \dfrac{x \tan^{-1} x}{\left(1 + x^2\right)^{3/2}}\, dx$
Q.355 marks $\vec{r} = (\hat{i} + \hat{j} - \hat{k}) + \lambda(3\hat{i} - \hat{j})$

Section E

An NGO organises a charity event in which they decide to distribute woollen caps to protect children from winter. The caps to be distributed are in three separate boxes, Box I has 30 red caps, Box II has 20 red and 10 green caps, and Box III has 30 green caps. The probability that a Box i is selected and a cap picked out is $\dfrac{i}{6}$ , where i = 1, 2, 3. Based on the above information, answer the following questions : A person selects a cap.
Q.36 (i)2 marks $What is the probability that he selects a red cap ?$
Q.36 (ii)2 marks $If he selects a green cap, what is the probability that the cap has come from Box II ?$
At a birthday party, children are being served orange juice in conical cups, as shown in the figure. Each cup is 15 cm deep and has a radius 5 cm. The juice is being poured into this cup at a rate of $0{\cdot}1$ cm $^3$ /s. On the basis of the above information, answer the following questions :
Q.37 (i)1 mark $\alpha$
Q.37 (ii)1 mark $At what rate is the juice level in the cup rising when the juice is 6 cm deep ?$
Q.37 (iii) (a)2 marks $When the juice is 6 cm deep, then find at what rate is the upper surface area of juice increasing ?$
OR
Q.37 (iii) (b)2 marks $When the juice is 6 cm deep, then find the rate at which the wetted surface area of the cup is increasing.$
A carpenter needs to design a wooden box in the shape of a cuboid such that the sum of its length and breadth is 3 cm more than its height. Twice of its length, thrice of its breadth and its height add up to 10 cm. Its breadth added to 7 times its height is 1 cm less than 3 times its length. On the basis of the above information, answer the following questions :
Q.38 (i)1 mark $AX = B$
Q.38 (ii)1 mark $A^{-1}$
Q.38 (iii) (a)2 marks $A^{-1}$
OR
Q.38 (iii) (b)2 marks $A^2 + 7\,I$

Section A

1 mark each

Q.11 mark $The following graph represents :$
Tap an option to check your answer.
Q.21 mark $A = [a_{ij}]$
Tap an option to check your answer.
Q.31 mark $f(x) = [x]$
Tap an option to check your answer.
Q.41 mark $y = x$
Tap an option to check your answer.
Q.51 mark $f(x) = \begin{cases} \dfrac{\sin x}{x} + \cos x, & x \neq 0 \\ k, & x = 0 \end{cases}$
Tap an option to check your answer.
Q.61 mark $^3$
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Q.71 mark $\sec^{-1}\left(\sqrt{2}\right) + 2\operatorname{cosec}^{-1}\left(-\sqrt{2}\right)$
Tap an option to check your answer.
Q.81 mark $If points (2, 3), (0, 4) and (p, 2) are collinear, then the value of p is :$
Tap an option to check your answer.
Q.91 mark $e^{e^x}$
Tap an option to check your answer.
Q.101 mark $\displaystyle\int \dfrac{dx}{\sqrt{25 - 16x^2}}$
Tap an option to check your answer.
Q.111 mark $\displaystyle\int_0^1 \dfrac{dx}{e^x + e^{-x}} = \tan^{-1} e + k$
Tap an option to check your answer.
Q.121 mark $y = x\dfrac{dy}{dx} + 3\left[2 - \left(\dfrac{dy}{dx}\right)^2\right]$
Tap an option to check your answer.
Q.131 mark $Z = px + qy$
Tap an option to check your answer.
Q.141 mark $-1$
Tap an option to check your answer.
Q.151 mark $(3\hat{i} - 2\hat{j} + 5\hat{k}) \times (4\hat{i} + p\hat{j} + q\hat{k}) = \vec{0}$
Tap an option to check your answer.
Q.161 mark $Which of the following is not a Linear Differential Equation ?$
Tap an option to check your answer.
Q.171 mark $3x + y \geq 3$
Tap an option to check your answer.
Q.181 mark $\dfrac{4}{5}$
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 $Assertion (A) : If A and B are two square matrices such that AB and BA are defined, then it is not necessary that AB = BA. Reason (R) : Product of two diagonal matrices of same order is commutative.$
Tap an option to check your answer.
Q.201 mark $f : N \to N$
Tap an option to check your answer.

Section B

2 marks each

Q.21 (a)2 marks $f(x) = \begin{cases} \dfrac{\cos x}{-x + \dfrac{\pi}{2}}, & x \neq \dfrac{\pi}{2} \\[2ex] 1, & x = \dfrac{\pi}{2} \end{cases}$
OR
Q.21 (b)2 marks $f(x) = \begin{cases} x - 1, & x < 2 \\ 2x - 3, & x \geq 2 \end{cases}$
Q.22 (a)2 marks $4\hat{i} - \hat{j} + 3\hat{k}$
OR
Q.22 (b)2 marks $\vec{a}$
Q.232 marks $f(x) = \dfrac{x}{3} + \dfrac{3}{x}$
Q.242 marks $\overrightarrow{PQ} = 2\hat{i} + 3\hat{j} + \hat{k}$
Q.252 marks $\tan^{-1}\left(-\dfrac{1}{\sqrt{3}}\right) + \cot^{-1}\left(\dfrac{1}{\sqrt{3}}\right) + \tan^{-1}\left(\sin\left(-\dfrac{\pi}{2}\right)\right) + \tan^{-1}\left(\tan\dfrac{2\pi}{3}\right)$

Section C

3 marks each

Q.26 (a)3 marks $\dfrac{x - 2}{3} = \dfrac{1 - y}{2} = \dfrac{z - 3}{2}$
OR
Q.26 (b)3 marks $\vec{r} = (4 + \lambda)\hat{i} + (2\lambda - 1)\hat{j} - 3\lambda\hat{k}$
Q.273 marks $A = \begin{bmatrix} 2 & 0 & 1 \\ 2 & 1 & 3 \\ 1 & -1 & 0 \end{bmatrix}$
Q.283 marks $\dfrac{1}{3}$
Q.293 marks $Z = 20x + 10y$
Q.30 (a)3 marks $A = \mathbb{R} - \{3\}$
OR
Q.30 (b)3 marks $\mathbb{Z}$
Q.31 (a)3 marks $xy = e^{x - y}$
OR
Q.31 (b)3 marks $\tan^{-1}\left(\dfrac{\sqrt{1 + x^2} - \sqrt{1 - x^2}}{\sqrt{1 + x^2} + \sqrt{1 - x^2}}\right)$

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 $\displaystyle\int_0^1 \dfrac{x \tan^{-1} x}{\left(1 + x^2\right)^{3/2}}\, dx$
Q.335 marks $y = 5x + 4$
Q.34 (a)5 marks $y e^y\, dx = \left(y^3 + 2x e^y\right) dy$
OR
Q.34 (b)5 marks $\left(x^3 - 3xy^2\right) dx = \left(y^3 - 3x^2 y\right) dy$
Q.355 marks $\dfrac{x - 6}{3} = \dfrac{y - 7}{2} = \dfrac{z - 7}{-2}$

Section E

A carpenter needs to design a wooden box in the shape of a cuboid such that the sum of its length and breadth is 3 cm more than its height. Twice of its length, thrice of its breadth and its height add up to 10 cm. Its breadth added to 7 times its height is 1 cm less than 3 times its length. On the basis of the above information, answer the following questions :
Q.36 (i)1 mark $AX = B$
Q.36 (ii)1 mark $A^{-1}$
Q.36 (iii) (a)2 marks $A^{-1}$
OR
Q.36 (iii) (b)2 marks $A^2 + 7\,I$
An NGO organises a charity event in which they decide to distribute woollen caps to protect children from winter. The caps to be distributed are in three separate boxes, Box I has 30 red caps, Box II has 20 red and 10 green caps, and Box III has 30 green caps. The probability that a Box i is selected and a cap picked out is $\dfrac{i}{6}$ , where i = 1, 2, 3. Based on the above information, answer the following questions : A person selects a cap.
Q.37 (i)2 marks $What is the probability that he selects a red cap ?$
Q.37 (ii)2 marks $If he selects a green cap, what is the probability that the cap has come from Box II ?$
At a birthday party, children are being served orange juice in conical cups, as shown in the figure. Each cup is 15 cm deep and has a radius 5 cm. The juice is being poured into this cup at a rate of $0{\cdot}1$ cm $^3$ /s. On the basis of the above information, answer the following questions :
Q.38 (i)1 mark $\alpha$
Q.38 (ii)1 mark $At what rate is the juice level in the cup rising when the juice is 6 cm deep ?$
Q.38 (iii) (a)2 marks $When the juice is 6 cm deep, then find at what rate is the upper surface area of juice increasing ?$
OR
Q.38 (iii) (b)2 marks $When the juice is 6 cm deep, then find the rate at which the wetted surface area of the cup is increasing.$