CBSE Class 12 Mathematics 2026 question paper (65/2)

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 $\{1, 2, 3\}$
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Q.21 mark $(A + B) (A - B) = A^2 - B^2$
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Q.31 mark $A = \begin{bmatrix} 1 & a & b \\ -1 & 2 & c \\ 0 & 5 & 3 \end{bmatrix}$
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Q.41 mark $A = \begin{bmatrix} \cos x & -\sin x \\ \sin x & \cos x \end{bmatrix}$
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Q.51 mark $(3A)^{-1} =$
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Q.61 mark $\begin{vmatrix} -1 & -2 & 5 \\ -2 & a & -1 \\ 0 & 4 & 2a \end{vmatrix} = -86$
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Q.71 mark $e^{-x} + e^{-y} = 2$
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Q.81 mark $f(x) = x + \dfrac{1}{x}$
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Q.91 mark $\displaystyle\int_{0}^{2a} \dfrac{1}{1 + 4x^2}\, dx = \dfrac{\pi}{6}$
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Q.101 mark $y = x^2$
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Q.111 mark $\dfrac{dy}{dx} = \dfrac{\sqrt{y}}{\sqrt{x}}$
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Q.121 mark $2x\dfrac{dy}{dx} - y = 3$
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Q.131 mark $|\vec{a}| = 5$
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Q.141 mark $x$
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Q.151 mark $\dfrac{2x - 1}{4} = \dfrac{1 - y}{3} = \dfrac{-z}{6}$
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Q.161 mark $In a linear programming problem, the linear function which has to be maximized or minimized is called$
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Q.171 mark $For the feasible region shown below, the non-trivial constraints of the linear programming problem are$
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Q.181 mark $P(A) \neq 0$
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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 codes (A), (B), (C) and (D) as given below.
Q.191 mark $\vec{a}$
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Q.201 mark $< 1, 1, 1 >$
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Section B

2 marks each

Q.21 (a)2 marks $f : R - \{3\} \to R$
OR
Q.21 (b)2 marks $f : Z \times Z \to Z \times Z$
Q.222 marks $x = a \sin^3 t$
Q.23 (a)2 marks $f(x) = \cos x + \sin^2 x$
OR
Q.23 (b)2 marks $If the volume of a solid hemisphere increases at a uniform rate, prove that its surface area varies inversely as its radius.$
Q.242 marks $\overrightarrow{AB} = \hat{j} + \hat{k}$
Q.252 marks $\vec{r} = -\hat{j} + 3\hat{k} + \lambda(2\hat{i} - 2\hat{j} + \hat{k})$

Section C

3 marks each

Q.263 marks $\displaystyle\int \dfrac{x + 2}{\sqrt{9x - x^2}}\, dx$
Q.27 (a)3 marks $\displaystyle\int_{\frac{\pi}{12}}^{\frac{5\pi}{12}} \dfrac{dx}{1 + \sqrt{\cot x}}$
OR
Q.27 (b)3 marks $\displaystyle\int_{\frac{-\pi}{6}}^{\frac{\pi}{2}} (\sin |x| + \cos |x|)\, dx$
Q.283 marks $\dfrac{d}{dx}(F(x)) = \dfrac{1}{e^x + 1}$
Q.29 (a)3 marks $x\dfrac{dy}{dx} = y - x \sin^2 \left(\dfrac{y}{x}\right)$
OR
Q.29 (b)3 marks $y \log y \dfrac{dx}{dy} + x = \dfrac{2}{y}$
Q.303 marks $Z = 10500x + 9000y$
Q.31 (a)3 marks $The probability of hitting the target by a trained sniper is three times the probability of not hitting the target on a stormy day due to high wind speed. The sniper fired two shots on the target on a stormy day when wind speed was very high. Find the probability that (i) target is hit (ii) atleast one shot misses the target.$
OR
Q.31 (b)3 marks $P(E/F)$

Section D

5 marks each

Q.32 (a)5 marks $P = \begin{bmatrix} 1 & -1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2 \end{bmatrix}$
OR
Q.32 (b)5 marks $\Delta = \begin{vmatrix} 1 + x & 1 & 1 \\ 1 & 1 + y & 1 \\ 1 & 1 & 1 + z \end{vmatrix}$
Q.33 (a)5 marks $f(x) = \cot^{-1}(\sin x + \cos x)$
OR
Q.33 (b)5 marks $A rectangle of perimeter 36 cm is revolved around one of its sides to sweep out a cylinder of maximum volume. Find the dimensions of the rectangle.$
Q.345 marks $g(x) = \cos^{-1}(x^2 - 1)$
Q.355 marks $\vec{r} = 4\hat{i} + \hat{j} + \lambda(5\hat{i} + 2\hat{j} + \hat{k})$

Section E

Smoking increases the risk of lung problems. A study revealed that 170 in 1000 males who smoke develop lung complications, while 120 out of 1000 females who smoke develop lung related problems. In a colony, 50 people were found to be smokers of which 30 are males. A person is selected at random from these 50 people and tested for lung related problems. Based on the given information, answer the following questions :
Q.36 (i)1 mark $What is the probability that selected person is a female ?$
Q.36 (ii)1 mark $If a male person is selected, what is the probability that he will not be suffering from lung problems ?$
Q.36 (iii) (a)2 marks $A person selected at random is detected with lung complications. Find the probability that selected person is a female.$
OR
Q.36 (iii) (b)2 marks $A person selected at random is not having lung problems, find the probability that the person is a male.$
A racing track is build around an elliptical ground whose equation is given by $9x^2 + 16y^2 = 144$ . The width of the track is 3 m as shown below : Based on given information, answer the following questions :
Q.37 (i)1 mark $x$
Q.37 (ii)1 mark $x$
Q.37 (iii) (a)2 marks $Find the area of the region enclosed within the elliptical ground excluding the track using integration.$
OR
Q.37 (iii) (b)2 marks $x$
Sports car racing is a form of motorsport which uses sports car prototypes. The competition is held on special tracks designed in various shapes. The equation of one such track is given as follows : $f(x) = \begin{cases} x^4 - 4x^2 + 4, & 0 \leq x < 3 \\ x^2 + 40, & x \geq 3 \end{cases}$ Based on given information, answer the following questions :
Q.38 (i)1 mark $f'(x)$
Q.38 (ii)1 mark $f'(4)$
Q.38 (iii) (a)2 marks $f(x)$
OR
Q.38 (iii) (b)2 marks $f(x)$

Section A

1 mark each

Q.11 mark $\displaystyle\int \dfrac{dx}{1 + \cos x}$
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Q.21 mark $f(x) = x + \dfrac{1}{x}$
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Q.31 mark $y = x^2$
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Q.41 mark $x\,dy - y\,dx = 0$
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Q.51 mark $2x\dfrac{dy}{dx} - y = 3$
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Q.61 mark $|\vec{a}| = 5$
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Q.71 mark $x$
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Q.81 mark $x = y = 1 - z$
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Q.91 mark $In a linear programming problem, the linear function which has to be maximized or minimized is called$
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Q.101 mark $For the feasible region shown below, the non-trivial constraints of the linear programming problem are$
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Q.111 mark $P(A) \neq 0$
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Q.121 mark $\{1, 2, 3\}$
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Q.131 mark $(A + B) (A - B) = A^2 - B^2$
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Q.141 mark $A = \begin{bmatrix} 1 & a & b \\ -1 & 2 & c \\ 0 & 5 & 3 \end{bmatrix}$
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Q.151 mark $A = \begin{bmatrix} \dfrac{1}{2}\cos x & -\sin x \\ \sin x & \dfrac{1}{2}\cos x \end{bmatrix}$
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Q.161 mark $(3A)^{-1} =$
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Q.171 mark $\begin{vmatrix} -1 & -2 & 5 \\ -2 & a & -1 \\ 0 & 4 & 2a \end{vmatrix} = -86$
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Q.181 mark $e^{x + y} = 3x$
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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 codes (A), (B), (C) and (D) as given below.
Q.191 mark $< 1, 1, 1 >$
Tap an option to check your answer.
Q.201 mark $\vec{a}$
Tap an option to check your answer.

Section B

2 marks each

Q.21 (a)2 marks $f(x) = \cos x + \sin^2 x$
OR
Q.21 (b)2 marks $If the volume of a solid hemisphere increases at a uniform rate, prove that its surface area varies inversely as its radius.$
Q.222 marks $\overrightarrow{AB} = \hat{j} + \hat{k}$
Q.232 marks $\dfrac{x}{1} = \dfrac{y + 1}{-1} = \dfrac{z - 3}{-2}$
Q.24 (a)2 marks $f : R - \{3\} \to R$
OR
Q.24 (b)2 marks $f : Z \times Z \to Z \times Z$
Q.252 marks $x = \sin t - \cos t$

Section C

3 marks each

Q.263 marks $\dfrac{d}{dx}(F(x)) = \dfrac{1}{e^x + 1}$
Q.27 (a)3 marks $x\dfrac{dy}{dx} = y - x \sin^2 \left(\dfrac{y}{x}\right)$
OR
Q.27 (b)3 marks $y \log y \dfrac{dx}{dy} + x = \dfrac{2}{y}$
Q.283 marks $Z = 8000x + 12000y$
Q.29 (a)3 marks $The probability of hitting the target by a trained sniper is three times the probability of not hitting the target on a stormy day due to high wind speed. The sniper fired two shots on the target on a stormy day when wind speed was very high. Find the probability that (i) target is hit (ii) atleast one shot misses the target.$
OR
Q.29 (b)3 marks $P(E/F)$
Q.303 marks $\displaystyle\int \dfrac{2x + 1}{\sqrt{6x + x^2}}\, dx$
Q.31 (a)3 marks $\displaystyle\int_{\frac{\pi}{12}}^{\frac{5\pi}{12}} \dfrac{dx}{1 + \sqrt{\cot x}}$
OR
Q.31 (b)3 marks $\displaystyle\int_{\frac{-\pi}{6}}^{\frac{\pi}{2}} (\sin |x| + \cos |x|)\, dx$

Section D

5 marks each

Q.325 marks $p(x) = \sin^{-1}(1 - 2x^2)$
Q.335 marks $\vec{r} = 4\hat{i} + \hat{j} + \lambda(5\hat{i} + 2\hat{j} + \hat{k})$
Q.34 (a)5 marks $P = \begin{bmatrix} 1 & -1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2 \end{bmatrix}$
OR
Q.34 (b)5 marks $\Delta = \begin{vmatrix} 1 + x & 1 & 1 \\ 1 & 1 + y & 1 \\ 1 & 1 & 1 + z \end{vmatrix}$
Q.35 (a)5 marks $(0, \pi)$
OR
Q.35 (b)5 marks $A rectangle of perimeter 24 cm is revolved along one of its sides to sweep out a cylinder of maximum volume. Find the dimensions of the rectangle.$

Section E

A racing track is build around an elliptical ground whose equation is given by $9x^2 + 16y^2 = 144$ . The width of the track is 3 m as shown below : Based on given information, answer the following questions :
Q.36 (i)1 mark $x$
Q.36 (ii)1 mark $x$
Q.36 (iii) (a)2 marks $Find the area of the region enclosed within the elliptical ground excluding the track using integration.$
OR
Q.36 (iii) (b)2 marks $x$
Sports car racing is a form of motorsport which uses sports car prototypes. The competition is held on special tracks designed in various shapes. The equation of one such track is given as follows : $f(x) = \begin{cases} x^4 - 4x^2 + 4, & 0 \leq x < 3 \\ x^2 + 40, & x \geq 3 \end{cases}$ Based on given information, answer the following questions :
Q.37 (i)1 mark $f'(x)$
Q.37 (ii)1 mark $f'(4)$
Q.37 (iii) (a)2 marks $f(x)$
OR
Q.37 (iii) (b)2 marks $f(x)$
Smoking increases the risk of lung problems. A study revealed that 170 in 1000 males who smoke develop lung complications, while 120 out of 1000 females who smoke develop lung related problems. In a colony, 50 people were found to be smokers of which 30 are males. A person is selected at random from these 50 people and tested for lung related problems. Based on the given information, answer the following questions :
Q.38 (i)1 mark $What is the probability that selected person is a female ?$
Q.38 (ii)1 mark $If a male person is selected, what is the probability that he will not be suffering from lung problems ?$
Q.38 (iii) (a)2 marks $A person selected at random is detected with lung complications. Find the probability that selected person is a female.$
OR
Q.38 (iii) (b)2 marks $A person selected at random is not having lung problems, find the probability that the person is a male.$

Section A

1 mark each

Q.11 mark $\dfrac{1 - x}{0} = y = z$
Tap an option to check your answer.
Q.21 mark $In a linear programming problem, the linear function which has to be maximized or minimized is called$
Tap an option to check your answer.
Q.31 mark $For the feasible region shown below, the non-trivial constraints of the linear programming problem are$
Tap an option to check your answer.
Q.41 mark $P(A) \neq 0$
Tap an option to check your answer.
Q.51 mark $\{1, 2, 3\}$
Tap an option to check your answer.
Q.61 mark $(A + B) (A - B) = A^2 - B^2$
Tap an option to check your answer.
Q.71 mark $A = \begin{bmatrix} 1 & a & b \\ -1 & 2 & c \\ 0 & 5 & 3 \end{bmatrix}$
Tap an option to check your answer.
Q.81 mark $A = \begin{bmatrix} \tan x & \cot x \\ -\cot x & \tan x \end{bmatrix}$
Tap an option to check your answer.
Q.91 mark $(3A)^{-1} =$
Tap an option to check your answer.
Q.101 mark $\begin{vmatrix} -1 & -2 & 5 \\ -2 & a & -1 \\ 0 & 4 & 2a \end{vmatrix} = -86$
Tap an option to check your answer.
Q.111 mark $x + y = xy$
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Q.121 mark $\displaystyle\int \dfrac{dx}{\sec x + \tan x}$
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Q.131 mark $f(x) = x + \dfrac{1}{x}$
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Q.141 mark $y = x^2$
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Q.151 mark $x^2 dy + y^2 dx = 0$
Tap an option to check your answer.
Q.161 mark $2x\dfrac{dy}{dx} - y = 3$
Tap an option to check your answer.
Q.171 mark $|\vec{a}| = 5$
Tap an option to check your answer.
Q.181 mark $x$
Tap an option to check your answer.
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 codes (A), (B), (C) and (D) as given below.
Q.191 mark $\vec{a}$
Tap an option to check your answer.
Q.201 mark $< 1, 1, 1 >$
Tap an option to check your answer.

Section B

2 marks each

Q.212 marks $x = \dfrac{y - 1}{2} = \dfrac{z - 2}{3}$
Q.22 (a)2 marks $f : R - \{3\} \to R$
OR
Q.22 (b)2 marks $f : Z \times Z \to Z \times Z$
Q.232 marks $x = e^{\sin^{-1} t}$
Q.24 (a)2 marks $f(x) = \cos x + \sin^2 x$
OR
Q.24 (b)2 marks $If the volume of a solid hemisphere increases at a uniform rate, prove that its surface area varies inversely as its radius.$
Q.252 marks $\overrightarrow{AB} = \hat{j} + \hat{k}$

Section C

3 marks each

Q.26 (a)3 marks $The probability of hitting the target by a trained sniper is three times the probability of not hitting the target on a stormy day due to high wind speed. The sniper fired two shots on the target on a stormy day when wind speed was very high. Find the probability that (i) target is hit (ii) atleast one shot misses the target.$
OR
Q.26 (b)3 marks $P(E/F)$
Q.273 marks $\displaystyle\int \dfrac{3x - 1}{\sqrt{x^2 - 4x}}\, dx$
Q.28 (a)3 marks $\displaystyle\int_{\frac{\pi}{12}}^{\frac{5\pi}{12}} \dfrac{dx}{1 + \sqrt{\cot x}}$
OR
Q.28 (b)3 marks $\displaystyle\int_{\frac{-\pi}{6}}^{\frac{\pi}{2}} (\sin |x| + \cos |x|)\, dx$
Q.293 marks $\dfrac{d}{dx}(F(x)) = \dfrac{1}{e^x + 1}$
Q.30 (a)3 marks $x\dfrac{dy}{dx} = y - x \sin^2 \left(\dfrac{y}{x}\right)$
OR
Q.30 (b)3 marks $y \log y \dfrac{dx}{dy} + x = \dfrac{2}{y}$
Q.313 marks $Z = 4500x + 5000y$

Section D

5 marks each

Q.32 (a)5 marks $\left(0, \dfrac{\pi}{2}\right)$
OR
Q.32 (b)5 marks $A rectangle of perimeter 30 cm is revolved along one of its sides to sweep out a cylinder of maximum volume. Find the dimensions of the rectangle.$
Q.335 marks $q(x) = \cos^{-1} (4x^2 - 3)$
Q.345 marks $\vec{r} = 4\hat{i} + \hat{j} + \lambda(5\hat{i} + 2\hat{j} + \hat{k})$
Q.35 (a)5 marks $P = \begin{bmatrix} 1 & -1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2 \end{bmatrix}$
OR
Q.35 (b)5 marks $\Delta = \begin{vmatrix} 1 + x & 1 & 1 \\ 1 & 1 + y & 1 \\ 1 & 1 & 1 + z \end{vmatrix}$

Section E

Sports car racing is a form of motorsport which uses sports car prototypes. The competition is held on special tracks designed in various shapes. The equation of one such track is given as follows : $f(x) = \begin{cases} x^4 - 4x^2 + 4, & 0 \leq x < 3 \\ x^2 + 40, & x \geq 3 \end{cases}$ Based on given information, answer the following questions :
Q.36 (i)1 mark $f'(x)$
Q.36 (ii)1 mark $f'(4)$
Q.36 (iii) (a)2 marks $f(x)$
OR
Q.36 (iii) (b)2 marks $f(x)$
Smoking increases the risk of lung problems. A study revealed that 170 in 1000 males who smoke develop lung complications, while 120 out of 1000 females who smoke develop lung related problems. In a colony, 50 people were found to be smokers of which 30 are males. A person is selected at random from these 50 people and tested for lung related problems. Based on the given information, answer the following questions :
Q.37 (i)1 mark $What is the probability that selected person is a female ?$
Q.37 (ii)1 mark $If a male person is selected, what is the probability that he will not be suffering from lung problems ?$
Q.37 (iii) (a)2 marks $A person selected at random is detected with lung complications. Find the probability that selected person is a female.$
OR
Q.37 (iii) (b)2 marks $A person selected at random is not having lung problems, find the probability that the person is a male.$
A racing track is build around an elliptical ground whose equation is given by $9x^2 + 16y^2 = 144$ . The width of the track is 3 m as shown below : Based on given information, answer the following questions :
Q.38 (i)1 mark $x$
Q.38 (ii)1 mark $x$
Q.38 (iii) (a)2 marks $Find the area of the region enclosed within the elliptical ground excluding the track using integration.$
OR
Q.38 (iii) (b)2 marks $x$