CBSE Class 12 Mathematics 2026 question paper (65/3)

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 $f(x) = \cos^{-1} (2x - 5)$
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Q.21 mark $A^2 = 4A + 3I$
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Q.31 mark $AB' + BA'$
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Q.41 mark $\begin{bmatrix} 4 \\ 1 \\ 3 \end{bmatrix} A = \begin{bmatrix} -4 & 8 & 4 \\ -1 & 2 & 1 \\ -3 & 6 & 3 \end{bmatrix}$
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Q.51 mark $A^2 = A$
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Q.61 mark $B(\text{adj } B) = \begin{bmatrix} \dfrac{1}{3} & 0 & 0 \\ 0 & \dfrac{1}{3} & 0 \\ 0 & 0 & \dfrac{1}{3} \end{bmatrix}$
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Q.71 mark $f(x) = \begin{cases} x^2 \sin \dfrac{1}{x}, & x \ne 0 \\ k(x + 1), & x = 0 \end{cases}$
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Q.81 mark $\sin^{-1} x = y$
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Q.91 mark $The rate of change of volume of a sphere with respect to its diameter, when its radius is 5 cm, is :$
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Q.101 mark $\int \dfrac{dx}{2^x + 2^{-x}}$
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Q.111 mark $\int_{-1}^{1} (1 - |x|)\, dx$
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Q.121 mark $The area of the shaded region of the circle given below is equal to :$
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Q.131 mark $\dfrac{dy}{dx} = F(x, y)$
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Q.141 mark $\vec{a}$
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Q.151 mark $(\vec{a} + \vec{b}) \cdot (\vec{a} - \vec{b}) = 198$
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Q.161 mark $l_1, m_1, n_1$
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Q.171 mark $l_1$
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Q.181 mark $P(E) = \dfrac{3}{10}$
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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 $\dfrac{dy}{dx} = e^{x+y}$
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Q.201 mark $\vec{a}$
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Section B

2 marks each

Q.212 marks $x = t + \dfrac{1}{t}$
Q.222 marks $\left(0, \dfrac{\pi}{2}\right)$
Q.23 (a)2 marks $\sin [\cot^{-1} \sqrt{2}\, (\cos (\tan^{-1} 1))]$
OR
Q.23 (b)2 marks $A = \{1, 2, 3\}$
Q.242 marks $\vec{a}$
Q.25 (a)2 marks $\dfrac{x - 3}{1} = \dfrac{1 - y}{1} = \dfrac{z + 2}{p}$
OR
Q.25 (b)2 marks $\vec{r} = 2\hat{i} - \hat{j} + 2\hat{k} + \lambda(3\hat{i} + 4\hat{j} + 2\hat{k})$

Section C

3 marks each

Q.26 (a)3 marks $\int \dfrac{dx}{x^{1/2} + x^{1/3}}$
OR
Q.26 (b)3 marks $\int \tan^{-1} \left(\dfrac{1 - x}{1 + x}\right) dx$
Q.273 marks $\int_{0}^{\pi} \dfrac{\sin^{2026} x}{\sin^{2026} x + \cos^{2026} x}\, dx$
Q.28 (a)3 marks $\int \dfrac{\cos x}{(2 + \sin x)\,(4 + \sin x)}\, dx$
OR
Q.28 (b)3 marks $\int \dfrac{x + 3}{x^2 + 4x + 5}\, dx$
Q.29 (a)3 marks $2x^2 \dfrac{dy}{dx} = y^2 + 2xy$
OR
Q.29 (b)3 marks $(x + 1) \dfrac{dy}{dx} = 2\,e^{-y} - 1$
Q.303 marks $(\sin x)^y = y^{\cos x}$
Q.313 marks $A survey was conducted on the patients who have undergone knee replacement surgeries. It was found that, Robotic Knee replacement surgeries have 90% success rate. On a particular day, robotic surgery was performed on three patients, A, B and C, one after the other. Assuming that the success and failure of each surgery is independent of each other, find the probability that : (i) exactly one surgery is successful, (ii) at most two surgeries are successful.$

Section D

5 marks each

Q.325 marks $f : R \to R$
Q.335 marks $Z = 600x + 400y$
Q.34 (a)5 marks $On the inauguration day of a new showroom, a lucky draw was organized and some vouchers of ₹ 1,000 and ₹ 500 were given to the lucky draw winners. A total of 60 vouchers were given on the day. The number of ₹ 1,000 vouchers added to 3 times the number of ₹ 500 vouchers, gives 100. Express the given information as a system of linear equations in two variables. Hence, find the number of vouchers of each type by matrix method.$
OR
Q.34 (b)5 marks $P = \begin{bmatrix} 2 & -1 \\ 3 & 4 \end{bmatrix}$
Q.35 (a)5 marks $l_1$
OR
Q.35 (b)5 marks $\vec{r} = \hat{i} + 2\hat{j} - 4\hat{k} + \lambda(2\hat{i} + 3\hat{j} + 6\hat{k})$

Section E

Two vertical light poles of height 22 m and 16 m stand on the opposite sides of a 20 m wide road as shown below in the figure. Two ladders of length $l_1$ and $l_2$ are placed from a common point R on the road at a distance of x m from the smaller pole. Based on the above information, answer the following questions :
Q.36 (i)1 mark $p(x) = l_1 + l_2$
Q.36 (ii)1 mark $p'(x)$
Q.36 (iii) (a)2 marks $l_1^2 + l_2^2$
OR
Q.36 (iii) (b)2 marks $If the 22 m long pole is also replaced by a 16 m long pole, at what distance from either pole should the ladders be kept so that the sum of squares of lengths of ladders needed to reach the top of the pole is minimum ?$
A survey was conducted to find out the success rate of students who qualified the entrance examination by dropping a year after class XII. As per the data collected, 40% students appearing in the examination were dropouts and the remaining students were regular students of class XII. Of the dropouts, 5% qualify the examination while 10% of the regular students qualify the examination. Based on the above information, answer the following questions.
Q.37 (i)1 mark $Find the probability that a student selected at random is a regular student.$
Q.37 (ii)1 mark $A student is selected at random from a group of dropout students. What is the probability that the student will not qualify the examination ?$
Q.37 (iii) (a)2 marks $A student selected at random qualified the examination. Find the probability that student is not a dropout.$
OR
Q.37 (iii) (b)2 marks $A student selected at random did not qualify the examination. Find the probability that the student was a regular student.$
There is a triangular park in the society. The park is divided into two sections as shown in the figure. In the region OAC, children are allowed to play games like cricket, football, while in the region AOB, activities which involve running are not allowed. The vertices of the triangular park ABC are $A(0, 4)$ , $B(-2, 0)$ and $C(3, 0)$ . Based on the above information, answer the following questions :
Q.38 (i)1 mark $Write the equation of the boundary line AB of the park.$
Q.38 (ii)1 mark $Write the equation of the boundary line AC of the park.$
Q.38 (iii) (a)2 marks $Using integration, find the area of region OAC, in which children are allowed to play cricket, football.$
OR
Q.38 (iii) (b)2 marks $Using integration, find the area of region AOB.$

Section A

1 mark each

Q.11 mark $\tan^{-1} x = y$
Tap an option to check your answer.
Q.21 mark $The rate of change of volume of a sphere with respect to its diameter, when its radius is 5 cm, is :$
Tap an option to check your answer.
Q.31 mark $\int \dfrac{dx}{\sqrt{e^{-2x} - 1}}$
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Q.41 mark $\int_{-1}^{1} (1 - |x|)\, dx$
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Q.51 mark $The area of the shaded region of the circle given below is equal to :$
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Q.61 mark $\dfrac{dy}{dx} = F(x, y)$
Tap an option to check your answer.
Q.71 mark $\vec{a}$
Tap an option to check your answer.
Q.81 mark $(\vec{a} + \vec{b}) \cdot (\vec{a} - \vec{b}) = 198$
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Q.91 mark $l_1, m_1, n_1$
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Q.101 mark $l_1$
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Q.111 mark $P(E) = \dfrac{3}{10}$
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Q.121 mark $\sin^{-1} (1 - 2x)$
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Q.131 mark $A^2 = 4A + 3I$
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Q.141 mark $AB' + BA'$
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Q.151 mark $B[1 \;\; 2 \;\; 3] = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 1 \\ 2 & 0 & 1 \end{bmatrix}$
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Q.161 mark $A^2 = A$
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Q.171 mark $A(\text{adj } A) = \begin{bmatrix} 2026 & 0 & 0 \\ 0 & 2026 & 0 \\ 0 & 0 & 2026 \end{bmatrix}$
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Q.181 mark $f(x) = \begin{cases} x^2 \sin \dfrac{1}{x}, & x \ne 0 \\ k(x + 1), & x = 0 \end{cases}$
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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 $\vec{a}$
Tap an option to check your answer.
Q.201 mark $\dfrac{dy}{dx} = e^{x+y}$
Tap an option to check your answer.

Section B

2 marks each

Q.21 (a)2 marks $\sin [\cot^{-1} \sqrt{2}\, (\cos (\tan^{-1} 1))]$
OR
Q.21 (b)2 marks $A = \{1, 2, 3\}$
Q.222 marks $\vec{a}$
Q.23 (a)2 marks $\dfrac{x - 3}{1} = \dfrac{1 - y}{1} = \dfrac{z + 2}{p}$
OR
Q.23 (b)2 marks $\vec{r} = 2\hat{i} - \hat{j} + 2\hat{k} + \lambda(3\hat{i} + 4\hat{j} + 2\hat{k})$
Q.242 marks $x = e^{t + \frac{1}{t}}$
Q.252 marks $(0, \infty)$

Section C

3 marks each

Q.26 (a)3 marks $\int \dfrac{\cos x}{(2 + \sin x)\,(4 + \sin x)}\, dx$
OR
Q.26 (b)3 marks $\int \dfrac{x + 3}{x^2 + 4x + 5}\, dx$
Q.27 (a)3 marks $(x^2 + y^2)\, dy = xy\, dx$
OR
Q.27 (b)3 marks $\dfrac{dy}{dx} - 3y \cot x = \sin 2x$
Q.283 marks $(\sin x)^y = y^{\cos x}$
Q.293 marks $A survey was conducted on the patients who have undergone knee replacement surgeries. It was found that, Robotic Knee replacement surgeries have 90% success rate. On a particular day, robotic surgery was performed on three patients, A, B and C, one after the other. Assuming that the success and failure of each surgery is independent of each other, find the probability that : (i) exactly one surgery is successful, (ii) at most two surgeries are successful.$
Q.30 (a)3 marks $\int \dfrac{dx}{x^{1/2} + x^{1/3}}$
OR
Q.30 (b)3 marks $\int \tan^{-1} \left(\dfrac{1 - x}{1 + x}\right) dx$
Q.313 marks $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \dfrac{\cos^2 x}{2^x + 1}\, dx$

Section D

5 marks each

Q.32 (a)5 marks $On the inauguration day of a new showroom, a lucky draw was organized and some vouchers of ₹ 1,000 and ₹ 500 were given to the lucky draw winners. A total of 60 vouchers were given on the day. The number of ₹ 1,000 vouchers added to 3 times the number of ₹ 500 vouchers, gives 100. Express the given information as a system of linear equations in two variables. Hence, find the number of vouchers of each type by matrix method.$
OR
Q.32 (b)5 marks $P = \begin{bmatrix} 2 & -1 \\ 3 & 4 \end{bmatrix}$
Q.33 (a)5 marks $l_1$
OR
Q.33 (b)5 marks $\vec{r} = \hat{i} + 2\hat{j} - 4\hat{k} + \lambda(2\hat{i} + 3\hat{j} + 6\hat{k})$
Q.345 marks $f : R_+ \to [-5, \infty)$
Q.355 marks $Z = 12x + 18y$

Section E

A survey was conducted to find out the success rate of students who qualified the entrance examination by dropping a year after class XII. As per the data collected, 40% students appearing in the examination were dropouts and the remaining students were regular students of class XII. Of the dropouts, 5% qualify the examination while 10% of the regular students qualify the examination. Based on the above information, answer the following questions.
Q.36 (i)1 mark $Find the probability that a student selected at random is a regular student.$
Q.36 (ii)1 mark $A student is selected at random from a group of dropout students. What is the probability that the student will not qualify the examination ?$
Q.36 (iii) (a)2 marks $A student selected at random qualified the examination. Find the probability that student is not a dropout.$
OR
Q.36 (iii) (b)2 marks $A student selected at random did not qualify the examination. Find the probability that the student was a regular student.$
There is a triangular park in the society. The park is divided into two sections as shown in the figure. In the region OAC, children are allowed to play games like cricket, football, while in the region AOB, activities which involve running are not allowed. The vertices of the triangular park ABC are $A(0, 4)$ , $B(-2, 0)$ and $C(3, 0)$ . Based on the above information, answer the following questions :
Q.37 (i)1 mark $Write the equation of the boundary line AB of the park.$
Q.37 (ii)1 mark $Write the equation of the boundary line AC of the park.$
Q.37 (iii) (a)2 marks $Using integration, find the area of region OAC, in which children are allowed to play cricket, football.$
OR
Q.37 (iii) (b)2 marks $Using integration, find the area of region AOB.$
Two vertical light poles of height 22 m and 16 m stand on the opposite sides of a 20 m wide road as shown below in the figure. Two ladders of length $l_1$ and $l_2$ are placed from a common point R on the road at a distance of x m from the smaller pole. Based on the above information, answer the following questions :
Q.38 (i)1 mark $p(x) = l_1 + l_2$
Q.38 (ii)1 mark $p'(x)$
Q.38 (iii) (a)2 marks $l_1^2 + l_2^2$
OR
Q.38 (iii) (b)2 marks $If the 22 m long pole is also replaced by a 16 m long pole, at what distance from either pole should the ladders be kept so that the sum of squares of lengths of ladders needed to reach the top of the pole is minimum ?$

Section A

1 mark each

Q.11 mark $(\vec{a} + \vec{b}) \cdot (\vec{a} - \vec{b}) = 198$
Tap an option to check your answer.
Q.21 mark $l_1, m_1, n_1$
Tap an option to check your answer.
Q.31 mark $l_1$
Tap an option to check your answer.
Q.41 mark $P(E) = \dfrac{3}{10}$
Tap an option to check your answer.
Q.51 mark $\cos^{-1} (4x + 1)$
Tap an option to check your answer.
Q.61 mark $A^2 = 4A + 3I$
Tap an option to check your answer.
Q.71 mark $AB' + BA'$
Tap an option to check your answer.
Q.81 mark $\begin{bmatrix} 2 & 1 \end{bmatrix} X = \begin{bmatrix} 3 & 4 & 5 \end{bmatrix}$
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Q.91 mark $A^2 = A$
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Q.101 mark $A(\text{adj } A) = \begin{bmatrix} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{bmatrix}$
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Q.111 mark $f(x) = \begin{cases} x^2 \sin \dfrac{1}{x}, & x \ne 0 \\ k(x + 1), & x = 0 \end{cases}$
Tap an option to check your answer.
Q.121 mark $2 \cos^{-1} x = y$
Tap an option to check your answer.
Q.131 mark $The rate of change of volume of a sphere with respect to its diameter, when its radius is 5 cm, is :$
Tap an option to check your answer.
Q.141 mark $\int \dfrac{\cos x}{\sqrt{\sin^2 x + 1}}\, dx$
Tap an option to check your answer.
Q.151 mark $\int_{-1}^{1} (1 - |x|)\, dx$
Tap an option to check your answer.
Q.161 mark $The area of the shaded region of the circle given below is equal to :$
Tap an option to check your answer.
Q.171 mark $\dfrac{dy}{dx} = F(x, y)$
Tap an option to check your answer.
Q.181 mark $\vec{a}$
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 $\dfrac{dy}{dx} = e^{x+y}$
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 $\dfrac{x - 3}{1} = \dfrac{1 - y}{1} = \dfrac{z + 2}{p}$
OR
Q.21 (b)2 marks $\vec{r} = 2\hat{i} - \hat{j} + 2\hat{k} + \lambda(3\hat{i} + 4\hat{j} + 2\hat{k})$
Q.222 marks $x = e^{\sin^{-1} t}$
Q.232 marks $(0, \infty)$
Q.24 (a)2 marks $\sin [\cot^{-1} \sqrt{2}\, (\cos (\tan^{-1} 1))]$
OR
Q.24 (b)2 marks $A = \{1, 2, 3\}$
Q.252 marks $\vec{a}$

Section C

3 marks each

Q.263 marks $A survey was conducted on the patients who have undergone knee replacement surgeries. It was found that, Robotic Knee replacement surgeries have 90% success rate. On a particular day, robotic surgery was performed on three patients, A, B and C, one after the other. Assuming that the success and failure of each surgery is independent of each other, find the probability that : (i) exactly one surgery is successful, (ii) at most two surgeries are successful.$
Q.27 (a)3 marks $\int \dfrac{dx}{x^{1/2} + x^{1/3}}$
OR
Q.27 (b)3 marks $\int \tan^{-1} \left(\dfrac{1 - x}{1 + x}\right) dx$
Q.283 marks $\int_{0}^{\pi/2} \dfrac{\sin^2 x}{1 + \sin 2x}\, dx$
Q.29 (a)3 marks $\int \dfrac{\cos x}{(2 + \sin x)\,(4 + \sin x)}\, dx$
OR
Q.29 (b)3 marks $\int \dfrac{x + 3}{x^2 + 4x + 5}\, dx$
Q.30 (a)3 marks $(x^2 - y^2)\, dx + 2xy\, dy = 0$
OR
Q.30 (b)3 marks $\sin x \cos y\, dx + \cos x \sin y\, dy = 0$
Q.313 marks $(\sin x)^y = y^{\cos x}$

Section D

5 marks each

Q.32 (a)5 marks $l_1$
OR
Q.32 (b)5 marks $\vec{r} = \hat{i} + 2\hat{j} - 4\hat{k} + \lambda(2\hat{i} + 3\hat{j} + 6\hat{k})$
Q.335 marks $f : R_+ \to A \subset N$
Q.345 marks $Z = 8x + 8y$
Q.35 (a)5 marks $On the inauguration day of a new showroom, a lucky draw was organized and some vouchers of ₹ 1,000 and ₹ 500 were given to the lucky draw winners. A total of 60 vouchers were given on the day. The number of ₹ 1,000 vouchers added to 3 times the number of ₹ 500 vouchers, gives 100. Express the given information as a system of linear equations in two variables. Hence, find the number of vouchers of each type by matrix method.$
OR
Q.35 (b)5 marks $P = \begin{bmatrix} 2 & -1 \\ 3 & 4 \end{bmatrix}$

Section E

There is a triangular park in the society. The park is divided into two sections as shown in the figure. In the region OAC, children are allowed to play games like cricket, football, while in the region AOB, activities which involve running are not allowed. The vertices of the triangular park ABC are $A(0, 4)$ , $B(-2, 0)$ and $C(3, 0)$ . Based on the above information, answer the following questions :
Q.36 (i)1 mark $Write the equation of the boundary line AB of the park.$
Q.36 (ii)1 mark $Write the equation of the boundary line AC of the park.$
Q.36 (iii) (a)2 marks $Using integration, find the area of region OAC, in which children are allowed to play cricket, football.$
OR
Q.36 (iii) (b)2 marks $Using integration, find the area of region AOB.$
Two vertical light poles of height 22 m and 16 m stand on the opposite sides of a 20 m wide road as shown below in the figure. Two ladders of length $l_1$ and $l_2$ are placed from a common point R on the road at a distance of x m from the smaller pole. Based on the above information, answer the following questions :
Q.37 (i)1 mark $p(x) = l_1 + l_2$
Q.37 (ii)1 mark $p'(x)$
Q.37 (iii) (a)2 marks $l_1^2 + l_2^2$
OR
Q.37 (iii) (b)2 marks $If the 22 m long pole is also replaced by a 16 m long pole, at what distance from either pole should the ladders be kept so that the sum of squares of lengths of ladders needed to reach the top of the pole is minimum ?$
A survey was conducted to find out the success rate of students who qualified the entrance examination by dropping a year after class XII. As per the data collected, 40% students appearing in the examination were dropouts and the remaining students were regular students of class XII. Of the dropouts, 5% qualify the examination while 10% of the regular students qualify the examination. Based on the above information, answer the following questions.
Q.38 (i)1 mark $Find the probability that a student selected at random is a regular student.$
Q.38 (ii)1 mark $A student is selected at random from a group of dropout students. What is the probability that the student will not qualify the examination ?$
Q.38 (iii) (a)2 marks $A student selected at random qualified the examination. Find the probability that student is not a dropout.$
OR
Q.38 (iii) (b)2 marks $A student selected at random did not qualify the examination. Find the probability that the student was a regular student.$