CBSE Class 12 Mathematics 2026 question paper (65/5)

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 $A = \begin{bmatrix} -p & q \\ r & p \end{bmatrix}$
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Q.21 mark $A^2 = A$
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Q.31 mark $For the inverse trigonometric functions, which of the following Principal Value Branch is not correctly defined ?$
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Q.41 mark $A = \begin{bmatrix} 0 & -3 & 4 \\ 1 & 0 & 2 \end{bmatrix}$
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Q.51 mark $If A is a non-singular matrix, then which of the following is not true ?$
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Q.61 mark $f(x) = \begin{cases} \dfrac{x^2 - 4x - 5}{x + 1}, & x \ne -1 \\[2mm] k, & x = -1 \end{cases}$
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Q.71 mark $\Delta$
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Q.81 mark $\cos^{-1}\left(\dfrac{\sin x + \cos x}{\sqrt{2}}\right)$
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Q.91 mark $f(x) = (x - 2)^2 + 5$
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Q.101 mark $\displaystyle\int \dfrac{1}{\sqrt{1 + \cos 2x}}\,dx$
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Q.111 mark $\displaystyle\int_{-5}^{-1} \dfrac{1}{x}\,dx$
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Q.121 mark $y = 2x - 4$
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Q.131 mark $1 + \left(\dfrac{d^3y}{dx^3}\right)^3 = \lambda\,\dfrac{d^2y}{dx^2}$
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Q.141 mark $\dfrac{dy}{dx} = e^{3x - y}$
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Q.151 mark $Z = 4x + 3y$
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Q.161 mark $\vec{p}$
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Q.171 mark $\vec{a} = 3\hat{i} + 2\hat{j} + \lambda\hat{k}$
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Q.181 mark $3P(A) = P(B) = \dfrac{3}{5}$
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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 $\{1,\ 2,\ 3\}$
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Q.201 mark $Z = x + 2y$
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Section B

2 marks each

Q.212 marks $\sin\left[\tan^{-1}\tan\left(\dfrac{3\pi}{4}\right)\right]$
Q.22 (a)2 marks $x^x$
OR
Q.22 (b)2 marks $y = P \cos ux + Q \sin ux$
Q.232 marks $f(x) = \dfrac{x - 3}{x + 1}$
Q.24 (a)2 marks $\vec{a} = 2\hat{i} - 3\hat{j} + \hat{k}$
OR
Q.24 (b)2 marks $\overrightarrow{AB} = \hat{i} + 2\hat{j} - \hat{k}$
Q.252 marks $\dfrac{x - 2}{3} = \dfrac{y + 5}{2} = \dfrac{1 - z}{-6}$

Section C

3 marks each

Q.263 marks $A spherical balloon loses its volume due to escape of air from it in such a way that decrease of volume at any instant is proportional to its surface area. Show that the radius is decreasing at a constant rate.$
Q.27 (a)3 marks $\displaystyle\int \dfrac{x - \sin x}{1 - \cos x}\,dx$
OR
Q.27 (b)3 marks $\displaystyle\int_{0}^{2} \dfrac{1}{\sqrt{x^2 + 2x + 3}}\,dx$
Q.283 marks $(x + 2y^3)\,dy = y\,dx$
Q.293 marks $Z = \dfrac{2x}{5} + \dfrac{3y}{10}$
Q.30 (a)3 marks $55\hat{i} - 2\hat{j}$
OR
Q.30 (b)3 marks $\vec{a}$
Q.31 (a)3 marks $A = \{1,\ 2,\ 5\}$
OR
Q.31 (b)3 marks $A box contains 6 cards numbered 1 to 6. A student is asked to pick up two cards, one by one after replacement and note down the numbers on the cards. Let A be the event of getting sum of the numbers on two cards as 10, and B, the event of a number other than 4 on the first card selected. Find P(A and B) and find whether the events A and B are independent events or not.$

Section D

5 marks each

Q.325 marks $A man goes to buy fruits from the market. The shopkeeper informs him that 4 apples, 3 oranges and 2 bananas cost ₹ 60; 2 apples, 4 oranges and 6 bananas cost ₹ 90; whereas 6 apples, 2 oranges and 3 bananas cost ₹ 70. Using matrix method, find the cost of one fruit of each kind.$
Q.33 (a)5 marks $y\sqrt{x^2 + 1} = \log \sqrt{x^2 + 1} - x$
OR
Q.33 (b)5 marks $x^{\cot x} + \dfrac{2x^2 - 3}{2x^2 - x + 2}$
Q.345 marks $\{(x,\ y) : 100x^2 + 25y^2 = 2500\}$
Q.35 (a)5 marks $\dfrac{-x - 3}{-5} = \dfrac{1 - y}{-2} = \dfrac{3z + 12}{9}$
OR
Q.35 (b)5 marks $\vec{r} = (\hat{i} + 2\hat{j} + \hat{k}) + \lambda(\hat{i} - \hat{j} + \hat{k})$

Section E

A school wants the students of class XII to do a project on ‘Sustainability’ keeping the world environment in mind. They select the student participants on the basis of an essay writing competition. 7 students out of 80 are selected for the project and are categorized into two sets such that : Girl students belong to Set $A = \{G_1,\ G_2,\ G_3,\ G_4\}$ , Boy students belong to Set $B = \{B_1,\ B_2,\ B_3\}$ . Based on the above information, answer the following questions :
Q.36 (i)1 mark $\to$
Q.36 (ii)1 mark $\to$
Q.36 (iii) (a)2 marks $\to$
OR
Q.36 (iii) (b)2 marks $f : B \to A$
There are three types of vaccines $A_1$ , $A_2$ , $A_3$ , available in the market to protect the population of the country from spread of certain infection. According to a survey conducted, it was found that 25% of the population was given Vaccine $A_1$ , 35% of the population was given Vaccine $A_2$ and 40% of the population was given Vaccine $A_3$ . The survey also stated that the probabilities that Vaccines $A_1$ , $A_2$ and $A_3$ would protect against the infection were 60%, 55% and 50% respectively. Based on the above information, answer the following questions : Find the probability that :
Q.37 (i)1 mark $A_2$
Q.37 (ii)1 mark $If a person is chosen randomly, he/she will be protected from the infection.$
Q.37 (iii) (a)2 marks $A_1$
OR
Q.37 (iii) (b)2 marks $A_3$
A company produces cylindrical tumblers, open from the top. Since they want uniformity in the product, they fix the surface area of the tumblers produced. Based on the above information, answer the following questions : If for a tumbler, V is its volume, h the height and r the radius of the circular base, then :
Q.38 (i)2 marks $Differentiate its volume with respect to radius of the base, where the surface area is constant.$
Q.38 (ii)2 marks $If the company wants to maximize the volume of each tumbler, then establish a relation between its height and the radius of the base.$

Section A

1 mark each

Q.11 mark $A = \begin{bmatrix} \sin\theta & \cos\theta \\ -\cos\theta & \sin\theta \end{bmatrix}$
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Q.21 mark $BAB'$
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Q.31 mark $A = \begin{bmatrix} 0 & -3 & 4 \\ 1 & 0 & 2 \end{bmatrix}$
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Q.41 mark $For the inverse trigonometric functions, which of the following Principal Value Branch is not correctly defined ?$
Tap an option to check your answer.
Q.51 mark $f(x) = \begin{cases} \dfrac{x^2 - 4x - 5}{x + 1}, & x \ne -1 \\[2mm] k, & x = -1 \end{cases}$
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Q.61 mark $If A is a non-singular matrix, then which of the following is not true ?$
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Q.71 mark $f(x) = (x - 2)^2 + 5$
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Q.81 mark $\Delta$
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Q.91 mark $\cos^{-1}\left(\dfrac{\sin x + \cos x}{\sqrt{2}}\right)$
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Q.101 mark $\displaystyle\int \sqrt{\dfrac{1 - \cos 2x}{1 + \cos 2x}}\,dx$
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Q.111 mark $\displaystyle\int_{0}^{1} \dfrac{1}{3x - 4}\,dx$
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Q.121 mark $\dfrac{dy}{dx} = e^{3x - y}$
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Q.131 mark $\dfrac{d^2y}{dx^2} = 1 - \left(\dfrac{d^3y}{dx^3}\right)^2$
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Q.141 mark $y = 2x - 4$
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Q.151 mark $\vec{p}$
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Q.161 mark $Z = 4x + 3y$
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Q.171 mark $3P(A) = P(B) = \dfrac{3}{5}$
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Q.181 mark $\vec{a} = 3\hat{i} + 2\hat{j} + \lambda\hat{k}$
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 $\{1,\ 2,\ 3\}$
Tap an option to check your answer.
Q.201 mark $Z = x + 2y$
Tap an option to check your answer.

Section B

2 marks each

Q.212 marks $\sin\left[\cos^{-1}\cos\left(\dfrac{7\pi}{6}\right)\right]$
Q.22 (a)2 marks $\vec{a} = 2\hat{i} - 3\hat{j} + \hat{k}$
OR
Q.22 (b)2 marks $\overrightarrow{AB} = \hat{i} + 2\hat{j} - \hat{k}$
Q.232 marks $f(x) = 5x^{3/2} - 3x^{5/2}$
Q.242 marks $\dfrac{x - 2}{3} = \dfrac{y + 5}{2} = \dfrac{1 - z}{-6}$
Q.25 (a)2 marks $x^x$
OR
Q.25 (b)2 marks $y = P \cos ux + Q \sin ux$

Section C

3 marks each

Q.263 marks $A thin metallic wire in the shape of a circular ring has its enclosed area increasing at a uniform rate when heated. Show that the rate of change of circumference varies inversely as the radius.$
Q.273 marks $Z = \dfrac{2x}{5} + \dfrac{3y}{10}$
Q.283 marks $y\,dx + (x - y^3)\,dy = 0$
Q.29 (a)3 marks $\displaystyle\int \dfrac{x - \sin x}{1 - \cos x}\,dx$
OR
Q.29 (b)3 marks $\displaystyle\int_{0}^{2} \dfrac{1}{\sqrt{x^2 + 2x + 3}}\,dx$
Q.30 (a)3 marks $A = \{1,\ 2,\ 5\}$
OR
Q.30 (b)3 marks $A box contains 6 cards numbered 1 to 6. A student is asked to pick up two cards, one by one after replacement and note down the numbers on the cards. Let A be the event of getting sum of the numbers on two cards as 10, and B, the event of a number other than 4 on the first card selected. Find P(A and B) and find whether the events A and B are independent events or not.$
Q.31 (a)3 marks $55\hat{i} - 2\hat{j}$
OR
Q.31 (b)3 marks $\vec{a}$

Section D

5 marks each

Q.325 marks $Find cost (per kg) of each fertilizer A, B and C that the farmer needs to buy, such that 1 kg each of fertilizer A and C added to 2 kg of B costs him ₹ 400. Also, cost of each kg of fertilizer B and C added together is equal to cost of 1 kg of fertilizer A. However, cost of 3 kg of fertilizer B added to ₹ 200 is the same as cost of 1 kg of fertilizer A and C together. Use matrix method to find the solution.$
Q.33 (a)5 marks $\dfrac{-x - 3}{-5} = \dfrac{1 - y}{-2} = \dfrac{3z + 12}{9}$
OR
Q.33 (b)5 marks $\vec{r} = (\hat{i} + 2\hat{j} + \hat{k}) + \lambda(\hat{i} - \hat{j} + \hat{k})$
Q.345 marks $\{(x,\ y) : 9x^2 + 16y^2 = 144\}$
Q.35 (a)5 marks $y\sqrt{x^2 + 1} = \log \sqrt{x^2 + 1} - x$
OR
Q.35 (b)5 marks $x^{\cot x} + \dfrac{2x^2 - 3}{2x^2 - x + 2}$

Section E

There are three types of vaccines $A_1$ , $A_2$ , $A_3$ , available in the market to protect the population of the country from spread of certain infection. According to a survey conducted, it was found that 25% of the population was given Vaccine $A_1$ , 35% of the population was given Vaccine $A_2$ and 40% of the population was given Vaccine $A_3$ . The survey also stated that the probabilities that Vaccines $A_1$ , $A_2$ and $A_3$ would protect against the infection were 60%, 55% and 50% respectively. Based on the above information, answer the following questions : Find the probability that :
Q.36 (i)1 mark $A_2$
Q.36 (ii)1 mark $If a person is chosen randomly, he/she will be protected from the infection.$
Q.36 (iii) (a)2 marks $A_1$
OR
Q.36 (iii) (b)2 marks $A_3$
A company produces cylindrical tumblers, open from the top. Since they want uniformity in the product, they fix the surface area of the tumblers produced. Based on the above information, answer the following questions : If for a tumbler, V is its volume, h the height and r the radius of the circular base, then :
Q.37 (i)2 marks $Differentiate its volume with respect to radius of the base, where the surface area is constant.$
Q.37 (ii)2 marks $If the company wants to maximize the volume of each tumbler, then establish a relation between its height and the radius of the base.$
A school wants the students of class XII to do a project on ‘Sustainability’ keeping the world environment in mind. They select the student participants on the basis of an essay writing competition. 7 students out of 80 are selected for the project and are categorized into two sets such that : Girl students belong to Set $A = \{G_1,\ G_2,\ G_3,\ G_4\}$ , Boy students belong to Set $B = \{B_1,\ B_2,\ B_3\}$ . Based on the above information, answer the following questions :
Q.38 (i)1 mark $\to$
Q.38 (ii)1 mark $\to$
Q.38 (iii) (a)2 marks $\to$
OR
Q.38 (iii) (b)2 marks $f : B \to A$

Section A

1 mark each

Q.11 mark $A + A'$
Tap an option to check your answer.
Q.21 mark $B = [b_{ij}]_{m \times m}$
Tap an option to check your answer.
Q.31 mark $If A is a non-singular matrix, then which of the following is not true ?$
Tap an option to check your answer.
Q.41 mark $f(x) = \begin{cases} \dfrac{x^2 - 4x - 5}{x + 1}, & x \ne -1 \\[2mm] k, & x = -1 \end{cases}$
Tap an option to check your answer.
Q.51 mark $For the inverse trigonometric functions, which of the following Principal Value Branch is not correctly defined ?$
Tap an option to check your answer.
Q.61 mark $A = \begin{bmatrix} 0 & -3 & 4 \\ 1 & 0 & 2 \end{bmatrix}$
Tap an option to check your answer.
Q.71 mark $\vec{a} = 3\hat{i} + 2\hat{j} + \lambda\hat{k}$
Tap an option to check your answer.
Q.81 mark $3P(A) = P(B) = \dfrac{3}{5}$
Tap an option to check your answer.
Q.91 mark $\Delta$
Tap an option to check your answer.
Q.101 mark $\displaystyle\int \sqrt{\dfrac{1 + \cos x}{1 - \cos x}}\,dx$
Tap an option to check your answer.
Q.111 mark $\displaystyle\int_{0}^{1} (6x^2 - 4x + k)\,dx = 0$
Tap an option to check your answer.
Q.121 mark $\vec{p}$
Tap an option to check your answer.
Q.131 mark $y = \left(\dfrac{d^2y}{dx^2}\right)^2 - \lambda\,\dfrac{dy}{dx}$
Tap an option to check your answer.
Q.141 mark $Z = 4x + 3y$
Tap an option to check your answer.
Q.151 mark $y = 2x - 4$
Tap an option to check your answer.
Q.161 mark $\dfrac{dy}{dx} = e^{3x - y}$
Tap an option to check your answer.
Q.171 mark $\cos^{-1}\left(\dfrac{\sin x + \cos x}{\sqrt{2}}\right)$
Tap an option to check your answer.
Q.181 mark $f(x) = (x - 2)^2 + 5$
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 $\{1,\ 2,\ 3\}$
Tap an option to check your answer.
Q.201 mark $Z = x + 2y$
Tap an option to check your answer.

Section B

2 marks each

Q.212 marks $\tan\left[\cos^{-1}\left(\tan \dfrac{3\pi}{4}\right)\right]$
Q.222 marks $\dfrac{x - 2}{3} = \dfrac{y + 5}{2} = \dfrac{1 - z}{-6}$
Q.232 marks $f(x) = \dfrac{x}{3} + \dfrac{3}{x}$
Q.24 (a)2 marks $x^x$
OR
Q.24 (b)2 marks $y = P \cos ux + Q \sin ux$
Q.25 (a)2 marks $\vec{a} = 2\hat{i} - 3\hat{j} + \hat{k}$
OR
Q.25 (b)2 marks $\overrightarrow{AB} = \hat{i} + 2\hat{j} - \hat{k}$

Section C

3 marks each

Q.263 marks $The volume of a wooden block in the shape of a cube increases at a constant rate as the air becomes moist during the rainy season. Show that the rate of change of its surface area varies inversely as the length of edge of the cube.$
Q.27 (a)3 marks $A = \{1,\ 2,\ 5\}$
OR
Q.27 (b)3 marks $A box contains 6 cards numbered 1 to 6. A student is asked to pick up two cards, one by one after replacement and note down the numbers on the cards. Let A be the event of getting sum of the numbers on two cards as 10, and B, the event of a number other than 4 on the first card selected. Find P(A and B) and find whether the events A and B are independent events or not.$
Q.283 marks $(x - \sin y)\,dy + \tan y\,dx = 0$
Q.29 (a)3 marks $55\hat{i} - 2\hat{j}$
OR
Q.29 (b)3 marks $\vec{a}$
Q.30 (a)3 marks $\displaystyle\int \dfrac{x - \sin x}{1 - \cos x}\,dx$
OR
Q.30 (b)3 marks $\displaystyle\int_{0}^{2} \dfrac{1}{\sqrt{x^2 + 2x + 3}}\,dx$
Q.313 marks $Z = \dfrac{2x}{5} + \dfrac{3y}{10}$

Section D

5 marks each

Q.325 marks $Three students A, B and C go to a book-store to buy art books, story books and puzzle solving books. A buys one of each type of book for a total of ₹ 21. B buys 4 art books, 3 story books and 2 puzzle solving books for ₹ 60. C buys 6 art books, 2 story books and 3 puzzle solving books and pays ₹ 10 more than B. Use matrix method to find the cost of each type of book.$
Q.33 (a)5 marks $y\sqrt{x^2 + 1} = \log \sqrt{x^2 + 1} - x$
OR
Q.33 (b)5 marks $x^{\cot x} + \dfrac{2x^2 - 3}{2x^2 - x + 2}$
Q.345 marks $\left\{(x,\ y) : \dfrac{x^2}{25} + \dfrac{y^2}{25} = 1\right\}$
Q.35 (a)5 marks $\dfrac{-x - 3}{-5} = \dfrac{1 - y}{-2} = \dfrac{3z + 12}{9}$
OR
Q.35 (b)5 marks $\vec{r} = (\hat{i} + 2\hat{j} + \hat{k}) + \lambda(\hat{i} - \hat{j} + \hat{k})$

Section E

A company produces cylindrical tumblers, open from the top. Since they want uniformity in the product, they fix the surface area of the tumblers produced. Based on the above information, answer the following questions : If for a tumbler, V is its volume, h the height and r the radius of the circular base, then :
Q.36 (i)2 marks $Differentiate its volume with respect to radius of the base, where the surface area is constant.$
Q.36 (ii)2 marks $If the company wants to maximize the volume of each tumbler, then establish a relation between its height and the radius of the base.$
There are three types of vaccines $A_1$ , $A_2$ , $A_3$ , available in the market to protect the population of the country from spread of certain infection. According to a survey conducted, it was found that 25% of the population was given Vaccine $A_1$ , 35% of the population was given Vaccine $A_2$ and 40% of the population was given Vaccine $A_3$ . The survey also stated that the probabilities that Vaccines $A_1$ , $A_2$ and $A_3$ would protect against the infection were 60%, 55% and 50% respectively. Based on the above information, answer the following questions : Find the probability that :
Q.37 (i)1 mark $A_2$
Q.37 (ii)1 mark $If a person is chosen randomly, he/she will be protected from the infection.$
Q.37 (iii) (a)2 marks $A_1$
OR
Q.37 (iii) (b)2 marks $A_3$
A school wants the students of class XII to do a project on ‘Sustainability’ keeping the world environment in mind. They select the student participants on the basis of an essay writing competition. 7 students out of 80 are selected for the project and are categorized into two sets such that : Girl students belong to Set $A = \{G_1,\ G_2,\ G_3,\ G_4\}$ , Boy students belong to Set $B = \{B_1,\ B_2,\ B_3\}$ . Based on the above information, answer the following questions :
Q.38 (i)1 mark $\to$
Q.38 (ii)1 mark $\to$
Q.38 (iii) (a)2 marks $\to$
OR
Q.38 (iii) (b)2 marks $f : B \to A$