CBSE Class 12 Mathematics 2024 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 $\begin{bmatrix} a & c & 0 \\ b & d & 0 \\ 0 & 0 & 5 \end{bmatrix}$
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Q.21 mark $A^{-1} = \dfrac{1}{7}\begin{bmatrix} 2 & 1 \\ -3 & 2 \end{bmatrix}$
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Q.31 mark $A = \begin{bmatrix} 2 & 1 \\ -4 & -2 \end{bmatrix}$
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Q.41 mark $A = \begin{bmatrix} -2 & 0 & 0 \\ 1 & 2 & 3 \\ 5 & 1 & -1 \end{bmatrix}$
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Q.51 mark $[1 \quad x]\begin{bmatrix} 4 & 0 \\ -2 & 0 \end{bmatrix} = 0$
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Q.61 mark $e^{2x}$
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Q.71 mark $x = 0$
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Q.81 mark $\displaystyle\int_{0}^{3} \dfrac{dx}{\sqrt{9 - x^2}}$
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Q.91 mark $x\,dy + y\,dx = 0$
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Q.101 mark $(x + 2y^2)\dfrac{dy}{dx} = y\ (y > 0)$
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Q.111 mark $\vec{a}$
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Q.121 mark $\vec{a} = 2\hat{i} - \hat{j} + \hat{k}$
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Q.131 mark $\vec{a}$
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Q.141 mark $(1, -1, 0)$
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Q.151 mark $\dfrac{1-x}{2} = \dfrac{y-1}{3} = \dfrac{z}{1}$
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Q.161 mark $Z = 4x + y$
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Q.171 mark $The probability distribution of a random variable X is : X 01234 P(X) 0.1 k 2k k 0.1 where k is some unknown constant. The probability that the random variable X takes the value 2 is :$
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Q.181 mark $f(x) = kx - \sin x$
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Questions No. 19 & 20, are Assertion (A) and Reason (R) based questions carrying 1 mark each. 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 $R = \{(x, y) : (x + y)$
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Q.201 mark $Z = x + 2y$
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X	0	1	2	3	4
P(X)	0.1	k	2k	k	0.1

Section B

2 marks each

Q.21 (a)2 marks $\tan^{-1}\left(\dfrac{\cos x}{1 - \sin x}\right)$
OR
Q.21 (b)2 marks $\tan^{-1}(1) + \cos^{-1}\left(-\dfrac{1}{2}\right) + \sin^{-1}\left(-\dfrac{1}{\sqrt{2}}\right)$
Q.22 (a)2 marks $y = \cos^{3}(\sec^{2} 2t)$
OR
Q.22 (b)2 marks $x^{y} = e^{x-y}$
Q.232 marks $f(x) = x^{4} - 4x^{3} + 10$
Q.242 marks $^{3}$
Q.252 marks $\displaystyle\int \dfrac{1}{x(x^{2} - 1)}\,dx$

Section C

3 marks each

Q.263 marks $y = (\sin x)^{x} \cdot x^{\sin x} + a^{x}$
Q.27 (a)3 marks $\displaystyle\int_{0}^{\pi/4} \dfrac{x\,dx}{1 + \cos 2x + \sin 2x}$
OR
Q.27 (b)3 marks $\displaystyle\int e^{x}\left[\dfrac{1}{(1 + x^{2})^{3/2}} + \dfrac{x}{\sqrt{1 + x^{2}}}\right] dx$
Q.283 marks $\displaystyle\int \dfrac{3x + 5}{\sqrt{x^{2} + 2x + 4}}\,dx$
Q.29 (a)3 marks $\dfrac{dy}{dx} = y \cot 2x$
OR
Q.29 (b)3 marks $\left(xe^{y/x} + y\right) dx = x\,dy$
Q.303 marks $Z = 2x + 3y$
Q.31 (a)3 marks $A card from a well shuffled deck of 52 playing cards is lost. From the remaining cards of the pack, a card is drawn at random and is found to be a King. Find the probability of the lost card being a King.$
OR
Q.31 (b)3 marks $A biased die is twice as likely to show an even number as an odd number. If such a die is thrown twice, find the probability distribution of the number of sixes. Also, find the mean of the distribution.$

Section D

5 marks each

Q.32 (a)5 marks $y = x\,|x|$
OR
Q.32 (b)5 marks $9x^{2} + 25y^{2} = 225$
Q.33 (a)5 marks $A = R - \{5\}$
OR
Q.33 (b)5 marks $S = \{(a, b) : \text{where } a - b + \sqrt{2} \text{ is an irrational number}\}$
Q.345 marks $A = \begin{bmatrix} 2 & 1 & -3 \\ 3 & 2 & 1 \\ 1 & 2 & -1 \end{bmatrix}$
Q.35 (a)5 marks $\dfrac{x}{2} = \dfrac{2y-6}{4} = \dfrac{1-z}{-1}$
OR
Q.35 (b)5 marks $\dfrac{x-1}{-3} = \dfrac{y-2}{2k} = \dfrac{z-3}{2}$

Section E

A store has been selling calculators at ₹ 350 each. A market survey indicates that a reduction in price (p) of calculator increases the number of units ( $x$ ) sold. The relation between the price and quantity sold is given by the demand function $p = 450 - \dfrac{1}{2}x$ . Based on the above information, answer the following questions :
Q.36 (i)2 marks $x$
Q.36 (ii)2 marks $What rebate in price of calculator should the store give to maximise the revenue ?$
An instructor at the astronomical centre shows three among the brightest stars in a particular constellation. Assume that the telescope is located at O(0, 0, 0) and the three stars have their locations at the points D, A and V having position vectors $2\hat{i} + 3\hat{j} + 4\hat{k}$ , $7\hat{i} + 5\hat{j} + 8\hat{k}$ and $-3\hat{i} + 7\hat{j} + 11\hat{k}$ respectively. Based on the above information, answer the following questions :
Q.37 (i)1 mark $How far is the star V from star A ?$
Q.37 (ii)1 mark $\overrightarrow{DA}$
Q.37 (iii) (a)2 marks $\angle VDA$
OR
Q.37 (iii) (b)2 marks $\overrightarrow{DV}$
Rohit, Jaspreet and Alia appeared for an interview for three vacancies in the same post. The probability of Rohit's selection is $\dfrac{1}{5}$ , Jaspreet's selection is $\dfrac{1}{3}$ and Alia's selection is $\dfrac{1}{4}$ . The event of selection is independent of each other. Based on the above information, answer the following questions :
Q.38 (i)1 mark $What is the probability that at least one of them is selected ?$
Q.38 (ii)1 mark $P(G\,|\,\overline{H})$
Q.38 (iii) (a)2 marks $Find the probability that exactly one of them is selected.$
OR
Q.38 (iii) (b)2 marks $Find the probability that exactly two of them are selected.$

Section A

1 mark each

Q.11 mark $\dfrac{1-x}{2} = \dfrac{y-1}{3} = \dfrac{z}{1}$
Tap an option to check your answer.
Q.21 mark $Z = 4x + y$
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Q.31 mark $The probability distribution of a random variable X is : X 01234 P(X) 0.1 k 2k k 0.1 where k is some unknown constant. The probability that the random variable X takes the value 2 is :$
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Q.41 mark $A = [a_{ij}] = \begin{bmatrix} 2 & -1 & 5 \\ 1 & 3 & 2 \\ 5 & 0 & 4 \end{bmatrix}$
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Q.51 mark $A = \begin{bmatrix} 1 & 3 \\ 3 & 4 \end{bmatrix}$
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Q.61 mark $e^{x^{2}y} = c$
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Q.71 mark $f(x) = \begin{cases} x^2 - c^2, & \text{if } x < 4 \\ cx + 20, & \text{if } x \geq 4 \end{cases}$
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Q.81 mark $\displaystyle\int_{-1}^{1} |x|\ dx$
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Q.91 mark $\log\left(\dfrac{dy}{dx}\right) = 3x + 4y;\ y(0) = 0$
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Q.101 mark $\begin{bmatrix} a & c & 0 \\ b & d & 0 \\ 0 & 0 & 5 \end{bmatrix}$
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Q.111 mark $A = \begin{bmatrix} 2 & 1 \\ -4 & -2 \end{bmatrix}$
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Q.121 mark $A^{-1} = \dfrac{1}{7}\begin{bmatrix} 2 & 1 \\ -3 & 2 \end{bmatrix}$
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Q.131 mark $(x + 2y^2)\dfrac{dy}{dx} = y\ (y > 0)$
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Q.141 mark $\vec{r} = \hat{i} + \hat{j} - \hat{k} + \lambda(3\hat{i} - \hat{j})$
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Q.151 mark $\vec{a} = 2\hat{i} - \hat{j} + \hat{k}$
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Q.161 mark $\vec{a}$
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Q.171 mark $\vec{a}$
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Q.181 mark $f(x) = kx - \sin x$
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Questions No. 19 & 20, are Assertion (A) and Reason (R) based questions carrying 1 mark each. 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 $Z = x + 2y$
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Q.201 mark $R = \{(x, y) : (x + y)$
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X	0	1	2	3	4
P(X)	0.1	k	2k	k	0.1

Section B

2 marks each

Q.212 marks $^{3}$
Q.22 (a)2 marks $\tan^{-1}\left(\dfrac{\cos x}{1 - \sin x}\right)$
OR
Q.22 (b)2 marks $\tan^{-1}(1) + \cos^{-1}\left(-\dfrac{1}{2}\right) + \sin^{-1}\left(-\dfrac{1}{\sqrt{2}}\right)$
Q.232 marks $f(x) = \dfrac{4 \sin x}{2 + \cos x} - x$
Q.24 (a)2 marks $y = \cos^{3}(\sec^{2} 2t)$
OR
Q.24 (b)2 marks $x^{y} = e^{x-y}$
Q.252 marks $\displaystyle\int_{-1/2}^{1/2} \cos x \cdot \log\left(\dfrac{1+x}{1-x}\right) dx$

Section C

3 marks each

Q.263 marks $x^{y} + y^{x} = a^{b}$
Q.27 (a)3 marks $\dfrac{dy}{dx} = y \cot 2x$
OR
Q.27 (b)3 marks $\left(xe^{y/x} + y\right) dx = x\,dy$
Q.283 marks $\displaystyle\int \dfrac{2x + 3}{x^{2}(x + 3)}\,dx$
Q.29 (a)3 marks $A card from a well shuffled deck of 52 playing cards is lost. From the remaining cards of the pack, a card is drawn at random and is found to be a King. Find the probability of the lost card being a King.$
OR
Q.29 (b)3 marks $A biased die is twice as likely to show an even number as an odd number. If such a die is thrown twice, find the probability distribution of the number of sixes. Also, find the mean of the distribution.$
Q.303 marks $Z = x + 3y$
Q.31 (a)3 marks $\displaystyle\int_{0}^{\pi/4} \dfrac{x\,dx}{1 + \cos 2x + \sin 2x}$
OR
Q.31 (b)3 marks $\displaystyle\int e^{x}\left[\dfrac{1}{(1 + x^{2})^{3/2}} + \dfrac{x}{\sqrt{1 + x^{2}}}\right] dx$

Section D

5 marks each

Q.32 (a)5 marks $A = R - \{5\}$
OR
Q.32 (b)5 marks $S = \{(a, b) : \text{where } a - b + \sqrt{2} \text{ is an irrational number}\}$
Q.33 (a)5 marks $\dfrac{x}{2} = \dfrac{2y-6}{4} = \dfrac{1-z}{-1}$
OR
Q.33 (b)5 marks $\dfrac{x-1}{-3} = \dfrac{y-2}{2k} = \dfrac{z-3}{2}$
Q.345 marks $\begin{bmatrix} 1 & 2 & -3 \\ 3 & 2 & -2 \\ 2 & -1 & 1 \end{bmatrix}\begin{bmatrix} 0 & 1 & 2 \\ -7 & 7 & -7 \\ -7 & 5 & -4 \end{bmatrix}$
Q.35 (a)5 marks $y = x\,|x|$
OR
Q.35 (b)5 marks $9x^{2} + 25y^{2} = 225$

Section E

An instructor at the astronomical centre shows three among the brightest stars in a particular constellation. Assume that the telescope is located at O(0, 0, 0) and the three stars have their locations at the points D, A and V having position vectors $2\hat{i} + 3\hat{j} + 4\hat{k}$ , $7\hat{i} + 5\hat{j} + 8\hat{k}$ and $-3\hat{i} + 7\hat{j} + 11\hat{k}$ respectively. Based on the above information, answer the following questions :
Q.36 (i)1 mark $How far is the star V from star A ?$
Q.36 (ii)1 mark $\overrightarrow{DA}$
Q.36 (iii) (a)2 marks $\angle VDA$
OR
Q.36 (iii) (b)2 marks $\overrightarrow{DV}$
Rohit, Jaspreet and Alia appeared for an interview for three vacancies in the same post. The probability of Rohit's selection is $\dfrac{1}{5}$ , Jaspreet's selection is $\dfrac{1}{3}$ and Alia's selection is $\dfrac{1}{4}$ . The event of selection is independent of each other. Based on the above information, answer the following questions :
Q.37 (i)1 mark $What is the probability that at least one of them is selected ?$
Q.37 (ii)1 mark $P(G\,|\,\overline{H})$
Q.37 (iii) (a)2 marks $Find the probability that exactly one of them is selected.$
OR
Q.37 (iii) (b)2 marks $Find the probability that exactly two of them are selected.$
A store has been selling calculators at ₹ 350 each. A market survey indicates that a reduction in price (p) of calculator increases the number of units ( $x$ ) sold. The relation between the price and quantity sold is given by the demand function $p = 450 - \dfrac{1}{2}x$ . Based on the above information, answer the following questions :
Q.38 (i)2 marks $x$
Q.38 (ii)2 marks $What rebate in price of calculator should the store give to maximise the revenue ?$

Section A

1 mark each

Q.11 mark $\vec{a}$
Tap an option to check your answer.
Q.21 mark $\vec{a} = 2\hat{i} - \hat{j} + \hat{k}$
Tap an option to check your answer.
Q.31 mark $\vec{a}$
Tap an option to check your answer.
Q.41 mark $A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}$
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Q.51 mark $A = \begin{bmatrix} 1 & -1 & 2 \\ 0 & 2 & -3 \\ 3 & -2 & 4 \end{bmatrix}$
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Q.61 mark $x^2$
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Q.71 mark $f(x) = |\,x\,| + |\,x - 2\,|$
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Q.81 mark $\displaystyle\int_{0}^{\pi} \tan^{2}\left(\dfrac{\theta}{3}\right) d\theta$
Tap an option to check your answer.
Q.91 mark $\dfrac{dy}{dx} + \dfrac{2}{x}y = 0$
Tap an option to check your answer.
Q.101 mark $\dfrac{1-x}{2} = \dfrac{y-1}{3} = \dfrac{z}{1}$
Tap an option to check your answer.
Q.111 mark $Z = 4x + y$
Tap an option to check your answer.
Q.121 mark $The probability distribution of a random variable X is : X 01234 P(X) 0.1 k 2k k 0.1 where k is some unknown constant. The probability that the random variable X takes the value 2 is :$
Tap an option to check your answer.
Q.131 mark $f(x) = kx - \sin x$
Tap an option to check your answer.
Q.141 mark $\vec{a} = \hat{i} - \hat{j}$
Tap an option to check your answer.
Q.151 mark $\begin{bmatrix} a & c & 0 \\ b & d & 0 \\ 0 & 0 & 5 \end{bmatrix}$
Tap an option to check your answer.
Q.161 mark $A^{-1} = \dfrac{1}{7}\begin{bmatrix} 2 & 1 \\ -3 & 2 \end{bmatrix}$
Tap an option to check your answer.
Q.171 mark $A = \begin{bmatrix} 2 & 1 \\ -4 & -2 \end{bmatrix}$
Tap an option to check your answer.
Q.181 mark $(x + 2y^2)\dfrac{dy}{dx} = y\ (y > 0)$
Tap an option to check your answer.
Questions No. 19 & 20, are Assertion (A) and Reason (R) based questions carrying 1 mark each. 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 $R = \{(x, y) : (x + y)$
Tap an option to check your answer.
Q.201 mark $Z = x + 2y$
Tap an option to check your answer.

X	0	1	2	3	4
P(X)	0.1	k	2k	k	0.1

Section B

2 marks each

Q.21 (a)2 marks $y = \cos^{3}(\sec^{2} 2t)$
OR
Q.21 (b)2 marks $x^{y} = e^{x-y}$
Q.222 marks $^{3}$
Q.232 marks $f(x) = \sin x + \cos x$
Q.24 (a)2 marks $\tan^{-1}\left(\dfrac{\cos x}{1 - \sin x}\right)$
OR
Q.24 (b)2 marks $\tan^{-1}(1) + \cos^{-1}\left(-\dfrac{1}{2}\right) + \sin^{-1}\left(-\dfrac{1}{\sqrt{2}}\right)$
Q.252 marks $\displaystyle\int \dfrac{2x}{(x^{2} + 1)(x^{2} - 4)}\,dx$

Section C

3 marks each

Q.263 marks $\dfrac{dy}{dx}$
Q.27 (a)3 marks $\dfrac{dy}{dx} = y \cot 2x$
OR
Q.27 (b)3 marks $\left(xe^{y/x} + y\right) dx = x\,dy$
Q.283 marks $\displaystyle\int \sec^{3}\theta\,d\theta$
Q.29 (a)3 marks $A card from a well shuffled deck of 52 playing cards is lost. From the remaining cards of the pack, a card is drawn at random and is found to be a King. Find the probability of the lost card being a King.$
OR
Q.29 (b)3 marks $A biased die is twice as likely to show an even number as an odd number. If such a die is thrown twice, find the probability distribution of the number of sixes. Also, find the mean of the distribution.$
Q.303 marks $Z = 3x - 4y$
Q.31 (a)3 marks $\displaystyle\int_{0}^{\pi/4} \dfrac{x\,dx}{1 + \cos 2x + \sin 2x}$
OR
Q.31 (b)3 marks $\displaystyle\int e^{x}\left[\dfrac{1}{(1 + x^{2})^{3/2}} + \dfrac{x}{\sqrt{1 + x^{2}}}\right] dx$

Section D

5 marks each

Q.32 (a)5 marks $A = R - \{5\}$
OR
Q.32 (b)5 marks $S = \{(a, b) : \text{where } a - b + \sqrt{2} \text{ is an irrational number}\}$
Q.33 (a)5 marks $\dfrac{x}{2} = \dfrac{2y-6}{4} = \dfrac{1-z}{-1}$
OR
Q.33 (b)5 marks $\dfrac{x-1}{-3} = \dfrac{y-2}{2k} = \dfrac{z-3}{2}$
Q.345 marks $A^{-1}$
Q.35 (a)5 marks $y = x\,|x|$
OR
Q.35 (b)5 marks $9x^{2} + 25y^{2} = 225$

Section E

Rohit, Jaspreet and Alia appeared for an interview for three vacancies in the same post. The probability of Rohit's selection is $\dfrac{1}{5}$ , Jaspreet's selection is $\dfrac{1}{3}$ and Alia's selection is $\dfrac{1}{4}$ . The event of selection is independent of each other. Based on the above information, answer the following questions :
Q.36 (i)1 mark $What is the probability that at least one of them is selected ?$
Q.36 (ii)1 mark $P(G\,|\,\overline{H})$
Q.36 (iii) (a)2 marks $Find the probability that exactly one of them is selected.$
OR
Q.36 (iii) (b)2 marks $Find the probability that exactly two of them are selected.$
A store has been selling calculators at ₹ 350 each. A market survey indicates that a reduction in price (p) of calculator increases the number of units ( $x$ ) sold. The relation between the price and quantity sold is given by the demand function $p = 450 - \dfrac{1}{2}x$ . Based on the above information, answer the following questions :
Q.37 (i)2 marks $x$
Q.37 (ii)2 marks $What rebate in price of calculator should the store give to maximise the revenue ?$
An instructor at the astronomical centre shows three among the brightest stars in a particular constellation. Assume that the telescope is located at O(0, 0, 0) and the three stars have their locations at the points D, A and V having position vectors $2\hat{i} + 3\hat{j} + 4\hat{k}$ , $7\hat{i} + 5\hat{j} + 8\hat{k}$ and $-3\hat{i} + 7\hat{j} + 11\hat{k}$ respectively. Based on the above information, answer the following questions :
Q.38 (i)1 mark $How far is the star V from star A ?$
Q.38 (ii)1 mark $\overrightarrow{DA}$
Q.38 (iii) (a)2 marks $\angle VDA$
OR
Q.38 (iii) (b)2 marks $\overrightarrow{DV}$