CBSE Class 12 Mathematics 2023 question paper (65/1)

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 $A^2 - 3A + I = O$
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Q.21 mark $|A| = 2$
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Q.31 mark $3 \times 3$
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Q.41 mark $A = \begin{bmatrix} 3 & 4 \\ 5 & 2 \end{bmatrix}$
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Q.51 mark $\frac{d}{dx}(f(x)) = \log x$
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Q.61 mark $\int\limits_{0}^{\frac{\pi}{6}} \sec^2\left(x - \frac{\pi}{6}\right) dx$
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Q.71 mark $\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^3 = \sin y$
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Q.81 mark $2\hat{i} + p\hat{j} + \hat{k}$
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Q.91 mark $(\hat{i} \times \hat{j}) \cdot \hat{j} + (\hat{j} \times \hat{i}) \cdot \hat{k}$
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Q.101 mark $\vec{a} + \vec{b} = \hat{i}$
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Q.111 mark $\frac{x - 1}{2} = \frac{1 - y}{3} = \frac{2z - 1}{12}$
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Q.121 mark $P\left(\frac{A}{B}\right) = 0{\cdot}3$
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Q.131 mark $f(x) = \begin{cases} 3x + 5, & x \ge 2 \\ kx^2, & x < 2 \end{cases}$
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Q.141 mark $A = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$
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Q.151 mark $x \, dy - (1 + x^2) \, dx = dx$
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Q.161 mark $f(x) = a(x - \cos x)$
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Q.171 mark $The corner points of the feasible region in the graphical representation of a linear programming problem are (2, 72), (15, 20) and (40, 15). If z = 18x + 9y be the objective function, then :$
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Q.181 mark $x - y \ge 0$
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Questions number 19 and 20 are Assertion and Reason 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 $f(x) = 2 \sin^{-1} x + \frac{3\pi}{2}$
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Q.201 mark $(3, -1, 3)$
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Section B

2 marks each

Q.21 (a)2 marks $f : A \to B$
OR
Q.21 (b)2 marks $\sin^{-1}\left(\sin \frac{3\pi}{4}\right) + \cos^{-1}\left(\cos \frac{3\pi}{4}\right) + \tan^{-1}(1)$
Q.222 marks $3\sqrt{3}$
Q.23 (a)2 marks $\vec{a}$
OR
Q.23 (b)2 marks $x = 2\lambda + 2$
Q.242 marks $y = \left(x + \sqrt{x^2 - 1}\right)^2$
Q.252 marks $f(x) = \frac{16 \sin x}{4 + \cos x} - x$

Section C

3 marks each

Q.263 marks $\int\limits_{0}^{\frac{\pi}{2}} [\log (\sin x) - \log (2 \cos x)] \, dx$
Q.273 marks $\int \frac{1}{\sqrt{x}\left(\sqrt{x} + 1\right)\left(\sqrt{x} + 2\right)} \, dx$
Q.28 (a)3 marks $\frac{dy}{dx} + \sec^2 x \cdot y = \tan x \cdot \sec^2 x$
OR
Q.28 (b)3 marks $x \, dy - y \, dx - \sqrt{x^2 + y^2} \, dx = 0$
Q.293 marks $4x + y \ge 80$
Q.30 (a)3 marks $\frac{k}{2}$
OR
Q.30 (b)3 marks $P(A \cap \overline{B}) = \frac{1}{4}$
Q.31 (a)3 marks $\int\limits_{0}^{\frac{\pi}{2}} e^x \sin x \, dx$
OR
Q.31 (b)3 marks $\int \frac{1}{\cos(x - a) \, \cos(x - b)} \, dx$

X	1	2	3
P(X)	$\frac{k}{2}$	$\frac{k}{3}$	$\frac{k}{6}$

Section D

5 marks each

Q.325 marks $\mathbb{R}$
Q.33 (a)5 marks $A = \begin{bmatrix} 1 & 2 & -2 \\ -1 & 3 & 0 \\ 0 & -2 & 1 \end{bmatrix}$
OR
Q.33 (b)5 marks $x + 2y + 3z = 6$
Q.34 (a)5 marks $(1, 2, -4)$
OR
Q.34 (b)5 marks $Find the equations of the line passing through the points A(1, 2, 3) and B(3, 5, 9). Hence, find the coordinates of the points on this line which are at a distance of 14 units from point B.$
Q.355 marks $x^2 = y$

Section E

There are different types of Yoga which involve the usage of different poses of Yoga Asanas, Meditation and Pranayam as shown in the figure below : The Venn diagram below represents the probabilities of three different types of Yoga, A, B and C performed by the people of a society. Further, it is given that probability of a member performing type C Yoga is $0{\cdot}44$ . On the basis of the above information, answer the following questions :
Q.36 (i)1 mark $Find the value of x.$
Q.36 (ii)1 mark $Find the value of y.$
Q.36 (iii) (a)2 marks $P\left(\frac{C}{B}\right)$
OR
Q.36 (iii) (b)2 marks $Find the probability that a randomly selected person of the society does Yoga of type A or B but not C.$
A tank, as shown in the figure below, formed using a combination of a cylinder and a cone, offers better drainage as compared to a flat bottomed tank. A tap is connected to such a tank whose conical part is full of water. Water is dripping out from a tap at the bottom at the uniform rate of $2 \text{ cm}^3/\text{s}$ . The semi-vertical angle of the conical tank is $45^\circ$ . On the basis of given information, answer the following questions :
Q.37 (i)1 mark $Find the volume of water in the tank in terms of its radius r.$
Q.37 (ii)1 mark $r = 2\sqrt{2}$
Q.37 (iii) (a)2 marks $r = 2\sqrt{2}$
OR
Q.37 (iii) (b)2 marks $Find the rate of change of height 'h' at an instant when slant height is 4 cm.$
The equation of the path traced by a roller-coaster is given by the polynomial $f(x) = a(x + 9)(x + 1)(x - 3)$ . If the roller-coaster crosses y-axis at a point $(0, -1)$ , answer the following :
Q.38 (i)2 marks $Find the value of 'a'.$
Q.38 (ii)2 marks $f''(x)$

Section A

1 mark each

Q.11 mark $\int \frac{1 + \tan x}{1 - \tan x} \, dx$
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Q.21 mark $\int\limits_{0}^{\frac{\pi}{6}} \sec^2\left(x - \frac{\pi}{6}\right) dx$
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Q.31 mark $\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^3 = \sin y$
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Q.41 mark $2\hat{i} + p\hat{j} + \hat{k}$
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Q.51 mark $\hat{i} - b\hat{j} + \hat{k}$
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Q.61 mark $\vec{a} + \vec{b} = \hat{i}$
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Q.71 mark $Direction cosines of a line perpendicular to both x-axis and z-axis are :$
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Q.81 mark $P\left(\frac{A}{B}\right) = 0{\cdot}3$
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Q.91 mark $f(x) = \begin{cases} k(3x^2 - 5x), & x \le 0 \\ \cos x, & x > 0 \end{cases}$
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Q.101 mark $A = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$
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Q.111 mark $x \, dy - (1 + x^2) \, dx = dx$
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Q.121 mark $f(x) = a(x - \cos x)$
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Q.131 mark $The corner points of the feasible region in the graphical representation of a linear programming problem are (2, 72), (15, 20) and (40, 15). If z = 18x + 9y be the objective function, then :$
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Q.141 mark $x - y \ge 0$
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Q.151 mark $A^2 - 3A + I = O$
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Q.161 mark $\left|\frac{A^{-1}}{2}\right| = \frac{1}{k|A|}$
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Q.171 mark $3 \times 3$
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Q.181 mark $A = \begin{bmatrix} 3 & 4 \\ 5 & 2 \end{bmatrix}$
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Questions number 19 and 20 are Assertion and Reason 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 $(3, -1, 3)$
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Q.201 mark $n^m$
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Section B

2 marks each

Q.21 (a)2 marks $\vec{a}$
OR
Q.21 (b)2 marks $x = 2\lambda + 2$
Q.222 marks $y = \left(x + \sqrt{x^2 - 1}\right)^2$
Q.232 marks $f(x) = \log (2 + x) - \frac{x}{2 + x}$
Q.24 (a)2 marks $f : A \to B$
OR
Q.24 (b)2 marks $\sin^{-1}\left(\sin \frac{3\pi}{4}\right) + \cos^{-1}\left(\cos \frac{3\pi}{4}\right) + \tan^{-1}(1)$
Q.252 marks $\vec{a}$

Section C

3 marks each

Q.26 (a)3 marks $\frac{dx}{dy} = \frac{e^{x/y}\left(\frac{x}{y} - 1\right)}{1 + e^{x/y}}$
OR
Q.26 (b)3 marks $\frac{dy}{dx} + \cot x \cdot y = \cos^2 x$
Q.273 marks $x + y \le 300$
Q.28 (a)3 marks $\frac{k}{2}$
OR
Q.28 (b)3 marks $P(A \cap \overline{B}) = \frac{1}{4}$
Q.29 (a)3 marks $\int\limits_{0}^{\frac{\pi}{2}} e^x \sin x \, dx$
OR
Q.29 (b)3 marks $\int \frac{1}{\cos(x - a) \, \cos(x - b)} \, dx$
Q.303 marks $\int\limits_{0}^{\frac{\pi}{2}} |\sin x - \cos x| \, dx$
Q.313 marks $\int \frac{1}{\sqrt{x}\left(\sqrt{x} + 1\right)\left(\sqrt{x} + 2\right)} \, dx$

X	1	2	3
P(X)	$\frac{k}{2}$	$\frac{k}{3}$	$\frac{k}{6}$

Section D

5 marks each

Q.32 (a)5 marks $(1, 2, -4)$
OR
Q.32 (b)5 marks $Find the equations of the line passing through the points A(1, 2, 3) and B(3, 5, 9). Hence, find the coordinates of the points on this line which are at a distance of 14 units from point B.$
Q.335 marks $\{(x, y) : x^2 + y^2 \le 1 \le x + y\}$
Q.345 marks $\mathbb{R}$
Q.35 (a)5 marks $A = \begin{bmatrix} 1 & 2 & -2 \\ -1 & 3 & 0 \\ 0 & -2 & 1 \end{bmatrix}$
OR
Q.35 (b)5 marks $x + 2y + 3z = 6$

Section E

A tank, as shown in the figure below, formed using a combination of a cylinder and a cone, offers better drainage as compared to a flat bottomed tank. A tap is connected to such a tank whose conical part is full of water. Water is dripping out from a tap at the bottom at the uniform rate of $2 \text{ cm}^3/\text{s}$ . The semi-vertical angle of the conical tank is $45^\circ$ . On the basis of given information, answer the following questions :
Q.36 (i)1 mark $Find the volume of water in the tank in terms of its radius r.$
Q.36 (ii)1 mark $r = 2\sqrt{2}$
Q.36 (iii) (a)2 marks $r = 2\sqrt{2}$
OR
Q.36 (iii) (b)2 marks $Find the rate of change of height 'h' at an instant when slant height is 4 cm.$
There are different types of Yoga which involve the usage of different poses of Yoga Asanas, Meditation and Pranayam as shown in the figure below : The Venn diagram below represents the probabilities of three different types of Yoga, A, B and C performed by the people of a society. Further, it is given that probability of a member performing type C Yoga is $0{\cdot}44$ . On the basis of the above information, answer the following questions :
Q.37 (i)1 mark $Find the value of x.$
Q.37 (ii)1 mark $Find the value of y.$
Q.37 (iii) (a)2 marks $P\left(\frac{C}{B}\right)$
OR
Q.37 (iii) (b)2 marks $Find the probability that a randomly selected person of the society does Yoga of type A or B but not C.$
The equation of the path traced by a roller-coaster is given by the polynomial $f(x) = a(x + 9)(x + 1)(x - 3)$ . If the roller-coaster crosses y-axis at a point $(0, -1)$ , answer the following :
Q.38 (i)2 marks $Find the value of 'a'.$
Q.38 (ii)2 marks $f''(x)$

Section A

1 mark each

Q.11 mark $\vec{a} + \vec{b} = \hat{i}$
Tap an option to check your answer.
Q.21 mark $The direction ratios of a line parallel to z-axis are :$
Tap an option to check your answer.
Q.31 mark $Ashima can hit a target 2 out of 3 times. She tried to hit the target twice. The probability that she missed the target exactly once is$
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Q.41 mark $f(x) = |x| - x$
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Q.51 mark $A = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$
Tap an option to check your answer.
Q.61 mark $x \, dy - (1 + x^2) \, dx = dx$
Tap an option to check your answer.
Q.71 mark $f(x) = a(x - \cos x)$
Tap an option to check your answer.
Q.81 mark $The corner points of the feasible region in the graphical representation of a linear programming problem are (2, 72), (15, 20) and (40, 15). If z = 18x + 9y be the objective function, then :$
Tap an option to check your answer.
Q.91 mark $x - y \ge 0$
Tap an option to check your answer.
Q.101 mark $A^2 - 3A + I = O$
Tap an option to check your answer.
Q.111 mark $\begin{bmatrix} x & 2 \\ 3 & x - 1 \end{bmatrix}$
Tap an option to check your answer.
Q.121 mark $3 \times 3$
Tap an option to check your answer.
Q.131 mark $A = \begin{bmatrix} 3 & 4 \\ 5 & 2 \end{bmatrix}$
Tap an option to check your answer.
Q.141 mark $\frac{2}{1 + \cos 2x}$
Tap an option to check your answer.
Q.151 mark $\int\limits_{0}^{\frac{\pi}{6}} \sec^2\left(x - \frac{\pi}{6}\right) dx$
Tap an option to check your answer.
Q.161 mark $\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^3 = \sin y$
Tap an option to check your answer.
Q.171 mark $2\hat{i} + p\hat{j} + \hat{k}$
Tap an option to check your answer.
Q.181 mark $\lambda$
Tap an option to check your answer.
Questions number 19 and 20 are Assertion and Reason 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 $f(x) = 2 \sin^{-1} x + \frac{3\pi}{2}$
Tap an option to check your answer.
Q.201 mark $(3, -1, 3)$
Tap an option to check your answer.

Section B

2 marks each

Q.212 marks $If the product of two positive numbers is 9, find the numbers so that the sum of their squares is minimum.$
Q.22 (a)2 marks $f : A \to B$
OR
Q.22 (b)2 marks $\sin^{-1}\left(\sin \frac{3\pi}{4}\right) + \cos^{-1}\left(\cos \frac{3\pi}{4}\right) + \tan^{-1}(1)$
Q.232 marks $3\sqrt{3}$
Q.24 (a)2 marks $\vec{a}$
OR
Q.24 (b)2 marks $x = 2\lambda + 2$
Q.252 marks $x = \sqrt{a^{\tan^{-1} t}}$

Section C

3 marks each

Q.26 (a)3 marks $\int\limits_{0}^{\frac{\pi}{2}} e^x \sin x \, dx$
OR
Q.26 (b)3 marks $\int \frac{1}{\cos(x - a) \, \cos(x - b)} \, dx$
Q.273 marks $\int\limits_{0}^{\frac{\pi}{2}} [\log (\sin x) - \log (2 \cos x)] \, dx$
Q.283 marks $\int \frac{x^2}{(x^2 + 4)(x^2 + 9)} \, dx$
Q.29 (a)3 marks $\frac{dy}{dx} - \frac{2y}{x} = \sin \frac{1}{x}$
OR
Q.29 (b)3 marks $\frac{dy}{dx} = \sin(x + y) + \sin(x - y)$
Q.303 marks $x + 2y \le 12$
Q.31 (a)3 marks $\frac{k}{2}$
OR
Q.31 (b)3 marks $P(A \cap \overline{B}) = \frac{1}{4}$

X	1	2	3
P(X)	$\frac{k}{2}$	$\frac{k}{3}$	$\frac{k}{6}$

Section D

5 marks each

Q.32 (a)5 marks $A = \begin{bmatrix} 1 & 2 & -2 \\ -1 & 3 & 0 \\ 0 & -2 & 1 \end{bmatrix}$
OR
Q.32 (b)5 marks $x + 2y + 3z = 6$
Q.33 (a)5 marks $(1, 2, -4)$
OR
Q.33 (b)5 marks $Find the equations of the line passing through the points A(1, 2, 3) and B(3, 5, 9). Hence, find the coordinates of the points on this line which are at a distance of 14 units from point B.$
Q.345 marks $Find the area of the region bounded by the lines y = 4x + 5, x + y = 5 and 4y = x + 5, using integration.$
Q.355 marks $\mathbb{R}$

Section E

A tank, as shown in the figure below, formed using a combination of a cylinder and a cone, offers better drainage as compared to a flat bottomed tank. A tap is connected to such a tank whose conical part is full of water. Water is dripping out from a tap at the bottom at the uniform rate of $2 \text{ cm}^3/\text{s}$ . The semi-vertical angle of the conical tank is $45^\circ$ . On the basis of given information, answer the following questions :
Q.36 (i)1 mark $Find the volume of water in the tank in terms of its radius r.$
Q.36 (ii)1 mark $r = 2\sqrt{2}$
Q.36 (iii) (a)2 marks $r = 2\sqrt{2}$
OR
Q.36 (iii) (b)2 marks $Find the rate of change of height 'h' at an instant when slant height is 4 cm.$
There are different types of Yoga which involve the usage of different poses of Yoga Asanas, Meditation and Pranayam as shown in the figure below : The Venn diagram below represents the probabilities of three different types of Yoga, A, B and C performed by the people of a society. Further, it is given that probability of a member performing type C Yoga is $0{\cdot}44$ . On the basis of the above information, answer the following questions :
Q.37 (i)1 mark $Find the value of x.$
Q.37 (ii)1 mark $Find the value of y.$
Q.37 (iii) (a)2 marks $P\left(\frac{C}{B}\right)$
OR
Q.37 (iii) (b)2 marks $Find the probability that a randomly selected person of the society does Yoga of type A or B but not C.$
The equation of the path traced by a roller-coaster is given by the polynomial $f(x) = a(x + 9)(x + 1)(x - 3)$ . If the roller-coaster crosses y-axis at a point $(0, -1)$ , answer the following :
Q.38 (i)2 marks $Find the value of 'a'.$
Q.38 (ii)2 marks $f''(x)$