CBSE Class 12 Mathematics 2023 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 $A = \begin{bmatrix} 1 & 4 & x \\ z & 2 & y \\ -3 & -1 & 3 \end{bmatrix}$
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Q.21 mark $A \cdot (\text{adj } A) = \begin{bmatrix} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{bmatrix}$
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Q.31 mark $A and B are skew-symmetric matrices of same order. AB is symmetric, if :$
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Q.41 mark $x \in \left[ 0, \frac{\pi}{2} \right]$
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Q.51 mark $(x_1, y_1)$
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Q.61 mark $\int 2^{x+2} \, dx$
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Q.71 mark $\int \frac{2 \cos 2x - 1}{1 + 2 \sin x} \, dx$
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Q.81 mark $\frac{dx}{x} + \frac{dy}{y} = 0$
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Q.91 mark $\frac{d^2 y}{dx^2} \sin y + \left( \frac{dy}{dx} \right)^3 \cos y = \sqrt{y}$
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Q.101 mark $\frac{\pi}{4}$
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Q.111 mark $\vec{a}$
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Q.121 mark $\Delta \, ABC$
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Q.131 mark $\lambda$
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Q.141 mark $P(A \cap B) = \frac{1}{8}$
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Q.151 mark $f(x) = \begin{cases} x^2, & x \geq 0 \\ kx, & x < 0 \end{cases}$
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Q.161 mark $y = \frac{\cos x - \sin x}{\cos x + \sin x}$
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Q.171 mark $3x + y \leq 12$
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Q.181 mark $The feasible region of a linear programming problem is shown in the figure below : Which of the following are the possible constraints ?$
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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 $\left[ \sin^{-1} x + 2 \cos^{-1} x \right]$
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Q.201 mark $(-1, -2, 1)$
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Section B

2 marks each

Q.212 marks $\vec{r} = 3\hat{i} - 2\hat{j} + 6\hat{k}$
Q.222 marks $\frac{x-5}{\alpha} = \frac{y+2}{-5} = \frac{z + \frac{24}{5}}{\beta}$
Q.232 marks $-$
Q.24 (a)2 marks $3 \sin^{-1} \left( \frac{1}{\sqrt{2}} \right) + 2 \cos^{-1} \left( \frac{\sqrt{3}}{2} \right) + \cos^{-1} (0)$
OR
Q.24 (b)2 marks $f(x) = \sin^{-1} x$
Q.25 (a)2 marks $y = x^{\frac{1}{x}}$
OR
Q.25 (b)2 marks $\frac{dy}{dx}$

Section C

3 marks each

Q.26 (a)3 marks $\frac{d}{dx} (x y^2) = 2y (1 + x^2)$
OR
Q.26 (b)3 marks $x e^{\frac{y}{x}} - y + x \frac{dy}{dx} = 0$
Q.273 marks $\int_{1}^{3} \frac{\sqrt{4-x}}{\sqrt{x} + \sqrt{4-x}} \, dx$
Q.283 marks $\int_{1}^{e} \frac{1}{\sqrt{4x^2 - (x \log x)^2}} \, dx$
Q.29 (a)3 marks $\int \frac{\cos x}{\sin 3x} \, dx$
OR
Q.29 (b)3 marks $\int x^2 \log (x^2 + 1) \, dx$
Q.303 marks $x + y \leq 200$
Q.31 (a)3 marks $A pair of dice is thrown simultaneously. If X denotes the absolute difference of numbers obtained on the pair of dice, then find the probability distribution of X.$
OR
Q.31 (b)3 marks $There are two coins. One of them is a biased coin such that P (head) : P (tail) is 1 : 3 and the other coin is a fair coin. A coin is selected at random and tossed once. If the coin showed head, then find the probability that it is a biased coin.$

Section D

5 marks each

Q.325 marks $\mathbb{R} \to \mathbb{R}$
Q.335 marks $x^2 + y^2 = 4$
Q.34 (a)5 marks $A = \begin{bmatrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{bmatrix}$
OR
Q.34 (b)5 marks $A = \begin{bmatrix} 3 & 2 \\ 5 & -7 \end{bmatrix}$
Q.35 (a)5 marks $\frac{x-1}{2} = \frac{y-b}{3} = \frac{z-3}{4}$
OR
Q.35 (b)5 marks $-1, -2, 1$

Section E

An octagonal prism is a three-dimensional polyhedron bounded by two octagonal bases and eight rectangular side faces. It has 24 edges and 16 vertices. The prism is rolled along the rectangular faces and number on the bottom face (touching the ground) is noted. Let X denote the number obtained on the bottom face and the following table give the probability distribution of X.
X : 1 2 3 4 5 6 7 8
P(X) : p 2p 2p p 2p $p^2$ $2p^2$ $7p^2 + p$
Q.36 (i)1 mark $Find the value of p.$
Q.36 (ii)1 mark $Find P(X > 6).$
Q.36 (iii) (a)2 marks $Find P(X = 3m), where m is a natural number.$
OR
Q.36 (iii) (b)2 marks $Find the mean E(X).$
In order to set up a rain water harvesting system, a tank to collect rain water is to be dug. The tank should have a square base and a capacity of 250 $\text{m}^3$ . The cost of land is ₹ 5,000 per square metre and cost of digging increases with depth and for the whole tank, it is ₹ 40,000 $h^2$ , where h is the depth of the tank in metres. x is the side of the square base of the tank in metres.
Q.37 (i)1 mark $Find the total cost C of digging the tank in terms of x.$
Q.37 (ii)1 mark $\frac{dC}{dx}$
Q.37 (iii) (a)2 marks $Find the value of x for which cost C is minimum.$
OR
Q.37 (iii) (b)2 marks $Check whether the cost function C(x) expressed in terms of x is increasing or not, where x > 0.$
A volleyball player serves the ball which takes a parabolic path given by the equation $h(t) = -\frac{7}{2} t^2 + \frac{13}{2} t + 1$ , where h(t) is the height of ball at any time t (in seconds), $(t \geq 0)$ .
Q.38 (i)2 marks $Is h(t) a continuous function ? Justify.$
Q.38 (ii)2 marks $Find the time at which the height of the ball is maximum.$

X :	1	2	3	4	5	6	7	8
P(X) :	p	2p	2p	p	2p	$p^2$	$2p^2$	$7p^2 + p$

Section A

1 mark each

Q.11 mark $\int 2^{x+2} \, dx$
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Q.21 mark $|A| = x$
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Q.31 mark $\int_{0}^{2} \sqrt{4 - x^2} \, dx$
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Q.41 mark $\frac{dx}{x} + \frac{dy}{y} = 0$
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Q.51 mark $\frac{d^2 y}{dx^2} \sin y + \left( \frac{dy}{dx} \right)^3 \cos y = \sqrt{y}$
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Q.61 mark $\overrightarrow{BA}$
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Q.71 mark $\vec{a}$
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Q.81 mark $\Delta \, ABC$
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Q.91 mark $\frac{x+1}{2} = \frac{y+2}{3} = \frac{z+3}{4}$
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Q.101 mark $P(\overline{A}) = \frac{1}{2}$
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Q.111 mark $f(x) = \begin{cases} x^2, & x \geq 0 \\ kx, & x < 0 \end{cases}$
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Q.121 mark $y = \frac{\cos x - \sin x}{\cos x + \sin x}$
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Q.131 mark $3x + y \leq 12$
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Q.141 mark $The feasible region of a linear programming problem is shown in the figure below : Which of the following are the possible constraints ?$
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Q.151 mark $A = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}$
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Q.161 mark $A \cdot (\text{adj } A) = \begin{bmatrix} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{bmatrix}$
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Q.171 mark $A and B are skew-symmetric matrices of same order. AB is symmetric, if :$
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Q.181 mark $x \in \left[ 0, \frac{\pi}{2} \right]$
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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 $(-1, -2, 1)$
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Q.201 mark $\left[ \sin^{-1} x + 2 \cos^{-1} x \right]$
Tap an option to check your answer.

Section B

2 marks each

Q.212 marks $b \in \mathbb{R}$
Q.22 (a)2 marks $3 \sin^{-1} \left( \frac{1}{\sqrt{2}} \right) + 2 \cos^{-1} \left( \frac{\sqrt{3}}{2} \right) + \cos^{-1} (0)$
OR
Q.22 (b)2 marks $f(x) = \sin^{-1} x$
Q.23 (a)2 marks $y = x^{\frac{1}{x}}$
OR
Q.23 (b)2 marks $\frac{dy}{dx}$
Q.242 marks $\vec{r} = 3\hat{i} - 2\hat{j} + 6\hat{k}$
Q.252 marks $\frac{x-1}{-2} = \frac{y-4}{3p} = \frac{z-3}{4}$

Section C

3 marks each

Q.263 marks $\int \frac{e^x}{\sqrt{e^{2x} - 4e^x - 5}} \, dx$
Q.27 (a)3 marks $\int \frac{\cos x}{\sin 3x} \, dx$
OR
Q.27 (b)3 marks $\int x^2 \log (x^2 + 1) \, dx$
Q.283 marks $x + y \geq 10$
Q.29 (a)3 marks $A pair of dice is thrown simultaneously. If X denotes the absolute difference of numbers obtained on the pair of dice, then find the probability distribution of X.$
OR
Q.29 (b)3 marks $There are two coins. One of them is a biased coin such that P (head) : P (tail) is 1 : 3 and the other coin is a fair coin. A coin is selected at random and tossed once. If the coin showed head, then find the probability that it is a biased coin.$
Q.30 (a)3 marks $\frac{d}{dx} (x y^2) = 2y (1 + x^2)$
OR
Q.30 (b)3 marks $x e^{\frac{y}{x}} - y + x \frac{dy}{dx} = 0$
Q.313 marks $\int_{-\pi/2}^{\pi/2} \frac{\sin^{100} x}{\sin^{100} x + \cos^{100} x} \, dx$

Section D

5 marks each

Q.32 (a)5 marks $A = \begin{bmatrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{bmatrix}$
OR
Q.32 (b)5 marks $A = \begin{bmatrix} 3 & 2 \\ 5 & -7 \end{bmatrix}$
Q.33 (a)5 marks $\frac{x-1}{2} = \frac{y-b}{3} = \frac{z-3}{4}$
OR
Q.33 (b)5 marks $-1, -2, 1$
Q.345 marks $[0, \infty) \to [-5, \infty)$
Q.355 marks $x^2 + y^2 = 4$

Section E

In order to set up a rain water harvesting system, a tank to collect rain water is to be dug. The tank should have a square base and a capacity of 250 $\text{m}^3$ . The cost of land is ₹ 5,000 per square metre and cost of digging increases with depth and for the whole tank, it is ₹ 40,000 $h^2$ , where h is the depth of the tank in metres. x is the side of the square base of the tank in metres.
Q.36 (i)1 mark $Find the total cost C of digging the tank in terms of x.$
Q.36 (ii)1 mark $\frac{dC}{dx}$
Q.36 (iii) (a)2 marks $Find the value of x for which cost C is minimum.$
OR
Q.36 (iii) (b)2 marks $Check whether the cost function C(x) expressed in terms of x is increasing or not, where x > 0.$
An octagonal prism is a three-dimensional polyhedron bounded by two octagonal bases and eight rectangular side faces. It has 24 edges and 16 vertices. The prism is rolled along the rectangular faces and number on the bottom face (touching the ground) is noted. Let X denote the number obtained on the bottom face and the following table give the probability distribution of X.
X : 1 2 3 4 5 6 7 8
P(X) : p 2p 2p p 2p $p^2$ $2p^2$ $7p^2 + p$
Q.37 (i)1 mark $Find the value of p.$
Q.37 (ii)1 mark $Find P(X > 6).$
Q.37 (iii) (a)2 marks $Find P(X = 3m), where m is a natural number.$
OR
Q.37 (iii) (b)2 marks $Find the mean E(X).$
A volleyball player serves the ball which takes a parabolic path given by the equation $h(t) = -\frac{7}{2} t^2 + \frac{13}{2} t + 1$ , where h(t) is the height of ball at any time t (in seconds), $(t \geq 0)$ .
Q.38 (i)2 marks $Is h(t) a continuous function ? Justify.$
Q.38 (ii)2 marks $Find the time at which the height of the ball is maximum.$

X :	1	2	3	4	5	6	7	8
P(X) :	p	2p	2p	p	2p	$p^2$	$2p^2$	$7p^2 + p$

Section A

1 mark each

Q.11 mark $\vec{a}$
Tap an option to check your answer.
Q.21 mark $\vec{a}$
Tap an option to check your answer.
Q.31 mark $\Delta \, ABC$
Tap an option to check your answer.
Q.41 mark $(2, -1, 0)$
Tap an option to check your answer.
Q.51 mark $P(X \cap \overline{Y}) = \frac{2}{5}$
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Q.61 mark $f(x) = \begin{cases} x^2, & x \geq 0 \\ kx, & x < 0 \end{cases}$
Tap an option to check your answer.
Q.71 mark $y = \frac{\cos x - \sin x}{\cos x + \sin x}$
Tap an option to check your answer.
Q.81 mark $3x + y \leq 12$
Tap an option to check your answer.
Q.91 mark $The feasible region of a linear programming problem is shown in the figure below : Which of the following are the possible constraints ?$
Tap an option to check your answer.
Q.101 mark $(A + B)^2 = A^2 + B^2$
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Q.111 mark $A \cdot (\text{adj } A) = \begin{bmatrix} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{bmatrix}$
Tap an option to check your answer.
Q.121 mark $A and B are skew-symmetric matrices of same order. AB is symmetric, if :$
Tap an option to check your answer.
Q.131 mark $x \in \left[ 0, \frac{\pi}{2} \right]$
Tap an option to check your answer.
Q.141 mark $(x_1, y_1)$
Tap an option to check your answer.
Q.151 mark $\int 2^{x+2} \, dx$
Tap an option to check your answer.
Q.161 mark $\int e^{-x} \left( \frac{x+1}{x^2} \right) \, dx$
Tap an option to check your answer.
Q.171 mark $\int_{0}^{\pi/2} \log \tan x \, dx$
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Q.181 mark $\frac{d^2 y}{dx^2} \sin y + \left( \frac{dy}{dx} \right)^3 \cos y = \sqrt{y}$
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 $\left[ \sin^{-1} x + 2 \cos^{-1} x \right]$
Tap an option to check your answer.
Q.201 mark $(-1, -2, 1)$
Tap an option to check your answer.

Section B

2 marks each

Q.21 (a)2 marks $y = x^{\frac{1}{x}}$
OR
Q.21 (b)2 marks $\frac{dy}{dx}$
Q.222 marks $\vec{r} = 3\hat{i} - 2\hat{j} + 6\hat{k}$
Q.232 marks $-$
Q.242 marks $6y = x^3 + 2$
Q.25 (a)2 marks $3 \sin^{-1} \left( \frac{1}{\sqrt{2}} \right) + 2 \cos^{-1} \left( \frac{\sqrt{3}}{2} \right) + \cos^{-1} (0)$
OR
Q.25 (b)2 marks $f(x) = \sin^{-1} x$

Section C

3 marks each

Q.26 (a)3 marks $A pair of dice is thrown simultaneously. If X denotes the absolute difference of numbers obtained on the pair of dice, then find the probability distribution of X.$
OR
Q.26 (b)3 marks $There are two coins. One of them is a biased coin such that P (head) : P (tail) is 1 : 3 and the other coin is a fair coin. A coin is selected at random and tossed once. If the coin showed head, then find the probability that it is a biased coin.$
Q.27 (a)3 marks $\frac{d}{dx} (x y^2) = 2y (1 + x^2)$
OR
Q.27 (b)3 marks $x e^{\frac{y}{x}} - y + x \frac{dy}{dx} = 0$
Q.283 marks $\int_{0}^{\pi/4} \log (1 + \tan x) \, dx$
Q.29 (a)3 marks $\int \frac{\cos x}{\sin 3x} \, dx$
OR
Q.29 (b)3 marks $\int x^2 \log (x^2 + 1) \, dx$
Q.303 marks $\int_{1}^{4} \frac{1}{\sqrt{2x+1} - \sqrt{2x-1}} \, dx$
Q.313 marks $2x + y \geq 3$

Section D

5 marks each

Q.32 (a)5 marks $\frac{x-1}{2} = \frac{y-b}{3} = \frac{z-3}{4}$
OR
Q.32 (b)5 marks $-1, -2, 1$
Q.335 marks $\mathbb{R} \to \left[ -\frac{1}{2}, \frac{1}{2} \right]$
Q.345 marks $x^2 + y^2 = 4$
Q.35 (a)5 marks $A = \begin{bmatrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{bmatrix}$
OR
Q.35 (b)5 marks $A = \begin{bmatrix} 3 & 2 \\ 5 & -7 \end{bmatrix}$

Section E

In order to set up a rain water harvesting system, a tank to collect rain water is to be dug. The tank should have a square base and a capacity of 250 $\text{m}^3$ . The cost of land is ₹ 5,000 per square metre and cost of digging increases with depth and for the whole tank, it is ₹ 40,000 $h^2$ , where h is the depth of the tank in metres. x is the side of the square base of the tank in metres.
Q.36 (i)1 mark $Find the total cost C of digging the tank in terms of x.$
Q.36 (ii)1 mark $\frac{dC}{dx}$
Q.36 (iii) (a)2 marks $Find the value of x for which cost C is minimum.$
OR
Q.36 (iii) (b)2 marks $Check whether the cost function C(x) expressed in terms of x is increasing or not, where x > 0.$
An octagonal prism is a three-dimensional polyhedron bounded by two octagonal bases and eight rectangular side faces. It has 24 edges and 16 vertices. The prism is rolled along the rectangular faces and number on the bottom face (touching the ground) is noted. Let X denote the number obtained on the bottom face and the following table give the probability distribution of X.
X : 1 2 3 4 5 6 7 8
P(X) : p 2p 2p p 2p $p^2$ $2p^2$ $7p^2 + p$
Q.37 (i)1 mark $Find the value of p.$
Q.37 (ii)1 mark $Find P(X > 6).$
Q.37 (iii) (a)2 marks $Find P(X = 3m), where m is a natural number.$
OR
Q.37 (iii) (b)2 marks $Find the mean E(X).$
A volleyball player serves the ball which takes a parabolic path given by the equation $h(t) = -\frac{7}{2} t^2 + \frac{13}{2} t + 1$ , where h(t) is the height of ball at any time t (in seconds), $(t \geq 0)$ .
Q.38 (i)2 marks $Is h(t) a continuous function ? Justify.$
Q.38 (ii)2 marks $Find the time at which the height of the ball is maximum.$