CBSE Class 12 Mathematics 2024 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 $f : R_+ \to R$
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Q.21 mark $If a matrix has 36 elements, the number of possible orders it can have, is :$
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Q.31 mark $f(x) = \begin{cases} x^2 + 3, & x \neq 0 \\ 1, & x = 0 \end{cases}$
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Q.41 mark $Let f(x) be a continuous function on [a, b] and differentiable on (a, b). Then, this function f(x) is strictly increasing in (a, b) if$
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Q.51 mark $\begin{bmatrix} x + y & 2 \\ 5 & xy \end{bmatrix} = \begin{bmatrix} 6 & 2 \\ 5 & 8 \end{bmatrix}$
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Q.61 mark $\int_a^b f(x)\,dx$
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Q.71 mark $\theta$
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Q.81 mark $(1 - x^2) \frac{dy}{dx} + xy = ax$
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Q.91 mark $\sqrt{3}\,k$
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Q.101 mark $A linear programming problem deals with the optimization of a/an :$
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Q.111 mark $P(A|B) = P(A'|B)$
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Q.121 mark $\begin{vmatrix} x + 1 & x - 1 \\ x^2 + x + 1 & x^2 - x + 1 \end{vmatrix}$
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Q.131 mark $\sin (x^2)$
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Q.141 mark $\left[ 1 + \left( \frac{dy}{dx} \right)^2 \right]^3 = \frac{d^2y}{dx^2}$
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Q.151 mark $-$
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Q.161 mark $The distance of point P(a, b, c) from y-axis is :$
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Q.171 mark $x \geq 0$
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Q.181 mark $(A + B)^2 = A^2 + B^2$
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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 $A = \begin{bmatrix} 1 & \cos \theta & 1 \\ -\cos \theta & 1 & \cos \theta \\ -1 & -\cos \theta & 1 \end{bmatrix}$
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Q.201 mark $\alpha$
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Section B

2 marks each

Q.21 (a)2 marks $f(x) = x^2 |x|$
OR
Q.21 (b)2 marks $y = \sqrt{\tan \sqrt{x}}$
Q.222 marks $f(x) = 4x^3 - 18x^2 + 27x - 7$
Q.23 (a)2 marks $\int x \sqrt{1 + 2x} \; dx$
OR
Q.23 (b)2 marks $\int_0^{\frac{\pi^2}{4}} \frac{\sin \sqrt{x}}{\sqrt{x}} \; dx$
Q.242 marks $\vec{a}$
Q.252 marks $\overrightarrow{AB} = 2\hat{i} - 4\hat{j} + 5\hat{k}$

Section C

3 marks each

Q.26 (a)3 marks $A = \{1, 2, 3, 4, 5\}$
OR
Q.26 (b)3 marks $R \to R$
Q.27 (a)3 marks $\sqrt{1 - x^2} + \sqrt{1 - y^2} = a\,(x - y)$
OR
Q.27 (b)3 marks $y = (\tan x)^x$
Q.28 (a)3 marks $\int \frac{x^2}{(x^2 + 4)\,(x^2 + 9)} \; dx$
OR
Q.28 (b)3 marks $\int_1^3 \left( |x - 1| + |x - 2| + |x - 3| \right) dx$
Q.293 marks $x^2 \frac{dy}{dx} - xy = x^2 \cos^2 \left( \frac{y}{2x} \right)$
Q.303 marks $z = 500x + 300y$
Q.313 marks $P(\overline{E}) = 0{\cdot}6$

Section D

5 marks each

Q.32 (a)5 marks $A = \begin{bmatrix} 1 & -2 & 0 \\ 2 & -1 & -1 \\ 0 & -2 & 1 \end{bmatrix}$
OR
Q.32 (b)5 marks $A = \begin{bmatrix} -1 & a & 2 \\ 1 & 2 & x \\ 3 & 1 & 1 \end{bmatrix}$
Q.33 (a)5 marks $\int_0^{\frac{\pi}{4}} \frac{\sin x + \cos x}{9 + 16 \sin 2x} \; dx$
OR
Q.33 (b)5 marks $\int_0^{\frac{\pi}{2}} \sin 2x \, \tan^{-1} (\sin x) \; dx$
Q.345 marks $\frac{x^2}{16} + \frac{y^2}{4} = 1$
Q.355 marks $\frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3}$

Section E

The traffic police has installed Over Speed Violation Detection (OSVD) system at various locations in a city. These cameras can capture a speeding vehicle from a distance of 300 m and even function in the dark. A camera is installed on a pole at the height of 5 m. It detects a car travelling away from the pole at the speed of 20 m/s. At any point, x m away from the base of the pole, the angle of elevation of the speed camera from the car C is $\theta$ . On the basis of the above information, answer the following questions :
Q.36 (i)1 mark $\theta$
Q.36 (ii)1 mark $\dfrac{d\theta}{dx}$
Q.36 (iii) (a)2 marks $Find the rate of change of angle of elevation with respect to time at an instant when the car is 50 m away from the pole.$
OR
Q.36 (iii) (b)2 marks $\dfrac{3}{101}$
According to recent research, air turbulence has increased in various regions around the world due to climate change. Turbulence makes flights bumpy and often delays the flights. Assume that, an airplane observes severe turbulence, moderate turbulence or light turbulence with equal probabilities. Further, the chance of an airplane reaching late to the destination are 55%, 37% and 17% due to severe, moderate and light turbulence respectively. On the basis of the above information, answer the following questions :
Q.37 (i)2 marks $Find the probability that an airplane reached its destination late.$
Q.37 (ii)2 marks $If the airplane reached its destination late, find the probability that it was due to moderate turbulence.$
If a function $f : X \to Y$ defined as $f(x) = y$ is one-one and onto, then we can define a unique function $g : Y \to X$ such that $g(y) = x$ , where $x \in X$ and $y = f(x)$ , $y \in Y$ . Function g is called the inverse of function f. The domain of sine function is R and function sine : $R \to R$ is neither one-one nor onto. The following graph shows the sine function. Let sine function be defined from set A to $[-1, 1]$ such that inverse of sine function exists, i.e., $\sin^{-1} x$ is defined from $[-1, 1]$ to A. On the basis of the above information, answer the following questions :
Q.38 (i)1 mark $If A is the interval other than principal value branch, give an example of one such interval.$
Q.38 (ii)1 mark $\sin^{-1} (x)$
Q.38 (iii) (a)2 marks $\sin^{-1} x$
OR
Q.38 (iii) (b)2 marks $f(x) = 2 \sin^{-1} (1 - x)$

Section A

1 mark each

Q.11 mark $\theta$
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Q.21 mark $x \frac{dy}{dx} - y = x^4 - 3x$
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Q.31 mark $\sqrt{3}\,k$
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Q.41 mark $A linear programming problem deals with the optimization of a/an :$
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Q.51 mark $P(A|B) = P(A'|B)$
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Q.61 mark $a_{ij}$
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Q.71 mark $\sin (x^2)$
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Q.81 mark $\left[ 1 + \left( \frac{dy}{dx} \right)^2 \right]^3 = \frac{d^2y}{dx^2}$
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Q.91 mark $-$
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Q.101 mark $The distance of point P(a, b, c) from y-axis is :$
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Q.111 mark $x \geq 0$
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Q.121 mark $(A + B)^2 = A^2 + B^2$
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Q.131 mark $A = \{x : x \in Z \text{ and } 0 \leq x \leq 10\}$
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Q.141 mark $If a matrix has 36 elements, the number of possible orders it can have, is :$
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Q.151 mark $f(x) = [x]$
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Q.161 mark $Let f(x) be a continuous function on [a, b] and differentiable on (a, b). Then, this function f(x) is strictly increasing in (a, b) if$
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Q.171 mark $\begin{bmatrix} x + y & 2 \\ 5 & xy \end{bmatrix} = \begin{bmatrix} 6 & 2 \\ 5 & 8 \end{bmatrix}$
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Q.181 mark $\int_0^a \frac{1}{4 + x^2} \; dx = \frac{\pi}{6}$
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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 $\alpha$
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Q.201 mark $A = \begin{bmatrix} 1 & \cos \theta & 1 \\ -\cos \theta & 1 & \cos \theta \\ -1 & -\cos \theta & 1 \end{bmatrix}$
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Section B

2 marks each

Q.21 (a)2 marks $\int x \sqrt{1 + 2x} \; dx$
OR
Q.21 (b)2 marks $\int_0^{\frac{\pi^2}{4}} \frac{\sin \sqrt{x}}{\sqrt{x}} \; dx$
Q.222 marks $\vec{a}$
Q.232 marks $\overrightarrow{AB} = 2\hat{i} - 4\hat{j} + 5\hat{k}$
Q.24 (a)2 marks $y = \sqrt{\cos x + y}$
OR
Q.24 (b)2 marks $f(x) = |x|^3$
Q.252 marks $f(x) = 12x^{4/3} - 6x^{1/3}$

Section C

3 marks each

Q.26 (a)3 marks $\int \frac{x^2}{(x^2 + 4)\,(x^2 + 9)} \; dx$
OR
Q.26 (b)3 marks $\int_1^3 \left( |x - 1| + |x - 2| + |x - 3| \right) dx$
Q.273 marks $\frac{dy}{dx} = \frac{x^2 + y^2}{2xy}$
Q.283 marks $z = 5x + 4y$
Q.293 marks $P(\overline{E}) = 0{\cdot}6$
Q.30 (a)3 marks $A = \{1, 2, 3, 4, 5\}$
OR
Q.30 (b)3 marks $R \to R$
Q.31 (a)3 marks $\sqrt{1 - x^2} + \sqrt{1 - y^2} = a\,(x - y)$
OR
Q.31 (b)3 marks $y = (\tan x)^x$

Section D

5 marks each

Q.32 (a)5 marks $\int_0^{\frac{\pi}{2}} e^x \left( \frac{1 + \sin x}{1 + \cos x} \right) dx$
OR
Q.32 (b)5 marks $\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sin x + \cos x}{\sqrt{\sin 2x}} \; dx$
Q.335 marks $\frac{x^2}{16} + \frac{y^2}{4} = 1$
Q.345 marks $\frac{x + 1}{1} = \frac{y - 2}{-2} = \frac{z - 1}{2}$
Q.35 (a)5 marks $A = \begin{bmatrix} 1 & -2 & 0 \\ 2 & -1 & -1 \\ 0 & -2 & 1 \end{bmatrix}$
OR
Q.35 (b)5 marks $A = \begin{bmatrix} -1 & a & 2 \\ 1 & 2 & x \\ 3 & 1 & 1 \end{bmatrix}$

Section E

According to recent research, air turbulence has increased in various regions around the world due to climate change. Turbulence makes flights bumpy and often delays the flights. Assume that, an airplane observes severe turbulence, moderate turbulence or light turbulence with equal probabilities. Further, the chance of an airplane reaching late to the destination are 55%, 37% and 17% due to severe, moderate and light turbulence respectively. On the basis of the above information, answer the following questions :
Q.36 (i)2 marks $Find the probability that an airplane reached its destination late.$
Q.36 (ii)2 marks $If the airplane reached its destination late, find the probability that it was due to moderate turbulence.$
If a function $f : X \to Y$ defined as $f(x) = y$ is one-one and onto, then we can define a unique function $g : Y \to X$ such that $g(y) = x$ , where $x \in X$ and $y = f(x)$ , $y \in Y$ . Function g is called the inverse of function f. The domain of sine function is R and function sine : $R \to R$ is neither one-one nor onto. The following graph shows the sine function. Let sine function be defined from set A to $[-1, 1]$ such that inverse of sine function exists, i.e., $\sin^{-1} x$ is defined from $[-1, 1]$ to A. On the basis of the above information, answer the following questions :
Q.37 (i)1 mark $If A is the interval other than principal value branch, give an example of one such interval.$
Q.37 (ii)1 mark $\sin^{-1} (x)$
Q.37 (iii) (a)2 marks $\sin^{-1} x$
OR
Q.37 (iii) (b)2 marks $f(x) = 2 \sin^{-1} (1 - x)$
The traffic police has installed Over Speed Violation Detection (OSVD) system at various locations in a city. These cameras can capture a speeding vehicle from a distance of 300 m and even function in the dark. A camera is installed on a pole at the height of 5 m. It detects a car travelling away from the pole at the speed of 20 m/s. At any point, x m away from the base of the pole, the angle of elevation of the speed camera from the car C is $\theta$ . On the basis of the above information, answer the following questions :
Q.38 (i)1 mark $\theta$
Q.38 (ii)1 mark $\dfrac{d\theta}{dx}$
Q.38 (iii) (a)2 marks $Find the rate of change of angle of elevation with respect to time at an instant when the car is 50 m away from the pole.$
OR
Q.38 (iii) (b)2 marks $\dfrac{3}{101}$

Section A

1 mark each

Q.11 mark $x = at$
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Q.21 mark $\frac{dy}{dx} = \frac{1}{\log y}$
Tap an option to check your answer.
Q.31 mark $-$
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Q.41 mark $The distance of point P(a, b, c) from y-axis is :$
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Q.51 mark $x \geq 0$
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Q.61 mark $1 \times 3$
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Q.71 mark $A relation R defined on a set of human beings as R = {(x, y) : x is 5 cm shorter than y} is :$
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Q.81 mark $If a matrix has 36 elements, the number of possible orders it can have, is :$
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Q.91 mark $f(x) = \begin{cases} x^2 + 3, & x \neq 0 \\ 1, & x = 0 \end{cases}$
Tap an option to check your answer.
Q.101 mark $Let f(x) be a continuous function on [a, b] and differentiable on (a, b). Then, this function f(x) is strictly increasing in (a, b) if$
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Q.111 mark $\begin{bmatrix} x + y & 2 \\ 5 & xy \end{bmatrix} = \begin{bmatrix} 6 & 2 \\ 5 & 8 \end{bmatrix}$
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Q.121 mark $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} f(x) \cos^3 x \; dx$
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Q.131 mark $\theta$
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Q.141 mark $(1 - x^2) \frac{dy}{dx} + xy = ax$
Tap an option to check your answer.
Q.151 mark $\sqrt{3}\,k$
Tap an option to check your answer.
Q.161 mark $A linear programming problem deals with the optimization of a/an :$
Tap an option to check your answer.
Q.171 mark $P(A|B) = P(A'|B)$
Tap an option to check your answer.
Q.181 mark $\begin{vmatrix} x + 1 & x - 1 \\ x^2 + x + 1 & x^2 - x + 1 \end{vmatrix}$
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 $A = \begin{bmatrix} 1 & \cos \theta & 1 \\ -\cos \theta & 1 & \cos \theta \\ -1 & -\cos \theta & 1 \end{bmatrix}$
Tap an option to check your answer.
Q.201 mark $\alpha$
Tap an option to check your answer.

Section B

2 marks each

Q.212 marks $\overrightarrow{AB} = 2\hat{i} - 4\hat{j} + 5\hat{k}$
Q.22 (a)2 marks $f(x) = [x]$
OR
Q.22 (b)2 marks $x^{1/3} + y^{1/3} = 1$
Q.232 marks $f(x) = 4x^2 + \frac{1}{x}$
Q.24 (a)2 marks $\int x \sqrt{1 + 2x} \; dx$
OR
Q.24 (b)2 marks $\int_0^{\frac{\pi^2}{4}} \frac{\sin \sqrt{x}}{\sqrt{x}} \; dx$
Q.252 marks $\vec{a}$

Section C

3 marks each

Q.263 marks $z = 5x - 2y$
Q.273 marks $P(\overline{E}) = 0{\cdot}6$
Q.28 (a)3 marks $A = \{1, 2, 3, 4, 5\}$
OR
Q.28 (b)3 marks $R \to R$
Q.29 (a)3 marks $\sqrt{1 - x^2} + \sqrt{1 - y^2} = a\,(x - y)$
OR
Q.29 (b)3 marks $y = (\tan x)^x$
Q.30 (a)3 marks $\int \frac{x^2}{(x^2 + 4)\,(x^2 + 9)} \; dx$
OR
Q.30 (b)3 marks $\int_1^3 \left( |x - 1| + |x - 2| + |x - 3| \right) dx$
Q.313 marks $(\tan^{-1} y - x) \; dy = (1 + y^2) \; dx$

Section D

5 marks each

Q.325 marks $l_2$
Q.33 (a)5 marks $A = \begin{bmatrix} 1 & -2 & 0 \\ 2 & -1 & -1 \\ 0 & -2 & 1 \end{bmatrix}$
OR
Q.33 (b)5 marks $A = \begin{bmatrix} -1 & a & 2 \\ 1 & 2 & x \\ 3 & 1 & 1 \end{bmatrix}$
Q.34 (a)5 marks $\int \frac{(3 \cos x - 2) \sin x}{5 - \sin^2 x - 4 \cos x} \; dx$
OR
Q.34 (b)5 marks $\int_{-2}^{2} \frac{x^3 + |x| + 1}{x^2 + 4|x| + 4} \; dx$
Q.355 marks $\frac{x^2}{16} + \frac{y^2}{4} = 1$

Section E

If a function $f : X \to Y$ defined as $f(x) = y$ is one-one and onto, then we can define a unique function $g : Y \to X$ such that $g(y) = x$ , where $x \in X$ and $y = f(x)$ , $y \in Y$ . Function g is called the inverse of function f. The domain of sine function is R and function sine : $R \to R$ is neither one-one nor onto. The following graph shows the sine function. Let sine function be defined from set A to $[-1, 1]$ such that inverse of sine function exists, i.e., $\sin^{-1} x$ is defined from $[-1, 1]$ to A. On the basis of the above information, answer the following questions :
Q.36 (i)1 mark $If A is the interval other than principal value branch, give an example of one such interval.$
Q.36 (ii)1 mark $\sin^{-1} (x)$
Q.36 (iii) (a)2 marks $\sin^{-1} x$
OR
Q.36 (iii) (b)2 marks $f(x) = 2 \sin^{-1} (1 - x)$
The traffic police has installed Over Speed Violation Detection (OSVD) system at various locations in a city. These cameras can capture a speeding vehicle from a distance of 300 m and even function in the dark. A camera is installed on a pole at the height of 5 m. It detects a car travelling away from the pole at the speed of 20 m/s. At any point, x m away from the base of the pole, the angle of elevation of the speed camera from the car C is $\theta$ . On the basis of the above information, answer the following questions :
Q.37 (i)1 mark $\theta$
Q.37 (ii)1 mark $\dfrac{d\theta}{dx}$
Q.37 (iii) (a)2 marks $Find the rate of change of angle of elevation with respect to time at an instant when the car is 50 m away from the pole.$
OR
Q.37 (iii) (b)2 marks $\dfrac{3}{101}$
According to recent research, air turbulence has increased in various regions around the world due to climate change. Turbulence makes flights bumpy and often delays the flights. Assume that, an airplane observes severe turbulence, moderate turbulence or light turbulence with equal probabilities. Further, the chance of an airplane reaching late to the destination are 55%, 37% and 17% due to severe, moderate and light turbulence respectively. On the basis of the above information, answer the following questions :
Q.38 (i)2 marks $Find the probability that an airplane reached its destination late.$
Q.38 (ii)2 marks $If the airplane reached its destination late, find the probability that it was due to moderate turbulence.$