CBSE Class 12 Mathematics 2024 question paper (65/2)

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 $3 \times 3$
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Q.21 mark $f : R_+ \to [-5, \infty)$
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Q.31 mark $\begin{vmatrix} -a & b & c \\ a & -b & c \\ a & b & -c \end{vmatrix} = kabc$
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Q.41 mark $f(x) = \begin{cases} |x| + 3, & \text{if } x \le -3 \\ -2x, & \text{if } -3 < x < 3 \\ 6x + 2, & \text{if } x \ge 3 \end{cases}$
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Q.51 mark $f(x) = x^3 - 3x^2 + 12x - 18$
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Q.61 mark $\displaystyle\int_0^{\pi/2} \dfrac{\sin x - \cos x}{1 + \sin x \cos x}\, dx$
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Q.71 mark $\dfrac{dy}{dx} = F(x, y)$
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Q.81 mark $\vec{a}$
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Q.91 mark $The coordinates of the foot of the perpendicular drawn from the point (0, 1, 2) on the x-axis are given by :$
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Q.101 mark $The common region determined by all the constraints of a linear programming problem is called :$
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Q.111 mark $P(S \,|\, E) =$
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Q.121 mark $A = [a_{ij}]$
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Q.131 mark $\tan^{-1}(x^2)$
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Q.141 mark $(y'')^2 + (y')^3 = x \sin (y')$
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Q.151 mark $\hat{i} + \hat{k}$
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Q.161 mark $\dfrac{x-1}{2} = -y = \dfrac{2z+1}{6}$
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Q.171 mark $F(x) = \begin{bmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{bmatrix}$
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Q.181 mark $30^\circ$
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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 $P' = -P$
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Q.201 mark $\vec{a}$
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Section B

2 marks each

Q.21 (a)2 marks $\tan^{-1}\left(-\dfrac{1}{\sqrt{3}}\right) + \cot^{-1}\left(\dfrac{1}{\sqrt{3}}\right) + \tan^{-1}\left[\sin\left(-\dfrac{\pi}{2}\right)\right]$
OR
Q.21 (b)2 marks $f(x) = \sin^{-1}(x^2 - 4)$
Q.22 (a)2 marks $f(x) = |\tan 2x|$
OR
Q.22 (b)2 marks $y = \text{cosec}\,(\cot^{-1} x)$
Q.232 marks $f(x) = x + \dfrac{1}{x}$
Q.242 marks $\displaystyle\int \dfrac{e^{4x} - 1}{e^{4x} + 1}\, dx$
Q.252 marks $f(x) = e^x - e^{-x} + x - \tan^{-1} x$

Section C

3 marks each

Q.26 (a)3 marks $x = e^{\cos 3t}$
OR
Q.26 (b)3 marks $\dfrac{d}{dx}\big(|x|\big) = \dfrac{x}{|x|}, x \ne 0$
Q.27 (a)3 marks $\displaystyle\int_{-2}^{2} \sqrt{\dfrac{2 - x}{2 + x}}\, dx$
OR
Q.27 (b)3 marks $\displaystyle\int \dfrac{1}{x\,[(\log x)^2 - 3 \log x - 4]}\, dx$
Q.28 (a)3 marks $2xy + y^2 - 2x^2 \dfrac{dy}{dx} = 0$
OR
Q.28 (b)3 marks $y\, dx = (x + 2y^2)\, dy$
Q.293 marks $\Delta$
Q.303 marks $A pair of dice is thrown simultaneously. If X denotes the absolute difference of the numbers appearing on top of the dice, then find the probability distribution of X.$
Q.313 marks $\displaystyle\int x^2 \cdot \sin^{-1}(x^{3/2})\, dx$

Section D

5 marks each

Q.32 (a)5 marks $f : R \to R$
OR
Q.32 (b)5 marks $N \times N$
Q.335 marks $\dfrac{x-8}{3} = \dfrac{y+19}{-16} = \dfrac{z-10}{7}$
Q.34 (a)5 marks $\dfrac{2}{x} + \dfrac{3}{y} + \dfrac{10}{z} = 4$
OR
Q.34 (b)5 marks $A = \begin{bmatrix} 1 & \cot x \\ -\cot x & 1 \end{bmatrix}$
Q.355 marks $A_1$

Section E

Overspeeding increases fuel consumption and decreases fuel economy as a result of tyre rolling friction and air resistance. While vehicles reach optimal fuel economy at different speeds, fuel mileage usually decreases rapidly at speeds above 80 km/h. The relation between fuel consumption F (l/100 km) and speed V (km/h) under some constraints is given as $F = \dfrac{V^2}{500} - \dfrac{V}{4} + 14$ . On the basis of the above information, answer the following questions :
Q.36 (i)1 mark $Find F, when V = 40 km/h.$
Q.36 (ii)1 mark $\dfrac{dF}{dV}$
Q.36 (iii) (a)2 marks $Find the speed V for which fuel consumption F is minimum.$
OR
Q.36 (iii) (b)2 marks $\dfrac{dF}{dV} = -0{\cdot}01$
The month of September is celebrated as the Rashtriya Poshan Maah across the country. Following a healthy and well-balanced diet is crucial in order to supply the body with the proper nutrients it needs. A balanced diet also keeps us mentally fit and promotes improved level of energy. A dietician wishes to minimize the cost of a diet involving two types of foods, food X (x kg) and food Y (y kg) which are available at the rate of ₹ 16/kg and ₹ 20/kg respectively. The feasible region satisfying the constraints is shown in Figure-2. On the basis of the above information, answer the following questions :
Q.37 (i)2 marks $Identify and write all the constraints which determine the given feasible region in Figure-2.$
Q.37 (ii)2 marks $Z = 16x + 20y$
Airplanes are by far the safest mode of transportation when the number of transported passengers are measured against personal injuries and fatality totals. Previous records state that the probability of an airplane crash is $0{\cdot}00001\%$ . Further, there are 95% chances that there will be survivors after a plane crash. Assume that in case of no crash, all travellers survive. Let $E_1$ be the event that there is a plane crash and $E_2$ be the event that there is no crash. Let A be the event that passengers survive after the journey. On the basis of the above information, answer the following questions :
Q.38 (i)1 mark $Find the probability that the airplane will not crash.$
Q.38 (ii)1 mark $P(A \,|\, E_1) + P(A \,|\, E_2)$
Q.38 (iii) (a)2 marks $Find P(A).$
OR
Q.38 (iii) (b)2 marks $P(E_2 \,|\, A)$

Section A

1 mark each

Q.11 mark $\dfrac{dy}{dx} - y = 1$
Tap an option to check your answer.
Q.21 mark $\vec{a}$
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Q.31 mark $The coordinates of the foot of the perpendicular drawn from the point (0, 1, 2) on the x-axis are given by :$
Tap an option to check your answer.
Q.41 mark $The common region determined by all the constraints of a linear programming problem is called :$
Tap an option to check your answer.
Q.51 mark $P(S \,|\, E) =$
Tap an option to check your answer.
Q.61 mark $-1$
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Q.71 mark $\dfrac{d}{dx} [\cos (\log x + e^x)]$
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Q.81 mark $(y'')^2 + (y')^3 = x \sin (y')$
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Q.91 mark $\hat{i} + \hat{k}$
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Q.101 mark $\dfrac{x-1}{2} = -y = \dfrac{2z+1}{6}$
Tap an option to check your answer.
Q.111 mark $F(x) = \begin{bmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{bmatrix}$
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Q.121 mark $30^\circ$
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Q.131 mark $3 \times 3$
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Q.141 mark $A_i$
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Q.151 mark $\begin{vmatrix} -a & b & c \\ a & -b & c \\ a & b & -c \end{vmatrix} = kabc$
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Q.161 mark $f(x) = \begin{cases} |x| + 3, & \text{if } x \le -3 \\ -2x, & \text{if } -3 < x < 3 \\ 6x + 2, & \text{if } x \ge 3 \end{cases}$
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Q.171 mark $f(x) = x^3 - 3x^2 + 12x - 18$
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Q.181 mark $\displaystyle\int_0^2 2 e^{2x}\, dx = \int_0^a e^x\, dx$
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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 $\vec{a}$
Tap an option to check your answer.
Q.201 mark $P' = -P$
Tap an option to check your answer.

Section B

2 marks each

Q.212 marks $f(x) = x + \dfrac{1}{x}$
Q.222 marks $\displaystyle\int_0^{a^3} \dfrac{x^2}{x^6 + a^6}\, dx$
Q.232 marks $f(x) = e^x - e^{-x} + x - \tan^{-1} x$
Q.24 (a)2 marks $\tan^{-1}\left(-\dfrac{1}{\sqrt{3}}\right) + \cot^{-1}\left(\dfrac{1}{\sqrt{3}}\right) + \tan^{-1}\left[\sin\left(-\dfrac{\pi}{2}\right)\right]$
OR
Q.24 (b)2 marks $f(x) = \sin^{-1}(x^2 - 4)$
Q.25 (a)2 marks $f(x) = |\cos x|$
OR
Q.25 (b)2 marks $y = A \sin 2x + B \cos 2x$

Section C

3 marks each

Q.26 (a)3 marks $2xy + y^2 - 2x^2 \dfrac{dy}{dx} = 0$
OR
Q.26 (b)3 marks $y\, dx = (x + 2y^2)\, dy$
Q.273 marks $\vec{a}$
Q.283 marks $A pair of dice is thrown simultaneously. If X denotes the absolute difference of the numbers appearing on top of the dice, then find the probability distribution of X.$
Q.293 marks $\displaystyle\int x^2 \cdot \sin^{-1}(x^{3/2})\, dx$
Q.30 (a)3 marks $x^{30} y^{20} = (x + y)^{50}$
OR
Q.30 (b)3 marks $\dfrac{dy}{dx}$
Q.31 (a)3 marks $\displaystyle\int_{-2}^{2} \sqrt{\dfrac{2 - x}{2 + x}}\, dx$
OR
Q.31 (b)3 marks $\displaystyle\int \dfrac{1}{x\,[(\log x)^2 - 3 \log x - 4]}\, dx$

Section D

5 marks each

Q.325 marks $\vec{r} = \lambda\hat{i} + (2\lambda + 1)\hat{j} + (3\lambda + 2)\hat{k}$
Q.33 (a)5 marks $A = \begin{bmatrix} 5 & 0 & 4 \\ 2 & 3 & 2 \\ 1 & 2 & 1 \end{bmatrix}$
OR
Q.33 (b)5 marks $A = \begin{bmatrix} 1 & 1 & 1 \\ 2 & 3 & 2 \\ 1 & 1 & 2 \end{bmatrix}$
Q.345 marks $A_1$
Q.35 (a)5 marks $f : R \to R$
OR
Q.35 (b)5 marks $N \times N$

Section E

The month of September is celebrated as the Rashtriya Poshan Maah across the country. Following a healthy and well-balanced diet is crucial in order to supply the body with the proper nutrients it needs. A balanced diet also keeps us mentally fit and promotes improved level of energy. A dietician wishes to minimize the cost of a diet involving two types of foods, food X (x kg) and food Y (y kg) which are available at the rate of ₹ 16/kg and ₹ 20/kg respectively. The feasible region satisfying the constraints is shown in Figure-2. On the basis of the above information, answer the following questions :
Q.36 (i)2 marks $Identify and write all the constraints which determine the given feasible region in Figure-2.$
Q.36 (ii)2 marks $Z = 16x + 20y$
Airplanes are by far the safest mode of transportation when the number of transported passengers are measured against personal injuries and fatality totals. Previous records state that the probability of an airplane crash is $0{\cdot}00001\%$ . Further, there are 95% chances that there will be survivors after a plane crash. Assume that in case of no crash, all travellers survive. Let $E_1$ be the event that there is a plane crash and $E_2$ be the event that there is no crash. Let A be the event that passengers survive after the journey. On the basis of the above information, answer the following questions :
Q.37 (i)1 mark $Find the probability that the airplane will not crash.$
Q.37 (ii)1 mark $P(A \,|\, E_1) + P(A \,|\, E_2)$
Q.37 (iii) (a)2 marks $Find P(A).$
OR
Q.37 (iii) (b)2 marks $P(E_2 \,|\, A)$
Overspeeding increases fuel consumption and decreases fuel economy as a result of tyre rolling friction and air resistance. While vehicles reach optimal fuel economy at different speeds, fuel mileage usually decreases rapidly at speeds above 80 km/h. The relation between fuel consumption F (l/100 km) and speed V (km/h) under some constraints is given as $F = \dfrac{V^2}{500} - \dfrac{V}{4} + 14$ . On the basis of the above information, answer the following questions :
Q.38 (i)1 mark $Find F, when V = 40 km/h.$
Q.38 (ii)1 mark $\dfrac{dF}{dV}$
Q.38 (iii) (a)2 marks $Find the speed V for which fuel consumption F is minimum.$
OR
Q.38 (iii) (b)2 marks $\dfrac{dF}{dV} = -0{\cdot}01$

Section A

1 mark each

Q.11 mark $y = \cos^{-1}(e^x)$
Tap an option to check your answer.
Q.21 mark $(y'')^2 + \log (y') = x^5$
Tap an option to check your answer.
Q.31 mark $\hat{i} + \hat{k}$
Tap an option to check your answer.
Q.41 mark $\dfrac{x-1}{2} = -y = \dfrac{2z+1}{6}$
Tap an option to check your answer.
Q.51 mark $A = \begin{bmatrix} \tan x & 1 \\ -1 & \tan x \end{bmatrix}$
Tap an option to check your answer.
Q.61 mark $30^\circ$
Tap an option to check your answer.
Q.71 mark $3 \times 3$
Tap an option to check your answer.
Q.81 mark $f : R_+ \to [-5, \infty)$
Tap an option to check your answer.
Q.91 mark $\begin{vmatrix} -a & b & c \\ a & -b & c \\ a & b & -c \end{vmatrix} = kabc$
Tap an option to check your answer.
Q.101 mark $f(x) = \begin{cases} |x| + 3, & \text{if } x \le -3 \\ -2x, & \text{if } -3 < x < 3 \\ 6x + 2, & \text{if } x \ge 3 \end{cases}$
Tap an option to check your answer.
Q.111 mark $f(x) = x^3 - 3x^2 + 12x - 18$
Tap an option to check your answer.
Q.121 mark $\sqrt{1 + \sin 2x}$
Tap an option to check your answer.
Q.131 mark $\dfrac{dy}{dx} = F(x, y)$
Tap an option to check your answer.
Q.141 mark $\vec{a}$
Tap an option to check your answer.
Q.151 mark $The coordinates of the foot of the perpendicular drawn from the point (0, 1, 2) on the x-axis are given by :$
Tap an option to check your answer.
Q.161 mark $The common region determined by all the constraints of a linear programming problem is called :$
Tap an option to check your answer.
Q.171 mark $P(S \,|\, E) =$
Tap an option to check your answer.
Q.181 mark $A = [a_{ij}]$
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 $P' = -P$
Tap an option to check your answer.
Q.201 mark $(\vec{b} \cdot \vec{c})\, \vec{a}$
Tap an option to check your answer.

Section B

2 marks each

Q.212 marks $f(x) = x^2 - 6x + 3$
Q.22 (a)2 marks $\sin^{-1}\left(-\dfrac{1}{2}\right) + \cos^{-1}\left(-\dfrac{\sqrt{3}}{2}\right) + \cot^{-1}\left(\tan \dfrac{4\pi}{3}\right)$
OR
Q.22 (b)2 marks $f(x) = \cos^{-1}(1 - x^2)$
Q.23 (a)2 marks $f(x) = |\tan 2x|$
OR
Q.23 (b)2 marks $y = \text{cosec}\,(\cot^{-1} x)$
Q.242 marks $f(x) = x + \dfrac{1}{x}$
Q.252 marks $\displaystyle\int \dfrac{e^{4x} - 1}{e^{4x} + 1}\, dx$

Section C

3 marks each

Q.263 marks $A pair of dice is thrown simultaneously. If X denotes the absolute difference of the numbers appearing on top of the dice, then find the probability distribution of X.$
Q.273 marks $\displaystyle\int x^2 \log (x^2 - 1)\, dx$
Q.28 (a)3 marks $y = (\log x)^2$
OR
Q.28 (b)3 marks $y = \sin (\tan^{-1} e^x)$
Q.29 (a)3 marks $\displaystyle\int_{-2}^{2} \sqrt{\dfrac{2 - x}{2 + x}}\, dx$
OR
Q.29 (b)3 marks $\displaystyle\int \dfrac{1}{x\,[(\log x)^2 - 3 \log x - 4]}\, dx$
Q.30 (a)3 marks $2xy + y^2 - 2x^2 \dfrac{dy}{dx} = 0$
OR
Q.30 (b)3 marks $y\, dx = (x + 2y^2)\, dy$
Q.313 marks $\Delta$

Section D

5 marks each

Q.325 marks $A_1$
Q.33 (a)5 marks $f : R \to R$
OR
Q.33 (b)5 marks $N \times N$
Q.345 marks $\Delta$
Q.35 (a)5 marks $\dfrac{2}{x} + \dfrac{3}{y} + \dfrac{10}{z} = 4$
OR
Q.35 (b)5 marks $A = \begin{bmatrix} 1 & \cot x \\ -\cot x & 1 \end{bmatrix}$

Section E

Airplanes are by far the safest mode of transportation when the number of transported passengers are measured against personal injuries and fatality totals. Previous records state that the probability of an airplane crash is $0{\cdot}00001\%$ . Further, there are 95% chances that there will be survivors after a plane crash. Assume that in case of no crash, all travellers survive. Let $E_1$ be the event that there is a plane crash and $E_2$ be the event that there is no crash. Let A be the event that passengers survive after the journey. On the basis of the above information, answer the following questions :
Q.36 (i)1 mark $Find the probability that the airplane will not crash.$
Q.36 (ii)1 mark $P(A \,|\, E_1) + P(A \,|\, E_2)$
Q.36 (iii) (a)2 marks $Find P(A).$
OR
Q.36 (iii) (b)2 marks $P(E_2 \,|\, A)$
Overspeeding increases fuel consumption and decreases fuel economy as a result of tyre rolling friction and air resistance. While vehicles reach optimal fuel economy at different speeds, fuel mileage usually decreases rapidly at speeds above 80 km/h. The relation between fuel consumption F (l/100 km) and speed V (km/h) under some constraints is given as $F = \dfrac{V^2}{500} - \dfrac{V}{4} + 14$ . On the basis of the above information, answer the following questions :
Q.37 (i)1 mark $Find F, when V = 40 km/h.$
Q.37 (ii)1 mark $\dfrac{dF}{dV}$
Q.37 (iii) (a)2 marks $Find the speed V for which fuel consumption F is minimum.$
OR
Q.37 (iii) (b)2 marks $\dfrac{dF}{dV} = -0{\cdot}01$
The month of September is celebrated as the Rashtriya Poshan Maah across the country. Following a healthy and well-balanced diet is crucial in order to supply the body with the proper nutrients it needs. A balanced diet also keeps us mentally fit and promotes improved level of energy. A dietician wishes to minimize the cost of a diet involving two types of foods, food X (x kg) and food Y (y kg) which are available at the rate of ₹ 16/kg and ₹ 20/kg respectively. The feasible region satisfying the constraints is shown in Figure-2. On the basis of the above information, answer the following questions :
Q.38 (i)2 marks $Identify and write all the constraints which determine the given feasible region in Figure-2.$
Q.38 (ii)2 marks $Z = 16x + 20y$