CBSE Class 12 Mathematics 2023 question paper (65/5)

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 $Let A = {3, 5}. Then number of reflexive relations on A is$
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Q.21 mark $\sin\left[\dfrac{\pi}{3} + \sin^{-1}\left(\dfrac{1}{2}\right)\right]$
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Q.31 mark $A^2 - A + I = O$
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Q.41 mark $A = \begin{bmatrix} 1 & 0 \\ 2 & 1 \end{bmatrix}$
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Q.51 mark $\begin{vmatrix} \alpha & 3 & 4 \\ 1 & 2 & 1 \\ 1 & 4 & 1 \end{vmatrix} = 0$
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Q.61 mark $x^{2x}$
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Q.71 mark $f(x) = [x]$
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Q.81 mark $x = A\cos 4t + B\sin 4t$
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Q.91 mark $f(x) = 2x^3 + 9x^2 + 12x - 1$
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Q.101 mark $\displaystyle\int \dfrac{\sec x}{\sec x - \tan x}\,dx$
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Q.111 mark $\displaystyle\int_{-1}^{1} \dfrac{|x - 2|}{x - 2}\,dx$
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Q.121 mark $\dfrac{d}{dx}\left(\left(\dfrac{dy}{dx}\right)^3\right)$
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Q.131 mark $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$
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Q.141 mark $6\hat{i} - 2\hat{j} + 3\hat{k}$
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Q.151 mark $90^{\circ}$
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Q.161 mark $2x = 3y = -z$
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Q.171 mark $P(A) = \dfrac{4}{5}$
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Q.181 mark $Five fair coins are tossed simultaneously. The probability of the events that atleast one head comes up is$
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In the following questions 19 and 20, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct answer out of the following choices :
Q.191 mark $\dfrac{1}{3}$
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Q.201 mark $\displaystyle\int_{2}^{8} \dfrac{\sqrt{10 - x}}{\sqrt{x} + \sqrt{10 - x}}\,dx = 3$
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Section B

2 marks each

Q.212 marks $f(x) = \tan^{-1} x$
Q.22 (a)2 marks $f(x) = \begin{cases} x^2, & \text{if } x \ge 1 \\ x, & \text{if } x < 1 \end{cases}$
OR
Q.22 (b)2 marks $\lambda$
Q.232 marks $2x + y = 8$
Q.24 (a)2 marks $\vec{a}$
OR
Q.24 (b)2 marks $\vec{a} = \hat{i} - \hat{j} + 3\hat{k}$
Q.252 marks $-1$

Section C

3 marks each

Q.263 marks $A = \begin{bmatrix} 1 & 2 & 3 \\ 3 & -2 & 1 \\ 4 & 2 & 1 \end{bmatrix}$
Q.27 (a)3 marks $\sec^{-1}\left(\dfrac{1}{\sqrt{1 - x^2}}\right)$
OR
Q.27 (b)3 marks $y = \tan x + \sec x$
Q.28 (a)3 marks $\displaystyle\int_{0}^{2\pi} \dfrac{1}{1 + e^{\sin x}}\,dx$
OR
Q.28 (b)3 marks $\displaystyle\int \dfrac{x^4}{(x - 1)(x^2 + 1)}\,dx$
Q.293 marks $\{(x, y) : y^2 \le 2x \text{ and } y \ge x - 4\}$
Q.30 (a)3 marks $\dfrac{x + 3}{5} = \dfrac{y - 1}{2} = \dfrac{z + 4}{3}$
OR
Q.30 (b)3 marks $\vec{a}$
Q.313 marks $\vec{r} = (\hat{i} + 2\hat{j} - 4\hat{k}) + \lambda(2\hat{i} + 3\hat{j} + 6\hat{k})$

Section D

5 marks each

Q.32 (a)5 marks $2\sqrt{3}$
OR
Q.32 (b)5 marks $Sum of two numbers is 5. If the sum of the cubes of these numbers is least, then find the sum of the squares of these numbers.$
Q.335 marks $\displaystyle\int_{0}^{\pi/2} \sin 2x \tan^{-1}(\sin x)\,dx$
Q.345 marks $P = 70x + 40y$
Q.35 (a)5 marks $\dfrac{3}{5}$
OR
Q.35 (b)5 marks $A box contains 10 tickets, 2 of which carry a prize of ₹ 8 each, 5 of which carry a prize of ₹ 4 each, and remaining 3 carry a prize of ₹ 2 each. If one ticket is drawn at random, find the mean value of the prize.$

Section E

An organization conducted bike race under two different categories – Boys and Girls. There were 28 participants in all. Among all of them, finally three from category 1 and two from category 2 were selected for the final race. Ravi forms two sets B and G with these participants for his college project. Let B = $\{b_1, b_2, b_3\}$ and G = $\{g_1, g_2\}$ , where B represents the set of Boys selected and G the set of Girls selected for the final race. Based on the above information, answer the following questions :
Q.36 (i)1 mark $How many relations are possible from B to G ?$
Q.36 (ii)1 mark $Among all the possible relations from B to G, how many functions can be formed from B to G ?$
Q.36 (iii) (a)2 marks $R : B \to B$
OR
Q.36 (iii) (b)2 marks $f : B \to G$
Gautam buys 5 pens, 3 bags and 1 instrument box and pays a sum of ₹ 160. From the same shop, Vikram buys 2 pens, 1 bag and 3 instrument boxes and pays a sum of ₹ 190. Also Ankur buys 1 pen, 2 bags and 4 instrument boxes and pays a sum of ₹ 250. Based on the above information, answer the following questions :
Q.37 (i)1 mark $Convert the given above situation into a matrix equation of the form AX = B.$
Q.37 (ii)1 mark $|A|$
Q.37 (iii) (a)2 marks $A^{-1}$
OR
Q.37 (iii) (b)2 marks $P = A^2 - 5A$
An equation involving derivatives of the dependent variable with respect to the independent variables is called a differential equation. A differential equation of the form $\dfrac{dy}{dx} = F(x, y)$ is said to be homogeneous if F(x, y) is a homogeneous function of degree zero, whereas a function F(x, y) is a homogenous function of degree n if $F(\lambda x, \lambda y) = \lambda^n F(x, y)$ . To solve a homogeneous differential equation of the type $\dfrac{dy}{dx} = F(x, y) = g\left(\dfrac{y}{x}\right)$ , we make the substitution $y = vx$ and then separate the variables. Based on the above, answer the following questions :
Q.38 (i)2 marks $(x^2 - y^2)\,dx + 2xy\,dy = 0$
Q.38 (ii)2 marks $Solve the above equation to find its general solution.$

Section A

1 mark each

Q.11 mark $\sin\left[\dfrac{\pi}{3} + \sin^{-1}\left(\dfrac{1}{2}\right)\right]$
Tap an option to check your answer.
Q.21 mark $Let A = {3, 5}. Then number of reflexive relations on A is$
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Q.31 mark $A = \begin{bmatrix} 1 & 0 \\ 2 & 1 \end{bmatrix}$
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Q.41 mark $A = [a_{ij}]$
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Q.51 mark $\begin{vmatrix} 6 & 0 & -1 \\ 2 & 1 & 4 \\ 1 & 1 & 3 \end{vmatrix}$
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Q.61 mark $f(x) = [x]$
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Q.71 mark $x^{2x}$
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Q.81 mark $f(x) = 2x^3 + 9x^2 + 12x - 1$
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Q.91 mark $f(x) = x\,|\,x\,|$
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Q.101 mark $\displaystyle\int \dfrac{\sec x}{\sec x - \tan x}\,dx$
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Q.111 mark $\displaystyle\int_{0}^{\pi/4} (\sin 2x)\,dx$
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Q.121 mark $\dfrac{d}{dx}\left(\left(\dfrac{dy}{dx}\right)^3\right)$
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Q.131 mark $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$
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Q.141 mark $\hat{a}$
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Q.151 mark $2x = 3y = -z$
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Q.161 mark $90^{\circ}$
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Q.171 mark $P(A) = \dfrac{4}{5}$
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Q.181 mark $P(A) = \dfrac{1}{3}$
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In the following questions 19 and 20, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct answer out of the following choices :
Q.191 mark $\displaystyle\int_{2}^{8} \dfrac{\sqrt{10 - x}}{\sqrt{x} + \sqrt{10 - x}}\,dx = 3$
Tap an option to check your answer.
Q.201 mark $\dfrac{1}{3}$
Tap an option to check your answer.

Section B

2 marks each

Q.212 marks $f(x) = \cos^{-1} x$
Q.22 (a)2 marks $\vec{a}$
OR
Q.22 (b)2 marks $\vec{a} = \hat{i} - \hat{j} + 3\hat{k}$
Q.23 (a)2 marks $f(x) = \begin{cases} x^2, & \text{if } x \ge 1 \\ x, & \text{if } x < 1 \end{cases}$
OR
Q.23 (b)2 marks $\lambda$
Q.242 marks $2x + y = 8$
Q.252 marks $\vec{r} = 2\hat{i} - 5\hat{j} + \hat{k} + \lambda(3\hat{i} + 2\hat{j} + 6\hat{k})$

Section C

3 marks each

Q.263 marks $\Delta PQR$
Q.27 (a)3 marks $\sec^{-1}\left(\dfrac{1}{\sqrt{1 - x^2}}\right)$
OR
Q.27 (b)3 marks $y = \tan x + \sec x$
Q.28 (a)3 marks $\displaystyle\int_{-\pi/4}^{\pi/4} \dfrac{\cos 2x}{1 + \cos 2x}\,dx$
OR
Q.28 (b)3 marks $\displaystyle\int e^{x^2}(x^5 + 2x^3)\,dx$
Q.293 marks $x^2 + y^2 = 4$
Q.303 marks $\vec{r} = (\hat{i} + 2\hat{j} - 4\hat{k}) + \lambda(2\hat{i} + 3\hat{j} + 6\hat{k})$
Q.31 (a)3 marks $\dfrac{x + 3}{5} = \dfrac{y - 1}{2} = \dfrac{z + 4}{3}$
OR
Q.31 (b)3 marks $\vec{a}$

Section D

5 marks each

Q.325 marks $\displaystyle\int_{0}^{\pi} \dfrac{x}{1 + \sin x}\,dx$
Q.33 (a)5 marks $2\sqrt{3}$
OR
Q.33 (b)5 marks $Sum of two numbers is 5. If the sum of the cubes of these numbers is least, then find the sum of the squares of these numbers.$
Q.34 (a)5 marks $\dfrac{3}{5}$
OR
Q.34 (b)5 marks $A box contains 10 tickets, 2 of which carry a prize of ₹ 8 each, 5 of which carry a prize of ₹ 4 each, and remaining 3 carry a prize of ₹ 2 each. If one ticket is drawn at random, find the mean value of the prize.$
Q.355 marks $P = 70x + 40y$

Section E

Gautam buys 5 pens, 3 bags and 1 instrument box and pays a sum of ₹ 160. From the same shop, Vikram buys 2 pens, 1 bag and 3 instrument boxes and pays a sum of ₹ 190. Also Ankur buys 1 pen, 2 bags and 4 instrument boxes and pays a sum of ₹ 250. Based on the above information, answer the following questions :
Q.36 (i)1 mark $Convert the given above situation into a matrix equation of the form AX = B.$
Q.36 (ii)1 mark $|A|$
Q.36 (iii) (a)2 marks $A^{-1}$
OR
Q.36 (iii) (b)2 marks $P = A^2 - 5A$
An organization conducted bike race under two different categories – Boys and Girls. There were 28 participants in all. Among all of them, finally three from category 1 and two from category 2 were selected for the final race. Ravi forms two sets B and G with these participants for his college project. Let B = $\{b_1, b_2, b_3\}$ and G = $\{g_1, g_2\}$ , where B represents the set of Boys selected and G the set of Girls selected for the final race. Based on the above information, answer the following questions :
Q.37 (i)1 mark $How many relations are possible from B to G ?$
Q.37 (ii)1 mark $Among all the possible relations from B to G, how many functions can be formed from B to G ?$
Q.37 (iii) (a)2 marks $R : B \to B$
OR
Q.37 (iii) (b)2 marks $f : B \to G$
An equation involving derivatives of the dependent variable with respect to the independent variables is called a differential equation. A differential equation of the form $\dfrac{dy}{dx} = F(x, y)$ is said to be homogeneous if F(x, y) is a homogeneous function of degree zero, whereas a function F(x, y) is a homogenous function of degree n if $F(\lambda x, \lambda y) = \lambda^n F(x, y)$ . To solve a homogeneous differential equation of the type $\dfrac{dy}{dx} = F(x, y) = g\left(\dfrac{y}{x}\right)$ , we make the substitution $y = vx$ and then separate the variables. Based on the above, answer the following questions :
Q.38 (i)2 marks $(x^2 - y^2)\,dx + 2xy\,dy = 0$
Q.38 (ii)2 marks $Solve the above equation to find its general solution.$

Section A

1 mark each

Q.11 mark $R = \{(a, b) : a = b - 2, b > 6\}$
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Q.21 mark $A = \begin{bmatrix} 5 & x \\ y & 0 \end{bmatrix}$
Tap an option to check your answer.
Q.31 mark $\sin\left[\dfrac{\pi}{3} + \sin^{-1}\left(\dfrac{1}{2}\right)\right]$
Tap an option to check your answer.
Q.41 mark $A^2 - A + I = O$
Tap an option to check your answer.
Q.51 mark $\begin{vmatrix} \alpha & 3 & 4 \\ 1 & 2 & 1 \\ 1 & 4 & 1 \end{vmatrix} = 0$
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Q.61 mark $f(x) = |\cos x|$
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Q.71 mark $x = A\cos 4t + B\sin 4t$
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Q.81 mark $f(x) = [x]$
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Q.91 mark $f(x) = x^3 + 3x$
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Q.101 mark $\displaystyle\int_{-1}^{1} \dfrac{|x - 2|}{x - 2}\,dx$
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Q.111 mark $\displaystyle\int \dfrac{\sec x}{\sec x - \tan x}\,dx$
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Q.121 mark $\left(1 + 3\dfrac{dy}{dx}\right)^2 = 4\dfrac{d^3y}{dx^3}$
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Q.131 mark $\vec{a} \cdot \hat{i} = \vec{a} \cdot (\hat{i} + \hat{j}) = \vec{a} \cdot (\hat{i} + \hat{j} + \hat{k}) = 1$
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Q.141 mark $Five fair coins are tossed simultaneously. The probability of the events that atleast one head comes up is$
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Q.151 mark $P(A) = \dfrac{4}{5}$
Tap an option to check your answer.
Q.161 mark $2x = 3y = -z$
Tap an option to check your answer.
Q.171 mark $90^{\circ}$
Tap an option to check your answer.
Q.181 mark $6\hat{i} - 2\hat{j} + 3\hat{k}$
Tap an option to check your answer.
In the following questions 19 and 20, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct answer out of the following choices :
Q.191 mark $\displaystyle\int_{2}^{8} \dfrac{\sqrt{10 - x}}{\sqrt{x} + \sqrt{10 - x}}\,dx = 3$
Tap an option to check your answer.
Q.201 mark $\dfrac{1}{3}$
Tap an option to check your answer.

Section B

2 marks each

Q.21 (a)2 marks $f(x) = \begin{cases} \dfrac{1 - \cos x}{2x^2}, & \text{if } x \ne 0 \\ k, & \text{if } x = 0 \end{cases}$
OR
Q.21 (b)2 marks $x = a\cos t$
Q.222 marks $\tan^{-1}\left[2\cos\left(2\sin^{-1}\dfrac{1}{2}\right)\right] + \tan^{-1} 1$
Q.232 marks $-1$
Q.242 marks $2x + y = 8$
Q.25 (a)2 marks $\vec{a}$
OR
Q.25 (b)2 marks $\vec{a} = \hat{i} - \hat{j} + 3\hat{k}$

Section C

3 marks each

Q.263 marks $\begin{vmatrix} x & \sin\theta & \cos\theta \\ -\sin\theta & -x & 1 \\ \cos\theta & 1 & x \end{vmatrix}$
Q.273 marks $y = mx$
Q.28 (a)3 marks $\dfrac{x - 15}{3} = \dfrac{y - 29}{8} = \dfrac{z - 5}{-5}$
OR
Q.28 (b)3 marks $\vec{a} = \hat{i} + \hat{j} + \hat{k}$
Q.293 marks $\vec{r} = (\hat{i} + 2\hat{j} - 4\hat{k}) + \lambda(2\hat{i} + 3\hat{j} + 6\hat{k})$
Q.30 (a)3 marks $\sec^{-1}\left(\dfrac{1}{\sqrt{1 - x^2}}\right)$
OR
Q.30 (b)3 marks $y = \tan x + \sec x$
Q.31 (a)3 marks $\displaystyle\int_{0}^{2\pi} \dfrac{1}{1 + e^{\sin x}}\,dx$
OR
Q.31 (b)3 marks $\displaystyle\int \dfrac{x^4}{(x - 1)(x^2 + 1)}\,dx$

Section D

5 marks each

Q.325 marks $Z = 60x + 80y$
Q.33 (a)5 marks $2\sqrt{3}$
OR
Q.33 (b)5 marks $Sum of two numbers is 5. If the sum of the cubes of these numbers is least, then find the sum of the squares of these numbers.$
Q.345 marks $\displaystyle\int_{0}^{\pi/2} \sin 2x \tan^{-1}(\sin x)\,dx$
Q.35 (a)5 marks $\dfrac{3}{5}$
OR
Q.35 (b)5 marks $A box contains 10 tickets, 2 of which carry a prize of ₹ 8 each, 5 of which carry a prize of ₹ 4 each, and remaining 3 carry a prize of ₹ 2 each. If one ticket is drawn at random, find the mean value of the prize.$

Section E

An organization conducted bike race under two different categories – Boys and Girls. There were 28 participants in all. Among all of them, finally three from category 1 and two from category 2 were selected for the final race. Ravi forms two sets B and G with these participants for his college project. Let B = $\{b_1, b_2, b_3\}$ and G = $\{g_1, g_2\}$ , where B represents the set of Boys selected and G the set of Girls selected for the final race. Based on the above information, answer the following questions :
Q.36 (i)1 mark $How many relations are possible from B to G ?$
Q.36 (ii)1 mark $Among all the possible relations from B to G, how many functions can be formed from B to G ?$
Q.36 (iii) (a)2 marks $R : B \to B$
OR
Q.36 (iii) (b)2 marks $f : B \to G$
Gautam buys 5 pens, 3 bags and 1 instrument box and pays a sum of ₹ 160. From the same shop, Vikram buys 2 pens, 1 bag and 3 instrument boxes and pays a sum of ₹ 190. Also Ankur buys 1 pen, 2 bags and 4 instrument boxes and pays a sum of ₹ 250. Based on the above information, answer the following questions :
Q.37 (i)1 mark $Convert the given above situation into a matrix equation of the form AX = B.$
Q.37 (ii)1 mark $|A|$
Q.37 (iii) (a)2 marks $A^{-1}$
OR
Q.37 (iii) (b)2 marks $P = A^2 - 5A$
An equation involving derivatives of the dependent variable with respect to the independent variables is called a differential equation. A differential equation of the form $\dfrac{dy}{dx} = F(x, y)$ is said to be homogeneous if F(x, y) is a homogeneous function of degree zero, whereas a function F(x, y) is a homogenous function of degree n if $F(\lambda x, \lambda y) = \lambda^n F(x, y)$ . To solve a homogeneous differential equation of the type $\dfrac{dy}{dx} = F(x, y) = g\left(\dfrac{y}{x}\right)$ , we make the substitution $y = vx$ and then separate the variables. Based on the above, answer the following questions :
Q.38 (i)2 marks $(x^2 - y^2)\,dx + 2xy\,dy = 0$
Q.38 (ii)2 marks $Solve the above equation to find its general solution.$