CBSE Class 12 Mathematics 2022 question paper (65/2)

Maximum marks 40 · Time 2 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

2 marks each

Q.12 marks $\left[ \frac{d}{dx} (xy^2) \right] \cdot \frac{dy}{dx} + y = 0$
Q.2 (a)2 marks $\int \frac{\sin 3x}{\sin x} \, dx$
OR
Q.2 (b)2 marks $\int_{0}^{\frac{1}{2} \log 3} \frac{e^x}{e^{2x} + 1} \, dx$
Q.32 marks $\vec{a}$
Q.42 marks $A pair of dice is thrown. It is given that the sum of numbers appearing on both dice is an even number. Find the probability that the number appearing on at least one die is 3.$
Q.52 marks $\frac{2}{3}$
Q.62 marks $P(2, 2, 1)$

3 marks each

Q.114 marks $(-1, -5, -10)$
Q.124 marks $\int_{0}^{1} x (1 - x)^n \, dx$
Q.13 (a)4 marks $4x^2 + 4y^2 = 9$
OR
Q.13 (b)4 marks $y^2 = 4ax$
At the start of a cricket match, a coin is tossed and the team winning the toss has the opportunity to choose to bat or bowl. Such a coin is unbiased with equal probabilities of getting head and tail. Based on the above information, answer the following questions :
Q.14 (a)2 marks $If such a coin is tossed 2 times, then find the probability distribution of number of tails.$
Q.14 (b)2 marks $Find the probability of getting at least one head in three tosses of such a coin.$

2 marks each

Q.12 marks $A pair of dice is thrown. It is given that the sum of numbers appearing on both dice is an even number. Find the probability that the number appearing on at least one die is 3.$
Q.22 marks $\vec{a}$
Q.3 (a)2 marks $\int \frac{\sin 3x}{\sin x} \, dx$
OR
Q.3 (b)2 marks $\int_{0}^{\frac{1}{2} \log 3} \frac{e^x}{e^{2x} + 1} \, dx$
Q.42 marks $\frac{2}{3}$
Q.52 marks $P(2, 2, 1)$
Q.62 marks $\frac{dy}{dx} = 2y^2$

3 marks each

Q.11 (a)4 marks $4x^2 + 4y^2 = 9$
OR
Q.11 (b)4 marks $y^2 = 4ax$
Q.124 marks $(-1, -5, -10)$
Q.134 marks $\int_{0}^{\pi/2} \frac{x}{\sin x + \cos x} \, dx$
At the start of a cricket match, a coin is tossed and the team winning the toss has the opportunity to choose to bat or bowl. Such a coin is unbiased with equal probabilities of getting head and tail. Based on the above information, answer the following questions :
Q.14 (a)2 marks $If such a coin is tossed 2 times, then find the probability distribution of number of tails.$
Q.14 (b)2 marks $Find the probability of getting at least one head in three tosses of such a coin.$

2 marks each

Q.12 marks $P(2, 2, 1)$
Q.22 marks $\frac{2}{3}$
Q.32 marks $A pair of dice is thrown. It is given that the sum of numbers appearing on both dice is an even number. Find the probability that the number appearing on at least one die is 3.$
Q.42 marks $\vec{a}$
Q.52 marks $\left[ \frac{d}{dx} (xy^2) \right] \cdot \frac{dy}{dx} + y = 0$
Q.6 (a)2 marks $\int \frac{\sqrt{\cot x}}{\sin x \cos x} \, dx$
OR
Q.6 (b)2 marks $\int \frac{1}{x (x^2 + 4)} \, dx$

3 marks each

Q.7 (a)3 marks $\int_{0}^{1} \tan^{-1} x \, dx$
OR
Q.7 (b)3 marks $\int \frac{2x}{x^2 + 3x + 2} \, dx$
Q.8 (a)3 marks $(2, 1, 0)$
OR
Q.8 (b)3 marks $x = \frac{y - 1}{2} = \frac{z - 2}{3}$
Q.93 marks $(1 + e^{y/x}) \, dy + e^{y/x} \left( 1 - \frac{y}{x} \right) dx = 0$
Q.103 marks $\vec{a}$

Q.114 marks $\int_{0}^{1} x (1 - x)^n \, dx$
Q.12 (a)4 marks $4x^2 + 4y^2 = 9$
OR
Q.12 (b)4 marks $y^2 = 4ax$
Q.134 marks $(5, 1, 6)$
At the start of a cricket match, a coin is tossed and the team winning the toss has the opportunity to choose to bat or bowl. Such a coin is unbiased with equal probabilities of getting head and tail. Based on the above information, answer the following questions :
Q.14 (a)2 marks $If such a coin is tossed 2 times, then find the probability distribution of number of tails.$
Q.14 (b)2 marks $Find the probability of getting at least one head in three tosses of such a coin.$