CBSE Class 12 Mathematics 2022 question paper (65/4)

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 $A bag contains 3 red and 4 white balls. Three balls are drawn at random, one-by-one without replacement from the bag. If the first ball drawn is red in colour, then find the probability that the remaining two balls drawn are also red in colour.$
Q.22 marks $A coin is tossed twice. The following table shows the probability distribution of number of tails : X 012 P(X) K 6K 9K (a) Find the value of K. (b) Is the coin tossed biased or unbiased ? Justify your answer.$
Q.32 marks $(-2, -1, -3)$
Q.4 (a)2 marks $|\vec{a} \times \vec{b}|^2 + |\vec{a} \cdot \vec{b}|^2 = 400$
OR
Q.4 (b)2 marks $5\sqrt{3}$
Q.52 marks $\sec^2 x \cdot \tan y \, dx + \sec^2 y \cdot \tan x \, dy = 0.$
Q.62 marks $\displaystyle\int_0^1 x^2 e^x \, dx$

X	0	1	2
P(X)	K	6K	9K

3 marks each

Q.114 marks $\vec{r} \cdot (\hat{i} + 3\hat{j}) = 6$
Q.12 (a)4 marks $x \dfrac{dy}{dx} + y + \dfrac{1}{1 + x^2} = 0$
OR
Q.12 (b)4 marks $x (y^3 + x^3) \, dy = (2y^4 + 5x^3 y) \, dx.$
Q.134 marks $\displaystyle\int_0^{\pi/2} (2 \log \cos x - \log \sin 2x) \, dx$
In a game of Archery, each ring of the Archery target is valued. The centremost ring is worth 10 points and rest of the rings are allotted points 9 to 1 in sequential order moving outwards. Archer A is likely to earn 10 points with a probability of $0{\cdot}8$ and Archer B is likely the earn 10 points with a probability of $0{\cdot}9$ . Based on the above information, answer the following questions : If both of them hit the Archery target, then find the probability that
Q.14 (a)2 marks $exactly one of them earns 10 points.$
Q.14 (b)2 marks $both of them earn 10 points.$

2 marks each

Q.12 marks $(-2, -1, -3)$
Q.22 marks $A coin is tossed twice. The following table shows the probability distribution of number of tails : X 012 P(X) K 6K 9K (a) Find the value of K. (b) Is the coin tossed biased or unbiased ? Justify your answer.$
Q.3 (a)2 marks $|\vec{a} \times \vec{b}|^2 + |\vec{a} \cdot \vec{b}|^2 = 400$
OR
Q.3 (b)2 marks $5\sqrt{3}$
Q.42 marks $\sec^2 x \cdot \tan y \, dx + \sec^2 y \cdot \tan x \, dy = 0.$
Q.52 marks $\displaystyle\int_0^1 x^2 e^x \, dx$
Q.62 marks $There are two bags. Bag I contains 1 red and 3 white balls, and Bag II contains 3 red and 5 white balls. A bag is selected at random and a ball is drawn from it. Find the probability that the ball so drawn is red in colour.$

X	0	1	2
P(X)	K	6K	9K

3 marks each

Q.11 (a)4 marks $x \dfrac{dy}{dx} + y + \dfrac{1}{1 + x^2} = 0$
OR
Q.11 (b)4 marks $x (y^3 + x^3) \, dy = (2y^4 + 5x^3 y) \, dx.$
Q.124 marks $\displaystyle\int_0^{\pi} \dfrac{x}{9 \sin^2 x + 16 \cos^2 x} \, dx$
Q.134 marks $\vec{r} \cdot (\hat{i} + 3\hat{j}) = 6$
In a game of Archery, each ring of the Archery target is valued. The centremost ring is worth 10 points and rest of the rings are allotted points 9 to 1 in sequential order moving outwards. Archer A is likely to earn 10 points with a probability of $0{\cdot}8$ and Archer B is likely the earn 10 points with a probability of $0{\cdot}9$ . Based on the above information, answer the following questions : If both of them hit the Archery target, then find the probability that
Q.14 (a)2 marks $exactly one of them earns 10 points.$
Q.14 (b)2 marks $both of them earn 10 points.$

2 marks each

Q.1 (a)2 marks $|\vec{a} \times \vec{b}|^2 + |\vec{a} \cdot \vec{b}|^2 = 400$
OR
Q.1 (b)2 marks $5\sqrt{3}$
Q.22 marks $\displaystyle\int_0^1 x^2 e^x \, dx$
Q.32 marks $Three friends A, B and C got their photograph clicked. Find the probability that B is standing at the central position, given that A is standing at the left corner.$
Q.42 marks $A coin is tossed twice. The following table shows the probability distribution of number of tails : X 012 P(X) K 6K 9K (a) Find the value of K. (b) Is the coin tossed biased or unbiased ? Justify your answer.$
Q.52 marks $(-2, -1, -3)$
Q.62 marks $\sec^2 x \cdot \tan y \, dx + \sec^2 y \cdot \tan x \, dy = 0.$

X	0	1	2
P(X)	K	6K	9K

3 marks each

Q.114 marks $\displaystyle\int_{-3}^{3} \dfrac{x^4}{1 + e^x} \, dx$
Q.124 marks $\vec{r} \cdot (\hat{i} + 3\hat{j}) = 6$
Q.13 (a)4 marks $x \dfrac{dy}{dx} + y + \dfrac{1}{1 + x^2} = 0$
OR
Q.13 (b)4 marks $x (y^3 + x^3) \, dy = (2y^4 + 5x^3 y) \, dx.$
In a game of Archery, each ring of the Archery target is valued. The centremost ring is worth 10 points and rest of the rings are allotted points 9 to 1 in sequential order moving outwards. Archer A is likely to earn 10 points with a probability of $0{\cdot}8$ and Archer B is likely the earn 10 points with a probability of $0{\cdot}9$ . Based on the above information, answer the following questions : If both of them hit the Archery target, then find the probability that
Q.14 (a)2 marks $exactly one of them earns 10 points.$
Q.14 (b)2 marks $both of them earn 10 points.$