Board Paper of Class 12-Humanities Term-II 2022 Math Delhi(Set 1) - Solutions

General Instructions:
Read the following instructions very carefully and strictly follow them:
1. This question paper contains three Sections - A, B and C.
2. Each section is compulsory.
3. Section - A has 6 short-answer type-I questions of 2 marks each.
4. Section - B has 4 short answer type-II questions of 3 marks each.
5. Section - C has 4 long-answer type questions of 4 marks each.
6. There is an internal choice in some questions.
7. Question 14 is a case study based question with two subparts of 2 marks each.

Question 1
Find: $\int \frac{d x}{\sqrt{4 x - x^{2}}}$ VIEW SOLUTION

Question 2
Find the general solution of the following differential equation:
$\frac{d y}{d x} = e^{x - y} + x^{2} e^{- y}$ VIEW SOLUTION

Question 3
Let X be a random variable which assumes values x₁, x₂, x₃, x₄ such that 2P(X = x₁) = 3P(X = x₂) = P(X = x₃) = 5P(X = x₄).
Find the probability distribution of X. VIEW SOLUTION

Question 4
If $\vec{a} = \hat{i} + \hat{j} + \hat{k}, \vec{a} . \vec{b} = 1$ and $\vec{a} \times \vec{b} = \hat{j} - \hat{k}, then find |\vec{b}|$ VIEW SOLUTION

Question 5
If a line makes an angle $α, β, γ$ with the coordinate axes, then find the value of $\cos 2 α + \cos 2 β + \cos 2 γ .$ VIEW SOLUTION

Question 6
Events A and B are such that
$P (A) = \frac{1}{2}, P (B) = \frac{7}{12} and P (\bar{A} \cup \bar{B}) = \frac{1}{4}$
Find whether the events A and B are independent or not.
OR

A box $B_{1}$ contains 1 white ball and 3 red balls. Another box $B_{2}$ contains 2 white balls and 3 red balls. If one ball is drawn at random from each of the boxes $B_{1} and B_{2}$ , then find the probability that the two balls drawn are of the same colour. VIEW SOLUTION

Question 11
Three persons A, B and C apply for a job of manager in a private company. Chances of their selection are in the ratio 1 : 2 : 4. The probability that A, B and C can introduce changes to increase the profits of a company are 0.8, 0.5 and 0.3 respectively. If increase in the profit does not take place, find the probability that it is due to the appointment of A. VIEW SOLUTION

Question 12
Find the area bounded by the curves y = |x – 1| and y = 1, using integration. VIEW SOLUTION

Question 13
Solve the following differential equation :
(y – sin²x)dx + tanx dy = 0
OR

Find the general solution of the differential equation:
(x³ + y³)dy = x²ydx VIEW SOLUTION

Question 14
Two motorcycles A and B are running at the speed more than the allowed speed on the roads represented by the lines $\vec{r} = λ (\hat{i} + 2 \hat{j} - \hat{k}) and \vec{r} = (3 \hat{i} + 3 \hat{j}) + µ (2 \hat{i} + \hat{j} + \hat{k})$ respectively.

Based on the above information, answer the following questions:
(a) Find the shortest distance between the given lines.
(b) Find the point at which the motorcycles may collide. VIEW SOLUTION

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