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

General Instructions:
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: $\int \frac{dx}{\sqrt{4x-{x}^{2}}}$ VIEW SOLUTION

• Question 2
Find the general solution of the following differential equation:
$\frac{dy}{dx}={e}^{x-y}+{x}^{2}{e}^{-y}$ 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 5
If a line makes an angle  with the coordinate axes, then find the value of  VIEW SOLUTION

• Question 6
Events A and B are such that

Find whether the events A and B are independent or not.

OR

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

• Question 7
Evaluate: ${\int }_{0}^{\frac{\pi }{4}}\frac{dx}{1+\mathrm{tan}x}$ VIEW SOLUTION

• Question 8
If $\stackrel{\to }{a}$ and $\stackrel{\to }{b}$ are unit vectors and θ is the angle between them, then prove that $\mathrm{sin}\frac{\mathrm{\theta }}{2}=\frac{1}{2}\left|\begin{array}{c}\stackrel{\to }{a}–\stackrel{\to }{b}\end{array}\right|.$

OR

If $\stackrel{\to }{a}$ and $\stackrel{\to }{b}$ are two vectors such that $\left|\stackrel{\to }{a}+\stackrel{\to }{b}\right|=\left|\stackrel{\to }{b}\right|,$ then prove that $\left(\stackrel{\to }{a}+2\stackrel{\to }{b}\right)$ is perpendicular to $\stackrel{\to }{a}.$ VIEW SOLUTION

• Question 9
Find the equation of the plane passing through the line of intersection of the planes   and passing through the point (–2, 3, 1). VIEW SOLUTION

• Question 10
Find:

OR

Find:
$\int \frac{2x}{\left({x}^{2}+1\right)\left({x}^{2}+2\right)}dx$ 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 = |– 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 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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