Evaluate : $$\int_\limits0^{\frac{\pi}{2}} \sin 2 x \tan ^{-1}(\sin x) d x$$

Solve the following Linear Programming Problem graphically :

Maximize: $$P=70 x+40 y$$

Subject to:

$$\begin{aligned} \mathrm{3 x + 2 y} & \leq \mathrm{9} \\ \mathrm{3 x + y} & \leq \mathrm{9} \\ x & \geq \mathrm{0}, y \geq \mathrm{0} \end{aligned}$$

(a) In answering a question on a multiple choice test, a student either knows the answer or guesses. Let $$\frac{3}{5}$$ be the probability that he knows the answer and $$\frac{2}{5}$$ be the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability $$\frac{1}{3}$$. What is the probability that the student knows the answer, given that he answered it correctly?

OR

(b) 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.

Case Study I

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 $$B=\left\{b_1, b_2, b_3\right\}$$ and $$G=\left\{g_1, g_2\right\}$$, 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:

(i) How many relations are possible from $$B$$ to $$G$$ ?

(ii) Among all the possible relations from $$B$$ to $$G$$, how many functions can be formed from $$B$$ to $$\mathrm{G}$$ ?

(iii) Let $$\mathrm{R}: \mathrm{B} \rightarrow \mathrm{B}$$ be defined by $$R=\{(x, y): x$$ and $$y$$ are students of the same sex $$\}$$. Check if $$R$$ is an equivalence relation.

OR

(iii) A function $$f: \mathrm{B} \rightarrow \mathrm{G}$$ be defined by $$f=\left\{\left(b_1, g_1\right)\right.$$, $$\left.\left(b_2, g_2\right),\left(b_3, g_1\right)\right\}$$.

Check if $$f$$ is bijective. Justify your answer.