(a) The median of an equilateral triangle is increasing at the rate of $$2 \sqrt{3} \mathrm{~cm} / \mathrm{s}$$. Find the rate at which its side is increasing.

OR

(b) Sum of two numbers is 5. If the sum of the cubes of these numbers is least, then find the sum of the squares of these numbers.

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.