Find the distance between the lines:

$$\begin{aligned} & \vec{r}=(\hat{i}+2 \hat{j}-4 \hat{k})+\lambda(2 \hat{i}+3 \hat{j}+6 \hat{k}) ; \\ & \vec{r}=(3 \hat{i}+3 \hat{j}-5 \hat{k})+\mu(4 \hat{i}+6 \hat{j}+12 \hat{k}) \end{aligned}$$

(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:

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