The shaded area in the figure given below is a solution set of a system of inequations. The minimum value of objective function $$3 x+5 y$$, subject to the linear constraints given by this system of inequations is

If $$\overline{\mathrm{a}}=\mathrm{m} \overline{\mathrm{b}}+\mathrm{nc}$$, where $$\overline{\mathrm{a}}=4 \hat{\mathrm{i}}+13 \hat{\mathrm{j}}-18 \hat{\mathrm{k}}, \overline{\mathrm{b}}=\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}, \overline{\mathrm{c}}=2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-4 \hat{\mathrm{k}}$$, then $$\mathrm{m}+\mathrm{n}=$$

In a game, 3 coins are tossed. A person is paid ₹ 7 /-, if he gets all heads or all tails; and he is supposed to pay ₹ 3 /-, if he gets one head or two heads. The amount he can expect to win on an average per game is ₹

$$\int \mathrm{e}^x\left(1-\cot x+\cot ^2 x\right) \mathrm{d} x=$$