Let $\overline{\mathrm{a}}, \overline{\mathrm{b}}$ and $\overline{\mathrm{c}}$ be vectors of magnitude 2,3 and 4 respectively. If $\bar{a}$ is perpendicular to $(\bar{b}+\bar{c}), \bar{b}$ is perpendicular to $(\bar{c}+\bar{a})$ and $\bar{c}$ is perpendicular to $(\bar{a}+\bar{b})$, then the magnitude of $\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}$ is equal to

A square plate is contracting at the uniform rate $3 \mathrm{~cm}^2 / \mathrm{sec}$, then the rate at which the perimeter is decreasing, when the side of the square is 15 cm , is

The area (in sq. units), in the first quadrant bounded by the curve $y=x^2+2$ and the lines $y=x+1, x=0$ and $x=2$, is

The vector $\bar{a}=\alpha \hat{i}+2 \hat{j}+\beta \hat{k}$ lies in the plane of the vectors $\overline{\mathrm{b}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}$ and $\overline{\mathrm{c}}=\hat{\mathrm{j}}+\hat{\mathrm{k}}$ and bisects the angle between $\overline{\mathrm{b}}$ and $\overline{\mathrm{c}}$. Then which one of the following gives possible values of $\alpha$ and $\beta$ ?