If $\mathrm{E}^{\cdot}\left(\mathrm{Mg}_{(\mathrm{aq})}^{+2} \mid \mathrm{Mg}_{(\mathrm{s})}\right)=-2.37 \mathrm{~V}$. What is potential for $\mathrm{Mg}_{(\mathrm{s})} \longrightarrow \mathrm{Mg}^{+2}(0.01 \mathrm{M})+2 \mathrm{e}^{-}$at 298 K ?

If $y=y(x)$ and $\left(\frac{2+\sin x}{y+1}\right) \frac{\mathrm{d} y}{\mathrm{~d} x}=-\cos x, y(0)=1$, then $y\left(\frac{\pi}{2}\right)=$

If $A=\left[\begin{array}{cc}1 & \cot \frac{\theta}{2} \\ -\cot \frac{\theta}{2} & 1\end{array}\right]$ then $A^{-1}=$

If $\bar{a}, \bar{b}, \bar{c}$ are three unit vectors such that $|\overline{\mathrm{a}}+\overline{\mathrm{b}}|^2+|\overline{\mathrm{a}}+\overline{\mathrm{c}}|^2=8$, then $|\overline{\mathrm{a}}+3 \overline{\mathrm{~b}}|^2+|\overline{\mathrm{a}}+3 \overline{\mathrm{c}}|^2=$