Let M and N be foots of the perpendiculars drawn from the point $\mathrm{P}(\mathrm{a}, \mathrm{a}, \mathrm{a})$ on the lines $x-y=0, \mathrm{z}=1$ and $x+y=0, \mathrm{z}=-1$ respectively and if $\angle \mathrm{MPN}=90^{\circ}$ then $\mathrm{a}^2=$

If $y=\log _3\left(\log _3 x\right)$ then $\frac{\mathrm{d} y}{\mathrm{~d} x}$ at $x=3$ is $\ldots \ldots$

If in triangle ABC , with usual notations $\sin \frac{\mathrm{A}}{2} \cdot \sin \frac{\mathrm{C}}{2}=\sin \frac{\mathrm{B}}{2}$ and 2 s is the perimeter of the triangle, then the value of $s$ is

Two planets A and B have densities ' $\rho_1$ ', ' $\rho_2$ ' and have radii ' $r_1$ ', ' $r_2$ ', respectively. The ratio of acceleration due to gravity on $A$ to that of $B$ is