If $5x - 12y + 10 = 0$ and $12y - 5x + 16 = 0$ are two tangents to a circle,then the radius of the circle is

If $u=\sin ^{-1}\left(\frac{x^4+y^4}{x+y}\right)$, then $x \frac{\partial u}{\partial x}+y \frac{\partial u}{\partial y}$ is equal to

$A$ parallel plate capacitor of capacity $100 \mu F$ is charged by a battery of $50 \text{ V}$. The battery remains connected,and if the plates of the capacitor are separated so that the distance between them becomes double the original distance,the additional energy supplied by the battery to the capacitor in joules is:

If $y_k$ is the $k$-th derivative of $y$ with respect to $x$, and $y = \cos(\sin x)$, then $y_1 \sin x + y_2 \cos x$ is equal to

If the lines $4x + 3y - 1 = 0$,$x - y + 5 = 0$ and $kx + 5y - 3 = 0$ are concurrent,then $k$ is equal to:

$A$ radioactive nucleus (initial mass number $A$ and atomic number $Z$) emits $3$ $\alpha$-particles and $2$ positrons. The ratio of neutrons to protons in the final nucleus will be: