The solution of the equation $\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{1}{x+y+1}$ is

A spherical balloon is filled with $4500 \pi$ cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of $72 \pi$ cubic meters per minute, then the rate (in meters per minute) at which the radius of the balloon decreases 49 minutes after the leakage has begun, is

If $x \cdot \log _e\left(\log _e x\right)-x^2+y^2=4(y>0)$, then $\frac{d y}{d x}$ at $x=\mathrm{e}$ is

If p : switch $\mathrm{S}_1$ is closed, q : switch $\mathrm{S}_2$ is closed then correct interpretation from the following circuit is