The abscissae of the two points A and B are the roots of the equation $x^2+2 a x-b^2=0$ and their ordinates are roots of the equation $y^2+2 p y-q^2=0$. Then the equation of the circle with AB as diameter is given by

Water is running in a hemispherical bowl of radius 180 cm at the rate of 108 cubic decimeters per minute. How fast the water level is rising when depth of the water level in the bowl is 120 cm ? ( 1 decimeter $=10 \mathrm{~cm}$)

$\mathrm{S}_1=\sum_\limits{\mathrm{r}=1}^{\mathrm{n}} \mathrm{r}, \mathrm{S}_2=\sum_\limits{\mathrm{r}=1}^{\mathrm{n}} \mathrm{r}^2$ and $\mathrm{S}_3=\sum_\limits{\mathrm{r}=1}^{\mathrm{n}} \mathrm{r}^3$, then the value of $\lim _\limits{n \rightarrow \infty} \frac{s_1\left(1+\frac{s_3}{4}\right)}{s_2^2}$ is

The numerical value of $\tan \left(2 \tan ^{-1}\left(\frac{1}{5}\right)+\frac{\pi}{4}\right)$