The equation of the tangent at $(-4, -4)$ on the curve $x^2 = -4y$ is

The equation of the circle whose diameter is the common chord of the circles $x^2+y^2+2x+3y+2=0$ and $x^2+y^2+2x-3y-4=0$ is

$A$ large open tank has two holes in the wall. One is a square hole of side $L$ at a depth $y$ from the top and the other is a circular hole of radius $R$ at a depth $4y$ from the top. When the tank is completely filled with water,the quantities of water flowing out per second from the two holes are the same. Then the value of $R$ is

The position of a particle is given by $\overrightarrow {r} = (\hat i + 2\hat j - \hat k)$ and its momentum is given by $\overrightarrow {P} = (3\hat i + 4\hat j - 2\hat k)$. The angular momentum is perpendicular to:

$A$ parallel plate capacitor has circular plates of $10\, cm$ radius separated by an air-gap of $1\, mm$. It is charged by connecting the plates to a $100\, V$ battery. What is the change in energy stored in the capacitor when the plates are moved to a distance of $1\, cm$ while remaining connected to the battery?

The set of solutions of the equation $(\sqrt{3}-1) \sin \theta+(\sqrt{3}+1) \cos \theta=2$ is