The minimum distance of a point on the curve $y = x^2 - 4$ from the origin is

If the charge on the capacitor is increased by $2 \ C$,the energy stored in it increases by $21 \%$. The total original charge on the capacitor is: (in $C$)

$A$ thin circular ring of mass $M$ and radius $R$ is rotating about its axis with a constant angular velocity $\omega$. Two objects,each of mass $m$,are attached gently to the opposite ends of a diameter of the ring. The ring now rotates with an angular velocity:

The foot of the perpendicular from $(0,2,3)$ to the line $\frac{x+3}{5}=\frac{y-1}{2}=\frac{z+4}{3}$ is

If $\cos ^{-1}\left(\frac{y}{b}\right)=2 \log \left(\frac{x}{2}\right)$,where $x>0$,then $x^2 \frac{d^2 y}{d x^2}+x \frac{d y}{d x}$ is equal to

Consider the circle $x^2+y^2-4x-2y+c=0$ whose centre is $A(2,1)$. If the point $P(10,7)$ is such that the line segment $PA$ meets the circle at $Q$ with $PQ=5$,then $c$ is equal to