The magnetic field at the centre of a circular current-carrying conductor of radius $r$ is $B_c$. The magnetic field on its axis at a distance $r$ from the centre is $B_a$. The value of $B_c : B_a$ will be

The number of points in $(-\infty, \infty)$ for which $x^2 - x \sin x - \cos x = 0$ is

In the interval $(-3, 3)$, the function $f(x) = \frac{x}{3} + \frac{3}{x}, x \neq 0$ is :

$\frac{3\sqrt{2}}{\sqrt{6} + \sqrt{3}} - \frac{4\sqrt{3}}{\sqrt{6} + \sqrt{2}} + \frac{\sqrt{6}}{\sqrt{3} + \sqrt{2}} = $

An $AC$ voltage source of variable angular frequency $\omega$ and fixed amplitude $V_0$ is connected in series with a capacitance $C$ and an electric bulb of resistance $R$ (inductance zero). When $\omega$ is increased:

$A$ thin flexible wire of length $L$ is connected to two adjacent fixed points and carries a current $I$ in the clockwise direction as shown. When the system is placed in a uniform magnetic field of strength $B$ directed into the plane of the paper,the wire takes the shape of a circle. The tension in the wire after acquiring the circular shape is