Two concentric hollow spherical shells have radii $r$ and $R$ $(R \gg r)$. $A$ charge $Q$ is distributed on them such that the surface charge densities are equal. The electric potential at the centre is

The locus of $z$ satisfying the inequality $\left|\frac{z+2 i}{2 z+i}\right| < 1$,where $z=x+i y$,is

If $(a + \sqrt{2} b \cos x)(a - \sqrt{2} b \cos y) = a^2 - b^2$,where $a > b > 0$,then $\frac{dx}{dy}$ at $(\frac{\pi}{4}, \frac{\pi}{4})$ is

$A$ sound wave passing through an ideal gas at $NTP$ produces a pressure change of $0.001 \text{ dyne}/cm^2$ during adiabatic compression. The corresponding change in temperature $(\gamma = 1.5$ for the gas and atmospheric pressure is $1.013 \times 10^6 \text{ dyne}/cm^2)$ is:

If $f(x)=(x-1)(x-2)(x-3)$ for $x \in[0,4]$,then the value of $c \in(0,4)$ satisfying Lagrange's mean value theorem is: