When an inductance '$L$' and resistor '$R$' are connected in series to $25$V, $50$ Hz supply, a current of $0.5$ A flows in the circuit. The current lags behind in phase from applied voltage by $\left(\dfrac{\pi}{3}\right)$ radian. The value of R is $\left(\cos 60^\circ = \dfrac{1}{2}\right)$

In the circuit shown charge q varies with time t as $q = t^2 - 5$, where q is in coulomb and t is in second. At time $t = 3$ second, voltage $V_{AB}$ in volt will be

The prism has refracting angle A. The second refracting surface of the prism is silvered. Light ray falling on first refracting surface with angle of incidence $2A$, reaches the second surface and returns back through same path due to reflection at the silvered surface. The refractive index of the material of prism is

Light of wavelength $580$ nm is incident normally on a slit of width '$a$'. The distance between slit and screen is $2.5$ m and the distance of the second order maximum from the centre of the screen is $14.5$ mm in a diffraction pattern. The value of '$a$' is