A nuclear reactor starts producing a radioactive nuclide X from t = 0, at a constant rate of α per second. Each decay of X produces energy E₀, which is utilized to heat a liquid of mass m and specific heat s. Assuming no heat loss from the liquid and taking λ as the decay constant of X, the rate of increase in the temperature of the liquid is :

A beam of polychromatic light passes through a thin prism of prism angle $6^{\circ}$. The refractive index of the material of the prism varies with wavelength $(\lambda)$ as $n(\lambda) = \alpha \lambda + \frac{\beta}{\lambda^2}$, where $\alpha = 3\ \mu m^{-1}$ and $\beta = 0.096\ \mu m^2$. If $\lambda_{min}$ is the wavelength at which the angle of minimum deviation $D_m$ is smallest, then the correct value of $D_m$ at $\lambda_{min}$ is :

A particle of mass m, and angular momentum ℓ is moving in a circular orbit of radius r₀ under the influence of an attractive force $\vec{F}(r)=-\frac{k}{r^2} \hat{r}$. Keeping its angular momentum unchanged, the particle is displaced radially by a small distance $\delta r \ll r_0$, due to which its radial distance varies periodically. The corresponding time period is :