One circular ring and one circular disc,both having the same mass and radius. The ratio of their moments of inertia about the axes passing through their centres and perpendicular to their planes will be:

The number of common tangents to the circles ${x^2 + y^2 - 4x - 6y - 12 = 0}$ and ${x^2 + y^2 + 6x + 18y + 26 = 0}$ is

Three processes form a thermodynamic cycle as shown on the $P-V$ diagram for an ideal gas. Process $1 \to 2$ takes place at a constant temperature $(300\,K)$. Process $2 \to 3$ takes place at a constant volume. During this process,$40\,J$ of heat leaves the system. Process $3 \to 1$ is adiabatic and the temperature $T_3$ is $275\,K$. The work done by the gas during the process $3 \to 1$ is...... $J$.

If $f(x) = \sqrt{ax} + \frac{a^2}{\sqrt{ax}}$,then $f^{\prime}(a)$ is equal to

If $5x - 12y + 10 = 0$ and $12y - 5x + 16 = 0$ are two tangents to a circle,then the radius of the circle is

Aluminium is extracted from alumina $(Al_2O_3)$ by electrolysis of a molten mixture of: