Two rings of radius $R$ and $nR$ made of the same material have a ratio of moment of inertia about an axis passing through their centers and perpendicular to their planes as $1:8$. The value of $n$ is (mass per unit length is constant).

If $f(1)=1$ and $f^{\prime}(1)=3$,then the derivative of $f(f(f(x)))+(f(x))^2$ at $x=1$ is

Which of the following has the highest electropositive character?

The transition from the state $n = 3$ to $n = 1$ in a hydrogen-like atom results in ultraviolet radiation. Infrared radiation will be obtained in the transition from

Consider the following gaseous equilibrium in a closed container of volume $V$ at $T \text{ K}$:
$P_{2(g)} + Q_{2(g)} \rightleftharpoons 2PQ_{(g)}$.
$2 \text{ moles}$ each of $P_{2(g)}$, $Q_{2(g)}$, and $PQ_{(g)}$ are present at equilibrium. Now, $1 \text{ mole}$ each of $P_{2}$ and $Q_{2}$ are added to the equilibrium while keeping the temperature at $T \text{ K}$. The number of moles of $P_{2}$, $Q_{2}$, and $PQ$ at the new equilibrium, respectively, are:

In the following circuit,$PN-$ junction diodes $D_1, D_2,$ and $D_3$ are ideal. For the following potentials of $A$ and $B,$ the correct increasing order of resistance between $A$ and $B$ will be:
$(i) -10 \ V, -5 \ V$
$(ii) -5 \ V, -10 \ V$
$(iii) -4 \ V, -12 \ V$