Two large, identical water tanks, 1 and 2 , kept on the top of a building of height $H$, are filled with water up to height $h$ in each tank. Both the tanks contain an identical hole of small radius on their sides, close to their bottom. A pipe of the same internal radius as that of the hole is connected to tank 2 , and the pipe ends at the ground level. When the water flows from the tanks 1 and 2 through the holes, the times taken to empty the tanks are $t_1$ and $t_2$, respectively. If $H=\left(\frac{16}{9}\right) h$, then the ratio $t_1 / t_2$ is ___________.

A thin uniform rod of length $L$ and certain mass is kept on a frictionless horizontal table with a massless string of length $L$ fixed to one end (top view is shown in the figure). The other end of the string is pivoted to a point $\mathrm{O}$. If a horizontal impulse $P$ is imparted to the rod at a distance $x={L \over n}$ from the mid-point of the rod (see figure), then the rod and string revolve together around the point $\mathrm{O}$, with the rod remaining aligned with the string. In such a case, the value of $n$ is ___________.

One mole of a monatomic ideal gas undergoes the cyclic process $\mathrm{J} \rightarrow \mathrm{K} \rightarrow \mathrm{L} \rightarrow \mathrm{M} \rightarrow \mathrm{J}$, as shown in the P-T diagram.

Match the quantities mentioned in List-I with their values in List-II and choose the correct option.

[ $\mathcal{R}$ is the gas constant.]

List-I	List-II
(P) Work done in the complete cyclic process	(1) $RT_0 - 4RT_0 \ln 2$
(Q) Change in the internal energy of the gas in the process JK	(2) $0$
(R) Heat given to the gas in the process KL	(3) $3RT_0$
(S) Change in the internal energy of the gas in the process MJ	(4) $-2RT_0 \ln 2$
	(5) $-3RT_0 \ln 2$

Four identical thin, square metal sheets, $S_1, S_2, S_3$ and $S_4$, each of side $a$ are kept parallel to each other with equal distance $d(\ll a)$ between them, as shown in the figure. Let $${C_0} = {{{\varepsilon _0}{a^2}} \over d}$$, where $\varepsilon_0$ is the permittivity of free space.

Match the quantities mentioned in List-I with their values in List-II and choose the correct option.

List-I	List-II
(P) The capacitance between $S_1$ and $S_4$, with $S_2$ and $S_3$ not connected, is	(1) $3C_0$
(Q) The capacitance between $S_1$ and $S_4$, with $S_2$ shorted to $S_3$, is	(2) $\frac{C_0}{2}$
(R) The capacitance between $S_1$ and $S_3$, with $S_2$ shorted to $S_4$, is	(3) $\frac{C_0}{3}$
(S) The capacitance between $S_1$ and $S_2$, with $S_3$ shorted to $S_1$, and $S_2$ shorted to $S_4$, is	(4) $\frac{2C_0}{3}$
	(5) $2C_0$