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Assertion $A$ : Body $'P'$ having mass $M$ moving with speed $'u'$ has head-on collision elastically with another body $'Q'$ having mass $'m'$ initially at rest. If $m< < M,$ body $'Q'$ will have a maximum speed equal to $'2u'$ after collision.
Reason $R$ : During elastic collision, the momentum and kinetic energy are both conserved.
In the light of the above statements, choose the most appropriate answer from the options given below:
$\begin{array}{|c|c|c|c|}\hline \mathrm{C_{1 }} & {\mathrm{C}_{2}} & {\mathrm{C}_{3}} & {\mathrm{T}} \\ \hline {1 l} & {2 l} & {-} & {60^{\circ} \mathrm{C}} \\ \hline {-} & {1 l} & {2 l} & {30^{\circ} \mathrm{C}} \\ \hline {2 l} & {-} & {1 l} & {60^{\circ} \mathrm{C}} \\ \hline {1 l} & {1 l} & {1 l} & {\theta} \\ \hline\end{array}$
The value of $\theta$ (in $^{\circ} \mathrm{C}$ to the nearest integer) is
Figure: $Image$
$1.$ As the bubble moves upwards, besides the buoyancy force the following forces are acting on it
$(A)$ Only the force of gravity
$(B)$ The force due to gravity and the force due to the pressure of the liquid
$(C)$ The force due to gravity, the force due to the pressure of the liquid and the force due to viscosity of the liquid
$(D)$ The force due to gravity and the force due to viscosity of the liquid
$2.$ When the gas bubble is at a height $\mathrm{y}$ from the bottom, its temperature is
$(A)$ $\mathrm{T}_0\left(\frac{\mathrm{P}_0+\rho_0 \mathrm{gH}}{\mathrm{P}_0+\rho_t \mathrm{gy}}\right)^{2 / 5}$
$(B)$ $T_0\left(\frac{P_0+\rho_t g(H-y)}{P_0+\rho_t g H}\right)^{2 / 5}$
$(C)$ $\mathrm{T}_0\left(\frac{\mathrm{P}_0+\rho_t \mathrm{gH}}{\mathrm{P}_0+\rho_t \mathrm{gy}}\right)^{3 / 5}$
$(D)$ $T_0\left(\frac{P_0+\rho_t g(H-y)}{P_0+\rho_t g H}\right)^{3 / 5}$
$3.$ The buoyancy force acting on the gas bubble is (Assume $R$ is the universal gas constant)
$(A)$ $\rho_t \mathrm{nRgT}_0 \frac{\left(\mathrm{P}_0+\rho_t \mathrm{gH}\right)^{2 / 5}}{\left(\mathrm{P}_0+\rho_t \mathrm{gy}\right)^{7 / 5}}$
$(B)$ $\frac{\rho_{\ell} \mathrm{nRgT}_0}{\left(\mathrm{P}_0+\rho_{\ell} \mathrm{gH}\right)^{2 / 5}\left[\mathrm{P}_0+\rho_{\ell} \mathrm{g}(\mathrm{H}-\mathrm{y})\right]^{3 / 5}}$
$(C)$ $\rho_t \mathrm{nRgT} \frac{\left(\mathrm{P}_0+\rho_t g \mathrm{H}\right)^{3 / 5}}{\left(\mathrm{P}_0+\rho_t g \mathrm{~g}\right)^{8 / 5}}$
$(D)$ $\frac{\rho_{\ell} \mathrm{nRgT}_0}{\left(\mathrm{P}_0+\rho_{\ell} \mathrm{gH}\right)^{3 / 5}\left[\mathrm{P}_0+\rho_t \mathrm{~g}(\mathrm{H}-\mathrm{y})\right]^{2 / 5}}$
Give the answer question $1,2,$ and $3.$