When $20 \ g$ of naphthoic acid $(C_{11}H_8O_2)$ is dissolved in $50 \ g$ of benzene $(K_f = 1.72 \ K \ kg \ mol^{-1})$,a freezing point depression of $2 \ K$ is observed. The Van't Hoff factor $(i)$ is:

$\log ab - \log |b| = $

$A$ black metal foil is warmed by radiation from a small sphere at temperature $T$ and at a distance $d$. It is found that the power received by the foil is $P$. If both the temperature and the distance are doubled,the power received by the foil will be

$A$ compound $X$ upon reaction with $H_{2}O$ produces a colorless gas $Y$ with a rotten fish smell. Gas $Y$ is absorbed in a solution of $CuSO_{4}$ to give $Cu_{3}P_{2}$ as one of the products. Predict the compound $X$.

Consider the following statements $A$ and $B$ and identify the correct choice in the given answers.
$A$. Tightly bound electrons of target material scatter $X$-ray photon,resulting in the Compton effect.
$B$. Photoelectric effect takes place with free electrons.

Two particles having position vectors $r_1 = (3 \hat{i} + 5 \hat{j}) \ m$ and $r_2 = (-5 \hat{i} - 3 \hat{j}) \ m$ are moving with velocities $v_1 = (4 \hat{i} + 3 \hat{j}) \ m/s$ and $v_2 = (a \hat{i} + 7 \hat{j}) \ m/s$. If they collide after $2 \ s$,then the value of $a$ is: