Correct Answer: 168

Solution : Let the length of train $x$ as $L_x$ and the length of train $y$ as $L_y$.
Given that $L_y$ is two-thirds that of $x$.
$⇒L_y = \frac{2}{3}L_x$
When two trains are moving in opposite directions, their relative speed is the sum of their individual speeds.
The relative speed of the two trains = 74 + 52 = 126 km/hr
$\text{Speed} = \frac{\text{Distance}}{\text{Time}}$
The total distance covered when the two trains cross each other is the sum of their lengths, i.e., $L_x + L_y$.
The time taken to cross each other is given as 12 seconds.
$⇒126 = \frac{L_x + L_y}{12}$
Convert the speed from km/hr to m/s by multiplying by $\frac{5}{18}$.
$⇒126 \times \frac{5}{18} = \frac{L_x + L_y}{12}$
$⇒35 = \frac{L_x + L_y}{12}$
$⇒L_x + L_y = 420$
Substituting $L_y = \frac{2}{3}L_x$ into the equation gives,
$⇒L_x + \frac{2}{3}L_x = 420$
$⇒\frac{5}{3}L_x = 420$
$⇒L_x = \frac{420 \times 3}{5} = 252 \, \text{m}$
Substituting $L_x = 252$ m into $L_y = \frac{2}{3}L_x$ gives:
$⇒L_y = \frac{2}{3} \times 252 = 168 \, \text{m}$
Hence, the correct answer is 168.