A train leaving Dhaka at 5 am reaches Khulna at 10 am and another train leaving Khulna at 6 am, and reaches Dhaka at 1 overline 2 noon. At what time, the two trains running in opposite direction should meet?

Let the distance between Dhaka and Khulna be \(D\) km.

For the first train (Dhaka to Khulna):

Departure time from Dhaka: 5 AM

Arrival time in Khulna: 10 AM

Time taken by the first train = 10 AM - 5 AM = 5 hours

Speed of the first train, \(S_1 = \frac{\text{Distance}}{\text{Time}} = \frac{D}{5}\) km/hr

For the second train (Khulna to Dhaka):

Departure time from Khulna: 6 AM

Arrival time in Dhaka: "1 overline 2 noon". Interpreting this as \(1 \frac{1}{2}\) noon, which means 1 hour and 30 minutes past noon, i.e., 1:30 PM.

Time taken by the second train = 1:30 PM - 6 AM = 7 hours 30 minutes = 7.5 hours

Speed of the second train, \(S_2 = \frac{\text{Distance}}{\text{Time}} = \frac{D}{7.5}\) km/hr

Distance covered by the first train before the second train starts:

The first train starts at 5 AM and the second train starts at 6 AM. So, the first train travels for 1 hour (from 5 AM to 6 AM) before the second train begins its journey.

Distance covered by the first train in 1 hour = \(S_1 \times 1 = \frac{D}{5} \times 1 = \frac{D}{5}\) km.

At 6 AM, the remaining distance between the two trains = \(D - \frac{D}{5} = \frac{4D}{5}\) km.

Relative speed of the two trains:

Since the trains are running in opposite directions, their relative speed is the sum of their individual speeds.

Relative Speed = \(S_1 + S_2 = \frac{D}{5} + \frac{D}{7.5}\)

\( = D \left( \frac{1}{5} + \frac{1}{7.5} \right) \)

\( = D \left( \frac{1.5}{7.5} + \frac{1}{7.5} \right) \)

\( = D \left( \frac{1.5 + 1}{7.5} \right) \)

\( = D \left( \frac{2.5}{7.5} \right) \)

\( = D \left( \frac{1}{3} \right) = \frac{D}{3}\) km/hr

Time taken for the trains to meet after 6 AM:

Time = \(\frac{\text{Remaining Distance}}{\text{Relative Speed}}\)

\( = \frac{\frac{4D}{5}}{\frac{D}{3}} \)

\( = \frac{4D}{5} \times \frac{3}{D} \)

\( = \frac{12}{5}\) hours

\( = 2 \frac{2}{5}\) hours

To convert the fractional part of an hour to minutes: \(\frac{2}{5} \times 60 = 24\) minutes.

So, the time taken to meet = 2 hours 24 minutes.

The exact time when the two trains meet:

The trains start moving towards each other simultaneously (relative to the remaining distance) from 6 AM.

Meeting time = 6 AM + 2 hours 24 minutes = 8:24 AM.

Therefore, the two trains should meet at 8:24 AM.