Let the time when the hour and the minute hand are in the same position between 2 o'clock and 3 o'clock be at T minutes past 2.
The hour hand moves 360/12 = 30 degrees in an hour. At 2, the hour hand makes an angle of 60 degrees with the line pointing towards 12.
At T past 2, the angle made by the hour hand with line pointing towards 12 is 30 + (30/60)*T. The angle made by the minute hand is (360/60)*T
As the two have to be at the same position, 30 + T/2 = (360/60)*T
=> 30 + T/2 = 6T
=> 11T/2 = 30
=> T = 60/11
=> T = 5.45
The time when the two hands are at the same position is 5.45 minutes past 2.