And aircraft travelling at 600 km/h accelerates steadily at 10 km/h per second taking the speed of sound as 1100 km/h at the aircrafts altitude how long will it take to reach the sound barrier

Use equation of motion:

Given:  (I converted everything to meters and seconds because of acceleration in your question)

Initial velocity (u) = 600km/h = 166.6667 m/s

Final velocity (v) = 1100 km/h = 305.5556 m/s

Acceleration (a) = 10km/h every second=2.7778 m/s^2

Formula: V=U+at

305.5556=166.6667+2.7778t

2.7778t =305.5556-166.667

t = (305.5556-166.667) / (2.778) = 49.9959

time it will take to reach sound barrier = 49.9959 seconds

You can round off your answer as you wish.