The radius of Mars is 3396 km. Rounding off to two significant figures, the radius of the Mars will be 3400 km.
A transverse direction signifies the angle is the right angle. So the distance that the thrusters have imparted a momentum sufficient to direct the space craft to the side of the original path would be,
d = 100 km+3400 km
 = 3500 km
As the space craft is half way through the six month journey, thus it has 3 months to move the 3500 km to the side.
So, time, t = 3 months
To obtain the corresponding transverse speed v, substitute 3500 km for d and 3 months for t in the equation v = d/t,
v = d/t
 =(3500 km/3 months)
 =(3500 km/3 months) (103 m/1 km) (1 month/30 days) (1 day/24 h) (1 h /3600 s)
 = 0.45 m/s.
To find the time Δt for the rocket to fire, substitute 5400 kg for the mass of spacecraft m, 0.45 m/s for speed v and 1200 N for F in the equation Δt = mv/F,
Δt = mv/F
 = (5400 kg) (0.45 m/s)/(1200 N)
 = (5400 kg) (0.45 m/s)/(1200 N) )(1 kg.m/s2/1 N)
 = 2.0s
From the above observation we conclude that, the required time for the rocket to fire would be 2.0s.