A particle of mass 1 gm starts from rest and moves under the action of a force of 30
Newtons defined in the rest frame. It will reach 99% the velocity of light in time
(a) 9.9 x 10³ sec ( 7 x 10⁴ sec © 0.999 sec (d) 0.7 sec

I can safely disregard © and (d)

( a ) can be found by kinematic calculation.But as the particle moves in relativistic speed, this classical calculation does not hold.

( b ) looks correct to me.

as speed increases,the value of m(v) also increases,thus reducing the acceleration.Thus, it takes longer time...In that respect ( b ) looks plausible.

But how to prove this?

Can anyone suggest anything?

30 Newtons is a force, force is change of momentum over time, momentum and time are stated "in the rest frame" so you need to solve

change-in-momentum / force = time

Since the starting momentum is zero this is

final-momentum / force = time

which is

final-momentum = (1 gm)(0.99 c)/√(1 - (0.99)²)
= (0.001 kg)(296794533 m/s)(7.0888)
= 2103920 kg m/s

time = (2103920 kg m/s)/ ( 30 kg m/s²) = 70131 s ≈ 7 x 10⁴s