A particle initially has a speed of 0.5c. (c= speed of light, 3 X
10^8) At what speed does its momentum increase by a) 1%, b) 10%, c)
100% ?

Initial momentum is p = mv = y.m_0.v (where y = gamma and m_0 is the
rest mass). For v = 0.5 we have y = 2/sqrt(3) = 1.155 (4 s.f.).

I'll sketch the solution for an increase of 40% and let you work out
the other answers. In the description below p' and y' refer to the new
values of momentum and gamma respectively.

Increased momentum is p' = 1.4 p so y' = 1.4 y = 1.617 (4 s.f.).
Plugging this into the formula for y gives (1 - v^2/c^2) = 0.3827 and
hence v = 0.786 c (3 s.f.).

To check your methodology, for the 100% case you should get v = 0.901 c (3 s.f.).