Would we need to know the experimental details of Boyles, Pressure law, Charles law. Not sure if they would ask something like draw a diagram with the apparatus to find out the relationship between ... pressure and temp etc
Theres more than 1 way to do this calculation however i think the easiest way to look at it is if the detector counts 0.62/s but this is only 1/400 of the photons emitted. So if the detector was on the source the count rate would be 248/s (0.62 x 400) this is the activity of the source however only a small part of it as the area of the detector is 1.5x10^-3 and the rays are emitted uniformly in all directions. The total activity of the source is how many emitted per second. As worked out in b(i) you know the ratio of the area of dectector to area of emitted rays. So use 248 ÷ this (3.684x10^-3) and you get 67000 Bq
ok, but what would be the point of a detector only detecting 1 in 400 photons that hit it?
Just that it's created at the end of the lifecycle of a star whose core is more than 3 times the sun's mass. This means that the neutrons can't withstand the gravitational forces and so it collapses to an infinitely dense point. This is the blackhole.