Edexcel AS unit 2 Past paper question mark scheme query

in a past paper question in january 2006 unit 2 i am told " the average KE of a molecule is directly proportional to the kelvin temp. T Pressure of an ideal gas given by : p=1/3p<c^2>

Use this information to show that p is directly proportional to T for a fixed mass of gas at constant volume.

I did : 1/3Nm<c^2> = nRT (*2/3)

giving 1/2Nm<c^2> = 2nRT/3

Then 1/2m<c^2> = 2nRT/N

And stated that this shows that 1/2m<c^2> is proportional to T (KE being the kKE of an average inidivdual particle)

however the mark scheme (q 7 part B, see attached file) shows something slighlty different and am not sure what its trying to get at or whether i would get the marks or not. If you could let me know what id have to write or whether what i wrote is correct id really appreciate it. Thankyou in advance.

From the attachment I can see

1. substitute for density in the pressure equation

so basically here, to get the mark, you need to quote (rho) = Nm/V

2. 0.5 * m * <c^2> = 3pV / 2N

so to prove this
pressure * volume = 1/3 * N * m * <c^2>
=> pV = 1/3 * N * m * <c^2>
=> pV = (2/3 * N) * (1/2 * m * <c^2>)
=> 3pV / 2N = 1/2 * m * <c^2>
Hence proved

3. Equate this to constant * T

This part is easy. We know that pV = nRT
so
3*nRT / 2N = 1/2 * m * <c^2>
For an ideal gas, n = 1
So
3RT / 2N = 1/2 * m * <c^2>
3R / 2N = k (Boltzmann constant)
[Boltzmann constant is actually for Na [avogadro constant thing] but here I'm writing N]
1/2 * m * <c^2> = constant (k)* T
hence equated