Edexcel IGCSE Maths P2

Two weird questions stood out to me:

First: the two cones question -- were we meant to assume the cones were similar? I think so, but idk why, because we don't know that the angle of the cones are the same (or the ratio height:radius are the same), unless we take the straight line at face value, but it says diagram not accurately drawn (instead of not to scale iirc) so idk

Second: probability -- I had to make a really weird argument (which I fucked up😔), idk what the intended solution is.

For those who didn't take it, the question was:

Bob (or someone with a different name) takes one bead out of a bag at random, puts it back, and then takes a bead again at random. The probability of him taking at least one red bead is 39/64. There are only red and green beads in the bag. Find the probability of taking only one red bead.

I said:

Let x, y ∈ ℤ be the number of red beads and total amount of beads in the bag respectively.

We can then see that x ≤ y (1) as otherwise it would be a very strange bag.

39/64 = 2x(y-x)/y² + x²/y² = (2xy - x²)/y² = x(2y - x)/y²

since 39 is a product of two primes, 3 × 13, either x = 3 (2) or x = 13 (3). [This argument is somewhat flawed. What I should have instead done was first establish there can be no common factor between the denominator and numerator (see appendix 1)]

Case 3:

x = 13 implies 2y - x = 2y - 13 = 3. Then, 2y = 16 => y = 8. But 13 > 8, so by (1) this is impossible.

Case 2:

x = 3 so 2y - x = 2y - 3 = 13 => 2y = 16 => y = 8 as before, but crucially 3 ≤ 8 so it's fine. After this it's trivial, so I won't bore you with the details; everyone I spoke to got 15/32 so I presume I got it right.

Appendix 1:

there can be no common factor between the denominator and numerator. We see this as follows:

Assume BWOC ∃k>1∈ℤ : 39k = x(2y - x) and 64k = y².

This implies ∃j∈ℤ : j² = k. So, y = 8j.

39k = x(2y - x) implies x/k∈ℤ or (2y - x)/k∈ℤ, but the second reduces to the first by distribution. So, ∃n>1, n∈ℤ : nk = x.

So, 39k = nk(128k - nk) = nk²(128-n) => 39 = nk(128 - n) = nj²(128 - n).

But 39 has no square factors except 1 since 39 = 3×13 and there are no repeated prime factors. So, k = 1. But k > 1 as assumed. QED

This took me like twenty minutes to come up with ts cannot be the intended solution😭