Q1 - You forgot/didn't bother to show that if (*) has a solution then n is even. Whether that is OK is up for debate because the question requires you to "deduce" that (*) has no solutions from these earlier parts. You may be able to get away with it for the first equation because it explicitly asks you to show that n^2 is a factor of 54, which is a fact you've based your argument upon but you certainly haven't deduced that (**) has no solutions from the fact that n must be even as requested so you may be punished quite heavily for that.
My mark: 12/20 (being generous - I'm not sure how lenient they would be)
Q2 - As you say, you've made quite the meal of the second part. After choosing a, you could have just noted that you need to choose (preferably small) N s.t.
for some integer n (as the denominator is already 5^3). Since you want N to be small, looking around the 60 mark for cubes would have been the next obvious step and you could have chosen N by inspection from there. You've also either made a small error in your division or a typo as there is one too many 9's in your approximation.
My mark: 18/20
Q3 - I think you have put in slightly too much effort in here too, you could just as easily have expressed
in partial fractions directly rather than doing it twice. The sum at the end should come to 1, rather than 1/2 (mistake is on the 4th line from the bottom, two equality signs in).
My mark: 19/20
Q12 - it's not one that I bothered doing so I can't really comment.
Q5 - If I'm going to be honest, I don't think this would have got you any marks at all. You don't seem to have used the geometrical interpretation of |z_1 - z_2| to prove the triangle inequality, as asked in the question and as you say, there isn't really much of an inductive step in your proof of the general case. If you need a hint, add
to both sides of your n=k assumed inequality and use the triangle inequality you proved at the start.
My mark: 1/20
Q7 - This was mostly fine, might be worth familiarising yourself with the definition of a strictly increasing function (at A-Level, a function f is said to be strictly increasing over a particular interval if
in that interval. Equivalently, with such functions,
(with equality iff
) for each pair of such
over said interval). This would have saved you from having to find the turning points. For the last part, you want to find an expression for F'(x) in terms of F(x) and other functions, and deduce it's sign from there.
My mark: 16/20
Overall opinion: Low grade 1/high grade 2 (66 marks) excluding Q12, which probably makes it a comfortable grade 1 overall.