Oxford University MAT 2015 Solved Papers

Squaring all answers results in $\frac{7}{4}$ which is larger than (b) $\frac{25}{16}$. After squaring (c) simplifies to $\frac{10 \times 9 \times 8 \times 7}{9 \times 6!}$, which further simplifies to $\frac{7}{9}$ which is smaller than $(a)$. $\log_2(30) \approx 5$ and $\log_3(85) \approx 4$, hence (d) is smaller than (a) (after squaring). Comparing (a) with (e) after squaring results in a comparison of $\frac{7}{4}$ and $\frac{7+2\sqrt{6}}{9}$. As $2 < \sqrt{6} < 3$, (e) squared must be less than $\frac{13}{9}$ and hence less than $\frac{7}{4}$. The answer is (a).