Year: 2023
Paper: 2
Question Number: 7
Course: LFM Stats And Pure
Section: Complex Numbers (L8th)
No solution available for this problem.
Many candidates were able to express their reasoning clearly and presented good solutions to the questions that they attempted. There were excellent solutions seen for all of the questions. An area where candidates struggled in several questions was in the direction of the logic that was required in a solution. Some candidates failed to appreciate that separate arguments may be needed for the "if" and "only if" parts of a question and, in some cases, candidates produced correct arguments, but for the wrong direction. In several questions it was clear that candidates who used sketches or diagrams generally performed much better that those who did not. Sketches often also helped to make the solution clearer and easier to understand. Several questions on the STEP papers ask candidates to show a given result. Candidates should be aware that there is a need to present sufficient detail in their solutions so that it is clear that the reasoning is well understood.
Difficulty Rating: 1500.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
\begin{questionparts}
\item The complex numbers $z$ and $w$ have real and imaginary parts given by $z = a + \mathrm{i}b$ and $w = c + \mathrm{i}d$. Prove that $|zw| = |z||w|$.
\item By considering the complex numbers $2 + \mathrm{i}$ and $10 + 11\mathrm{i}$, find positive integers $h$ and $k$ such that $h^2 + k^2 = 5 \times 221$.
\item Find positive integers $m$ and $n$ such that $m^2 + n^2 = 8045$.
\item You are given that $102^2 + 201^2 = 50805$.
Find positive integers $p$ and $q$ such that $p^2 + q^2 = 36 \times 50805$.
\item Find three distinct pairs of positive integers $r$ and $s$ such that $r^2 + s^2 = 25 \times 1002082$ and $r < s$.
\item You are given that $109 \times 9193 = 1002037$.
Find positive integers $t$ and $u$ such that $t^2 + u^2 = 9193$.
\end{questionparts}
Most candidates were successful in the first two parts, with marks being lost mostly due to the small inaccuracy of forgetting the square root in the expression for the modulus of a complex number. Part (iii) was also typically done well, with most candidates picking up the idea of dividing by 5, however with mixed accuracy on the other factor. The candidates who picked up that the other factor can be written as a sum of squares were mostly successful in this part, as were almost all the candidates who attempted part (iv). Parts (v) and (vi) discriminated between candidates, with many successfully getting through (i)-(iv) with full marks but unfortunately making little to no progress on these two. Many failed to spot the decompositions 1001² + 9² in (v) and 1001² + 6² in (vi). The candidates who found these got access to the marks, though many didn't manage to find three solutions in part (v). This was from either overlooking the Pythagorean triple of (3, 4, 5) or the simpler solution obtained by noting that 25 is the square of 5. In part (vi), many candidates either chose the wrong complex number and did not try another one or failed to notice that 10028 or 2943 are divisible by 109.