Year: 2023
Paper: 3
Question Number: 3
Course: UFM Pure
Section: Complex numbers 2
No solution available for this problem.
The total entry was a marginal increase on that of 2022 (by just over 1%). Two questions were attempted by more than 90% of candidates, another two by 80%, and another two by about two thirds. The least popular questions were attempted by more than a sixth of candidates. All the questions were perfectly answered by at least three candidates (but mostly more than this), with one being perfectly answered by eighty candidates. Very nearly 90% of candidates attempted no more than 7 questions. One general comment regarding all the questions is that candidates need to make sure that they read the question carefully, paying particular attention to command words such as "hence" and "show that".
Difficulty Rating: 1500.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
\begin{questionparts}
\item Show that, if $a$ and $b$ are complex numbers, with $b \neq 0$, and $s$ is a positive real number, then the points in the Argand diagram representing the complex numbers $a + sbi$, $a - sbi$ and $a + b$ form an isosceles triangle.
Given three points which form an isosceles triangle in the Argand diagram, explain with the aid of a diagram how to determine the values of $a$, $b$ and $s$ so that the vertices of the triangle represent complex numbers $a + sbi$, $a - sbi$ and $a + b$.
\item Show that, if the roots of the equation $z^3 + pz + q = 0$, where $p$ and $q$ are complex numbers, are represented in the Argand diagram by the vertices of an isosceles triangle, then there is a non-zero real number $s$ such that
\[\frac{p^3}{q^2} = \frac{27(3s^2 - 1)^3}{4(9s^2 + 1)^2}\,.\]
\item Sketch the graph $y = \dfrac{(3x-1)^3}{(9x+1)^2}$, identifying any stationary points.
\item Show that if the roots of the equation $z^3 + pz + q = 0$ are represented in the Argand diagram by the vertices of an isosceles triangle then $\dfrac{p^3}{q^2}$ is a real number and $\dfrac{p^3}{q^2} > -\dfrac{27}{4}$.
\end{questionparts}
This was the least popular of the Pure questions, being attempted by just under 45% of the candidates. Furthermore, it was not well answered yielding a mean score of 6/20. Many incorrectly treated a and b as real numbers in part (i) which rendered the question very simple. On the other hand, there were some that correctly simplified their working by 'ignoring' a in each number and then translating the triangle by a after. The second part of (i) was often well-answered. Most attempts at part (ii) were decent. Students that attempted it recognised that the roots should be written using a, b, s from part (i) and wrote down the sum/product of roots formulae for p and q. A few did not write down the equation for the coefficient of z2, and without this it was not possible for them to earn further credit for simplifying p and q. Sign errors in q were not uncommon. In part (iii), there was a large variety among the sketches seen. Only a few candidates specified the leading order behaviour at infinity. A fair number of candidates did not reflect the nature of the point of inflection in their drawing. Some did not specify intercepts. Pretty nearly all recognised the asymptote at -1/9. In part (iv), the majority that had successfully drawn the sketch in part (iii) managed to successfully satisfy the logic, although some failed to obtain the reality of the expression even though this was explicitly required.