Year: 2020
Paper: 2
Question Number: 7
Course: UFM Pure
Section: Complex numbers 2
No solution available for this problem.
There were just over 800 entries for this paper, and good solutions were seen to all of the questions. Candidates should be aware of the need to provide clear explanations of their reasoning throughout the paper (and particularly in questions where the result to be shown is given in the question). Short explanatory comments at key points in solutions can be very helpful in this regard, as can clearly drawn diagrams of the situation described in the question. The paper included a few questions where a statement of the form "A if and only if B" needed to be proven – candidates should be aware of the meaning of such statements and make sure that both directions of the implication are covered clearly. In general, candidates who performed better on the questions in this paper recognised the relationships between the different parts of each question and were able to adapt methods used in earlier parts when working on the later sections of the question.
Difficulty Rating: 1500.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
In this question, $w = \dfrac{2}{z-2}$.
\begin{questionparts}
\item Let $z$ be the complex number $3 + t\mathrm{i}$, where $t \in \mathbb{R}$. Show that $|w - 1|$ is independent of $t$. Hence show that, if $z$ is a complex number on the line $\operatorname{Re}(z) = 3$ in the Argand diagram, then $w$ lies on a circle in the Argand diagram with centre $1$.
Let $V$ be the line $\operatorname{Re}(z) = p$, where $p$ is a real constant not equal to $2$. Show that, if $z$ lies on $V$, then $w$ lies on a circle whose centre and radius you should give in terms of $p$. For which $z$ on $V$ is $\operatorname{Im}(w) > 0$?
\item Let $H$ be the line $\operatorname{Im}(z) = q$, where $q$ is a non-zero real constant. Show that, if $z$ lies on $H$, then $w$ lies on a circle whose centre and radius you should give in terms of $q$. For which $z$ on $H$ is $\operatorname{Re}(w) > 0$?
\end{questionparts}
This was the least popular of the pure questions and also the one on which marks were lowest on average. Many candidates were able to show the first result, that |w − 1| is independent of t. However, candidates often did not explain well enough the connection between the form of z and the line Re(z) = 3. The next part of part (i) required noting that the centre lies on the real axis and working out |w − c|. Some candidates guessed the value of c. Common mistakes here included guessing c = 1, p − 2, or failing to note conditions in which |w − c| is independent of t. In many solutions the absolute value sign on the radius was forgotten. Part (ii) was similar to the previous part but required noting that the centre lies on the imaginary axis and working out |w − ci|. In both parts a common attempt was to guess the centre to be at a point z = a + bi, few candidates were successful using this method. Again, absolute value signs on the radius were regularly forgotten. Another successful method employed by candidates in all parts of the question was to use the substitution t = tan(θ/2), t = (p − 2)tan(θ/2), t = q tan(θ/2) + 2 and using various trig identities to achieve the centre and radius. A few students also expressed t in terms of Re(w) and Im(w) and used that to obtain the equation of a circle in ℝ².