Year: 2022
Paper: 2
Question Number: 7
Course: UFM Pure
Section: Complex numbers 2
No solution available for this problem.
Candidates appeared to be generally well prepared for most topics within the examination, but there were a few situations in questions where some did not appear to be as proficient in standard techniques as needed. In particular, the method for finding invariant lines required in question 8 and the manipulation of trigonometric functions that were needed in question 10 caused considerable difficulties for some candidates. An additional issue that occurred at numerous points in the paper relates to the direction in which a deduction is required. It is important that candidates make sure that they know which statement is the one that they should start from as they deduce the other and that it is clear in their solution that the logic has gone in the correct direction. Clarity of solution is also an issue that candidates should be aware of, especially in the situations where the result to be reached has been given. It is important to check that there are no special cases that need to be considered separately, and when dividing by a function it is necessary to confirm that the function cannot be equal to 0 (and in the case of inequalities that the function always has the same sign). When drawing diagrams and sketching graphs it is useful if significant points that need to be clear are not drawn over the lines on the page as these can be difficult to interpret during the marking process.
Difficulty Rating: 1500.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
Let $\mathrm{h}(z) = nz^6 + z^5 + z + n$, where $z$ is a complex number and $n \geqslant 2$ is an integer.
\begin{questionparts}
\item Let $w$ be a root of the equation $\mathrm{h}(z) = 0$.
\begin{enumerate}
\item Show that $|w^5| = \sqrt{\dfrac{\mathrm{f}(w)}{\mathrm{g}(w)}}$, where
\[\mathrm{f}(z) = n^2 + 2n\operatorname{Re}(z) + |z|^2 \quad \text{and} \quad \mathrm{g}(z) = n^2|z|^2 + 2n\operatorname{Re}(z) + 1.\]
\item By considering $\mathrm{f}(w) - \mathrm{g}(w)$, prove by contradiction that $|w| \geqslant 1$.
\item Show that $|w| = 1$.
\end{enumerate}
\item It is given that the equation $\mathrm{h}(z) = 0$ has six distinct roots, none of which is purely real.
\begin{enumerate}
\item Show that $\mathrm{h}(z)$ can be written in the form
\[\mathrm{h}(z) = n(z^2 - a_1 z + 1)(z^2 - a_2 z + 1)(z^2 - a_3 z + 1),\]
where $a_1$, $a_2$ and $a_3$ are real constants.
\item Find $a_1 + a_2 + a_3$ in terms of $n$.
\item By considering the coefficient of $z^3$ in $\mathrm{h}(z)$, find $a_1 a_2 a_3$ in terms of $n$.
\item How many of the six roots of the equation $\mathrm{h}(z) = 0$ have a negative real part? Justify your answer.
\end{enumerate}
\end{questionparts}
Only a small number of candidates attempted this question, and many of those struggled to achieve good marks. In part (i)(a) many candidates failed to spot the useful way of writing w⁵ and were unable to secure any marks. Many of those who achieved few marks overall were able to obtain a mark by writing down an expression for f(w) − g(w). Of those who did score well, many lost a mark for not stating that g(w) > 0 when dividing through by it in their inequalities. While there were some successful alternative solutions to part (i)(c), many of those who did not mimic part (i)(b) and tried to deduce the result directly were unsuccessful. In part (ii)(a) some candidates failed to observe that the coefficients of the polynomial were real when stating that the roots occur in complex conjugate pairs. The remainder of part (ii) was dealt with easily by most candidates, including many of those who had otherwise obtained few parts.