| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2013 |
| Session | June |
| Marks | 13 |
| Topic | Complex numbers 2 |
| Type | Solve polynomial equations with complex roots |
| Difficulty | Standard +0.3 This is a straightforward multi-part question on complex numbers requiring standard techniques: finding square roots of complex numbers by equating real/imaginary parts, using Vieta's formulas to find polynomial coefficients from roots, and differentiating then solving a quadratic. All parts are routine applications of A-level/Further Maths methods with no novel insight required, making it slightly easier than average. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02j Cubic/quartic equations: conjugate pairs and factor theorem4.05a Roots and coefficients: symmetric functions |
**(i)** $(p+\mathrm{i}q)^2 = (p^2-q^2) + \mathrm{i}\cdot 2pq$ **B1**
Comparing real and imaginary parts: $p^2 - q^2 = 63$ and $2pq = -16$ **M1**
Solving simultaneously: $p = \pm 8,\ q = \mp 1$ i.e. $\pm(8-\mathrm{i})^2 = 63 - 16\mathrm{i}$ **M1 A1**
**[4]**
**(ii)(a)** Use of $z^3 - (\alpha+\beta+\gamma)z^2 + (\alpha\beta+\beta\gamma+\gamma\alpha)z - (\alpha\beta\gamma) = 0$ **M1**
$A = 4-4\mathrm{i},\ B = 21-16\mathrm{i},\ C = 84$ **A1 A1**
i.e. $\mathrm{f}(z) = z^3 - (4-4\mathrm{i})z^2 + (21-16\mathrm{i})z - 84 = 0$ **A1**
**[4]**
**(b)** Differentiating to get $f'(z) = 3z^2 - 8(1-\mathrm{i})z + (21-16\mathrm{i})$ **OR** $3z^2 - 2Az + B = 0$ **ft** **B1**
Solving $z = \dfrac{8-8\mathrm{i} \pm \sqrt{64(1-2\mathrm{i}-1) - 12(21-16\mathrm{i})}}{6}$ using the quadratic formula **M1**
$z = \frac{1}{3}\left(4-4\mathrm{i} \pm \sqrt{16\mathrm{i}-63}\right) = \frac{1}{3}\left(4-4\mathrm{i} \pm \mathrm{i}\sqrt{63-16\mathrm{i}}\right)$ **A1**
Use of (i)'s result (on the right thing): $z = \frac{1}{3}\left(4-4\mathrm{i} \pm \mathrm{i}(8-\mathrm{i})\right) = \frac{5}{3}+\frac{4}{3}\mathrm{i}$ or $1-4\mathrm{i}$ **M1 A1**
**[5]**11 (i) Determine $p$ and $q$ given that $( p + \mathrm { i } q ) ^ { 2 } = 63 - 16 \mathrm { i }$ and that $p$ and $q$ are real.\\
(ii) Let $\mathrm { f } ( z ) = z ^ { 3 } - A z ^ { 2 } + B z - C$ for complex numbers $A , B$ and $C$.
\begin{enumerate}[label=(\alph*)]
\item Given that the cubic equation $\mathrm { f } ( z ) = 0$ has roots $\alpha = - 7 \mathrm { i } , \beta = 3 \mathrm { i }$ and $\gamma = 4$, determine each of $A , B$ and $C$.
\item Find the roots of the equation $\mathrm { f } ^ { \prime } ( z ) = 0$.
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2013 Q11 [13]}}