| Exam Board | WJEC |
|---|---|
| Module | Further Unit 1 (Further Unit 1) |
| Year | 2018 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Convert to exponential/polar form |
| Difficulty | Moderate -0.3 This is a straightforward Further Maths question testing standard modulus-argument form conversions and multiplication. Part (a) requires calculating r = 5 and θ = arctan(4/-3) in the correct quadrant (standard technique), part (a)(ii) simply negates the argument, and part (b) applies the multiplication rule for polar form (multiply moduli, add arguments). All steps are routine applications of well-practiced procedures with no problem-solving or novel insight required, making it slightly easier than an average A-level question overall. |
| Spec | 4.02b Express complex numbers: cartesian and modulus-argument forms4.02e Arithmetic of complex numbers: add, subtract, multiply, divide |
| Answer | Marks | Guidance |
|---|---|---|
| Answer: (i) \( | z | = \sqrt{(-3)^2 + 4^2} = 5\); \(\tan^{-1}\frac{4}{-3} = -0.927\) or \(\tan^{-1}-\frac{4}{3} = 0.927\) |
| Answer: \(\arg(z) = 2.21\) | B1 | Accept awrt 2.2 |
| Answer: \(\therefore z = 5(\cos 2.21 + i\sin 2.21)\) | B1 | |
| Answer: (ii) \(z = 5(\cos(-2.21) + i\sin(-2.21))\) | B1 | Accept 4.07 FT (i) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer: \( | zw | = 5 \times \sqrt{5} (= 5\sqrt{5})\) |
| Answer: \(\arg(zw) = 2.21 + 2.68 = 4.89\) or \(-1.39\) | M1 | |
| Answer: \(\therefore zw = 5\sqrt{5}(\cos(-1.39) + i\sin(-1.39))\) | A1 | Accept awrt \(-1.4\) or \(4.9\) |
**Part a)**
Answer: (i) $|z| = \sqrt{(-3)^2 + 4^2} = 5$; $\tan^{-1}\frac{4}{-3} = -0.927$ or $\tan^{-1}-\frac{4}{3} = 0.927$ | B1 |
Answer: $\arg(z) = 2.21$ | B1 | Accept awrt 2.2
Answer: $\therefore z = 5(\cos 2.21 + i\sin 2.21)$ | B1 |
Answer: (ii) $z = 5(\cos(-2.21) + i\sin(-2.21))$ | B1 | Accept 4.07 FT (i)
**Part b)**
Answer: $|zw| = 5 \times \sqrt{5} (= 5\sqrt{5})$ | M1 | FT (a)
Answer: $\arg(zw) = 2.21 + 2.68 = 4.89$ or $-1.39$ | M1 |
Answer: $\therefore zw = 5\sqrt{5}(\cos(-1.39) + i\sin(-1.39))$ | A1 | Accept awrt $-1.4$ or $4.9$A complex number is defined by $z = -3 + 4\mathrm{i}$.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Express $z$ in the form $r(\cos\theta + \mathrm{i}\sin\theta)$, where $-\pi \leqslant \theta \leqslant \pi$.
\item Express $\bar{z}$, the complex conjugate of $z$, in the form $r(\cos\theta + \mathrm{i}\sin\theta)$. [4]
\end{enumerate}
\end{enumerate}
Another complex number is defined as $w = \sqrt{5}(\cos 2.68 + \mathrm{i}\sin 2.68)$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Express $zw$ in the form $r(\cos\theta + \mathrm{i}\sin\theta)$. [3]
\end{enumerate}
\hfill \mbox{\textit{WJEC Further Unit 1 2018 Q4 [7]}}