| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2013 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Division plus modulus/argument |
| Difficulty | Moderate -0.3 This is a straightforward Further Maths FP1 question testing standard complex number techniques: multiplying by conjugate to divide, squaring a complex number, finding modulus, and finding argument. All parts are routine applications of well-practiced methods with no problem-solving required, making it slightly easier than average despite being Further Maths content. |
| Spec | 4.02c Complex notation: z, z*, Re(z), Im(z), |z|, arg(z)4.02e Arithmetic of complex numbers: add, subtract, multiply, divide |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(\frac{50}{3+4i} = \frac{50(3-4i)}{(3+4i)(3-4i)} = \frac{50(3-4i)}{25} = 6 - 8i\) | M1 A1cao | M for \(\times \frac{3-4i}{3-4i}\) (accept use of -3+4i) and attempt to expand using \(i^2=-1\); A for 6-8i only |
| (b) \(z^2 = (6-8i)^2 = 36 - 64 - 96i = -28 - 96i\) | M1 A1 | M for attempting to expand their \(z^2\) using \(i^2=-1\); A for -28-96i only. If using original \(z\) then must attempt to multiply top and bottom by conjugate and use \(i^2=-1\) |
| (c) \( | z | = \sqrt{6^2 + (-8)^2} = 10\) |
| (d) \(\tan \alpha = \frac{-96}{-28}\) | M1 | |
| \(\alpha = -106.3°\) or \(253.7°\) | A1cao | |
| [8] |
| Answer | Marks | Guidance |
|---|---|---|
| (c) \( | z | = \frac{50}{ |
| (d) \(\arg(3+4i) = 53.13...\), so \(\arg\left(\frac{50}{3+4i}\right)^2 = -2 \times 53.13... = -106.3\) | M1 A1 |
(a) $\frac{50}{3+4i} = \frac{50(3-4i)}{(3+4i)(3-4i)} = \frac{50(3-4i)}{25} = 6 - 8i$ | M1 A1cao | M for $\times \frac{3-4i}{3-4i}$ (accept use of -3+4i) and attempt to expand using $i^2=-1$; A for 6-8i only
(b) $z^2 = (6-8i)^2 = 36 - 64 - 96i = -28 - 96i$ | M1 A1 | M for attempting to expand their $z^2$ using $i^2=-1$; A for -28-96i only. If using original $z$ then must attempt to multiply top and bottom by conjugate and use $i^2=-1$
(c) $|z| = \sqrt{6^2 + (-8)^2} = 10$ | M1 A1ft | M for $\sqrt{a^2 + b^2}$; A for 'their 10'
(d) $\tan \alpha = \frac{-96}{-28}$ | M1 |
$\alpha = -106.3°$ or $253.7°$ | A1cao |
| [8] |
**Alternatives:**
(c) $|z| = \frac{50}{|3+4i|} = 10$ | M1 A1 |
(d) $\arg(3+4i) = 53.13...$, so $\arg\left(\frac{50}{3+4i}\right)^2 = -2 \times 53.13... = -106.3$ | M1 A1 |
**Notes:**
(a) M for $\times \frac{3-4i}{3-4i}$ (accept use of -3+4i) and attempt to expand using $i^2=-1$; A for 6-8i only
(b) M for attempting to expand their $z^2$ using $i^2=-1$; A for -28-96i only. If using original $z$ then must attempt to multiply top and bottom by conjugate and use $i^2=-1$
(c) M for $\sqrt{a^2+b^2}$; A for 'their 10'
(d) M for use of tan or $\tan^{-1}$ and values from their $z^2$ either way up ignoring signs. Radians score A0.
---2.
$$z = \frac { 50 } { 3 + 4 \mathrm { i } }$$
Find, in the form $a + \mathrm { i } b$ where $a , b \in \mathbb { R }$,
\begin{enumerate}[label=(\alph*)]
\item $z$,
\item $z ^ { 2 }$.
Find
\item $| z |$,
\item $\arg z ^ { 2 }$, giving your answer in degrees to 1 decimal place.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 2013 Q2 [8]}}