| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2011 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Modulus-argument form conversion |
| Difficulty | Moderate -0.8 This is a straightforward Further Maths question testing basic complex number operations: plotting on Argand diagram, calculating argument using arctan, converting from modulus-argument to Cartesian form using standard formulas, and applying the modulus product rule. All parts are routine applications of standard techniques with no problem-solving or insight required. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02b Express complex numbers: cartesian and modulus-argument forms |
| Answer | Marks | Guidance |
|---|---|---|
| \( | w | =4\), \(\arg w = \frac{5\pi}{6}\), \(w = a+ib\) |
| Working | Mark | Guidance |
| \( | w | =4 \Rightarrow a^2+b^2=16\) |
| \(\arg w = \frac{5\pi}{6} \Rightarrow \arctan\left(\frac{b}{a}\right)=\frac{5\pi}{6} \Rightarrow \frac{b}{a}=-\frac{1}{\sqrt{3}}\) | A1 | Either \(a^2+b^2=16\) or \(\frac{b}{a}=-\frac{1}{\sqrt{3}}\) |
| \(a=-\sqrt{3}\,b \Rightarrow a^2=3b^2\); so \(3b^2+b^2=16 \Rightarrow b^2=4 \Rightarrow b=\pm2\), \(a=\mp2\sqrt{3}\) | ||
| As \(w\) is in the second quadrant: \(w = -2\sqrt{3}+2i\), so \(a=-2\sqrt{3},\ b=2\) | A1 | Either \(-2\sqrt{3}+2i\) or awrt \(-3.5+2i\) |
## Question 7(c) (Aliter Way 2):
$|w|=4$, $\arg w = \frac{5\pi}{6}$, $w = a+ib$
| Working | Mark | Guidance |
|---------|------|----------|
| $|w|=4 \Rightarrow a^2+b^2=16$ | M1 | Attempts to write down an equation in terms of $a$ and $b$ for either the modulus or the argument of $w$ |
| $\arg w = \frac{5\pi}{6} \Rightarrow \arctan\left(\frac{b}{a}\right)=\frac{5\pi}{6} \Rightarrow \frac{b}{a}=-\frac{1}{\sqrt{3}}$ | A1 | Either $a^2+b^2=16$ or $\frac{b}{a}=-\frac{1}{\sqrt{3}}$ |
| $a=-\sqrt{3}\,b \Rightarrow a^2=3b^2$; so $3b^2+b^2=16 \Rightarrow b^2=4 \Rightarrow b=\pm2$, $a=\mp2\sqrt{3}$ | | |
| As $w$ is in the second quadrant: $w = -2\sqrt{3}+2i$, so $a=-2\sqrt{3},\ b=2$ | A1 | Either $-2\sqrt{3}+2i$ or awrt $-3.5+2i$ |
**Subtotal: 3 marks**7.
$$z = - 24 - 7 i$$
\begin{enumerate}[label=(\alph*)]
\item Show $z$ on an Argand diagram.
\item Calculate $\arg z$, giving your answer in radians to 2 decimal places.
It is given that
$$w = a + b \mathrm { i } , \quad a \in \mathbb { R } , b \in \mathbb { R }$$
Given also that $| w | = 4$ and $\arg w = \frac { 5 \pi } { 6 }$,
\item find the values of $a$ and $b$,
\item find the value of $| z w |$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 2011 Q7 [9]}}