| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2014 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Parameter from argument condition |
| Difficulty | Standard +0.3 This is a straightforward FP1 complex numbers question testing basic operations (modulus, multiplication, division) and argument. Parts (a)-(c) are routine calculations using standard formulas. Part (d) requires finding a real parameter using the argument condition, which involves tan(π/3) and simple algebra. All techniques are standard textbook exercises with no novel insight required, making it slightly easier than average. |
| 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 |
|---|---|---|
| \(\ | w\ | = \left\{\sqrt{(\sqrt{3})^2 + (-1)^2}\right\} = 2\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(zw = (5 + i\sqrt{3})(\sqrt{3} - i) = 5\sqrt{3} - 5i + 3i + \sqrt{3} = 6\sqrt{3} - 2i\) | M1, A1 | Either the real or imaginary part is correct; \(6\sqrt{3} - 2i\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{z}{w} = \frac{(5 + i\sqrt{3})}{(\sqrt{3} - i)} \times \frac{(\sqrt{3} + i)}{(\sqrt{3} + i)}\) | M1 | Multiplies by \(\frac{(\sqrt{3} + i)}{(\sqrt{3} + i)}\) |
| \(= \frac{5\sqrt{3} + 5i + 3i - \sqrt{3}}{3 + 1}\) | M1 | Simplifies realising that a real number is needed on the denominator and applies \(i^2 = -1\) on their numerator expression and denominator expression |
| \(\left\{= \frac{4\sqrt{3} + 8i}{4}\right\} = \sqrt{3} + 2i\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(z + \lambda = 5 + i\sqrt{3} + \lambda = (5 + \lambda) + i\sqrt{3}\) | ||
| \(\arg(z + \lambda) = \frac{\pi}{3} \Rightarrow \frac{\sqrt{3}}{5 + \lambda} = \tan\left(\frac{\pi}{3}\right)\) | M1 oe | their combined real part \(= \tan\left(\frac{\pi}{3}\right)\) |
| \(\frac{\sqrt{3}}{5 + \lambda} = \frac{\sqrt{3}}{1} \Rightarrow 5 + \lambda = 1 \Rightarrow \lambda = -4\) | A1 |
**(a)**
| $\|w\| = \left\{\sqrt{(\sqrt{3})^2 + (-1)^2}\right\} = 2$ | B1 | |
**(b)**
| $zw = (5 + i\sqrt{3})(\sqrt{3} - i) = 5\sqrt{3} - 5i + 3i + \sqrt{3} = 6\sqrt{3} - 2i$ | M1, A1 | Either the real or imaginary part is correct; $6\sqrt{3} - 2i$ |
**(c)**
| $\frac{z}{w} = \frac{(5 + i\sqrt{3})}{(\sqrt{3} - i)} \times \frac{(\sqrt{3} + i)}{(\sqrt{3} + i)}$ | M1 | Multiplies by $\frac{(\sqrt{3} + i)}{(\sqrt{3} + i)}$ |
| $= \frac{5\sqrt{3} + 5i + 3i - \sqrt{3}}{3 + 1}$ | M1 | Simplifies realising that a real number is needed on the denominator and applies $i^2 = -1$ on their numerator expression and denominator expression |
| $\left\{= \frac{4\sqrt{3} + 8i}{4}\right\} = \sqrt{3} + 2i$ | A1 | |
**(d)**
| $z + \lambda = 5 + i\sqrt{3} + \lambda = (5 + \lambda) + i\sqrt{3}$ | | |
| $\arg(z + \lambda) = \frac{\pi}{3} \Rightarrow \frac{\sqrt{3}}{5 + \lambda} = \tan\left(\frac{\pi}{3}\right)$ | M1 oe | their combined real part $= \tan\left(\frac{\pi}{3}\right)$ |
| $\frac{\sqrt{3}}{5 + \lambda} = \frac{\sqrt{3}}{1} \Rightarrow 5 + \lambda = 1 \Rightarrow \lambda = -4$ | A1 | |
---5.
$$z = 5 + \mathrm { i } \sqrt { 3 } , \quad w = \sqrt { 3 } - \mathrm { i }$$
\begin{enumerate}[label=(\alph*)]
\item Find the value of $| w |$.
Find in the form $a + \mathrm { i } b$, where $a$ and $b$ are real constants,
\item $z w$, showing clearly how you obtained your answer,
\item $\frac { z } { w }$, showing clearly how you obtained your answer.
Given that
$$\arg ( z + \lambda ) = \frac { \pi } { 3 } , \quad \text { where } \lambda \text { is a real constant, }$$
\item find the value of $\lambda$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 2014 Q5 [8]}}