| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2015 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Complex arithmetic operations |
| Difficulty | Moderate -0.8 This is a straightforward FP1 complex numbers question requiring standard techniques: rationalizing a complex denominator (multiplying by conjugate), finding modulus and argument using standard formulas, and plotting points on an Argand diagram. All parts are routine exercises with no problem-solving or novel insight required, making it easier than the average A-level question. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02b Express complex numbers: cartesian and modulus-argument forms4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02f Convert between forms: cartesian and modulus-argument4.02k Argand diagrams: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(z_2 = \frac{6(1-i\sqrt{3})}{(1+i\sqrt{3})(1-i\sqrt{3})} = \frac{6(1-i\sqrt{3})}{4}\) | M1 | Multiplies numerator and denominator by \(1-i\sqrt{3}\) |
| \(z_2 = \frac{6(1-i\sqrt{3})}{4} = \frac{3}{2}-i\frac{3}{2}\sqrt{3}\) | A1 (2) | Any correct equivalent with real denominator |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \( | z_2 | = \sqrt{\left(\frac{3}{2}\right)^2+\left(\frac{3\sqrt{3}}{2}\right)^2} = \sqrt{\frac{9}{4}+\frac{27}{4}}\) |
| The modulus of \(z_2\) is \(3\) | A1 | A1: for 3 only |
| \(\tan\theta = (\pm)\sqrt{3}\) and attempts to find \(\theta\) | M1 | Uses tan or inverse tan |
| Argument is \(-\frac{\pi}{3}\) | A1 (4) | A1: \(-\frac{\pi}{3}\); accept \(\frac{5\pi}{3}\). NB answers only then award 4/4 but arg must be in terms of \(\pi\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Argand diagram with \(z_1\) on positive imaginary axis, \(z_2\) in 4th quadrant, \(z_1+z_2\) in first quadrant | M1 A1 (2)(8 marks) | M1: Either \(z_1\) on imaginary axis labelled \(z_1\) or \(3i\) or \((0,3)\) or axis labelled 3; or \(z_2\) in correct quadrant labelled appropriately. A1: All 3 correct: \(z_1\) on positive imaginary axis, \(z_2\) in 4th quadrant, \(z_1+z_2\) in first quadrant. Accept points or lines; arrows not required |
# Question 4:
## Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $z_2 = \frac{6(1-i\sqrt{3})}{(1+i\sqrt{3})(1-i\sqrt{3})} = \frac{6(1-i\sqrt{3})}{4}$ | M1 | Multiplies numerator and denominator by $1-i\sqrt{3}$ |
| $z_2 = \frac{6(1-i\sqrt{3})}{4} = \frac{3}{2}-i\frac{3}{2}\sqrt{3}$ | A1 (2) | Any correct equivalent with real denominator |
## Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $|z_2| = \sqrt{\left(\frac{3}{2}\right)^2+\left(\frac{3\sqrt{3}}{2}\right)^2} = \sqrt{\frac{9}{4}+\frac{27}{4}}$ | M1 | Uses correct method for modulus for their $z_2$ from part (a) |
| The modulus of $z_2$ is $3$ | A1 | A1: for 3 only |
| $\tan\theta = (\pm)\sqrt{3}$ and attempts to find $\theta$ | M1 | Uses tan or inverse tan |
| Argument is $-\frac{\pi}{3}$ | A1 (4) | A1: $-\frac{\pi}{3}$; accept $\frac{5\pi}{3}$. NB answers only then award 4/4 but arg must be in terms of $\pi$ |
## Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Argand diagram with $z_1$ on positive imaginary axis, $z_2$ in 4th quadrant, $z_1+z_2$ in first quadrant | M1 A1 (2)(8 marks) | M1: Either $z_1$ on imaginary axis labelled $z_1$ or $3i$ or $(0,3)$ or axis labelled 3; **or** $z_2$ in correct quadrant labelled appropriately. A1: All 3 correct: $z_1$ on positive imaginary axis, $z_2$ in 4th quadrant, $z_1+z_2$ in first quadrant. Accept points or lines; arrows not required |
---4.
$$z _ { 1 } = 3 \mathrm { i } \text { and } z _ { 2 } = \frac { 6 } { 1 + \mathrm { i } \sqrt { 3 } }$$
\begin{enumerate}[label=(\alph*)]
\item Express $z _ { 2 }$ in the form $a + \mathrm { i } b$, where $a$ and $b$ are real numbers.
\item Find the modulus and the argument of $z _ { 2 }$, giving the argument in radians in terms of $\pi$.
\item Show the three points representing $z _ { 1 } , z _ { 2 }$ and $\left( z _ { 1 } + z _ { 2 } \right)$ respectively, on a single Argand diagram.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 2015 Q4 [8]}}