| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Session | Specimen |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Complex arithmetic operations |
| Difficulty | Moderate -0.8 This is a straightforward Further Pure 1 question testing basic complex number operations: finding modulus and argument using standard formulas, performing complex division by multiplying by the conjugate, and plotting points on an Argand diagram. All parts are routine textbook exercises requiring direct application of learned techniques with no problem-solving or insight needed. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02k Argand diagrams: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| \( | z | = \sqrt{3^2+4^2} = 5\) |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| \(\arg z = \pi - \arctan\frac{4}{3} = 2.21\) | M1 A1 | (2 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| \(w = \frac{-14+2i}{-3+4i} = \frac{(-14+2i)(-3-4i)}{(-3+4i)(-3-4i)}\) | M1 | Multiply by conjugate |
| \(= \frac{(42+8)+i(-6+56)}{9+16}\) | A1 A1 | |
| \(= \frac{50+50i}{25} = 2+2i\) | A1 | (4 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| Point \(A(-3,4)\) and point \(B(2,2)\) correctly plotted on Argand diagram | B1 B1 | (2 marks) |
# Question 6:
## Part (a):
| Working/Answer | Marks | Guidance |
|---|---|---|
| $|z| = \sqrt{3^2+4^2} = 5$ | M1 A1 | **(2 marks)** |
## Part (b):
| Working/Answer | Marks | Guidance |
|---|---|---|
| $\arg z = \pi - \arctan\frac{4}{3} = 2.21$ | M1 A1 | **(2 marks)** |
## Part (c):
| Working/Answer | Marks | Guidance |
|---|---|---|
| $w = \frac{-14+2i}{-3+4i} = \frac{(-14+2i)(-3-4i)}{(-3+4i)(-3-4i)}$ | M1 | Multiply by conjugate |
| $= \frac{(42+8)+i(-6+56)}{9+16}$ | A1 A1 | |
| $= \frac{50+50i}{25} = 2+2i$ | A1 | **(4 marks)** |
## Part (d):
| Working/Answer | Marks | Guidance |
|---|---|---|
| Point $A(-3,4)$ and point $B(2,2)$ correctly plotted on Argand diagram | B1 B1 | **(2 marks)** |
---6. Given that $z = - 3 + 4 \mathrm { i }$,
\begin{enumerate}[label=(\alph*)]
\item find the modulus of $z$,
\item the argument of $z$ in radians to 2 decimal places.
Given also that $w = \frac { - 14 + 2 \mathrm { i } } { z }$,
\item use algebra to find $w$, giving your answers in the form $a + \mathrm { i } b$, where $a$ and $b$ are real.
The complex numbers $z$ and $w$ are represented by points $A$ and $B$ on an Argand diagram.
\item Show the points $A$ and $B$ on an Argand diagram.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 Q6 [10]}}