| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2021 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Parameter from modulus condition |
| Difficulty | Standard +0.3 This is a straightforward Further Maths question requiring multiplication of complex conjugates (which simplifies nicely), division by a complex number using conjugate multiplication, and solving a modulus equation. While it involves multiple steps, each technique is standard and the algebra is manageable. It's slightly easier than average even for Further Maths because the conjugate pair z₂z₃ simplifies immediately to p²+1, making the subsequent work cleaner. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02c Complex notation: z, z*, Re(z), Im(z), |z|, arg(z)4.02e Arithmetic of complex numbers: add, subtract, multiply, divide |
2. The complex numbers $z _ { 1 } , z _ { 2 }$ and $z _ { 3 }$ are given by
$$\mathrm { z } _ { 1 } = 2 - \mathrm { i } \quad \mathrm { z } _ { 2 } = p - \mathrm { i } \quad \mathrm { z } _ { 3 } = p + \mathrm { i }$$
where $p$ is a real number.
\begin{enumerate}[label=(\alph*)]
\item Find $\frac { z _ { 2 } z _ { 3 } } { z _ { 1 } }$ in the form $a + b \mathrm { i }$ where $a$ and $b$ are real. Give your answer in its simplest form in terms of $p$.
Given that $\left| \frac { z _ { 2 } z _ { 3 } } { z _ { 1 } } \right| = 2 \sqrt { 5 }$
\item find the possible values of $p$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel F1 2021 Q2 [7]}}