| Exam Board | OCR |
|---|---|
| Module | FP1 AS (Further Pure 1 AS) |
| Year | 2021 |
| Session | June |
| Marks | 13 |
| Topic | Complex Numbers Arithmetic |
| Type | Modulus and argument with operations |
| Difficulty | Moderate -0.3 This is a standard FP1 complex numbers question testing routine techniques: conjugate arithmetic, squaring, division by multiplying by conjugate, modulus-argument form, and using properties |zω|=|z||ω| and arg(zω)=arg(z)+arg(ω). All parts follow textbook methods with no novel insight required. While it's Further Maths content (inherently harder), these are foundational FP1 skills making it slightly easier than an average A-level question overall. |
| Spec | 4.02b Express complex numbers: cartesian and modulus-argument forms4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02f Convert between forms: cartesian and modulus-argument |
2 In this question you must show detailed reasoning.\\
The complex number $7 - 4 \mathrm { i }$ is denoted by $z$.
\begin{enumerate}[label=(\alph*)]
\item Giving your answers in the form $a + b \mathrm { i }$, where $a$ and $b$ are rational numbers, find the following.
\begin{enumerate}[label=(\roman*)]
\item $3 z - 4 z ^ { * }$
\item $( z + 1 - 3 \mathrm { i } ) ^ { 2 }$
\item $\frac { z + 1 } { z - 1 }$
\end{enumerate}\item Express $z$ in modulus-argument form giving the modulus exactly and the argument correct to 3 significant figures.
\item The complex number $\omega$ is such that $z \omega = \sqrt { 585 } ( \cos ( 0.5 ) + \mathrm { i } \sin ( 0.5 ) )$.
Find the following.
\begin{itemize}
\item $| \omega |$
\item $\arg ( \omega )$, giving your answer correct to 3 significant figures
\end{itemize}
\end{enumerate}
\hfill \mbox{\textit{OCR FP1 AS 2021 Q2 [13]}}