| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2022 |
| Session | June |
| Marks | 7 |
| Topic | Complex Numbers Argand & Loci |
| Type | Argument calculations and identities |
| Difficulty | Moderate -0.3 This is a straightforward Further Maths complex numbers question testing standard techniques: modulus/argument calculation (routine), algebraic manipulation with complex numbers (multiply out and divide), and using argument conditions (simple angle geometry). All parts are textbook exercises requiring no problem-solving insight, though the multi-part structure and FM context place it slightly below average difficulty overall. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02e Arithmetic of complex numbers: add, subtract, multiply, divide |
The complex number $w$ is given by
$$w = 10 - 5i$$
\begin{enumerate}[label=(\alph*)]
\item Find $|w|$. [1]
\item Find $\arg w$, giving your answer in radians to 2 decimal places. [1]
\end{enumerate}
The complex numbers $z$ and $w$ satisfy the equation
$$(2 + i)(z + 3i) = w$$
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Use algebra to find $z$, giving your answer in the form $a + bi$, where $a$ and $b$ are real numbers. [4]
\end{enumerate}
Given that
$$\arg(\lambda + 9i + w) = \frac{\pi}{4}$$
where $\lambda$ is a real constant,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item find the value of $\lambda$. [1]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2022 Q6 [7]}}