| Exam Board | SPS |
|---|---|
| Module | SPS ASFM Statistics (SPS ASFM Statistics) |
| Year | 2021 |
| Session | May |
| Marks | 10 |
| Topic | Complex Numbers Argand & Loci |
| Type | Perpendicular bisector locus |
| Difficulty | Moderate -0.8 This question tests basic complex number operations: finding modulus (straightforward formula), argument of conjugate (routine calculation), solving a linear equation with conjugates (algebraic manipulation), and sketching a perpendicular bisector locus (standard geometric interpretation). All parts are direct applications of fundamental techniques with no problem-solving insight required, making it easier than average but not trivial due to the multi-part nature. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02b Express complex numbers: cartesian and modulus-argument forms4.02k Argand diagrams: geometric interpretation |
\begin{enumerate}[label=(\alph*)]
\item The complex number $3 + 2i$ is denoted by $w$ and the complex conjugate of $w$ is denoted by $w^*$.
Find
\begin{enumerate}[label=(\roman*)]
\item the modulus of $w$, [1]
\item the argument of $w^*$, giving your answer in radians, correct to 2 decimal places. [3]
\end{enumerate}
\item Find the complex number $u$ given that $u + 2u^* = 3 + 2i$. [4]
\item Sketch, on an Argand diagram, the locus given by $|z + 1| = |z|$. [2]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS ASFM Statistics 2021 Q1 [10]}}