| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2021 |
| Session | April |
| Marks | 4 |
| Topic | Complex Numbers Arithmetic |
| Type | Real and imaginary part expressions |
| Difficulty | Standard +0.3 This is a straightforward Further Maths complex numbers question requiring standard algebraic manipulation. Part (i) involves multiplying by the conjugate to separate real and imaginary parts—a routine technique. Part (ii) uses the result from (i) to verify that the modulus equals 1, which is direct calculation. Both parts are mechanical applications of standard methods with no conceptual difficulty or novel insight required. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02e Arithmetic of complex numbers: add, subtract, multiply, divide |
The function f is defined, for any complex number $z$, by
$$\text{f}(z) = \frac{iz - 1}{iz + 1}.$$
Suppose throughout that $x$ is a real number.
\begin{enumerate}[label=(\roman*)]
\item Show that
$$\text{Re f}(x) = \frac{x^2 - 1}{x^2 + 1} \quad \text{and} \quad \text{Im f}(x) = \frac{2x}{x^2 + 1}.$$ [2]
\item Show that $\text{f}(x)\text{f}(x)^* = 1$, where $\text{f}(x)^*$ is the complex conjugate of $\text{f}(x)$. [2]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2021 Q8 [4]}}