| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2018 |
| Session | June |
| Marks | 7 |
| Topic | Complex numbers 2 |
| Type | Modulus and argument calculations |
| Difficulty | Standard +0.3 This is a straightforward application of modulus-argument rules for division and powers of complex numbers. Part (i) uses standard formulas |z₁/z₂| = |z₁|/|z₂| and arg(z₁/z₂) = arg(z₁) - arg(z₂). Part (ii) requires finding when the argument becomes a multiple of 2π (for positive real), which involves solving a simple linear equation in n, then computing the modulus raised to that power. All steps are routine for Further Maths students with no novel problem-solving required. |
| Spec | 4.02c Complex notation: z, z*, Re(z), Im(z), |z|, arg(z)4.02m Geometrical effects: multiplication and division |
The complex numbers $z_1$ and $z_2$ are such that $|z_1| = 2$, $\arg(z_1) = \frac{7}{12}\pi$, $|z_2| = \sqrt{2}$ and $\arg(z_2) = -\frac{1}{8}\pi$.
\begin{enumerate}[label=(\roman*)]
\item Find, in exact form, the modulus and argument of $\frac{z_1}{z_2}$. [3]
\item Let $z_3 = \left(\frac{z_1}{z_2}\right)^n$. It is given that $n$ is the least positive integer for which $z_3$ is a positive real number. Find this value of $n$ and the exact value of $z_3$. [4]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2018 Q3 [7]}}