| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Year | 2016 |
| Session | June |
| Marks | 6 |
| Topic | Complex Numbers Argand & Loci |
| Type | Single locus sketching |
| Difficulty | Moderate -0.8 This question tests basic complex number operations: calculating moduli (using Pythagoras), adding complex numbers, and recognizing that |z - z_1| = 2 represents a circle. Part (i) is straightforward verification requiring no insight (just compute three moduli and check an inequality). Part (ii) is standard bookwork on loci. The triangle inequality verification and circle sketching are routine exercises with no problem-solving demand. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02k Argand diagrams: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \( | z_1 | = \sqrt{5}\), \( |
| \( | z_1 + z_2 | = \sqrt{50}\); \(\sqrt{5} + 5 > \sqrt{50}\); \(\sqrt{5} + 5 > \sqrt{50}\) |
**(i)** $|z_1| = \sqrt{5}$, $|z_2| = 5$; $z_1 + z_2 = 5 + 5i$ | B1, M1, A1 | Both correct. Attempt $z_1 + z_2$. Could be implied by attempt at $|z_1 + z_2|$.
$|z_1 + z_2| = \sqrt{50}$; $\sqrt{5} + 5 > \sqrt{50}$; $\sqrt{5} + 5 > \sqrt{50}$ | A1, A1 [4] | Obtain $\sqrt{50}$ oe. Conclude by approximating to sufficient accuracy or comparing surds – A0 if no clear comparison.
Could also use geometrical argument.The complex numbers $z_1$ and $z_2$ are given by $z_1 = 2 + i$ and $z_2 = 3 + 4i$.
\begin{enumerate}[label=(\roman*)]
\item Verify that $|z_1| + |z_2| > |z_1 + z_2|$. [4]
\item Sketch on an Argand diagram the locus $|z - z_1| = 2$. [2]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2016 Q4 [6]}}