| Exam Board | Edexcel |
|---|---|
| Module | F2 (Further Pure Mathematics 2) |
| Year | 2016 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Cartesian equation from argument condition |
| Difficulty | Challenging +1.2 This is a standard Further Maths locus question requiring recognition that arg((z-a)/(z-b))=θ gives a circular arc, sketching it, then optimizing |z| geometrically. While it involves multiple steps and Further Maths content, the techniques are well-practiced and the optimization is straightforward once the locus is identified—moderately above average difficulty. |
| Spec | 4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| A circle or arc of a circle drawn anywhere | M1 | Allow dotted or dashed |
| Circle/arc passing through or touching at \(2\) and \(5\) on positive real axis | A1 | Imaginary axis may be missing |
| Fully correct diagram: \(2\) and \(5\) marked correctly, no part of circle below real axis, must be major arc (not semi-circle), imaginary axis present, arc must not cross or touch imaginary axis | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Centre \(C(x_c, y_c)\) is at \((3.5,\ 1.5)\) | B1 | May be implied; can appear on diagram or from circle equation |
| \(r = \sqrt{1.5^2 + 1.5^2} = \frac{3}{\sqrt{2}} = \frac{3\sqrt{2}}{2}\) | M1 | \(r = \sqrt{2 \times y_c^2}\) or equivalent e.g. \(r = \frac{1.5}{\cos45°}\), \(\frac{1.5}{\sin45°}\) or \(\frac{3\sqrt{2}}{2}\) seen |
| \(\max | z | = OC + r = \sqrt{3.5^2 + 1.5^2} + r\) |
| \(= \frac{\sqrt{58}}{2} + \frac{3}{\sqrt{2}}\) | A1 | Accept e.g. \(\sqrt{14.5} + \sqrt{4.5}\), \(\frac{\sqrt{58}+3\sqrt{2}}{2}\) |
# Question 9(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| A circle or arc of a circle drawn anywhere | M1 | Allow dotted or dashed |
| Circle/arc passing through or touching at $2$ and $5$ on positive real axis | A1 | Imaginary axis may be missing |
| Fully correct diagram: $2$ and $5$ marked correctly, no part of circle below real axis, must be major arc (not semi-circle), imaginary axis present, arc must not cross or touch imaginary axis | A1 | |
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# Question 9(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Centre $C(x_c, y_c)$ is at $(3.5,\ 1.5)$ | B1 | May be implied; can appear on diagram or from circle equation |
| $r = \sqrt{1.5^2 + 1.5^2} = \frac{3}{\sqrt{2}} = \frac{3\sqrt{2}}{2}$ | M1 | $r = \sqrt{2 \times y_c^2}$ or equivalent e.g. $r = \frac{1.5}{\cos45°}$, $\frac{1.5}{\sin45°}$ or $\frac{3\sqrt{2}}{2}$ seen |
| $\max|z| = OC + r = \sqrt{3.5^2 + 1.5^2} + r$ | M1 | |
| $= \frac{\sqrt{58}}{2} + \frac{3}{\sqrt{2}}$ | A1 | Accept e.g. $\sqrt{14.5} + \sqrt{4.5}$, $\frac{\sqrt{58}+3\sqrt{2}}{2}$ |
*Special Case (arc below real axis): Centre $(3.5,\ -1.5)$ scores B0, but M1 M1 A1 still available for same working.*9. The complex number $z$ is represented by the point $P$ in an Argand diagram.
Given that $\arg \left( \frac { z - 5 } { z - 2 } \right) = \frac { \pi } { 4 }$
\begin{enumerate}[label=(\alph*)]
\item sketch the locus of $P$ as $z$ varies,
\item find the exact maximum value of $| z |$.\\
VILM SIHI NITIIIUMI ON OC\\
VILV SIHI NI III HM ION OC\\
VALV SIHI NI JIIIM ION OO
\begin{center}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{Edexcel F2 2016 Q9 [7]}}