| Exam Board | Edexcel |
|---|---|
| Module | CP AS (Core Pure AS) |
| Session | Specimen |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Region shading with multiple inequalities |
| Difficulty | Challenging +1.2 This question requires understanding of loci in the complex plane (circle and argument rays) and finding an optimization point geometrically. Part (a) is standard shading of intersecting regions. Part (b) requires identifying that maximum arg(w) occurs at a tangent point from the origin to the circle, then using trigonometry to calculate the angle. While it involves multiple concepts and geometric insight, the techniques are well-practiced in Core Pure AS and the calculation is straightforward once the geometry is understood. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Circle drawn | M1 | Circle |
| Centre \((0, 4)\) and above the real axis | A1 | Centre \((0,4)\) and above real axis |
| Half-line drawn | M1 | Half-line |
| \((-3, 4)\) positioned correctly and half-line intersects top of circle on \(y\)-axis | A1 | \((-3,4)\) positioned correctly; half-line intersects top of circle on \(y\)-axis |
| Depends on both previous M marks; shades region inside circle and below half-line | M1 | Shades correct region |
| cso | A1 | cso (dependent on all previous marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((\arg w)_{\max} = \frac{\pi}{2} + \arcsin\!\left(\frac{3}{4}\right)\) | M1 | Uses trigonometry to give an expression for an angle in the range \(\left(\frac{\pi}{2}, \pi\right)\) or \((90°, 180°)\) |
| \(= 2.42\) (2dp) cao | A1 | \(2.42\) cao |
## Question 8:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Circle drawn | M1 | Circle |
| Centre $(0, 4)$ and above the real axis | A1 | Centre $(0,4)$ and above real axis |
| Half-line drawn | M1 | Half-line |
| $(-3, 4)$ positioned correctly and half-line intersects top of circle on $y$-axis | A1 | $(-3,4)$ positioned correctly; half-line intersects top of circle on $y$-axis |
| Depends on both previous M marks; shades region inside circle and below half-line | M1 | Shades correct region |
| cso | A1 | cso (dependent on all previous marks) |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(\arg w)_{\max} = \frac{\pi}{2} + \arcsin\!\left(\frac{3}{4}\right)$ | M1 | Uses trigonometry to give an expression for an angle in the range $\left(\frac{\pi}{2}, \pi\right)$ or $(90°, 180°)$ |
| $= 2.42$ (2dp) cao | A1 | $2.42$ cao |\begin{enumerate}
\item (a) Shade on an Argand diagram the set of points
\end{enumerate}
$$\{ z \in \mathbb { C } : | z - 4 i | \leqslant 3 \} \cap \left\{ z \in \mathbb { C } : - \frac { \pi } { 2 } < \arg ( z + 3 - 4 i ) \leqslant \frac { \pi } { 4 } \right\}$$
The complex number $w$ satisfies
$$| w - 4 \mathrm { i } | = 3$$
(b) Find the maximum value of $\arg w$ in the interval $( - \pi , \pi ]$.
Give your answer in radians correct to 2 decimal places.
\hfill \mbox{\textit{Edexcel CP AS Q8 [8]}}