| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2021 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Region shading with multiple inequalities |
| Difficulty | Standard +0.8 This FP2 question requires converting a distance ratio locus to circle form (algebraically intensive), then interpreting the intersection of a circular inequality with an argument wedge—both standard Further Maths techniques but requiring careful algebraic manipulation and geometric visualization across multiple steps. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(z=x+iy \Rightarrow | x+9+iy | =4 |
| \(\Rightarrow (x+9)^2+y^2=16(x^2+(y-12)^2)\) | M1, A1 | Squares and uses modulus to achieve \((x+a)^2+y^2=K(x^2+(y+b)^2)\); correct equation |
| \(\Rightarrow 15x^2+15y^2-18x-384y=81-16\times12^2\) \(\Rightarrow x^2+y^2-\frac{6}{5}x-\frac{128}{5}y=-\frac{741}{5}\) | Expands, gathers terms, completes the square | |
| \(\Rightarrow \left(x-\frac{3}{5}\right)^2+\left(y-\frac{64}{5}\right)^2=16\) | M1 | Completes the square correctly |
| Centre \(\frac{3}{5}+\frac{64}{5}i\) or radius 4 | A1 | Either centre or radius correct |
| Centre \(\frac{3}{5}+\frac{64}{5}i\) and radius 4 | A1 | Both centre and radius correct |
| (6) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| Circle with centre in correct quadrant per their answer to (a) | M1 | Sketches circle on Argand diagram |
| Pair of rays at roughly 45° to horizontal, with source in first quadrant OR on the circle | M1 | Rays at angles \(\frac{\pi}{4}\) above and below horizontal with vertex in first quadrant or on circle |
| Correct circle and rays; circle with centre in first quadrant spanning only quadrants 1 and 2; pair of rays at roughly 45° to horizontal, meeting at the bottom point of the circle | A1 | Circle in correct position, centre in first quadrant spanning quadrants 1 and 2; rays meeting at bottom of circle |
| Region between rays and outside circle shaded | B1ft | Area outside circle and between rays (minor sector ~90°) shaded |
| (4) |
# Question 5:
## Part (a):
| Working | Mark | Guidance |
|---------|------|----------|
| $z=x+iy \Rightarrow |x+9+iy|=4|x+(y-12)i|$ | **M1** | Applies $z=x+iy$ to given equation |
| $\Rightarrow (x+9)^2+y^2=16(x^2+(y-12)^2)$ | **M1, A1** | Squares and uses modulus to achieve $(x+a)^2+y^2=K(x^2+(y+b)^2)$; correct equation |
| $\Rightarrow 15x^2+15y^2-18x-384y=81-16\times12^2$ $\Rightarrow x^2+y^2-\frac{6}{5}x-\frac{128}{5}y=-\frac{741}{5}$ | | Expands, gathers terms, completes the square |
| $\Rightarrow \left(x-\frac{3}{5}\right)^2+\left(y-\frac{64}{5}\right)^2=16$ | **M1** | Completes the square correctly |
| Centre $\frac{3}{5}+\frac{64}{5}i$ **or** radius 4 | **A1** | Either centre or radius correct |
| Centre $\frac{3}{5}+\frac{64}{5}i$ **and** radius 4 | **A1** | Both centre and radius correct |
| | **(6)** | |
## Part (b):
| Working | Mark | Guidance |
|---------|------|----------|
| Circle with centre in correct quadrant per their answer to (a) | **M1** | Sketches circle on Argand diagram |
| Pair of rays at roughly 45° to horizontal, with source in first quadrant OR on the circle | **M1** | Rays at angles $\frac{\pi}{4}$ above and below horizontal with vertex in first quadrant or on circle |
| Correct circle and rays; circle with centre in first quadrant spanning only quadrants 1 and 2; pair of rays at roughly 45° to horizontal, meeting at the bottom point of the circle | **A1** | Circle in correct position, centre in first quadrant spanning quadrants 1 and 2; rays meeting at bottom of circle |
| Region between rays and outside circle shaded | **B1ft** | Area outside circle and between rays (minor sector ~90°) shaded |
| | **(4)** | |
---\begin{enumerate}
\item The point $P$ in the complex plane represents a complex number $z$ such that
\end{enumerate}
$$| z + 9 | = 4 | z - 12 i |$$
Given that, as $z$ varies, the locus of $P$ is a circle,\\
(a) determine the centre and radius of this circle.\\
(b) Shade on an Argand diagram the region defined by the set
$$\{ z \in \mathbb { C } : | z + 9 | < 4 | z - 12 i | \} \cap \left\{ z \in \mathbb { C } : - \frac { \pi } { 4 } < \arg \left( z - \frac { 3 + 44 i } { 5 } \right) < \frac { \pi } { 4 } \right\}$$
\hfill \mbox{\textit{Edexcel FP2 2021 Q5 [10]}}