| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2006 |
| Session | June |
| Marks | 14 |
| 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 complex modulus equation to Cartesian form to identify a circle (algebraically intensive), interpreting an argument condition as a half-line, then combining inequalities to shade a region. While methodical, it demands strong algebraic manipulation, geometric interpretation of complex loci, and careful handling of the inequality directions—more challenging than standard C3/C4 but typical for Further Maths. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| (a) Let \(z = x + iy\) | ||
| \((x-6)^2 + (y+3)^2 = 9[(x+2)^2 + (y-1)^2]\) | M1 | |
| Leading to \(8x^2 + 8y^2 + 48x - 24y = 0\) | M1 A1 | |
| This is a circle; the coefficients of \(x^2\) and \(y^2\) are the same and there is no \(xy\) term. | ||
| Allow equivalent arguments and fit their f, (x, y) if appropriate. | A1 ft | |
| \((x^2 + 6x + y^2 - 3y = 0)\) | ||
| Leading to \((x+3)^2 + \left(y - \frac{3}{2}\right)^2 = \frac{45}{4}\) | M1 | |
| Centre: \((-3, \frac{3}{2})\) | A1 | |
| Radius: \(\frac{3}{2}\sqrt{5}\) or equivalent | A1 | 7 marks |
| Answer | Marks | Guidance |
|---|---|---|
| The locus of P is a Circle of Apollonius, which is a circle with diameter XY, where the points X and Y cut (6, -3) and (-2, 1) internally and externally in the ratio 3 : 1. | M1 A1 | |
| X: (0, 0) Y: (-6, 3) | M1 A1 | |
| Centre: \((-3, \frac{3}{2})\) | M1 A1 | |
| Radius: \(\frac{3}{2}\sqrt{5}\) or equivalent | A1 | 7 marks |
| (b) Circle | B1 | |
| centre in correct quadrant | B1 ft | |
| through origin | B1 | |
| Line cuts -ve x and +ve y axes | B1 | |
| intersects with circle on axes and all correct | B1 | 5 marks |
| (c) Shading inside circle and above line with all correct | B1 B1 | 2 marks |
| Having 3 instead of 9 in first equation gains maximum of M1 M1 A0 A1 ft M1 A0 B1 B0 B1 B0 8/14 | [14] |
**(a)** Let $z = x + iy$ | |
$(x-6)^2 + (y+3)^2 = 9[(x+2)^2 + (y-1)^2]$ | M1 |
Leading to $8x^2 + 8y^2 + 48x - 24y = 0$ | M1 A1 |
This is a circle; the coefficients of $x^2$ and $y^2$ are the same and there is no $xy$ term. | |
Allow equivalent arguments and fit their f, (x, y) if appropriate. | A1 ft |
$(x^2 + 6x + y^2 - 3y = 0)$ | |
Leading to $(x+3)^2 + \left(y - \frac{3}{2}\right)^2 = \frac{45}{4}$ | M1 |
Centre: $(-3, \frac{3}{2})$ | A1 |
Radius: $\frac{3}{2}\sqrt{5}$ or equivalent | A1 | 7 marks
**Alternative:** Accept the following argument:-
The locus of P is a Circle of Apollonius, which is a circle with diameter XY, where the points X and Y cut (6, -3) and (-2, 1) internally and externally in the ratio 3 : 1. | M1 A1 |
X: (0, 0) Y: (-6, 3) | M1 A1 |
Centre: $(-3, \frac{3}{2})$ | M1 A1 |
Radius: $\frac{3}{2}\sqrt{5}$ or equivalent | A1 | 7 marks
**(b)** Circle | B1 |
centre in correct quadrant | B1 ft |
through origin | B1 |
Line cuts -ve x and +ve y axes | B1 |
intersects with circle on axes and all correct | B1 | 5 marks
**(c)** Shading inside circle and above line with all correct | B1 B1 | 2 marks
Having 3 instead of 9 in first equation gains maximum of M1 M1 A0 A1 ft M1 A0 B1 B0 B1 B0 8/14 | | [14]8. The point $P$ represents a complex number $z$ on an Argand diagram, where
$$| z - 6 + 3 i | = 3 | z + 2 - i |$$
\begin{enumerate}[label=(\alph*)]
\item Show that the locus of $P$ is a circle, giving the coordinates of the centre and the radius of this circle.
The point $Q$ represents a complex number $z$ on an Argand diagram, where
$$\tan [ \arg ( z + 6 ) ] = \frac { 1 } { 2 }$$
\item On the same Argand diagram, sketch the locus of $P$ and the locus of $Q$.
\item On your diagram, shade the region which satisfies both
$$| z - 6 + 3 \mathrm { i } | > 3 | z + 2 - \mathrm { i } | \text { and } \tan [ \arg ( z + 6 ) ] > \frac { 1 } { 2 }$$
(2)(Total 14 marks)
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 2006 Q8 [14]}}