| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2011 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Complex transformations and mappings |
| Difficulty | Challenging +1.2 This is a multi-part Further Maths question involving standard locus identification (circle), Möbius transformation properties, and proving a geometric relationship. Parts (a)-(b) are routine FP2 content. Part (c) requires algebraic manipulation to show the image condition implies the original locus, which is a standard technique but requires careful complex number algebra across multiple steps. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks |
|---|---|
| \(x^2 + (y-1)^2 = 4\) | M1 A1 |
| (2) |
| Answer | Marks |
|---|---|
| M1: Sketch of circle | M1 |
| A1: Evidence of correct centre and radius | A1 |
| (2) |
| Answer | Marks |
|---|---|
| \(w = \frac{(x+iy)+i}{3+i(x+iy)} = \frac{x+i(y+1)}{(3-y)+ix}\) | M1 |
| \(= \frac{[x+i(y+1)][(3-y)-ix]}{[(3-y)+ix][(3-y)-ix]}\) | M1 |
| On x-axis, so imaginary part = 0: \((y+1)(3-y) - x^2 = 0\) | M1 A1 |
| \((y+1)(3-y) - x^2 = 0 \Rightarrow x^2 + (y-1)^2 = 4\), so \(Q\) is on \(C\) | A1cso |
| (5) 9 | M1 Use of \(z = x + iy\) and find modulus. A0 if circle doesn't intersect x-axis twice. 1st M for subbing \(z = x+iy\) and collecting real and imaginary parts. 2nd M for multiply numerator and denominator by their complex conjugate. 3rd M for equating imaginary parts of numerator to 0. Award A1 for equation matching part (a), statement not required. |
| Answer | Marks | Guidance |
|---|---|---|
| Let \(w = u + iv\): \(u = \frac{z+i}{3+iz}\) (since \(v=0\)) | M1 | |
| \(z = \frac{3u - i}{1 - ui}\) | dM1 | |
| \(z - i = \frac{3u - i - i - ui}{1 - ui} = \frac{2(u-i)}{1 - ui}\) | M1 A1 | |
| \(\ | z - i\ | = \frac{2\sqrt{u^2 + 1}}{\sqrt{u^2 + 1}} = 2\), so \(Q\) is on \(C\) |
**Part (a):**
| $x^2 + (y-1)^2 = 4$ | M1 A1 | |
| | (2) | |
**Part (b):**
| M1: Sketch of circle | M1 | |
| A1: Evidence of correct centre and radius | A1 | |
| | (2) | |
**Part (c):**
| $w = \frac{(x+iy)+i}{3+i(x+iy)} = \frac{x+i(y+1)}{(3-y)+ix}$ | M1 | |
| $= \frac{[x+i(y+1)][(3-y)-ix]}{[(3-y)+ix][(3-y)-ix]}$ | M1 | |
| On x-axis, so imaginary part = 0: $(y+1)(3-y) - x^2 = 0$ | M1 A1 | |
| $(y+1)(3-y) - x^2 = 0 \Rightarrow x^2 + (y-1)^2 = 4$, so $Q$ is on $C$ | A1cso | |
| | (5) 9 | M1 Use of $z = x + iy$ and find modulus. A0 if circle doesn't intersect x-axis twice. 1st M for subbing $z = x+iy$ and collecting real and imaginary parts. 2nd M for multiply numerator and denominator by their complex conjugate. 3rd M for equating imaginary parts of numerator to 0. Award A1 for equation matching part (a), statement not required. |
**Alternative (c):**
| Let $w = u + iv$: $u = \frac{z+i}{3+iz}$ (since $v=0$) | M1 | |
| $z = \frac{3u - i}{1 - ui}$ | dM1 | |
| $z - i = \frac{3u - i - i - ui}{1 - ui} = \frac{2(u-i)}{1 - ui}$ | M1 A1 | |
| $\|z - i\| = \frac{2\sqrt{u^2 + 1}}{\sqrt{u^2 + 1}} = 2$, so $Q$ is on $C$ | A1cso | |
| | | |
**Guidance Notes for Part (c):**
M1 Use of $z = x+iy$ and find modulus. A0 if circle doesn't intersect x-axis twice. 1st M for subbing $z = x+iy$ and collecting real and imaginary parts. 2nd M for multiply numerator and denominator by their complex conjugate. 3rd M for equating imaginary parts of numerator to 0. Award A1 for equation matching part (a), statement not required.\begin{enumerate}
\item The point $P$ represents the complex number $z$ on an Argand diagram, where
\end{enumerate}
$$| z - \mathrm { i } | = 2$$
The locus of $P$ as $z$ varies is the curve $C$.\\
(a) Find a cartesian equation of $C$.\\
(b) Sketch the curve $C$.
A transformation $T$ from the $z$-plane to the $w$-plane is given by
$$w = \frac { z + \mathrm { i } } { 3 + \mathrm { i } z } , \quad z \neq 3 \mathrm { i }$$
The point $Q$ is mapped by $T$ onto the point $R$. Given that $R$ lies on the real axis,\\
(c) show that $Q$ lies on $C$.\\
\hfill \mbox{\textit{Edexcel FP2 2011 Q5 [9]}}