Edexcel FP2 2020 June — Question 5 10 marks

Exam BoardEdexcel
ModuleFP2 (Further Pure Mathematics 2)
Year2020
SessionJune
Marks10
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicComplex Numbers Argand & Loci
TypeComplex transformations and mappings
DifficultyChallenging +1.2 This is a Further Maths FP2 question on complex transformations, requiring substitution of a circle equation into a Möbius transformation and algebraic manipulation. While it involves multiple steps (substituting, manipulating moduli, squaring, expanding), the techniques are standard for FP2 and the question provides significant scaffolding by giving the intermediate form in part (a). The algebraic manipulation is moderately involved but follows predictable patterns for this topic.
Spec4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines
  1. A transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by
$$w = \frac { 1 - 3 z } { z + 2 i } \quad z \neq - 2 i$$ The circle with equation \(| z + \mathrm { i } | = 3\) is mapped by \(T\) onto the circle \(C\).
  1. Show that the equation for \(C\) can be written as $$3 | w + 3 | = | 1 + ( 3 - w ) \mathrm { i } |$$
  2. Hence find
    1. a Cartesian equation for \(C\),
    2. the centre and radius of \(C\).
Question 5:
Part (a):
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(w = \frac{1-3z}{z+2\text{i}} \Rightarrow w(z+2\text{i}) = 1-3z \Rightarrow z = \ldots\) or \((z+\text{i}) = \ldots\)M1 Attempts to make \(z\) the subject or extract \(z+\text{i}\) as a term
\(z = \frac{1-2\text{i}w}{w+3}\) or \((w+3)(z+\text{i}) = 1 - 2\text{i}w + (w+3)\text{i}\ (= 1+(3-w)\text{i})\)A1 Correct expression for \(z\) or correct equation with \(z+\text{i}\) as only term in \(z\)
\(\left\\frac{1-2\text{i}w}{w+3} + \text{i}\right\ = 3 \Rightarrow
Part (b)(i):
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(w = u+\text{i}v \Rightarrow(3-u)\text{i} + (v+1) = 3
\(\Rightarrow (3-u)^2 + (v+1)^2 = 9[(u+3)^2 + v^2]\)M1, A1 M1: Squares and applies Pythagoras; no \(\text{i}\)'s, sum of squares each side. A1: Correct Cartesian equation
Part (b)(ii):
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(\Rightarrow 8u^2 + 60u + 8v^2 - 2v + 71 = 0 \Rightarrow (u+\ldots)^2 + (v+\ldots)^2 = \ldots\)M1 Expands, gathers terms, completes the square
\(\left(u+\frac{15}{4}\right)^2 + \left(v - \frac{1}{8}\right)^2 = \frac{333}{64}\); Centre is \(\left(-\frac{15}{4}, \frac{1}{8}\right)\)A1 Accept as coordinates or complex number; allow \(\left(-\frac{15}{4}, \frac{1}{8}\text{i}\right)\)
Radius is \(\dfrac{3\sqrt{37}}{8}\)A1 
## Question 5:

### Part (a):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $w = \frac{1-3z}{z+2\text{i}} \Rightarrow w(z+2\text{i}) = 1-3z \Rightarrow z = \ldots$ or $(z+\text{i}) = \ldots$ | **M1** | Attempts to make $z$ the subject or extract $z+\text{i}$ as a term |
| $z = \frac{1-2\text{i}w}{w+3}$ or $(w+3)(z+\text{i}) = 1 - 2\text{i}w + (w+3)\text{i}\ (= 1+(3-w)\text{i})$ | **A1** | Correct expression for $z$ or correct equation with $z+\text{i}$ as only term in $z$ |
| $\left\|\frac{1-2\text{i}w}{w+3} + \text{i}\right\| = 3 \Rightarrow |1 - 2\text{i}w + \text{i}(w+3)| = 3|w+3|$ or $3|w+3| = |1+(3-w)\text{i}|$* | **M1, A1\*** | M1: Applies $|z+\text{i}|=3$; A1\*: Correctly completes to given result with no errors |

### Part (b)(i):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $w = u+\text{i}v \Rightarrow |(3-u)\text{i} + (v+1)| = 3|u+3+\text{i}v|$ | **M1** | Uses $w = u+\text{i}v$ in the given equation |
| $\Rightarrow (3-u)^2 + (v+1)^2 = 9[(u+3)^2 + v^2]$ | **M1, A1** | M1: Squares and applies Pythagoras; no $\text{i}$'s, sum of squares each side. A1: Correct Cartesian equation |

### Part (b)(ii):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $\Rightarrow 8u^2 + 60u + 8v^2 - 2v + 71 = 0 \Rightarrow (u+\ldots)^2 + (v+\ldots)^2 = \ldots$ | **M1** | Expands, gathers terms, completes the square |
| $\left(u+\frac{15}{4}\right)^2 + \left(v - \frac{1}{8}\right)^2 = \frac{333}{64}$; Centre is $\left(-\frac{15}{4}, \frac{1}{8}\right)$ | **A1** | Accept as coordinates or complex number; allow $\left(-\frac{15}{4}, \frac{1}{8}\text{i}\right)$ |
| Radius is $\dfrac{3\sqrt{37}}{8}$ | **A1** | |
\begin{enumerate}
  \item A transformation $T$ from the $z$-plane to the $w$-plane is given by
\end{enumerate}

$$w = \frac { 1 - 3 z } { z + 2 i } \quad z \neq - 2 i$$

The circle with equation $| z + \mathrm { i } | = 3$ is mapped by $T$ onto the circle $C$.\\
(a) Show that the equation for $C$ can be written as

$$3 | w + 3 | = | 1 + ( 3 - w ) \mathrm { i } |$$

(b) Hence find\\
(i) a Cartesian equation for $C$,\\
(ii) the centre and radius of $C$.

\hfill \mbox{\textit{Edexcel FP2 2020 Q5 [10]}}
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