Moderate -0.5 This is a straightforward algebraic manipulation question requiring cross-multiplication, expanding brackets with complex numbers, and collecting real/imaginary parts. While it involves complex arithmetic, it's a routine procedural exercise with no conceptual difficulty or problem-solving insight required, making it slightly easier than average.
4 Find the complex number \(z\) satisfying the equation
$$\frac { z - 3 \mathrm { i } } { z + 3 \mathrm { i } } = \frac { 2 - 9 \mathrm { i } } { 5 }$$
Give your answer in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
Substitute \(z = x + iy\) and obtain a horizontal equation; do not allow if \(xy\) terms do not cancel
*M1
E.g. \(5(x+(y-3)i) = (2-9i)(x+(y+3)i)\)
Use \(i^2 = -1\) anywhere
M1
Obtain e.g. \(5x + 5(y-3)i = (2x+9y+27) + i(2y+6-9x)\) or \(3x^2+3y^2-12y-63+(9x^2+9y^2-30x+54y+81)i = 0\)
A1
Or equivalent expression free of products of complex numbers; terms can be in any order
Obtain simultaneous equations by equating real and imaginary parts
DM1
E.g. \(3x-9y=27\) and \(3y+9x=21\); or \(3x^2+3y^2-12y-63=0\) and \(9x^2+9y^2-30x+54y+81=0\)
Obtain \(z = 3-2i\) only
A1
Alternative Method: Obtain horizontal equation in \(z\); do not allow if \(z^2\) terms with \(xy\) do not cancel
*M1
E.g. \(5z-15i = 2z+6i-27i^2-9iz\)
Use \(i^2=-1\) anywhere
M1
Obtain \(z = \dfrac{9+7i}{1+3i}\)
A1
OE (might have uncancelled factor of 3)
Multiply top and bottom by \(1-3i\) or equivalent for their \(z\)
DM1
Must see working for numerator or denominator, e.g. \(9-27i+21+7i\) or \(1+9\) or 10
Obtain \(z = 3-2i\) only
A1
Total
5
## Question 4:
| Answer | Marks | Guidance |
|--------|-------|----------|
| Substitute $z = x + iy$ and obtain a horizontal equation; do not allow if $xy$ terms do not cancel | *M1 | E.g. $5(x+(y-3)i) = (2-9i)(x+(y+3)i)$ |
| Use $i^2 = -1$ anywhere | M1 | |
| Obtain e.g. $5x + 5(y-3)i = (2x+9y+27) + i(2y+6-9x)$ or $3x^2+3y^2-12y-63+(9x^2+9y^2-30x+54y+81)i = 0$ | A1 | Or equivalent expression free of products of complex numbers; terms can be in any order |
| Obtain simultaneous equations by equating real and imaginary parts | DM1 | E.g. $3x-9y=27$ and $3y+9x=21$; or $3x^2+3y^2-12y-63=0$ and $9x^2+9y^2-30x+54y+81=0$ |
| Obtain $z = 3-2i$ only | A1 | |
| **Alternative Method:** Obtain horizontal equation in $z$; do not allow if $z^2$ terms with $xy$ do not cancel | *M1 | E.g. $5z-15i = 2z+6i-27i^2-9iz$ |
| Use $i^2=-1$ anywhere | M1 | |
| Obtain $z = \dfrac{9+7i}{1+3i}$ | A1 | OE (might have uncancelled factor of 3) |
| Multiply top and bottom by $1-3i$ or equivalent for their $z$ | DM1 | Must see working for numerator or denominator, e.g. $9-27i+21+7i$ or $1+9$ or 10 |
| Obtain $z = 3-2i$ only | A1 | |
| **Total** | **5** | |
---
4 Find the complex number $z$ satisfying the equation
$$\frac { z - 3 \mathrm { i } } { z + 3 \mathrm { i } } = \frac { 2 - 9 \mathrm { i } } { 5 }$$
Give your answer in the form $x + \mathrm { i } y$, where $x$ and $y$ are real.\\
\hfill \mbox{\textit{CAIE P3 2024 Q4 [5]}}