Standard +0.8 This requires applying the quadratic formula with complex coefficients, then performing multiple complex arithmetic operations (division by complex numbers, simplification of nested complex expressions). While the method is standard, the algebraic manipulation is substantially more involved than typical A-level questions, placing it moderately above average difficulty.
6 Solve the quadratic equation \(( 1 - 3 \mathrm { i } ) z ^ { 2 } - ( 2 + \mathrm { i } ) z + \mathrm { i } = 0\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
SC M1: For substitution of \(x+iy\) and multiplying out
Use \(i^2 = -1\) throughout
M1
SC M1: Use \(i^2 = -1\) throughout
Obtain correct answer in any form
A1
SC A1: For two correct equations \(x^2 - y^2 + 6xy - 2x + y = 0\) and \(-3(x^2-y^2)+2xy-x-2y+1=0\)
Multiply numerator and denominator by \((1+3i)\), or equivalent
M1
Obtain final answer, e.g. \(-\dfrac{1}{2}+\dfrac{1}{2}i\)
A1
Obtain second final answer, e.g. \(\dfrac{2}{5}+\dfrac{1}{5}i\)
A1
## Question 6:
| Answer | Mark | Guidance |
|--------|------|----------|
| Use quadratic formula to solve for $z$ | M1 | **SC M1**: For substitution of $x+iy$ and multiplying out |
| Use $i^2 = -1$ throughout | M1 | **SC M1**: Use $i^2 = -1$ throughout |
| Obtain correct answer in any form | A1 | **SC A1**: For two correct equations $x^2 - y^2 + 6xy - 2x + y = 0$ and $-3(x^2-y^2)+2xy-x-2y+1=0$ |
| Multiply numerator and denominator by $(1+3i)$, or equivalent | M1 | |
| Obtain final answer, e.g. $-\dfrac{1}{2}+\dfrac{1}{2}i$ | A1 | |
| Obtain second final answer, e.g. $\dfrac{2}{5}+\dfrac{1}{5}i$ | A1 | |
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6 Solve the quadratic equation $( 1 - 3 \mathrm { i } ) z ^ { 2 } - ( 2 + \mathrm { i } ) z + \mathrm { i } = 0$, giving your answers in the form $x + \mathrm { i } y$, where $x$ and $y$ are real.\\
\hfill \mbox{\textit{CAIE P3 2022 Q6 [6]}}