Question 1 - A-Level Maths

AQA FP1 2008 January — Question 1 4 marks

Exam Board	AQA
Module	FP1 (Further Pure Mathematics 1)
Year	2008
Session	January
Marks	4
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Complex Numbers Arithmetic
Type	Real and imaginary part expressions
Difficulty	Moderate -0.8 This is a straightforward Further Maths question requiring only basic complex number operations: finding a conjugate, performing scalar multiplication and addition, then equating real and imaginary parts. It's routine manipulation with no problem-solving insight needed, though being FP1 places it slightly above typical Core content.
Spec	4.02a Complex numbers: real/imaginary parts, modulus, argument 4.02e Arithmetic of complex numbers: add, subtract, multiply, divide

1 It is given that $z _ { 1 } = 2 + \mathrm { i }$ and that $z _ { 1 } { } ^ { * }$ is the complex conjugate of $z _ { 1 }$.
Find the real numbers $x$ and $y$ such that $$x + 3 \mathrm { i } y = z _ { 1 } + 4 \mathrm { i } z _ { 1 } *$$

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Question 1:

Answer	Marks	Guidance
Working/Answer	Marks	Guidance
$z_1 + 4i\,z_1^* = (2+i) + 4i(2-i)$	M1	Use of conjugate
$= (2+i) + (8i+4)$	M1	Use of $i^2 = -1$
$= 6 + 9i$, so $x = 6$ and $y = 3$	M1A1	M1 for equating real and imaginary parts
Total	4

## Question 1:

| Working/Answer | Marks | Guidance |
|---|---|---|
| $z_1 + 4i\,z_1^* = (2+i) + 4i(2-i)$ | M1 | Use of conjugate |
| $= (2+i) + (8i+4)$ | M1 | Use of $i^2 = -1$ |
| $= 6 + 9i$, so $x = 6$ and $y = 3$ | M1A1 | M1 for equating real and imaginary parts |
| **Total** | **4** | |

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1 It is given that $z _ { 1 } = 2 + \mathrm { i }$ and that $z _ { 1 } { } ^ { * }$ is the complex conjugate of $z _ { 1 }$.\\
Find the real numbers $x$ and $y$ such that

$$x + 3 \mathrm { i } y = z _ { 1 } + 4 \mathrm { i } z _ { 1 } *$$

\hfill \mbox{\textit{AQA FP1 2008 Q1 [4]}}

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Q1 4 Q2 5 Q3 5 Q4 7 Q5 8 Q6 10 Q7 12 Q8 12 Q9 12