Question 6 - A-Level Maths

Edexcel FP1 2011 June — Question 6 7 marks

Exam Board	Edexcel
Module	FP1 (Further Pure Mathematics 1)
Year	2011
Session	June
Marks	7
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Complex Numbers Arithmetic
Type	Linear equations in z and z*
Difficulty	Moderate -0.5 This is a straightforward algebraic manipulation question requiring substitution of z = x + iy and z* = x - iy, then equating real and imaginary parts to solve simultaneous equations. While it's a Further Maths topic, the technique is mechanical and requires no insight beyond standard complex number manipulation.
Spec	4.02a Complex numbers: real/imaginary parts, modulus, argument

6. Given that $z = x + \mathrm { i } y$, find the value of $x$ and the value of $y$ such that $$z + 3 \mathrm { i } z ^ { * } = - 1 + 13 \mathrm { i }$$ where $z ^ { * }$ is the complex conjugate of $z$.

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Question 6:

Answer	Marks	Guidance
Answer/Working	Marks	Guidance
$z^* = x - iy$	B1
$(x + iy) + 3i(x - iy)$	M1	Substituting $z = x+iy$ and their $z^$ into $z + 3iz^$
$x + iy + 3ix + 3y = -1 + 13i$	A1	Correct equation in $x$ and $y$ with $i^2 = -1$. Can be implied
Re part: $x + 3y = -1$	M1	Attempt to equate real and imaginary parts
Im part: $y + 3x = 13$	A1	Correct equations
$8y = -16 \Rightarrow y = -2$	M1	Attempt to solve simultaneous equations. At least one equation must contain both $x$ and $y$ terms
$x = 5,\ y = -2$	A1	Both $x=5$ and $y=-2$
$z = 5 - 2i$	—

(7 marks)

## Question 6:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $z^* = x - iy$ | B1 | |
| $(x + iy) + 3i(x - iy)$ | M1 | Substituting $z = x+iy$ and their $z^*$ into $z + 3iz^*$ |
| $x + iy + 3ix + 3y = -1 + 13i$ | A1 | Correct equation in $x$ and $y$ with $i^2 = -1$. Can be implied |
| Re part: $x + 3y = -1$ | M1 | Attempt to equate real **and** imaginary parts |
| Im part: $y + 3x = 13$ | A1 | Correct equations |
| $8y = -16 \Rightarrow y = -2$ | M1 | Attempt to solve simultaneous equations. At least one equation must contain both $x$ and $y$ terms |
| $x = 5,\ y = -2$ | A1 | Both $x=5$ and $y=-2$ |
| $z = 5 - 2i$ | — | |

**(7 marks)**

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6. Given that $z = x + \mathrm { i } y$, find the value of $x$ and the value of $y$ such that

$$z + 3 \mathrm { i } z ^ { * } = - 1 + 13 \mathrm { i }$$

where $z ^ { * }$ is the complex conjugate of $z$.\\

\hfill \mbox{\textit{Edexcel FP1 2011 Q6 [7]}}

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