AQA Further Paper 2 2023 June — Question 10 8 marks

Exam BoardAQA
ModuleFurther Paper 2 (Further Paper 2)
Year2023
SessionJune
Marks8
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicComplex Numbers Argand & Loci
TypeRegion shading with multiple inequalities
DifficultyChallenging +1.2 This is a standard Further Maths complex numbers loci question requiring sketching the intersection of a circle and angular sector, then finding maximum modulus. Part (a) is routine visualization; part (b) requires recognizing the maximum occurs at a boundary intersection and applying cosine rule or coordinate geometry, but follows a well-established method with no novel insight needed.
Spec4.02k Argand diagrams: geometric interpretation4.02p Set notation: for loci
The region \(R\) on an Argand diagram satisfies both \(|z + 2\text{i}| \leq 3\) and \(-\frac{\pi}{6} \leq \arg(z) \leq \frac{\pi}{2}\)
  1. Sketch \(R\) on the Argand diagram below. [3 marks] \includegraphics{figure_10a}
  2. Find the maximum value of \(|z|\) in the region \(R\), giving your answer in exact form. [5 marks]
Question 10:
AnswerMarks
10(a)Draws correct arc or circle,
intersecting the imaginary axis
AnswerMarks Guidance
at 1.1.1b B1
Draws correct half-line or line
π
at an angle between − and
4
AnswerMarks Guidance
0.1.1b B1
Shades or clearly labels
AnswerMarks Guidance
correct region.1.1b B1
Subtotal3
QMarking Instructions AO
AnswerMarks
10(b)Deduces that the maximum
value occurs where the half-
π
line argz =− and the circle
6
intersect.
AnswerMarks Guidance
PI2.2a M1
circle and half-line intersect.
x= − 3y
x2 +(y+2 )2 =9
3y2 + y2 +4y+4−9=0
4y2 +4y−5=0
=−1± 6
y
2 2
y=−1− 6
y <0 so
2 2
π
y = z sin
6
z =2 y
z =1+ 6
Selects a method to form a
AnswerMarks Guidance
quadratic equation in x, y or z3.1a M1
Forms a correct quadratic in x,
AnswerMarks Guidance
y or z2.2a A1
Obtains an expression for the
AnswerMarks Guidance
maximum value of z1.1a M1
Obtains the correct exact value
for the maximum value of z
AnswerMarks Guidance
ACF e.g. 7+2 61.1b A1
Subtotal5
Question total8
QMarking Instructions AO 
Question 10:
--- 10(a) ---
10(a) | Draws correct arc or circle,
intersecting the imaginary axis
at 1. | 1.1b | B1
Draws correct half-line or line
π
at an angle between − and
4
0. | 1.1b | B1
Shades or clearly labels
correct region. | 1.1b | B1
Subtotal | 3
Q | Marking Instructions | AO | Marks | Typical solution
--- 10(b) ---
10(b) | Deduces that the maximum
value occurs where the half-
π
line argz =− and the circle
6
intersect.
PI | 2.2a | M1 | Maximum value of z occurs where
circle and half-line intersect.
x= − 3y
x2 +(y+2 )2 =9
3y2 + y2 +4y+4−9=0
4y2 +4y−5=0
=−1± 6
y
2 2
y=−1− 6
y <0 so
2 2
π
y = z sin
6
z =2 y
z =1+ 6
Selects a method to form a
quadratic equation in x, y or z | 3.1a | M1
Forms a correct quadratic in x,
y or z | 2.2a | A1
Obtains an expression for the
maximum value of z | 1.1a | M1
Obtains the correct exact value
for the maximum value of z
ACF e.g. 7+2 6 | 1.1b | A1
Subtotal | 5
Question total | 8
Q | Marking Instructions | AO | Marks | Typical solution
The region $R$ on an Argand diagram satisfies both $|z + 2\text{i}| \leq 3$ and $-\frac{\pi}{6} \leq \arg(z) \leq \frac{\pi}{2}$

\begin{enumerate}[label=(\alph*)]
\item Sketch $R$ on the Argand diagram below.
[3 marks]

\includegraphics{figure_10a}

\item Find the maximum value of $|z|$ in the region $R$, giving your answer in exact form.
[5 marks]
\end{enumerate}

\hfill \mbox{\textit{AQA Further Paper 2 2023 Q10 [8]}}
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