AQA Further Paper 1 2022 June — Question 8 11 marks

Exam BoardAQA
ModuleFurther Paper 1 (Further Paper 1)
Year2022
SessionJune
Marks11
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicComplex Numbers Argand & Loci
TypeRegion shading with multiple inequalities
DifficultyStandard +0.8 This is a multi-part Further Maths loci question requiring geometric interpretation of complex number conditions, sketching regions, and optimization. Part (a) is routine (converting argument to Cartesian form). Part (b) requires understanding the intersection of an angular sector and a circular region. Part (c) involves finding the point in the region closest to the origin, requiring geometric insight about perpendicular distances. While systematic, it demands solid visualization skills and careful geometric reasoning across multiple steps, placing it moderately above average difficulty.
Spec4.02a Complex numbers: real/imaginary parts, modulus, argument4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines
  1. The complex number \(w\) is such that $$\arg(w + 2i) = \tan^{-1}\frac{1}{2}$$ It is given that \(w = x + iy\), where \(x\) and \(y\) are real and \(x > 0\) Find an equation for \(y\) in terms of \(x\) [2 marks]
  2. The complex number \(z\) satisfies both $$-\frac{\pi}{2} \leq \arg(z + 2i) \leq \tan^{-1}\frac{1}{2} \quad \text{and} \quad |z - 2 + 3i| \leq 2$$ The region \(R\) is the locus of \(z\) Sketch the region \(R\) on the Argand diagram below. [4 marks] \includegraphics{figure_1}
  3. \(z_1\) is the point in \(R\) at which \(|z|\) is minimum.
    1. Calculate the exact value of \(|z_1|\) [3 marks]
    2. Express \(z_1\) in the form \(a + ib\), where \(a\) and \(b\) are real. [2 marks]
Question 8:
AnswerMarks Guidance
8(a)Deduces correct gradient or
intercept PI2.2a M1
1
2
Line passes through (0, –2)
1
y= x−2
2
Obtains correct equation with
AnswerMarks Guidance
y as the subject1.1b A1
Total2
QMarking instructions AO
AnswerMarks
8(b)Draws a half line from –2i
passing through 4.
Condone full line
AnswerMarks Guidance
ft their linear equation in part (a)1.1b B1F
Draws circle or arc of a circle,
AnswerMarks Guidance
with centre at 2–3i or radius 22.2a M1
Draws circle or arc of a circle,
AnswerMarks Guidance
centre at 2–3i and radius 21.1b A1
Correct region indicated2.2a A1
Total4
QMarking instructions AO
AnswerMarks
8(c)(i)Identifies the point in their region
nearest to the origin.
For example draws the
perpendicular from the half-line
to the origin
or
AnswerMarks Guidance
Finds y = –2x3.1a B1F
sin𝛼𝛼 =
5
AnswerMarks
𝑧𝑧1
=sin𝛼𝛼
4
4 5
√
AnswerMarks
𝑧𝑧1=
5
Finds the distance between their
valid point and the origin
For example uses sin (tan–1 )
1
or
2
Finds the distance between the
origin and the point of
intersection of the lines
y = x –2 and y = –2x
AnswerMarks Guidance
13.1a M1
2
Obtains correct exact value of
z
AnswerMarks Guidance
11.1b A1
Total3
QMarking instructions AO
AnswerMarks Guidance
8(c)(ii)Uses their values from part (c)(i)
to obtain value of a or b1.1a M1
1 1
4 5 5 −4 5 2 5
= × = ×
5 5 5 5
4 −8
= =
5 5
4 8
z = − i
1 5 5
Obtains correct value of z
1
AnswerMarks Guidance
CSO1.1b A1
Total2
Question total11
QMarking instructions AO 
Question 8:
--- 8(a) ---
8(a) | Deduces correct gradient or
intercept PI | 2.2a | M1 | Gradient =
1
2
Line passes through (0, –2)
1
y= x−2
2
Obtains correct equation with
y as the subject | 1.1b | A1
Total | 2
Q | Marking instructions | AO | Marks | Typical solution
--- 8(b) ---
8(b) | Draws a half line from –2i
passing through 4.
Condone full line
ft their linear equation in part (a) | 1.1b | B1F
Draws circle or arc of a circle,
with centre at 2–3i or radius 2 | 2.2a | M1
Draws circle or arc of a circle,
centre at 2–3i and radius 2 | 1.1b | A1
Correct region indicated | 2.2a | A1
Total | 4
Q | Marking instructions | AO | Marks | Typical solution
--- 8(c)(i) ---
8(c)(i) | Identifies the point in their region
nearest to the origin.
For example draws the
perpendicular from the half-line
to the origin
or
Finds y = –2x | 3.1a | B1F | √5
sin𝛼𝛼 =
5
|𝑧𝑧1
|
=sin𝛼𝛼
4
4 5
√
|𝑧𝑧1 | =
5
Finds the distance between their
valid point and the origin
For example uses sin (tan–1 )
1
or
2
Finds the distance between the
origin and the point of
intersection of the lines
y = x –2 and y = –2x
1 | 3.1a | M1
2
Obtains correct exact value of
z
1 | 1.1b | A1
Total | 3
Q | Marking instructions | AO | Marks | Typical solution
--- 8(c)(ii) ---
8(c)(ii) | Uses their values from part (c)(i)
to obtain value of a or b | 1.1a | M1 | a= z sinα b=− z cosα
1 1
4 5 5 −4 5 2 5
= × = ×
5 5 5 5
4 −8
= =
5 5
4 8
z = − i
1 5 5
Obtains correct value of z
1
CSO | 1.1b | A1
Total | 2
Question total | 11
Q | Marking instructions | AO | Marks | Typical solution
\begin{enumerate}[label=(\alph*)]
\item The complex number $w$ is such that
$$\arg(w + 2i) = \tan^{-1}\frac{1}{2}$$
It is given that $w = x + iy$, where $x$ and $y$ are real and $x > 0$

Find an equation for $y$ in terms of $x$ [2 marks]

\item The complex number $z$ satisfies both
$$-\frac{\pi}{2} \leq \arg(z + 2i) \leq \tan^{-1}\frac{1}{2} \quad \text{and} \quad |z - 2 + 3i| \leq 2$$

The region $R$ is the locus of $z$

Sketch the region $R$ on the Argand diagram below. [4 marks]

\includegraphics{figure_1}

\item $z_1$ is the point in $R$ at which $|z|$ is minimum.

\begin{enumerate}[label=(\roman*)]
\item Calculate the exact value of $|z_1|$ [3 marks]

\item Express $z_1$ in the form $a + ib$, where $a$ and $b$ are real. [2 marks]
\end{enumerate}
\end{enumerate}

\hfill \mbox{\textit{AQA Further Paper 1 2022 Q8 [11]}}
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