AQA Further Paper 1 2022 June — Question 9 14 marks

Exam BoardAQA
ModuleFurther Paper 1 (Further Paper 1)
Year2022
SessionJune
Marks14
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicPolar coordinates
TypeArea enclosed by polar curve
DifficultyChallenging +1.8 This is a substantial Further Maths question requiring multiple polar coordinate techniques: sketching a lemniscate, identifying Roberto's error (wrong limits and using rΒ² directly), computing area correctly with proper limits, finding maxima, and applying matrix transformations to polar coordinates. The final part requires understanding that shear transformations preserve area (det M = 1). While each individual part is accessible, the combination of polar curves, calculus, and linear transformations with justification makes this significantly above average difficulty.
Spec4.03h Determinant 2x2: calculation4.03i Determinant: area scale factor and orientation4.09a Polar coordinates: convert to/from cartesian4.09b Sketch polar curves: r = f(theta)4.09c Area enclosed: by polar curve
Roberto is solving this mathematics problem:
The curve \(C_1\) has polar equation
\(r^2 = 9\sin 2\theta\)
for all possible values of \(\theta\)
Find the area enclosed by \(C_1\)
Roberto's solution is as follows:
\(A = \frac{1}{2}\int_{-\pi}^{\pi} 9\sin 2\theta \, d\theta\)
\(= \left[-\frac{9}{4}\cos 2\theta\right]_{-\pi}^{\pi}\)
\(= 0\)
  1. Sketch the curve \(C_1\) [2 marks]
  2. Explain what Roberto has done wrong. [2 marks]
  3. Find the area enclosed by \(C_1\) [2 marks]
  4. \(P\) and \(Q\) are distinct points on \(C_1\) for which \(r\) is a maximum. \(P\) is above the initial line. Find the polar coordinates of \(P\) and \(Q\) [2 marks]
  5. The matrix \(\mathbf{M} = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}\) represents the transformation T T maps \(C_1\) onto a curve \(C_2\)
    1. T maps \(P\) onto the point \(P'\) Find the polar coordinates of \(P'\) [4 marks]
    2. Find the area enclosed by \(C_2\) Fully justify your answer. [2 marks]
Question 9:
AnswerMarks Guidance
9(a)Draws at least one loop in the
correct place1.1b B1
Draws both loops correctly
(approx equal size) and no
AnswerMarks Guidance
others1.1b B1
Total2
QMarking instructions AO
AnswerMarks
9(b)Criticises limits of integration
used by Roberto
AnswerMarks Guidance
PI2.3 M1
integration.
He should only have included
values of which make
positive, because must be
positive πœƒπœƒ 2
sin2πœƒπœƒ π‘Ÿπ‘Ÿ
Explains what is wrong with
Roberto’s range of values,
AnswerMarks Guidance
including reference to r2 β‰₯02.4 R1
Total2
QMarking instructions AO
AnswerMarks
9(c)Forms an expression for an area
using valid limits
AnswerMarks Guidance
PI by 9/4, 9/2 or 92.2a M1
βˆ’
2
1
𝐴𝐴 = οΏ½ 9sin2πœƒπœƒπ‘‘π‘‘πœƒπœƒ
2
βˆ’πœ‹πœ‹
πœ‹πœ‹
2
1
+ οΏ½9sin2πœƒπœƒπ‘‘π‘‘πœƒπœƒ
2
0
πœ‹πœ‹
2
by symme=tryοΏ½ 9sin2πœƒπœƒπ‘‘π‘‘πœƒπœƒ
0
πœ‹πœ‹
2
9
𝐴𝐴 = οΏ½βˆ’ cos2πœƒπœƒοΏ½
2 0
9
=βˆ’ (cosπœ‹πœ‹βˆ’cos0)
2
AnswerMarks Guidance
Obtains correct answer1.1b A1
Total2 = 9
QMarking instructions AO
AnswerMarks Guidance
9(d)Deduces at least one correct
value of2.2a M1
For r maximum,
π‘Ÿπ‘Ÿ = 3
sin2πœƒπœƒ = 1
πœ‹πœ‹ 3πœ‹πœ‹
2πœƒπœƒ = ,βˆ’
2 2
πœ‹πœ‹ 3πœ‹πœ‹
P and πœƒπœƒQ= ,βˆ’
πœ‹πœ‹ 4 3πœ‹πœ‹ 4
οΏ½3,4οΏ½ οΏ½3,βˆ’ 4οΏ½
πœƒπœƒ
Obtains both correct solutions
(not
Condone r and transposed.
π‘Ÿπ‘Ÿ =5βˆ’Ο€3)
accept etc
πœƒπœƒ
4
OE decimals to 3sig fig or
AnswerMarks Guidance
better1.1b A1
Total2
QMarking instructions AO
AnswerMarks Guidance
9(e)(i)Finds cartesian coordinates or
position vector of their P, ACF1.1b B1F
3√2 3√2
οΏ½ 2 , 2 οΏ½
3√2 9√2
⎑ ⎀ ⎑ ⎀
1 2 ⎒ 2 βŽ₯ ⎒ 2 βŽ₯
οΏ½ οΏ½ =
⎒ βŽ₯ ⎒ βŽ₯
0 1 3√2 3√2
For P’, ⎒ βŽ₯ ⎒ βŽ₯
2 81 ⎣ 29 ⎦ ⎣ 2 ⎦
π‘Ÿπ‘Ÿ = 2 +2 = 45β‡’π‘Ÿπ‘Ÿ = 3 √5
1 βˆ’1 1
P’ tanπœƒπœƒ = β‡’ πœƒπœƒ = tan οΏ½ οΏ½
3 3
βˆ’1 1
�3√5,tan �3��
Multiplies their cartesian position
vector by matrix M, must be Mv,
AnswerMarks Guidance
to obtain an image3.1a M1
Obtains their correct value for r
AnswerMarks Guidance
or , ACF, for their image1.1a M1
πœƒπœƒ
Obtains their correct answer,
either exact or to at least 2 sig
fig AWRT (6.7, 0.32)
AnswerMarks Guidance
Must have M1M11.1b A1F
Total4
QMarking instructions AO
AnswerMarks Guidance
9(e)(ii)Explains correctly why the area
is unchanged2.4 E1
Area enclosed by C2 = (Area
enclosed by C1) Γ— det M
= 9 Γ— 1 = 9
Obtains their correct area from
AnswerMarks Guidance
their part (c) and their det2.2a B1F
Total2
Question total14
QMarking instructions AO 
Question 9:
--- 9(a) ---
9(a) | Draws at least one loop in the
correct place | 1.1b | B1
Draws both loops correctly
(approx equal size) and no
others | 1.1b | B1
Total | 2
Q | Marking instructions | AO | Marks | Typical solution
--- 9(b) ---
9(b) | Criticises limits of integration
used by Roberto
PI | 2.3 | M1 | Roberto has used incorrect limits of
integration.
He should only have included
values of which make
positive, because must be
positive πœƒπœƒ 2
sin2πœƒπœƒ π‘Ÿπ‘Ÿ
Explains what is wrong with
Roberto’s range of values,
including reference to r2 β‰₯0 | 2.4 | R1
Total | 2
Q | Marking instructions | AO | Marks | Typical solution
--- 9(c) ---
9(c) | Forms an expression for an area
using valid limits
PI by 9/4, 9/2 or 9 | 2.2a | M1 | πœ‹πœ‹
βˆ’
2
1
𝐴𝐴 = οΏ½ 9sin2πœƒπœƒπ‘‘π‘‘πœƒπœƒ
2
βˆ’πœ‹πœ‹
πœ‹πœ‹
2
1
+ οΏ½9sin2πœƒπœƒπ‘‘π‘‘πœƒπœƒ
2
0
πœ‹πœ‹
2
by symme=tryοΏ½ 9sin2πœƒπœƒπ‘‘π‘‘πœƒπœƒ
0
πœ‹πœ‹
2
9
𝐴𝐴 = οΏ½βˆ’ cos2πœƒπœƒοΏ½
2 0
9
=βˆ’ (cosπœ‹πœ‹βˆ’cos0)
2
Obtains correct answer | 1.1b | A1
Total | 2 | = 9
Q | Marking instructions | AO | Marks | Typical solution
--- 9(d) ---
9(d) | Deduces at least one correct
value of | 2.2a | M1 | Max. value of
For r maximum,
π‘Ÿπ‘Ÿ = 3
sin2πœƒπœƒ = 1
πœ‹πœ‹ 3πœ‹πœ‹
2πœƒπœƒ = ,βˆ’
2 2
πœ‹πœ‹ 3πœ‹πœ‹
P and πœƒπœƒQ= ,βˆ’
πœ‹πœ‹ 4 3πœ‹πœ‹ 4
οΏ½3,4οΏ½ οΏ½3,βˆ’ 4οΏ½
πœƒπœƒ
Obtains both correct solutions
(not
Condone r and transposed.
π‘Ÿπ‘Ÿ =5βˆ’Ο€3)
accept etc
πœƒπœƒ
4
OE decimals to 3sig fig or
better | 1.1b | A1
Total | 2
Q | Marking instructions | AO | Marks | Typical solution
--- 9(e)(i) ---
9(e)(i) | Finds cartesian coordinates or
position vector of their P, ACF | 1.1b | B1F | Cartesian coordinates of P are
3√2 3√2
οΏ½ 2 , 2 οΏ½
3√2 9√2
⎑ ⎀ ⎑ ⎀
1 2 ⎒ 2 βŽ₯ ⎒ 2 βŽ₯
οΏ½ οΏ½ =
⎒ βŽ₯ ⎒ βŽ₯
0 1 3√2 3√2
For P’, ⎒ βŽ₯ ⎒ βŽ₯
2 81 ⎣ 29 ⎦ ⎣ 2 ⎦
π‘Ÿπ‘Ÿ = 2 +2 = 45β‡’π‘Ÿπ‘Ÿ = 3 √5
1 βˆ’1 1
P’ tanπœƒπœƒ = β‡’ πœƒπœƒ = tan οΏ½ οΏ½
3 3
βˆ’1 1
�3√5,tan �3��
Multiplies their cartesian position
vector by matrix M, must be Mv,
to obtain an image | 3.1a | M1
Obtains their correct value for r
or , ACF, for their image | 1.1a | M1
πœƒπœƒ
Obtains their correct answer,
either exact or to at least 2 sig
fig AWRT (6.7, 0.32)
Must have M1M1 | 1.1b | A1F
Total | 4
Q | Marking instructions | AO | Marks | Typical solution
--- 9(e)(ii) ---
9(e)(ii) | Explains correctly why the area
is unchanged | 2.4 | E1 | Det M = 1
Area enclosed by C2 = (Area
enclosed by C1) Γ— det M
= 9 Γ— 1 = 9
Obtains their correct area from
their part (c) and their det | 2.2a | B1F
Total | 2
Question total | 14
Q | Marking instructions | AO | Marks | Typical solution
Roberto is solving this mathematics problem:

\begin{center}
\begin{tabular}{|l|}
\hline
The curve $C_1$ has polar equation \\
$r^2 = 9\sin 2\theta$ \\
for all possible values of $\theta$ \\
Find the area enclosed by $C_1$ \\
\hline
\end{tabular}
\end{center}

Roberto's solution is as follows:

\begin{center}
\begin{tabular}{|l|}
\hline
$A = \frac{1}{2}\int_{-\pi}^{\pi} 9\sin 2\theta \, d\theta$ \\
\\
$= \left[-\frac{9}{4}\cos 2\theta\right]_{-\pi}^{\pi}$ \\
\\
$= 0$ \\
\hline
\end{tabular}
\end{center}

\begin{enumerate}[label=(\alph*)]
\item Sketch the curve $C_1$ [2 marks]

\item Explain what Roberto has done wrong. [2 marks]

\item Find the area enclosed by $C_1$ [2 marks]

\item $P$ and $Q$ are distinct points on $C_1$ for which $r$ is a maximum.
$P$ is above the initial line.

Find the polar coordinates of $P$ and $Q$ [2 marks]

\item The matrix $\mathbf{M} = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}$ represents the transformation T

T maps $C_1$ onto a curve $C_2$

\begin{enumerate}[label=(\roman*)]
\item T maps $P$ onto the point $P'$

Find the polar coordinates of $P'$ [4 marks]

\item Find the area enclosed by $C_2$

Fully justify your answer. [2 marks]
\end{enumerate}
\end{enumerate}

\hfill \mbox{\textit{AQA Further Paper 1 2022 Q9 [14]}}
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