AQA Further Paper 1 2024 June — Question 16 9 marks

Exam BoardAQA
ModuleFurther Paper 1 (Further Paper 1)
Year2024
SessionJune
Marks9
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicPolar coordinates
TypeArea of region with line boundary
DifficultyChallenging +1.8 This is a Further Maths polar coordinates question requiring: (a) straightforward triangle area calculation using r₁r₂sin(θ)/2, and (b) integration of ½r² with tan θ, requiring substitution and careful handling of bounds. The integration is non-trivial but follows standard Further Maths techniques. The 7-mark allocation and polar integration with tan θ place it well above average difficulty but not at the extreme end for Further Maths.
Spec4.09b Sketch polar curves: r = f(theta)4.09c Area enclosed: by polar curve
The curve \(C\) has polar equation \(r = 2 + \tan \theta\) The curve \(C\) meets the line \(\theta = \frac{\pi}{4}\) at the point \(A\) The point \(B\) has polar coordinates \((4, 0)\) The diagram shows part of the curve \(C\), and the points \(A\) and \(B\) \includegraphics{figure_16}
  1. Show that the area of triangle \(OAB\) is \(3\sqrt{2}\) units. [2 marks]
  2. Find the area of the shaded region. Give your answer in an exact form. [7 marks]
Question 16:
AnswerMarks
16(a)
Obtains 2 + t a n
AnswerMarks Guidance
42.2a M1
At A, r = 2 + tan = 3
4
Area of triangle OAB
1 π
= × 4 × 3 × sin
2 4
= 3 2
Completes a reasoned
argument to obtain
A r e a = 3 2
AnswerMarks Guidance
AG2.1 R1
Subtotal2
QMarking instructions AO
AnswerMarks
16(b)Writes down
1
 ( 2 + t a n θ ) 2 d θ
AnswerMarks Guidance
21.1a M1
π4
1
= + ( 2 θ t a n ) 2 d θ
2
0
π4
1
= + ( 4 θ 4 t a n + t a n 2 θ d ) θ
2
0
π4
1
= + ( 3 θ 4 t a n + s e c 2 θ d ) θ
2
0
1 π40
 
= θ + 3 4 ln ( s e θ c ) + θ t a n
2
1   3 
=  + 4 ln ( 2 ) + 1 – 0 
2 4
3  1
= + ln + 2
8 2
Shaded area
3  1
= 3 2 – – ln 2 –
8 2
Obtains a correct integral for
polar area.
1
(4+4tan θ+tan2 θ)dθ
AnswerMarks Guidance
21.1b A1
Replaces t a n 2 θ in the
integrand with s e c 2 θ − 1
AnswerMarks Guidance
Condone  s e c 2 θ  13.1a M1
Obtains
AnswerMarks Guidance
3 θ + 4 ln ( s e c θ ) + t a n θ1.1b A1
 3  
Obtains k + 2 ln 2 + 1 OE
AnswerMarks Guidance
41.1b A1
Obtains a numerical value for
the area of the shaded region
− 2 th 3 e ir a r e a o f
using
AnswerMarks Guidance
u n s h a d e d r e g io n2.4 M1
Completes a reasoned
argument to obtain
3 1
=3 2– –ln2–
AnswerMarks Guidance
8 22.1 R1
Subtotal7
Question total9
QMarking instructions AO 
Question 16:
--- 16(a) ---
16(a) | 
Obtains 2 + t a n
4 | 2.2a | M1 | π
At A, r = 2 + tan = 3
4
Area of triangle OAB
1 π
= × 4 × 3 × sin
2 4
= 3 2
Completes a reasoned
argument to obtain
A r e a = 3 2
AG | 2.1 | R1
Subtotal | 2
Q | Marking instructions | AO | Marks | Typical solution
--- 16(b) ---
16(b) | Writes down
1
 ( 2 + t a n θ ) 2 d θ
2 | 1.1a | M1 | Area enclosed by curve
π4
1
= + ( 2 θ t a n ) 2 d θ
2
0
π4
1
= + ( 4 θ 4 t a n + t a n 2 θ d ) θ
2
0
π4
1
= + ( 3 θ 4 t a n + s e c 2 θ d ) θ
2
0
1 π40
 
= θ + 3 4 ln ( s e θ c ) + θ t a n
2
1   3 
=  + 4 ln ( 2 ) + 1 – 0 
2 4
3  1
= + ln + 2
8 2
Shaded area
3  1
= 3 2 – – ln 2 –
8 2
Obtains a correct integral for
polar area.
1
(4+4tan θ+tan2 θ)dθ
2 | 1.1b | A1
Replaces t a n 2 θ in the
integrand with s e c 2 θ − 1
Condone  s e c 2 θ  1 | 3.1a | M1
Obtains
3 θ + 4 ln ( s e c θ ) + t a n θ | 1.1b | A1
 3  
Obtains k + 2 ln 2 + 1 OE
4 | 1.1b | A1
Obtains a numerical value for
the area of the shaded region
− 2 th 3 e ir a r e a o f
using
u n s h a d e d r e g io n | 2.4 | M1
Completes a reasoned
argument to obtain
3 1
=3 2– –ln2–
8 2 | 2.1 | R1
Subtotal | 7
Question total | 9
Q | Marking instructions | AO | Marks | Typical solution
The curve $C$ has polar equation $r = 2 + \tan \theta$

The curve $C$ meets the line $\theta = \frac{\pi}{4}$ at the point $A$

The point $B$ has polar coordinates $(4, 0)$

The diagram shows part of the curve $C$, and the points $A$ and $B$

\includegraphics{figure_16}

\begin{enumerate}[label=(\alph*)]
\item Show that the area of triangle $OAB$ is $3\sqrt{2}$ units. [2 marks]

\item Find the area of the shaded region.

Give your answer in an exact form. [7 marks]
\end{enumerate}

\hfill \mbox{\textit{AQA Further Paper 1 2024 Q16 [9]}}
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