CAIE FP1 2018 November — Question 3 8 marks

Exam BoardCAIE
ModuleFP1 (Further Pure Mathematics 1)
Year2018
SessionNovember
Marks8
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicPolar coordinates
TypeSketch polar curve
DifficultyStandard +0.8 This is a multi-part Further Maths polar coordinates question requiring sketching a rose curve, applying the polar area formula with integration of cos²(3θ) using double angle identities, and converting to Cartesian form using a triple angle identity. While systematic, it demands fluency with multiple techniques (polar curves, integration identities, coordinate conversion) and careful algebraic manipulation, placing it moderately above average difficulty.
Spec1.05l Double angle formulae: and compound angle formulae4.09a Polar coordinates: convert to/from cartesian4.09c Area enclosed: by polar curve
The curve \(C\) has polar equation \(r = a \cos 3\theta\), for \(-\frac{1}{6}\pi \leqslant \theta \leqslant \frac{1}{6}\pi\), where \(a\) is a positive constant.
  1. Sketch \(C\). [2]
  2. Find the area of the region enclosed by \(C\), showing full working. [3]
  3. Using the identity \(\cos 3\theta \equiv 4\cos^3 \theta - 3\cos \theta\), find a cartesian equation of \(C\). [3]
Question 3:
AnswerMarks Guidance
3(i)a B1
B1Coorrect position inclluding (a, 0) labellled
or iin table.
2
AnswerMarks Guidance
QuestionAnswer Marks
AnswerMarks
3(ii)π
1 6
∫a2cos23θ dθ
2
π
−
AnswerMarks Guidance
6M1 For using correct formula
π
a2 6
∫ ( cos6θ+1 ) dθ
4
π
−
AnswerMarks Guidance
6M1 Using double angle formula correctly
π
a2 1 6 πa2
= sin6θ+θ =
 
4 6  π 12
−
AnswerMarks
6A1
3
AnswerMarks
3(iii)( )  x  x 2 
r =acosθ 4cos2θ−3 ⇒r =a   4  −3
r   r  
AnswerMarks Guidance
 B1 Uses x=rcosθ and x2 + y2 =r2.
( ) ( )2 ( ( ))
AnswerMarks Guidance
⇒r4 =ax 4x2 −3r2 ⇒ x2 + y2 =ax 4x2 −3 x2 + y2M1 For eliminating θ
( )2 ( )
AnswerMarks Guidance
⇒ x2 + y2 =ax x2 −3y2A1 Any equivalent cartesian form without
fractions.
3
AnswerMarks Guidance
QuestionAnswer Marks 
Question 3:
--- 3(i) ---
3(i) | a | B1 | Jusst one loop, correcct shape at extremmities
B1 | Coorrect position inclluding (a, 0) labellled
or iin table.
2
Question | Answer | Marks | Guidance
--- 3(ii) ---
3(ii) | π
1 6
∫a2cos23θ dθ
2
π
−
6 | M1 | For using correct formula
π
a2 6
∫ ( cos6θ+1 ) dθ
4
π
−
6 | M1 | Using double angle formula correctly
π
a2 1 6 πa2
= sin6θ+θ =
 
4 6  π 12
−
6 | A1
3
--- 3(iii) ---
3(iii) | ( )  x  x 2 
r =acosθ 4cos2θ−3 ⇒r =a   4  −3
r   r  
  | B1 | Uses x=rcosθ and x2 + y2 =r2.
( ) ( )2 ( ( ))
⇒r4 =ax 4x2 −3r2 ⇒ x2 + y2 =ax 4x2 −3 x2 + y2 | M1 | For eliminating θ
( )2 ( )
⇒ x2 + y2 =ax x2 −3y2 | A1 | Any equivalent cartesian form without
fractions.
3
Question | Answer | Marks | Guidance
The curve $C$ has polar equation $r = a \cos 3\theta$, for $-\frac{1}{6}\pi \leqslant \theta \leqslant \frac{1}{6}\pi$, where $a$ is a positive constant.

\begin{enumerate}[label=(\roman*)]
\item Sketch $C$. [2]

\item Find the area of the region enclosed by $C$, showing full working. [3]

\item Using the identity $\cos 3\theta \equiv 4\cos^3 \theta - 3\cos \theta$, find a cartesian equation of $C$. [3]
\end{enumerate}

\hfill \mbox{\textit{CAIE FP1 2018 Q3 [8]}}
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