OCR MEI Further Pure Core 2020 November — Question 5 8 marks

Exam BoardOCR MEI
ModuleFurther Pure Core (Further Pure Core)
Year2020
SessionNovember
Marks8
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicPolar coordinates
TypeArea enclosed by polar curve
DifficultyStandard +0.3 This is a straightforward Further Maths polar area question requiring standard formula application. Part (a) involves simple substitution (θ=0, π), part (b) tests understanding of cosine symmetry, and part (c) uses the standard polar area integral ½∫r²dθ with a simple polynomial integrand. While it's Further Maths content, it's a routine textbook exercise with no novel problem-solving required.
Spec4.09c Area enclosed: by polar curve
5 Fig. 5 shows the curve with polar equation \(r = a ( 3 + 2 \cos \theta )\) for \(- \pi \leqslant \theta \leqslant \pi\), where \(a\) is a constant. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c2be8838-50ec-4e82-b203-4608ab56c110-3_607_718_351_244} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure}
  1. Write down the polar coordinates of the points A and B .
  2. Explain why the curve is symmetrical about the initial line.
  3. In this question you must show detailed reasoning. Find in terms of \(a\) the exact area of the region enclosed by the curve.
Question 5:
AnswerMarks Guidance
5(a) A is [5a, 0],
​ ​
B is [3a, ½π]
AnswerMarks
​ ​ ​ ​B1
B1
AnswerMarks
[2]1.1
1.1SC Coordinates reversed
(θ, r ) award B1B0
AnswerMarks Guidance
5(b) cos (−θ) = cos θ
​ ​ ​
so the value of r for −θ is the same as for θ
AnswerMarks
​​ ​ ​ ​M1
A1
AnswerMarks
[2]2.4
2.2aaccept even function
PPMMTT
Y420/01 Mark SchemeNovember 2020
5 (c) DR
B1 1.1 or limits seen in work
M1 3.1a substitute an expression
involving cos2θ for cos2θ​
​
A1 1.1
A1cao 1.1
[4]
6 a2 ​− b2 ​+ 2abi − 4i(a − ib) + 11 = 0 B1 3.1a substitution for z and z* soi
​ ​ ​ ​ ​ ​ ​ ​ ​
⇒ a2 ​− b2 ​− 4b + 11 = 0, 2ab − 4a = 0 M1 1.1 put Re and Im parts equal to 0
​ ​ ​ ​ ​ ​ ​ ​ ​ ​
⇒ b = 2 B1 1.1
​ ​
a2 ​= 1, a = 1 so z = 1 + 2i A1 3.2a
​ ​ ​ ​​
[4]
7
B1 1.1
When n = 1,
​ ​
Assume true for n = k so M1 1.1
​ ​ ​​
M1 2.1
then
Or target seen
A1* 2.1
B1dep* 2.2a
So if true for n = k then true for n = k+1
​ ​ ​​ ​ ​ ​​ B1 2.4 Final B mark only awarded if
As true for n = 1, true for all positive n
​ ​ ​ [6] all previous marks awarded
9
AnswerMarks Guidance
5(c) DR
M1
A1
A1cao
AnswerMarks
[4]1.1
3.1a
1.1
AnswerMarks
1.1or
substitute an expression
involving cos2θ for cos2θ​
AnswerMarks
​limits seen in work 
Question 5:
5 | (a) | A is [5a, 0],
​ ​
B is [3a, ½π]
​ ​ ​ ​ | B1
B1
[2] | 1.1
1.1 | SC Coordinates reversed
(θ, r ) award B1B0
5 | (b) | cos (−θ) = cos θ
​ ​ ​
so the value of r for −θ is the same as for θ
​​ ​ ​ ​ | M1
A1
[2] | 2.4
2.2a | accept even function
PPMMTT
Y420/01 Mark SchemeNovember 2020
5 (c) DR
B1 1.1 or limits seen in work
M1 3.1a substitute an expression
involving cos2θ for cos2θ​
​
A1 1.1
A1cao 1.1
[4]
6 a2 ​− b2 ​+ 2abi − 4i(a − ib) + 11 = 0 B1 3.1a substitution for z and z* soi
​ ​ ​ ​ ​ ​ ​ ​ ​
⇒ a2 ​− b2 ​− 4b + 11 = 0, 2ab − 4a = 0 M1 1.1 put Re and Im parts equal to 0
​ ​ ​ ​ ​ ​ ​ ​ ​ ​
⇒ b = 2 B1 1.1
​ ​
a2 ​= 1, a = 1 so z = 1 + 2i A1 3.2a
​ ​ ​ ​​
[4]
7
B1 1.1
When n = 1,
​ ​
Assume true for n = k so M1 1.1
​ ​ ​​
M1 2.1
then
Or target seen
A1* 2.1
B1dep* 2.2a
So if true for n = k then true for n = k+1
​ ​ ​​ ​ ​ ​​ B1 2.4 Final B mark only awarded if
As true for n = 1, true for all positive n
​ ​ ​ [6] all previous marks awarded
9
5 | (c) | DR | B1
M1
A1
A1cao
[4] | 1.1
3.1a
1.1
1.1 | or
substitute an expression
involving cos2θ for cos2θ​
​ | limits seen in work
5 Fig. 5 shows the curve with polar equation $r = a ( 3 + 2 \cos \theta )$ for $- \pi \leqslant \theta \leqslant \pi$, where $a$ is a constant.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{c2be8838-50ec-4e82-b203-4608ab56c110-3_607_718_351_244}
\captionsetup{labelformat=empty}
\caption{Fig. 5}
\end{center}
\end{figure}
\begin{enumerate}[label=(\alph*)]
\item Write down the polar coordinates of the points A and B .
\item Explain why the curve is symmetrical about the initial line.
\item In this question you must show detailed reasoning.

Find in terms of $a$ the exact area of the region enclosed by the curve.
\end{enumerate}

\hfill \mbox{\textit{OCR MEI Further Pure Core 2020 Q5 [8]}}
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