| Exam Board | OCR MEI |
|---|---|
| Module | Further Pure Core (Further Pure Core) |
| Year | 2022 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Area enclosed by polar curve |
| Difficulty | Standard +0.3 This is a standard Further Maths polar coordinates question involving a cardioid. Part (a) requires sketching a well-known polar curve, and part (b) applies the standard polar area formula ½∫r²dθ with straightforward integration using double angle formulas. While polar coordinates is a Further Maths topic, this is a textbook exercise with no novel problem-solving required, making it slightly easier than average overall. |
| Spec | 4.09b Sketch polar curves: r = f(theta)4.09c Area enclosed: by polar curve |
| Answer | Marks | Guidance |
|---|---|---|
| 5 | (a) | O |
| Answer | Marks |
|---|---|
| [2] | 1.1 |
| 1.1 | symmetrical loop about the initial line |
| Answer | Marks | Guidance |
|---|---|---|
| 5 | (b) | A 2 1 a 2 (1 c o s ) 2 d = − |
| Answer | Marks |
|---|---|
| 2 | M1 |
| Answer | Marks |
|---|---|
| [5] | 1.1a |
| Answer | Marks |
|---|---|
| 1.1 | correct integral and limits, condone missing 𝑑𝜃 |
Question 5:
5 | (a) | O | M1
A1
[2] | 1.1
1.1 | symmetrical loop about the initial line
correct shape with cusp at O
5 | (b) | A 2 1 a 2 (1 c o s ) 2 d = −
2
0
1 a 2 2 ( 1 2 c o s 1 [1 c o s 2 ] ) d = − + +
2 2
0
2𝜋
= 1 𝑎2[3𝜃−4sin𝜃+ 1 sin2𝜃]
4 2 0
3
= 𝜋𝑎2
2 | M1
M1
M1
B1
A1cao
[5] | 1.1a
1.1
3.1a
1.1
1.1 | correct integral and limits, condone missing 𝑑𝜃
limits can be soi by later work
may see ∫ 𝜋 𝑎2(1−cos𝜃)2𝑑𝜃
0
Expanding correctly
substituting for cos2
1
𝑘[3𝜃−4sin𝜃+ sin2𝜃]
25
\begin{enumerate}[label=(\alph*)]
\item Sketch the polar curve $\mathrm { r } = \mathrm { a } ( 1 - \cos \theta ) , 0 \leqslant \theta < 2 \pi$, where $a$ is a positive constant.
\item Determine the exact area of the region enclosed by the curve.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Pure Core 2022 Q5 [7]}}