| Exam Board | OCR MEI |
|---|---|
| Module | Further Pure Core (Further Pure Core) |
| Year | 2024 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Area enclosed by polar curve |
| Difficulty | Challenging +1.2 This is a standard Further Maths polar curves question requiring a sketch of r = a sin 3θ and finding the area of one loop using the polar area formula. While it requires understanding of when r is negative and setting up the correct integral limits (0 to π/3 for one loop), the integration itself is routine using the double angle formula. The 'show detailed reasoning' requirement and multi-step nature elevate it slightly above average difficulty, but it follows a well-practiced template for polar area questions. |
| Spec | 4.09b Sketch polar curves: r = f(theta)4.09c Area enclosed: by polar curve |
| Answer | Marks | Guidance |
|---|---|---|
| 9 | (a) | B1* |
| Answer | Marks |
|---|---|
| [3] | 1.1 |
| Answer | Marks |
|---|---|
| 1.1 | One loop in correct position (requires initial line drawn) |
| Answer | Marks | Guidance |
|---|---|---|
| 9 | (b) | DR |
| Answer | Marks |
|---|---|
| 2 | M1 |
| Answer | Marks |
|---|---|
| [5] | 1.1 |
| Answer | Marks |
|---|---|
| 1.1 | A 12 a 2 s i n 2 3 d = . Condone missing d𝜃. |
Question 9:
9 | (a) | B1*
B1*
B1dep
[3] | 1.1
1.1
1.1 | One loop in correct position (requires initial line drawn)
Exactly three loops in correct position
Lower loop only shown with a broken line. Any
coordinates on the curve must be correct (either polar or
cartesian)
9 | (b) | DR
13
A 12 a 2 s i n 2 3 d =
0
1
A=3 1a 2 (1−cos6)d
0 4
13
14 a 2 16 s i n 6 = −
0
11 a 2 =
2 | M1
A1
M1
A1
A1
[5] | 1.1
1.1
3.1a
1.1
1.1 | A 12 a 2 s i n 2 3 d = . Condone missing d𝜃.
Limits correct. Accept alternative limits between 0 and 𝜋
provided correct multiplication or division of integral is
seen or implied at some stage.
Double angle formula used correctly in their integral.
Condone missing d𝜃.
1
𝑘[𝜃− sin6𝜃]. Condone incorrect or missing limits
6
SC B4 for an otherwise fully correct answer using limits
outside the range 0 to 𝜋.9 A curve has polar equation $r = \operatorname { asin } 3 \theta$, for $0 \leqslant \theta \leqslant \pi$, where $a$ is a positive constant.
\begin{enumerate}[label=(\alph*)]
\item Sketch the curve. Indicate the parts of the curve where $r$ is negative by using a broken line.
\item In this question you must show detailed reasoning.
Determine the area of one of the loops of the curve.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Pure Core 2024 Q9 [8]}}