| Exam Board | OCR MEI |
|---|---|
| Module | Further Pure Core (Further Pure Core) |
| Session | Specimen |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Area enclosed by polar curve |
| Difficulty | Challenging +1.3 This is a Further Maths polar curves question requiring knowledge of the polar area formula and integration of sin²(3θ). While the topic is advanced, the execution is relatively standard: sketch a three-petaled rose (restricted domain shows one petal), identify loop bounds (0 to π/3), apply ½∫r²dθ, and integrate using double-angle formula. The 5-mark allocation and 'show detailed reasoning' indicate moderate algebraic work, but no novel insight is required—it's a textbook polar area application that's harder than typical A-level due to the Further Maths content but straightforward within that context. |
| Spec | 4.09b Sketch polar curves: r = f(theta)4.09c Area enclosed: by polar curve |
| Answer | Marks | Guidance |
|---|---|---|
| 9 | (i) | B2 |
| [2] | 1.1 | |
| 2.5 | B1 for loop in first quadrant |
| Answer | Marks | Guidance |
|---|---|---|
| 9 | (ii) | DR |
| Answer | Marks |
|---|---|
| 12 | M1* |
| Answer | Marks |
|---|---|
| [5] | m2.1 |
| Answer | Marks |
|---|---|
| 1.1 | n |
Question 9:
9 | (i) | B2
[2] | 1.1
2.5 | B1 for loop in first quadrant
B1 for loop in second
quadrant with indication (e.g.
dotted line) of r negative
9 | (ii) | DR
(cid:83)
1
Area is (cid:179)3a2sin23(cid:84)d(cid:84)
2 0
Use cos6(cid:84)(cid:32)1(cid:16)2sin23(cid:84) to obtain
(cid:83)
1
(cid:179)3a2(cid:11)1(cid:16)cos6(cid:84)(cid:12)d(cid:84)
4 0
(cid:83) e
a2 (cid:170) 1 (cid:186)3
(cid:32) (cid:84)(cid:16) sin6(cid:84)
(cid:171) (cid:187)
4 (cid:172) 6 (cid:188)
0 p
(cid:83)a2
(cid:32)
12 | M1*
M1*
A1
c
A1 FT
A1
[5] | m2.1
3.1a
i
1.1
1.1
1.1 | n
e
Must be seen
Must be seen
Must be seen
dep*A curve has polar equation $r = a \sin 3\theta$ for $-\frac{1}{4}\pi \leq \theta \leq \frac{1}{4}\pi$, where $a$ is a positive constant.
\begin{enumerate}[label=(\roman*)]
\item Sketch the curve. [2]
\item In this question you must show detailed reasoning.
Find, in terms of $a$ and $\pi$, the area enclosed by one of the loops of the curve. [5]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Pure Core Q9 [7]}}