| Exam Board | OCR |
|---|---|
| Module | Further Pure Core 1 (Further Pure Core 1) |
| Year | 2021 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Area enclosed by polar curve |
| Difficulty | Standard +0.8 This is a Further Maths FP1 polar coordinates question requiring multiple techniques: finding where r=0, maximizing r, computing area using the polar area integral (∫½r²dθ), and converting to Cartesian form using a triple angle identity. While the individual steps are methodical, the polar area integration and coordinate conversion require solid understanding beyond standard A-level, placing it moderately above average difficulty. |
| Spec | 4.09b Sketch polar curves: r = f(theta)4.09c Area enclosed: by polar curve |
| Answer | Marks | Guidance |
|---|---|---|
| 7 | (a) | r =0⇒sin3θ=0 π |
| Answer | Marks | Guidance |
|---|---|---|
| ⇒3θ=0,π 3 | B1 | 1.1 |
| Answer | Marks |
|---|---|
| (b) | 3π π π |
| Answer | Marks |
|---|---|
| 6 6 6 | B1 |
| B1 | 1.1 |
| 1.1 | For r |
| Answer | Marks |
|---|---|
| (c) | DR |
| Answer | Marks |
|---|---|
| 4 3 12 | M1 |
| Answer | Marks |
|---|---|
| A1 | 1.1 |
| Answer | Marks |
|---|---|
| 1.1 | Correct use of formula – ignore limits |
| Answer | Marks |
|---|---|
| (d) | sin3θ=3sinθ−4sin3θ |
| Answer | Marks |
|---|---|
| e.g. x2 + y2 =3x2y− y3 | M1 |
| A1 | 1.1 |
| 1.1 | Using triple angle formula and y =rsinθ |
Question 7:
7 | (a) | r =0⇒sin3θ=0 π
⇒θ=0,
⇒3θ=0,π 3 | B1 | 1.1 | Both required
Don’t give if any extras within range.
Ignore values outside range
[1]
(b) | 3π π π
sin , i.e. 1,
6 6 6 | B1
B1 | 1.1
1.1 | For r
For θ
[2]
(c) | DR
π π
1 3 1 3
Area = ∫ r2dθ= ∫ sin23θdθ
2 2
0 0
π
1 3
= ∫ ( 1−cos6θ) dθ
4
0
π
1 1 3
= θ− sin6θ
4 6
0
1π
= −0 = 1π
4 3 12 | M1
M1*
DepM1
A1 | 1.1
3.1a
1.1
1.1 | Correct use of formula – ignore limits
Attempt to use double angle formula
(Could be wrong way round, 2 missing or sign wrong)
Integrate their integrand
Use correct limits, must be seen
[4]
(d) | sin3θ=3sinθ−4sin3θ
3
3y y
⇒r = −4
r r
⇒r4 =3r2y−4y3
( )2 ( )
x2 + y2 =3y x2 + y2 −4y3 oe
( )2
e.g. x2 + y2 =3x2y− y3 | M1
A1 | 1.1
1.1 | Using triple angle formula and y =rsinθ
isw
[2]The diagram below shows the curve with polar equation $r = \sin 3\theta$ for $0 \leqslant \theta \leqslant \frac{1}{3}\pi$.
\includegraphics{figure_7}
\begin{enumerate}[label=(\alph*)]
\item Find the values of $\theta$ at the pole. [1]
\item Find the polar coordinates of the point on the curve where $r$ takes its maximum value. [2]
\item \textbf{In this question you must show detailed reasoning.}
Find the exact area enclosed by the curve. [4]
\item Given that $\sin 3\theta = 3 \sin \theta - 4 \sin^3 \theta$, find a cartesian equation for the curve. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core 1 2021 Q7 [9]}}