| Exam Board | OCR |
|---|---|
| Module | Further Pure Core 1 (Further Pure Core 1) |
| Year | 2020 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Convert Cartesian to polar equation |
| Difficulty | Standard +0.8 This is a multi-part Further Maths question requiring conversion to polar form (non-trivial due to cubic terms), algebraic manipulation with trigonometric identities, symmetry analysis, and curve sketching insight. While systematic, it demands more sophistication than standard A-level questions and tests understanding beyond routine procedures. |
| Spec | 4.09a Polar coordinates: convert to/from cartesian4.09b Sketch polar curves: r = f(theta) |
| Answer | Marks | Guidance |
|---|---|---|
| 11 | (a) | x=rcosθ,y=rsinθ⇒(rcosθ)3+(rsinθ)3 =2rcosθ.rsinθ |
| Answer | Marks |
|---|---|
| cos3θ+sin3θ | M1 |
| A1 | 3.1a |
| 1.1 | Substitution |
| Answer | Marks |
|---|---|
| (b) | 1 1 |
| Answer | Marks |
|---|---|
| sin3θ+cos3θ | M1 |
| A1 | 1.1a |
| 1.1 | Correct substitution into their f(θ) |
| Answer | Marks |
|---|---|
| (c) | π |
| Answer | Marks | Guidance |
|---|---|---|
| 4 | B1 | 2.2a |
| Answer | Marks | Guidance |
|---|---|---|
| (d) | r =f(1π)= 2 | |
| 4 | B1 | 1.1 |
| Answer | Marks |
|---|---|
| (e) | r =0 when θ=0. |
| Answer | Marks |
|---|---|
| So there is a loop | B1 |
| B1 | 3.1a |
| 2.4 | For both, ignore extras. |
Question 11:
11 | (a) | x=rcosθ,y=rsinθ⇒(rcosθ)3+(rsinθ)3 =2rcosθ.rsinθ
⇒r ( cos3θ+sin3θ ) =2cosθsinθ
2cosθsinθ
⇒r= oe
cos3θ+sin3θ | M1
A1 | 3.1a
1.1 | Substitution
May see “or r = 0” but not required.
[2]
(b) | 1 1
2cos π−θ sin π−θ
1 2 2
f( π−θ)=
2 1 1
cos3 π−θ +sin3 π−θ
2 2
2sinθcosθ
=
sin3θ+cos3θ | M1
A1 | 1.1a
1.1 | Correct substitution into their f(θ)
[2]
(c) | π
So the line of symmetry is θ=
4 | B1 | 2.2a | Allow y = x.
Must have θ =
[1]
(d) | r =f(1π)= 2
4 | B1 | 1.1 | BC
[1]
(e) | r =0 when θ=0.
π
r =0 also when θ=
2
π
In range 0<θ< ,r >0 and is continuous
2
So there is a loop | B1
B1 | 3.1a
2.4 | For both, ignore extras.
Conclusion - both statements for r need to
be mentioned
[2]11 A curve has cartesian equation $x ^ { 3 } + y ^ { 3 } = 2 x y$.\\
$C$ is the portion of the curve for which $x \geqslant 0$ and $y \geqslant 0$. The equation of $C$ in polar form is given by $r = \mathrm { f } ( \theta )$ for $0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$.
\begin{enumerate}[label=(\alph*)]
\item Find $f ( \theta )$.
\item Find an expression for $\mathrm { f } \left( \frac { 1 } { 2 } \pi - \theta \right)$, giving your answer in terms of $\sin \theta$ and $\cos \theta$.
\item Hence find the line of symmetry of $C$.
\item Find the value of $r$ when $\theta = \frac { 1 } { 4 } \pi$.
\item By finding values of $\theta$ when $r = 0$, show that $C$ has a loop.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core 1 2020 Q11 [8]}}