| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2018 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Area of region with line boundary |
| Difficulty | Standard +0.8 This Further Maths polar coordinates question requires multiple techniques: area integration with a non-standard integrand (√cot θ), optimization to find maximum distance from initial line, and understanding polar curve behavior. While individual parts use standard methods, the combination of √cot θ (requiring trigonometric manipulation), finding extrema in polar form, and sketching an unfamiliar curve elevates this above routine exercises. |
| Spec | 4.09a Polar coordinates: convert to/from cartesian4.09b Sketch polar curves: r = f(theta)4.09c Area enclosed: by polar curve |
| Answer | Marks |
|---|---|
| 9(i) | π |
| Answer | Marks | Guidance |
|---|---|---|
| 0.01 | M1 | 1 |
| Answer | Marks |
|---|---|
| 2 0.01 | A1 |
| Answer | Marks |
|---|---|
| 2 | A1 |
| Answer | Marks |
|---|---|
| 9(ii) | 1 1 5 1 |
| Answer | Marks | Guidance |
|---|---|---|
| 2 | M1 | Uses y=rsinθ. |
| θ=0.01⇒ y≈0.5 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| 9(iii) | dy = 5 sin − 1 2 2θcos2θ=0 or max ( sin2θ )=1 | |
| dθ 2 | M1 A1 | dy |
| Answer | Marks |
|---|---|
| 2 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
| Answer | Marks | Guidance |
|---|---|---|
| 9(iv) | B1 | Intersecting the initial line only when x = 0 and y = 0. |
| B1 | Correct shape. |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 9:
--- 9(i) ---
9(i) | π
25 2
∫ cotθ dθ
2
0.01 | M1 | 1
Uses ∫r2dθ.
2
π
25[ ]
= lnsinθ 2
2 0.01 | A1
25
=− lnsin0.01≈57.6
2 | A1
3
--- 9(ii) ---
9(ii) | 1 1 5 1
y=5cos2θsin2θ= sin2 2θ
2 | M1 | Uses y=rsinθ.
θ=0.01⇒ y≈0.5 | A1
2
--- 9(iii) ---
9(iii) | dy = 5 sin − 1 2 2θcos2θ=0 or max ( sin2θ )=1
dθ 2 | M1 A1 | dy
Sets =0 or considers max (AEF).
dθ
⇒y= 5 2 (=3.54 )
2 | A1
3
Question | Answer | Marks | Guidance
--- 9(iv) ---
9(iv) | B1 | Intersecting the initial line only when x = 0 and y = 0.
B1 | Correct shape.
2
Question | Answer | Marks | GuidanceThe curve $C$ has polar equation
$$r = 5\sqrt{\cot \theta},$$
where $0.01 \leqslant \theta \leqslant \frac{1}{2}\pi$.
\begin{enumerate}[label=(\roman*)]
\item Find the area of the finite region bounded by $C$ and the line $\theta = 0.01$, showing full working. Give your answer correct to $1$ decimal place. [3]
\end{enumerate}
Let $P$ be the point on $C$ where $\theta = 0.01$.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item Find the distance of $P$ from the initial line, giving your answer correct to $1$ decimal place. [2]
\item Find the maximum distance of $C$ from the initial line. [3]
\item Sketch $C$. [2]
\end{enumerate}
\hfill \mbox{\textit{CAIE FP1 2018 Q9 [10]}}