Question 5 - A-Level Maths

CAIE FP1 2015 June — Question 5 9 marks

Exam Board	CAIE
Module	FP1 (Further Pure Mathematics 1)
Year	2015
Session	June
Marks	9
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Polar coordinates
Type	Area between two polar curves
Difficulty	Standard +0.8 This Further Maths polar coordinates question requires finding intersections by solving a trigonometric equation, sketching two curves (circle and less common polar curve), and computing area using polar integration with careful region identification. The multi-step nature, need for trigonometric manipulation, and integration of √(sin(θ/2)) place it moderately above average difficulty.
Spec	4.09a Polar coordinates: convert to/from cartesian 4.09b Sketch polar curves: r = f(theta)4.09c Area enclosed: by polar curve

5 The curves $C _ { 1 }$ and $C _ { 2 }$ have polar equations $$\begin{array} { l l } C _ { 1 } : & r = \frac { 1 } { \sqrt { 2 } } , \quad \text { for } 0 \leqslant \theta < 2 \pi \\ C _ { 2 } : & r = \sqrt { } \left( \sin \frac { 1 } { 2 } \theta \right) , \quad \text { for } 0 \leqslant \theta \leqslant \pi \end{array}$$ Find the polar coordinates of the point of intersection of $C _ { 1 }$ and $C _ { 2 }$. Sketch $C _ { 1 }$ and $C _ { 2 }$ on the same diagram. Find the exact value of the area of the region enclosed by $C _ { 1 } , C _ { 2 }$ and the half-line $\theta = 0$.

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Question 5:

Answer	Marks	Guidance
Working	Marks	Notes
$\sin\frac{1}{2}\theta = \frac{1}{2} \Rightarrow \theta = \frac{1}{3}\pi$; Intersection at $\left(\frac{1}{\sqrt{2}}, \frac{1}{3}\pi\right)$	M1A1	(2) Accept 1.05 for $\frac{1}{3}\pi$
$C_1$: Circle centre at pole and radius $1/\sqrt{2}$	B1
$C_2$: Curve, approx. correct orientation, from $(0,0)$ to $(1,\pi)$	B1
Completely correct shape	B1	(3)
$\frac{1}{6} \times \pi \times \frac{1}{2} - \frac{1}{2}\int_0^{\frac{1}{3}\pi} \sin\frac{1}{2}\theta \, d\theta$	M1A1
$= \frac{1}{12}\pi \times \frac{1}{2}\left[-2\cos\frac{1}{2}\theta\right]_0^{\frac{1}{3}\pi} = \frac{1}{12}\pi + \frac{\sqrt{3}}{2} - 1$	M1A1	(4)
Total: 9

## Question 5:

| Working | Marks | Notes |
|---------|-------|-------|
| $\sin\frac{1}{2}\theta = \frac{1}{2} \Rightarrow \theta = \frac{1}{3}\pi$; Intersection at $\left(\frac{1}{\sqrt{2}}, \frac{1}{3}\pi\right)$ | M1A1 | (2) Accept 1.05 for $\frac{1}{3}\pi$ |
| $C_1$: Circle centre at pole and radius $1/\sqrt{2}$ | B1 | |
| $C_2$: Curve, approx. correct orientation, from $(0,0)$ to $(1,\pi)$ | B1 | |
| Completely correct shape | B1 | (3) |
| $\frac{1}{6} \times \pi \times \frac{1}{2} - \frac{1}{2}\int_0^{\frac{1}{3}\pi} \sin\frac{1}{2}\theta \, d\theta$ | M1A1 | |
| $= \frac{1}{12}\pi \times \frac{1}{2}\left[-2\cos\frac{1}{2}\theta\right]_0^{\frac{1}{3}\pi} = \frac{1}{12}\pi + \frac{\sqrt{3}}{2} - 1$ | M1A1 | (4) |
| | **Total: 9** | |

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5 The curves $C _ { 1 }$ and $C _ { 2 }$ have polar equations

$$\begin{array} { l l } 
C _ { 1 } : & r = \frac { 1 } { \sqrt { 2 } } , \quad \text { for } 0 \leqslant \theta < 2 \pi \\
C _ { 2 } : & r = \sqrt { } \left( \sin \frac { 1 } { 2 } \theta \right) , \quad \text { for } 0 \leqslant \theta \leqslant \pi
\end{array}$$

Find the polar coordinates of the point of intersection of $C _ { 1 }$ and $C _ { 2 }$.

Sketch $C _ { 1 }$ and $C _ { 2 }$ on the same diagram.

Find the exact value of the area of the region enclosed by $C _ { 1 } , C _ { 2 }$ and the half-line $\theta = 0$.

\hfill \mbox{\textit{CAIE FP1 2015 Q5 [9]}}

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