| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2008 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Tangent meets circle problems |
| Difficulty | Standard +0.3 This is a straightforward multi-part question testing standard circle geometry and arc length formulas. Part (i) uses s=rθ directly, (ii) applies basic trigonometry in a right-angled triangle, and (iii) combines sector and triangle areas. All steps are routine applications of well-known formulas with no novel insight required, making it slightly easier than average. |
| Spec | 1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
| Answer | Marks | Guidance |
|---|---|---|
| Using \(s = r\theta, 9 = 5\theta \rightarrow \theta = 1.8 \text{ rad.}\) | M1, A1 | Use of formula. co |
| Answer | Marks | Guidance |
|---|---|---|
| Uses \(POT\). Halves the angle. Uses tangent in \(POT\). | M1, M1, A1 | Realises the need to have. Use of tangent – even if angle not halved. co |
| Answer | Marks | Guidance |
|---|---|---|
| Area of sector = \(\frac{1}{2} \times 5^2 \times 1.8 (22.5)\) | M1 | Use of \(A = \frac{1}{2}r^2\theta\) with 1.8 or 0.9. |
| Area of \(POT = \frac{1}{2} \times 5 \times 6.30 (15.75)\) | M1 | Use of \(\frac{1}{2}bh\) and (2 triangles – sector) |
| Shaded area = 2 triangles – sector \(\rightarrow 9.00\) (allow 8.95 to 9.05) | A1 | co |
**(i)**
Using $s = r\theta, 9 = 5\theta \rightarrow \theta = 1.8 \text{ rad.}$ | M1, A1 | Use of formula. co
**(ii)**
Uses $POT$. Halves the angle. Uses tangent in $POT$. | M1, M1, A1 | Realises the need to have. Use of tangent – even if angle not halved. co
$PT = 5\tan 9° = 6.30 \text{ cm (not 6.31)}$
**(iii)**
Area of sector = $\frac{1}{2} \times 5^2 \times 1.8 (22.5)$ | M1 | Use of $A = \frac{1}{2}r^2\theta$ with 1.8 or 0.9.
Area of $POT = \frac{1}{2} \times 5 \times 6.30 (15.75)$ | M1 | Use of $\frac{1}{2}bh$ and (2 triangles – sector)
Shaded area = 2 triangles – sector $\rightarrow 9.00$ (allow 8.95 to 9.05) | A1 | co
---6\\
\includegraphics[max width=\textwidth, alt={}, center]{08729aab-586b-4210-94c9-77b1f6b1d873-3_597_417_274_865}
In the diagram, the circle has centre $O$ and radius 5 cm . The points $P$ and $Q$ lie on the circle, and the arc length $P Q$ is 9 cm . The tangents to the circle at $P$ and $Q$ meet at the point $T$. Calculate\\
(i) angle $P O Q$ in radians,\\
(ii) the length of $P T$,\\
(iii) the area of the shaded region.
\hfill \mbox{\textit{CAIE P1 2008 Q6 [8]}}