| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2006 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Radians, Arc Length and Sector Area |
| Type | Tangent and sector - two tangents from external point |
| Difficulty | Standard +0.3 This is a standard tangent-circle geometry problem requiring basic trigonometry (tan θ = 15/8 to find angle OAT, then doubling for AOB), arc length formula (rθ), and sector area formula (½r²θ). All techniques are routine applications with clear structure and no novel insight required, making it slightly easier than average. |
| Spec | 1.03f Circle properties: angles, chords, tangents1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta1.05g Exact trigonometric values: for standard angles |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\tan(\frac{x}{5}) = \frac{15}{8} = 1.875\) | M1 | Uses correct 90° triangle and sine. Realises the need to ÷ 2 |
| \(\frac{1}{2}x = 1.081\) | A1 | |
| → \(x = 2.16\) | A1 | co |
| (ii) \(P = 15 + 15 + r\theta = 30 + 17.3\) | M1 | For \(r\theta\) only – \(\theta\) must be in radians. |
| → \(47.3\) | A1 | co. |
| (iii) Sector area \(= \frac{1}{2}r^2\theta = 69.1\) | M1 | For use of \(\frac{1}{2}r^2\theta\). |
| Area of \(AOB T = 2 \times \frac{1}{2} \times 8 \times 15 = 120\) | M1 | For use of 2 triangles or equivalent. |
| Shaded area \(= 120 - 69.1\) | ||
| → \(50.8\) or \(50.9\) | A1 | co. |
**(i)** $\tan(\frac{x}{5}) = \frac{15}{8} = 1.875$ | M1 | Uses correct 90° triangle and sine. Realises the need to ÷ 2
$\frac{1}{2}x = 1.081$ | A1
→ $x = 2.16$ | A1 | co
**(ii)** $P = 15 + 15 + r\theta = 30 + 17.3$ | M1 | For $r\theta$ only – $\theta$ must be in radians.
→ $47.3$ | A1 | co.
**(iii)** Sector area $= \frac{1}{2}r^2\theta = 69.1$ | M1 | For use of $\frac{1}{2}r^2\theta$.
Area of $AOB T = 2 \times \frac{1}{2} \times 8 \times 15 = 120$ | M1 | For use of 2 triangles or equivalent.
Shaded area $= 120 - 69.1$ |
→ $50.8$ or $50.9$ | A1 | co.
**Total: [3] [2] [3]**
---7\\
\includegraphics[max width=\textwidth, alt={}, center]{cbcb15b4-1870-4dfd-b6e9-839aa4601511-3_545_759_269_694}
The diagram shows a circle with centre $O$ and radius 8 cm . Points $A$ and $B$ lie on the circle. The tangents at $A$ and $B$ meet at the point $T$, and $A T = B T = 15 \mathrm {~cm}$.\\
(i) Show that angle $A O B$ is 2.16 radians, correct to 3 significant figures.\\
(ii) Find the perimeter of the shaded region.\\
(iii) Find the area of the shaded region.
\hfill \mbox{\textit{CAIE P1 2006 Q7 [8]}}