| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2007 |
| Session | June |
| Marks | 5 |
| 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-sector problem requiring basic trigonometry (tan 30° in a right triangle) and area calculation (sector minus triangle). The angle is given in radians and the exact value work is routine for A-level, making it slightly easier than average but still requiring multiple steps and careful geometry. |
| 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 |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\tan\frac{1}{6}\pi = AX \div 12\) or other valid method | M1 | Use of trig with tangent in correct \(\Delta\) |
| \(\tan\frac{1}{6}\pi = \sqrt{3} \div 3 \rightarrow AX = 4\sqrt{3}\) | A1 [2] | Co (\(12 \div \sqrt{3}\) ok) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Area \(AOC = \frac{1}{2}r^2\theta\) \((= 24\pi)\) | M1 | Correct formula + attempt with radians |
| Area of \(\Delta AOX = \frac{1}{2} \times AX \times 12\) | M1 | Use of \(\frac{1}{2}bh\) in correct \(\Delta\) (once ok) |
| \(\rightarrow\) shaded area \(= 48\sqrt{3} - 24\pi\) | A1 [3] | co (\(144 \div \sqrt{3}\) ok) |
## Question 5(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\tan\frac{1}{6}\pi = AX \div 12$ or other valid method | M1 | Use of trig with tangent in correct $\Delta$ |
| $\tan\frac{1}{6}\pi = \sqrt{3} \div 3 \rightarrow AX = 4\sqrt{3}$ | A1 [2] | Co ($12 \div \sqrt{3}$ ok) |
## Question 5(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Area $AOC = \frac{1}{2}r^2\theta$ $(= 24\pi)$ | M1 | Correct formula + attempt with radians |
| Area of $\Delta AOX = \frac{1}{2} \times AX \times 12$ | M1 | Use of $\frac{1}{2}bh$ in correct $\Delta$ (once ok) |
| $\rightarrow$ shaded area $= 48\sqrt{3} - 24\pi$ | A1 [3] | co ($144 \div \sqrt{3}$ ok) |
---5\\
\includegraphics[max width=\textwidth, alt={}, center]{b24ed4c7-ab07-45f4-adf2-027734c36b62-2_586_682_1726_733}
In the diagram, $O A B$ is a sector of a circle with centre $O$ and radius 12 cm . The lines $A X$ and $B X$ are tangents to the circle at $A$ and $B$ respectively. Angle $A O B = \frac { 1 } { 3 } \pi$ radians.\\
(i) Find the exact length of $A X$, giving your answer in terms of $\sqrt { } 3$.\\
(ii) Find the area of the shaded region, giving your answer in terms of $\pi$ and $\sqrt { } 3$.
\hfill \mbox{\textit{CAIE P1 2007 Q5 [5]}}