| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2011 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Radians, Arc Length and Sector Area |
| Type | Tangent and sector - single tangent line |
| Difficulty | Standard +0.3 This is a standard tangent-sector problem requiring basic trigonometry (tan 60° = √3), sector area formula, and arc length formula. All techniques are routine for P1 level with straightforward application of formulas. The 'show that' part guides students to the key step, 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 theta |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(AX = 6\tan\frac{\pi}{3} = 6\sqrt{3}\) | B1 | ag |
| [1] | ||
| (ii) Area of triangle \(= \frac{1}{2} \times 6 \times 6\sqrt{3}\) | M1 | Use of \(\frac{1}{2}bh\) |
| Area of sector \(= \frac{1}{2} 6^2 \times \frac{\pi}{3}\) | M1 | Use of \(\frac{1}{2}r^2\theta\) |
| Area shaded \(= 18\sqrt{3} - 6\pi\) | A1 | co |
| [3] | ||
| (iii) Arc \(AB = 6 \times \frac{\pi}{3} = 2\pi\) | M1 | Use of \(r\theta\) |
| \(OX = 6 \cos\frac{\pi}{3} = 12\), \(BX = 6\) | B1 | Use of trig to find (OX and then) BX. |
| Perimeter \(= 6\sqrt{3} + 2\pi + 6\) | M1 A1 | |
| [4] |
(i) $AX = 6\tan\frac{\pi}{3} = 6\sqrt{3}$ | B1 | ag
| [1] |
(ii) Area of triangle $= \frac{1}{2} \times 6 \times 6\sqrt{3}$ | M1 | Use of $\frac{1}{2}bh$
Area of sector $= \frac{1}{2} 6^2 \times \frac{\pi}{3}$ | M1 | Use of $\frac{1}{2}r^2\theta$
Area shaded $= 18\sqrt{3} - 6\pi$ | A1 | co
| [3] |
(iii) Arc $AB = 6 \times \frac{\pi}{3} = 2\pi$ | M1 | Use of $r\theta$
$OX = 6 \cos\frac{\pi}{3} = 12$, $BX = 6$ | B1 | Use of trig to find (OX and then) BX.
Perimeter $= 6\sqrt{3} + 2\pi + 6$ | M1 A1 |
| [4] |7\\
\includegraphics[max width=\textwidth, alt={}, center]{d68c82ec-8c85-40b9-8e81-bd53c7f8dafe-3_462_956_258_593}
In the diagram, $A B$ is an arc of a circle, centre $O$ and radius 6 cm , and angle $A O B = \frac { 1 } { 3 } \pi$ radians. The line $A X$ is a tangent to the circle at $A$, and $O B X$ is a straight line.\\
(i) Show that the exact length of $A X$ is $6 \sqrt { } 3 \mathrm {~cm}$.
Find, in terms of $\pi$ and $\sqrt { } 3$,\\
(ii) the area of the shaded region,\\
(iii) the perimeter of the shaded region.
\hfill \mbox{\textit{CAIE P1 2011 Q7 [8]}}