| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2012 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Radians, Arc Length and Sector Area |
| Type | Tangent and sector - two tangents from external point |
| Difficulty | Moderate -0.3 This is a straightforward application of basic circle geometry, arc length, and sector area formulas. Part (i) requires simple angle calculation using tangent properties and the given constraint that A is the midpoint of OX. Parts (ii) and (iii) are direct formula applications once the angle is known. The question is slightly easier than average as it's highly structured with clear steps and uses standard techniques without requiring problem-solving insight. |
| Spec | 1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(OBX = 90°\), \(\cos \theta = \frac{r}{2r} \to \theta = \frac{1}{3}\pi\) | M1, A1 | Needs \(90° +\) cos (or Pyth \(+\) sin or tan); co ag |
| (ii) Arc length \(AB = \frac{1}{3}r\pi\); \(BX = r\tan(\frac{1}{3}\pi) = r\sqrt{3}\); \(P = r + (\frac{1}{3}r\pi + r\sqrt{3})\) | B1, B1, B1 | \(r +\) sum of other two |
| (iii) Area \(= \frac{1}{2}r^2\sqrt{3} - \frac{1}{6}r^2\pi\) | B1, B1 | ✓ on \(\tan(\frac{1}{3}\pi)\); co |
**(i)** $OBX = 90°$, $\cos \theta = \frac{r}{2r} \to \theta = \frac{1}{3}\pi$ | M1, A1 | Needs $90° +$ cos (or Pyth $+$ sin or tan); co ag
**(ii)** Arc length $AB = \frac{1}{3}r\pi$; $BX = r\tan(\frac{1}{3}\pi) = r\sqrt{3}$; $P = r + (\frac{1}{3}r\pi + r\sqrt{3})$ | B1, B1, B1 | $r +$ sum of other two
**(iii)** Area $= \frac{1}{2}r^2\sqrt{3} - \frac{1}{6}r^2\pi$ | B1, B1 | ✓ on $\tan(\frac{1}{3}\pi)$; co\includegraphics{figure_8}
In the diagram, $AB$ is an arc of a circle with centre $O$ and radius $r$. The line $XB$ is a tangent to the circle at $B$ and $A$ is the mid-point of $OX$.
\begin{enumerate}[label=(\roman*)]
\item Show that angle $AOB = \frac{1}{3}\pi$ radians. [2]
\end{enumerate}
Express each of the following in terms of $r$, $\pi$ and $\sqrt{3}$:
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item the perimeter of the shaded region, [3]
\item the area of the shaded region. [2]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2012 Q8 [7]}}