| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2014 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Radians, Arc Length and Sector Area |
| Type | Proving angle relationships |
| Difficulty | Standard +0.3 This is a straightforward application of basic trigonometry (finding an angle using tan^(-1)) and sector area formulas. Part (i) is pure recall requiring one calculation, while part (ii) involves identifying that the shaded area equals triangle AOB minus two circular sectors, then applying standard formulas. The geometry is clearly presented and the method is routine for students who have practiced sector problems. |
| Spec | 1.05c Area of triangle: using 1/2 ab sin(C)1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
| Answer | Marks | Guidance |
|---|---|---|
| \(\rightarrow (\theta = 0.3948)\) | M1 [1] | Any valid trig method ag |
| Answer | Marks | Guidance |
|---|---|---|
| \(= 28.43 + 14.70 - 30 = 13.1\) | B1, B1, M1, DM1, A1 [5] | Unsimplified OK; co; With \(\theta\) in radians and \(r = 5\) or 12; Sum of 2 sectors – triangle or any other valid method using the given angle and a different one.; co |
**(i)** $\tan\theta = \frac{5}{12}$
$\rightarrow (\theta = 0.3948)$ | M1 [1] | Any valid trig method ag
**(ii)** Other angle in triangle $= \frac{7}{2}\pi - 0.3948$
Area of triangle $AOB = \frac{1}{2} \times 12 \times 5 (= 30)$
Use of $\frac{1}{2}r^2\theta$ once
Shaded area $= \text{sector} + \text{sector} - \text{triangle}$
$= \frac{1}{2} \times 12^2 \times 0.3948 + \frac{5}{2}5\theta - 30$
$= 28.43 + 14.70 - 30 = 13.1$ | B1, B1, M1, DM1, A1 [5] | Unsimplified OK; co; With $\theta$ in radians and $r = 5$ or 12; Sum of 2 sectors – triangle or any other valid method using the given angle and a different one.; co\includegraphics{figure_2}
The diagram shows a triangle $AOB$ in which $OA$ is 12 cm, $OB$ is 5 cm and angle $AOB$ is a right angle. Point $P$ lies on $AB$ and $OP$ is an arc of a circle with centre $A$. Point $Q$ lies on $AB$ and $OQ$ is an arc of a circle with centre $B$.
\begin{enumerate}[label=(\roman*)]
\item Show that angle $BAO$ is 0.3948 radians, correct to 4 decimal places. [1]
\item Calculate the area of the shaded region. [5]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2014 Q2 [6]}}