| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2002 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Circle sector area and angle |
| Difficulty | Standard +0.3 This is a straightforward multi-part question involving basic circle geometry and sector area calculations. Part (i) uses cosine rule (standard technique), part (ii) is direct sector area formula application, and part (iii) requires subtracting a rectangle area. All steps are routine with no novel insight required, making it slightly easier than average. |
| Spec | 1.05b Sine and cosine rules: including ambiguous case1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
| Answer | Marks | Guidance |
|---|---|---|
| Required angle \(= 1.855\) radians | M1, A1 | Sine in \(90°\) triangle – or cosine rule. Correct only (answer was given) |
| (ii) Area of sector \(= \frac{1}{2}r^2\theta = 371\) cm² | M1, A1 | Correct formula used. Correct only |
| Answer | Marks | Guidance |
|---|---|---|
| \(= 502\) cm² (accept 501) | M1, DM1, A1 | Correct logic – independent of method. Correct attempt at all parts. Correct only |
**(i)** $\sin(\frac{1}{2}\text{angle}) = \frac{16}{20}$
Required angle $= 1.855$ radians | M1, A1 | Sine in $90°$ triangle – or cosine rule. Correct only (answer was given)
**(ii)** Area of sector $= \frac{1}{2}r^2\theta = 371$ cm² | M1, A1 | Correct formula used. Correct only
**(iii)** Area = Circle – rectangle – sector + triangle
$= \pi r^2 - lxb - \frac{1}{2}r^2\theta + \frac{1}{2}bh$ (or $\frac{1}{2}absin C$)
$= 502$ cm² (accept 501) | M1, DM1, A1 | Correct logic – independent of method. Correct attempt at all parts. Correct only7\\
\includegraphics[max width=\textwidth, alt={}, center]{b2cefbd6-6e89-495a-9f42-60f76c8c5975-4_556_524_255_813}
The diagram shows the circular cross-section of a uniform cylindrical log with centre $O$ and radius 20 cm . The points $A , X$ and $B$ lie on the circumference of the cross-section and $A B = 32 \mathrm {~cm}$.\\
(i) Show that angle $A O B = 1.855$ radians, correct to 3 decimal places.\\
(ii) Find the area of the sector $A X B O$.
The section $A X B C D$, where $A B C D$ is a rectangle with $A D = 18 \mathrm {~cm}$, is removed.\\
(iii) Find the area of the new cross-section (shown shaded in the diagram).
\hfill \mbox{\textit{CAIE P1 2002 Q7 [7]}}