| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2012 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Radians, Arc Length and Sector Area |
| Type | Compound shape perimeter |
| Difficulty | Standard +0.3 This is a standard geometry problem combining right-angled trigonometry (finding AC = r cos θ) with arc length formulas. Part (i) requires basic trig in a right triangle, and part (ii) involves calculating a perimeter using the arc length formula with given values. The multi-step nature and combination of concepts makes it slightly above average, but all techniques are routine for P1 level. |
| Spec | 1.05a Sine, cosine, tangent: definitions for all arguments1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(AC = r - r\cos\theta\) | B1 | [1] |
| (ii) arc \(AB = \frac{4\pi}{3}\) | B1 | |
| arc \(AD = \frac{\pi}{2} \times\) their \(AC = \frac{\pi}{2} \times(4 - 4\cos\frac{\pi}{3}) = \pi\) | M1A1 | Allow \(\pi \times\) their \(AC\) for M1. Allow 3.14 |
| \(BD = 4\sin\frac{\pi}{3} -\) their \(AC = 2\sqrt{3} - 2\) | M1A1 | Allow 1.46 |
| Perimeter \(= \frac{7\pi}{3} + 2\sqrt{3} - 2\) | A1 | cao. Accept √12 |
**(i)** $AC = r - r\cos\theta$ | B1 | [1]
**(ii)** arc $AB = \frac{4\pi}{3}$ | B1 |
arc $AD = \frac{\pi}{2} \times$ their $AC = \frac{\pi}{2} \times(4 - 4\cos\frac{\pi}{3}) = \pi$ | M1A1 | Allow $\pi \times$ their $AC$ for M1. Allow 3.14
$BD = 4\sin\frac{\pi}{3} -$ their $AC = 2\sqrt{3} - 2$ | M1A1 | Allow 1.46
Perimeter $= \frac{7\pi}{3} + 2\sqrt{3} - 2$ | A1 | cao. Accept √12 | [6]6\\
\includegraphics[max width=\textwidth, alt={}, center]{e69332d0-2e45-4a86-a1f9-5d83bca1ad9b-2_526_659_1336_742}
The diagram shows a sector $O A B$ of a circle with centre $O$ and radius $r$. Angle $A O B$ is $\theta$ radians. The point $C$ on $O A$ is such that $B C$ is perpendicular to $O A$. The point $D$ is on $B C$ and the circular arc $A D$ has centre $C$.\\
(i) Find $A C$ in terms of $r$ and $\theta$.\\
(ii) Find the perimeter of the shaded region $A B D$ when $\theta = \frac { 1 } { 3 } \pi$ and $r = 4$, giving your answer as an exact value.
\hfill \mbox{\textit{CAIE P1 2012 Q6 [7]}}