| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2016 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Radians, Arc Length and Sector Area |
| Type | Shaded region between arcs |
| Difficulty | Standard +0.3 This is a straightforward application of basic radian geometry. Part (i) requires recognizing that the tangent is perpendicular to the radius (giving r = 6cos(0.9) ≈ 3.73), which is a standard result. Part (ii) involves calculating two sector areas and subtracting—routine bookwork with no novel insight required. Slightly easier than average due to the guided structure and standard techniques. |
| 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 |
|---|---|
| (ii) | π |
| Answer | Marks |
|---|---|
| Total area = 80.7(0) + 12.5(2) = 93.2 | M1 |
| Answer | Marks |
|---|---|
| A1 | [2] |
| [4] | Other methods possible |
Question 5:
--- 5 (i)
(ii) ---
5 (i)
(ii) | π
cos0.9=OE/6 or = sin −0.9 oe
2
OE =6cos0.9=3.73 oe AG
Use of (2π−1.8) or equivalent method
Area of large sector =½×62×(2π−1.8) oe
Area of small sector ½×3.732×1.8
Total area = 80.7(0) + 12.5(2) = 93.2 | M1
A1
M1
M1
M1
A1 | [2]
[4] | Other methods possible
Expect 4.48
Or π62 −½621.8. Expect 80.70
Expect 12.52
Other methods possible\includegraphics{figure_1}
The diagram shows a major arc $AB$ of a circle with centre $O$ and radius 6 cm. Points $C$ and $D$ on $OA$ and $OB$ respectively are such that the line $AB$ is a tangent at $E$ to the arc $CED$ of a smaller circle also with centre $O$. Angle $COD = 1.8$ radians.
\begin{enumerate}[label=(\roman*)]
\item Show that the radius of the arc $CED$ is 3.73 cm, correct to 3 significant figures. [2]
\item Find the area of the shaded region. [4]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2016 Q5 [6]}}