| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2003 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circles |
| Type | Sector and arc length |
| Difficulty | Moderate -0.8 This is a straightforward sector problem requiring basic recall of arc length and sector area formulas. Part (i) is simple algebraic manipulation of perimeter = 2r + rθ, part (ii) substitutes into area = ½r²θ, and part (iii) uses basic trigonometry (cosine rule or isosceles triangle). All steps are routine textbook exercises with no problem-solving insight required. |
| Spec | 1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
| Answer | Marks | Guidance |
|---|---|---|
| \(\rightarrow \theta = (20/r) -2\) | M1, A1 [2] | Eqn formed + use of \(r\theta\) + at least one \(r\). Answer given. |
| Answer | Marks | Guidance |
|---|---|---|
| \(\rightarrow A = 10r -r^2\) | M1, A1 [2] | Appropriate use of \(\frac{1}{2}r^2\theta\). Co – but ok unsimplified –eg \(\frac{1}{2}r^2(20/r-2)\). |
| Answer | Marks | Guidance |
|---|---|---|
| \(\rightarrow PQ = 3.96\) (allow 3.95). | M1, A1, A1 [3] | Recognition of "chord" +any attempt at trigonometry in triangle. Correct expression for PQ or \(PQ^2\). Co. |
**(i)** $20 = 2r + r\theta$
$\rightarrow \theta = (20/r) -2$ | M1, A1 [2] | Eqn formed + use of $r\theta$ + at least one $r$. Answer given.
**(ii)** $A = \frac{1}{2}r^2\theta$
$\rightarrow A = 10r -r^2$ | M1, A1 [2] | Appropriate use of $\frac{1}{2}r^2\theta$. Co – but ok unsimplified –eg $\frac{1}{2}r^2(20/r-2)$.
**(iii)** Cos rule $PQ^2 = 8^2+8^2-2.8.8\cos 0.5$
Or trig $PQ = 2 \times 8\sin 0.25$
$\rightarrow PQ = 3.96$ (allow 3.95). | M1, A1, A1 [3] | Recognition of "chord" +any attempt at trigonometry in triangle. Correct expression for PQ or $PQ^2$. Co.6\\
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The diagram shows the sector $O P Q$ of a circle with centre $O$ and radius $r \mathrm {~cm}$. The angle $P O Q$ is $\theta$ radians and the perimeter of the sector is 20 cm .\\
(i) Show that $\theta = \frac { 20 } { r } - 2$.\\
(ii) Hence express the area of the sector in terms of $r$.\\
(iii) In the case where $r = 8$, find the length of the chord $P Q$.
\hfill \mbox{\textit{CAIE P1 2003 Q6 [7]}}