| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2013 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circles |
| Type | Sector and arc length |
| Difficulty | Moderate -0.3 This is a straightforward application of circle geometry with arc length and sector area formulas. Part (i) uses inverse tangent to find an angle (routine trigonometry), while parts (ii) and (iii) apply standard formulas for arc length and sector area. The setup is clear, all necessary information is given, and the methods are direct applications of standard techniques with no novel problem-solving required. Slightly easier than average due to its procedural nature. |
| Spec | 1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(BOC = 2\tan^{-1}\frac{1}{2} = 0.9273\) | M1 A1 | Correct trigonometry. (ans given) |
| Answer | Marks | Guidance |
|---|---|---|
| Arc \(BXC = \sqrt{125} \times 0.9273\) → Perimeter = 20.4 cm | B1 M1 A1 | Use of trig (or Pyth) for the \(OB = \sqrt{125}\). Use of \(s = r\theta\) with \(\theta\) in rads, \(r \neq 10\) |
| (iii) Area \(= \frac{1}{2}r^2\theta\) → \(\frac{1}{2}.10.10 \rightarrow 7.96 \text{ cm}^2\) | M1 A1 | Correct formula used with rads, \(r \neq 10\). Allow 7.95 or 7.96 |
| [3] | [2] |
(i) $BOC = 2\tan^{-1}\frac{1}{2} = 0.9273$ | M1 A1 | Correct trigonometry. (ans given)
(ii) $OB = \sqrt{(10^2 + 5^2)} \text{ or } 11.2 = r$
Arc $BXC = \sqrt{125} \times 0.9273$ → Perimeter = 20.4 cm | B1 M1 A1 | Use of trig (or Pyth) for the $OB = \sqrt{125}$. Use of $s = r\theta$ with $\theta$ in rads, $r \neq 10$
(iii) Area $= \frac{1}{2}r^2\theta$ → $\frac{1}{2}.10.10 \rightarrow 7.96 \text{ cm}^2$ | M1 A1 | Correct formula used with rads, $r \neq 10$. Allow 7.95 or 7.96
[3] | [2] |
---4\\
\includegraphics[max width=\textwidth, alt={}, center]{fe4c3555-5736-48c4-b61a-9f6b9a1ee46e-2_645_652_1023_744}
The diagram shows a square $A B C D$ of side 10 cm . The mid-point of $A D$ is $O$ and $B X C$ is an arc of a circle with centre $O$.\\
(i) Show that angle $B O C$ is 0.9273 radians, correct to 4 decimal places.\\
(ii) Find the perimeter of the shaded region.\\
(iii) Find the area of the shaded region.
\hfill \mbox{\textit{CAIE P1 2013 Q4 [7]}}