| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2005 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Radians, Arc Length and Sector Area |
| Type | Compound shape area |
| Difficulty | Standard +0.3 This is a straightforward application of basic radian geometry combining right-angled triangle trigonometry (finding BD using sine) with standard arc length and sector area formulas. Part (i) is given as a 'show that' with the answer provided, parts (ii) and (iii) require routine calculations with no novel problem-solving. Slightly easier than average due to the scaffolded structure and standard techniques. |
| 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) \(BD = 9\sin(\pi - 2.4) = 6.08\) cm | M1 A1 [2] | Any valid method for \(BD\) (ans given) |
| (ii) \(OD = 9\cos(\pi - 2.4)\) or Pyth \((6.64)\) | M1 | Any valid method - not DM mark - this could come in part (iii). |
| Arc \(AB = 9 \times 2.4\) | M1 | Correct use of \(s = r\theta\). |
| Perimeter \(= 21.6 + 6.08 + 9 + 6.64 \Rightarrow 43.3\) cm | A1 [3] | CAO |
| (iii) Area of sector \(= \frac{1}{2} 9^2 \times 2.4\) | M1 | Correct use of \(\frac{1}{2}r^2\theta\). |
| Area of triangle \(= \frac{1}{2} \times 6.08 \times 6.64 \Rightarrow 117\) cm\(^2\) | M1 A1 [3] | Use of \(\frac{1}{2}bh\). CAO |
(i) $BD = 9\sin(\pi - 2.4) = 6.08$ cm | M1 A1 [2] | Any valid method for $BD$ (ans given)
(ii) $OD = 9\cos(\pi - 2.4)$ or Pyth $(6.64)$ | M1 | Any valid method - not DM mark - this could come in part (iii).
Arc $AB = 9 \times 2.4$ | M1 | Correct use of $s = r\theta$.
Perimeter $= 21.6 + 6.08 + 9 + 6.64 \Rightarrow 43.3$ cm | A1 [3] | CAO
(iii) Area of sector $= \frac{1}{2} 9^2 \times 2.4$ | M1 | Correct use of $\frac{1}{2}r^2\theta$.
Area of triangle $= \frac{1}{2} \times 6.08 \times 6.64 \Rightarrow 117$ cm$^2$ | M1 A1 [3] | Use of $\frac{1}{2}bh$. CAO8\\
\includegraphics[max width=\textwidth, alt={}, center]{e439eea6-76f0-41eb-aa91-bd0f3e4e1a07-3_438_805_849_669}
In the diagram, $A B C$ is a semicircle, centre $O$ and radius 9 cm . The line $B D$ is perpendicular to the diameter $A C$ and angle $A O B = 2.4$ radians.\\
(i) Show that $B D = 6.08 \mathrm {~cm}$, correct to 3 significant figures.\\
(ii) Find the perimeter of the shaded region.\\
(iii) Find the area of the shaded region.
\hfill \mbox{\textit{CAIE P1 2005 Q8 [8]}}