| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2018 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Radians, Arc Length and Sector Area |
| Type | Tangent and sector - single tangent line |
| Difficulty | Standard +0.3 This is a standard sector and tangent problem requiring area formula (sector minus triangle) and perimeter calculation using arc length and tangent properties. The setup is straightforward with clear geometric relationships, though part (ii) requires careful identification of all perimeter components including the tangent length using right-angled triangle properties. |
| Spec | 1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
| Answer | Marks | Guidance |
|---|---|---|
| \((\tan\theta = \frac{AT}{r}) \rightarrow AT = r\tan\theta\) or \(OT = \frac{r}{\cos\theta}\) SOI | B1 | CAO |
| \(\rightarrow A = \frac{1}{2}r^2\tan\theta \quad - \frac{1}{2}r^2\theta\) | B1 B1 | B1 for \(\frac{1}{2}r^2\tan\theta\). B1 for \(-\frac{1}{2}r^2\theta\). If Pythagoras used may see area of triangle as \(\frac{1}{2}r\sqrt{r^2 + r^2\tan^2\theta}\) or \(\frac{1}{2}r\left(\frac{r}{\cos\theta}\right)\sin\theta\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\tan\theta = \frac{AT}{3} \rightarrow AT = 7.716\) | M1 | Correct use of trigonometry and radians in right angle triangle |
| Arc length \(= r\theta = 3.6\) | B1 | Accept \(3 \times 1.2\) |
| \(OT\) by Pythagoras or \(\cos 1.2 = \frac{3}{OT}\ (= 8.279)\) | M1 | Correct method for \(OT\) |
| Perimeter \(= AT +\) arc \(+ OT -\) radius \(= 16.6\) | A1 | CAO, www |
## Question 6(i):
| $(\tan\theta = \frac{AT}{r}) \rightarrow AT = r\tan\theta$ or $OT = \frac{r}{\cos\theta}$ SOI | B1 | CAO |
|---|---|---|
| $\rightarrow A = \frac{1}{2}r^2\tan\theta \quad - \frac{1}{2}r^2\theta$ | B1 B1 | B1 for $\frac{1}{2}r^2\tan\theta$. B1 for $-\frac{1}{2}r^2\theta$. If Pythagoras used may see area of triangle as $\frac{1}{2}r\sqrt{r^2 + r^2\tan^2\theta}$ or $\frac{1}{2}r\left(\frac{r}{\cos\theta}\right)\sin\theta$ |
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## Question 6(ii):
| $\tan\theta = \frac{AT}{3} \rightarrow AT = 7.716$ | M1 | Correct use of trigonometry and radians in right angle triangle |
|---|---|---|
| Arc length $= r\theta = 3.6$ | B1 | Accept $3 \times 1.2$ |
| $OT$ by Pythagoras or $\cos 1.2 = \frac{3}{OT}\ (= 8.279)$ | M1 | Correct method for $OT$ |
| Perimeter $= AT +$ arc $+ OT -$ radius $= 16.6$ | A1 | CAO, www |
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\includegraphics[max width=\textwidth, alt={}, center]{5df7bd9f-31cc-41a3-b1c0-3ee9366e6d8a-10_499_922_262_607}
The diagram shows a circle with centre $O$ and radius $r \mathrm {~cm}$. The points $A$ and $B$ lie on the circle and $A T$ is a tangent to the circle. Angle $A O B = \theta$ radians and $O B T$ is a straight line.\\
(i) Express the area of the shaded region in terms of $r$ and $\theta$.\\
(ii) In the case where $r = 3$ and $\theta = 1.2$, find the perimeter of the shaded region.\\
\hfill \mbox{\textit{CAIE P1 2018 Q6 [7]}}