| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2019 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Radians, Arc Length and Sector Area |
| Type | Tangent and sector - two tangents from external point |
| Difficulty | Standard +0.3 This is a straightforward application of arc length formula (s = rθ) to find the radius, followed by standard tangent-from-external-point geometry and sector area formulas. All steps are routine with no novel insight required, making it slightly easier than average. |
| 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 | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(OA \times \frac{3}{8}\pi = 6\) | M1 | |
| \(OA = \frac{16}{\pi} = 5.093(0)\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(AB = their\ 5.0930 \times \tan\frac{3}{16}\pi\) | M1 | |
| Perimeter \(= 2\times 3.4030 + 6 = 12.8\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Area \(OABC = (2\times\frac{1}{2})\times their\ 5.0930 \times their\ 3.4030\) | M1 | |
| Area sector \(= \frac{1}{2}\times(their\ 5.0930)^2\times\frac{3}{8}\pi\) | M1 | |
| Shaded area \(= their\ 17.331 - their\ 15.279 = 2.05\) | M1A1 |
## Question 8(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $OA \times \frac{3}{8}\pi = 6$ | M1 | |
| $OA = \frac{16}{\pi} = 5.093(0)$ | A1 | |
## Question 8(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $AB = their\ 5.0930 \times \tan\frac{3}{16}\pi$ | M1 | |
| Perimeter $= 2\times 3.4030 + 6 = 12.8$ | A1 | |
## Question 8(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Area $OABC = (2\times\frac{1}{2})\times their\ 5.0930 \times their\ 3.4030$ | M1 | |
| Area sector $= \frac{1}{2}\times(their\ 5.0930)^2\times\frac{3}{8}\pi$ | M1 | |
| Shaded area $= their\ 17.331 - their\ 15.279 = 2.05$ | M1A1 | |8\\
\includegraphics[max width=\textwidth, alt={}, center]{0e4a249a-9e6a-49d4-996c-fe07b7730f59-12_560_574_258_781}
The diagram shows a sector $O A C$ of a circle with centre $O$. Tangents $A B$ and $C B$ to the circle meet at $B$. The arc $A C$ is of length 6 cm and angle $A O C = \frac { 3 } { 8 } \pi$ radians.\\
(i) Find the length of $O A$ correct to 4 significant figures.\\
(ii) Find the perimeter of the shaded region.\\
(iii) Find the area of the shaded region.\\
\hfill \mbox{\textit{CAIE P1 2019 Q8 [8]}}