| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2017 |
| Session | March |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Radians, Arc Length and Sector Area |
| Type | Compound shape perimeter |
| Difficulty | Standard +0.3 This is a straightforward application of radian measure and arc length formulas. Part (i) involves angle calculation (likely using given information), and part (ii) requires computing a perimeter using arc lengths and straight lines. These are standard P1 exercises requiring direct formula application rather than problem-solving insight, making it slightly easier than average. |
| Spec | 1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(ABC = \frac{\pi}{2} - \frac{\pi}{7} = \frac{5\pi}{14}\), \(CBD = \pi - \frac{5\pi}{14} = \frac{9\pi}{14}\) | B1 | AG Or other valid exact method |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\sin\frac{\pi}{7} = \frac{\frac{1}{2}BC}{8}\) or \(\frac{BC}{\sin\frac{2\pi}{7}} = \frac{8}{\sin\frac{5\pi}{14}}\) or \(BC^2 = 8^2 + 8^2 - 2(8)(8)\cos\frac{2\pi}{7}\) | M1 | |
| \(BC = 6.94(2)\) | A1 | |
| arc \(CD =\) their \(6.94 \times \frac{9\pi}{14}\) | M1 | Expect 14.02(0) |
| arc \(CB = 8 \times \frac{2\pi}{7}\) | M1 | Expect 7.18(1) |
| perimeter \(= 6.94 + 14.02 + 7.18 = 28.1\) | A1 |
## Question 4(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $ABC = \frac{\pi}{2} - \frac{\pi}{7} = \frac{5\pi}{14}$, $CBD = \pi - \frac{5\pi}{14} = \frac{9\pi}{14}$ | B1 | AG Or other valid exact method |
**Total: 1 mark**
## Question 4(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\sin\frac{\pi}{7} = \frac{\frac{1}{2}BC}{8}$ or $\frac{BC}{\sin\frac{2\pi}{7}} = \frac{8}{\sin\frac{5\pi}{14}}$ or $BC^2 = 8^2 + 8^2 - 2(8)(8)\cos\frac{2\pi}{7}$ | M1 | |
| $BC = 6.94(2)$ | A1 | |
| arc $CD =$ their $6.94 \times \frac{9\pi}{14}$ | M1 | Expect 14.02(0) |
| arc $CB = 8 \times \frac{2\pi}{7}$ | M1 | Expect 7.18(1) |
| perimeter $= 6.94 + 14.02 + 7.18 = 28.1$ | A1 | |
**Total: 5 marks**(i) Show that angle $C B D = \frac { 9 } { 14 } \pi$ radians.\\
(ii) Find the perimeter of the shaded region.\\
\hfill \mbox{\textit{CAIE P1 2017 Q4 [6]}}