| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2022 |
| Session | March |
| Marks | 8 |
| 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 standard compound shape problem requiring arc length and sector area formulas with radians. While it involves multiple steps and careful identification of components, it's a routine application of well-practiced formulas with no conceptual difficulty or novel insight required—slightly easier than the average A-level question. |
| Spec | 1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\tan A = \frac{12}{5}\) or \(\cos A = \frac{5}{13}\) or \(\sin A = \frac{12}{13}\) | M1 | OR \(\tan B = \frac{5}{12}\) or \(\cos B = \frac{12}{13}\) or \(\sin B = \frac{5}{13}\) |
| \(A = 1.176\), \(B = 0.3948\) | A1 | Allow 1.18 or 67.4°, Allow 0.395 or 22.6°. May be implied by \(\frac{\pi}{2} - 1.176\) |
| \(DE = 4\) | B1 | If trigonometry used accept AWRT 4.00 |
| Arcs \(= 5 \times their\, 1.176\) and \(8 \times their\, 0.3948\) | M1 | Or corresponding calculations in degrees. |
| Perimeter \(= 5.880 + 3.158 + 4 = 13.0\) | A1 | Accept 13. If \(DE\) is outside the given range this mark cannot be awarded. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Area of triangle \(= \frac{1}{2} \times 5 \times their\, 12\ [= 30]\) | B1 FT | |
| Area of sectors \(= \frac{1}{2} \times 5^2 \times their\, 1.176 + \frac{1}{2} \times 8^2 \times their\, 0.3948\) | M1 | Or corresponding calculations in degrees |
| Area \(= 30 - 14.70 - 12.63 = 2.67\) | A1 | Allow 2.66 to 2.67 |
## Question 10(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\tan A = \frac{12}{5}$ or $\cos A = \frac{5}{13}$ or $\sin A = \frac{12}{13}$ | M1 | OR $\tan B = \frac{5}{12}$ or $\cos B = \frac{12}{13}$ or $\sin B = \frac{5}{13}$ |
| $A = 1.176$, $B = 0.3948$ | A1 | Allow 1.18 or 67.4°, Allow 0.395 or 22.6°. May be implied by $\frac{\pi}{2} - 1.176$ |
| $DE = 4$ | B1 | If trigonometry used accept AWRT 4.00 |
| Arcs $= 5 \times their\, 1.176$ and $8 \times their\, 0.3948$ | M1 | Or corresponding calculations in degrees. |
| Perimeter $= 5.880 + 3.158 + 4 = 13.0$ | A1 | Accept 13. If $DE$ is outside the given range this mark cannot be awarded. |
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## Question 10(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Area of triangle $= \frac{1}{2} \times 5 \times their\, 12\ [= 30]$ | B1 FT | |
| Area of sectors $= \frac{1}{2} \times 5^2 \times their\, 1.176 + \frac{1}{2} \times 8^2 \times their\, 0.3948$ | M1 | Or corresponding calculations in degrees |
| Area $= 30 - 14.70 - 12.63 = 2.67$ | A1 | Allow 2.66 to 2.67 |
---\begin{enumerate}[label=(\alph*)]
\item Find the perimeter of the shaded region.
\item Find the area of the shaded region.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2022 Q10 [8]}}