| Exam Board | Edexcel |
|---|---|
| Module | PMT Mocks (PMT Mocks) |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Radians, Arc Length and Sector Area |
| Type | Multiple circles or sectors |
| Difficulty | Challenging +1.2 This question requires finding the angle in a triangle using the cosine rule, then calculating areas of two circular segments and subtracting from a quadrilateral. While it involves multiple steps and coordinate geometry setup, the techniques are standard A-level methods (cosine rule, sector area formula, triangle area) applied systematically without requiring novel insight. |
| 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 theta1.08e Area between curve and x-axis: using definite integrals |
| Answer | Marks | Guidance |
|---|---|---|
| \(O_2 = (6.0)\); \(AO_2 = BO_2 = 10\); \(AB = 16\), as it is the diameter of \(C_1\) | ||
| \(\frac{AO_2B}{2} = \sin^{-1}\frac{8}{10} = 0.927\) | M1 | Uses the radii of the circles \(C_1\) and \(C_2\) and correct attempt to find angle \(AO_2B\) in circle \(C_2\). e.g. Attempts \(\sin AO_2O = \frac{8}{10}\) to find \(AO_2O\) then \(\times 2\); Alternatively uses \(\cos AO_2O = \frac{6}{10}\) to find \(AO_2O\) then \(\times 2\); Or uses \(\tan AO_2O = \frac{8}{6}\) to find \(AO_2O\) then \(\times 2\); OR uses cosine rule \(\cos AO_2B = \frac{10^2 + 10^2 - 16^2}{2 \times 10 \times 10} = \frac{-56}{200} \Rightarrow AO_2B = \cos^{-1}\left(-\frac{56}{200}\right) = \ldots\) |
| \(AO_2B = 1.8545\text{ rad} = 1.85\) to 3 sig fig as required | A1 | Correct and careful work in proceeding to the given answer: i.e. 1.85 radians |
| Answer | Marks | Guidance |
|---|---|---|
| Area of sector \(AO_2B\) – area of triangle \(AO_2B = \frac{1}{2} \times 10^2 \times (1.85) - \frac{1}{2} \times 10^2 \times \sin 1.85 = 44.436\) | ||
| Area of the region shaded grey = Area of semicircle with centre \(O_1\) – Area of segment \(= \frac{\pi x 8^2}{2} - 44.436 = 56.1\) | b: M1 | Attempts to use the correct formula to find the area of the segment shaded black with centre \(O_2\) |
| M1 | Attempts to use the correct method in order to find area of the region shaded grey | |
| A1 | 56.1 |
**Part a:** Show that angle $AO_2B = 1.85$ radians to 3 significant figures
| $O_2 = (6.0)$; $AO_2 = BO_2 = 10$; $AB = 16$, as it is the diameter of $C_1$ | | |
| --- | --- | --- |
| $\frac{AO_2B}{2} = \sin^{-1}\frac{8}{10} = 0.927$ | M1 | Uses the radii of the circles $C_1$ and $C_2$ and correct attempt to find angle $AO_2B$ in circle $C_2$. e.g. Attempts $\sin AO_2O = \frac{8}{10}$ to find $AO_2O$ then $\times 2$; Alternatively uses $\cos AO_2O = \frac{6}{10}$ to find $AO_2O$ then $\times 2$; Or uses $\tan AO_2O = \frac{8}{6}$ to find $AO_2O$ then $\times 2$; OR uses cosine rule $\cos AO_2B = \frac{10^2 + 10^2 - 16^2}{2 \times 10 \times 10} = \frac{-56}{200} \Rightarrow AO_2B = \cos^{-1}\left(-\frac{56}{200}\right) = \ldots$ |
| $AO_2B = 1.8545\text{ rad} = 1.85$ to 3 sig fig as required | A1 | Correct and careful work in proceeding to the given answer: i.e. 1.85 radians |
**Part b:** Find the area of the shaded region, giving your answer correct to 1 decimal place
| Area of sector $AO_2B$ – area of triangle $AO_2B = \frac{1}{2} \times 10^2 \times (1.85) - \frac{1}{2} \times 10^2 \times \sin 1.85 = 44.436$ | | |
| --- | --- | --- |
| Area of the region shaded grey = Area of semicircle with centre $O_1$ – Area of segment $= \frac{\pi x 8^2}{2} - 44.436 = 56.1$ | b: M1 | Attempts to use the correct formula to find the area of the segment shaded black with centre $O_2$ |
| | M1 | Attempts to use the correct method in order to find area of the region shaded grey |
| | A1 | 56.1 |
**(3) marks + (3) marks = (6) marks total for Question 10**
---10.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{802e56f7-5cff-491a-b90b-0759a9b35778-16_1116_1433_360_420}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{center}
\end{figure}
Circle $C _ { 1 }$ has equation $x ^ { 2 } + y ^ { 2 } = 64$ with centre $O _ { 1 }$.\\
Circle $C _ { 2 }$ has equation $( x - 6 ) ^ { 2 } + y ^ { 2 } = 100$ with centre $O _ { 2 }$.\\
The circles meet at points $A$ and $B$ as shown in Figure 3.\\
a. Show that angle $A O _ { 2 } B = 1.85$ radians to 3 significant figures.\\
(3)\\
b. Find the area of the shaded region, giving your answer correct to 1 decimal place.\\
\hfill \mbox{\textit{Edexcel PMT Mocks Q10 [6]}}