| Exam Board | OCR |
|---|---|
| Module | H240/01 (Pure Mathematics) |
| Year | 2018 |
| Session | September |
| Marks | 7 |
| Topic | Radians, Arc Length and Sector Area |
| Type | Compound shape area |
| Difficulty | Standard +0.3 This is a straightforward application of standard formulas for triangle area (½ab sin C) and sector area (½r²θ). Part (i) requires setting up and solving a simple equation, while part (ii) involves calculating arc length and using the cosine rule for the triangle side. All techniques are routine for this topic with no novel insight required, making it slightly easier than average. |
| Spec | 1.05c Area of triangle: using 1/2 ab sin(C)1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
| Answer | Marks | Guidance |
|---|---|---|
| (i) | \(0.5 \times 10 \times 7 \times \sin 0.8 = 25.107\) | M1 |
| (i) | \(0.5 \times r^2 \times 0.8 = 12.55\) | M1 |
| (i) | \(r = 5.6\) cm | A1 |
Total: [3]
| Answer | Marks | Guidance |
|---|---|---|
| (ii) | \(CD = 5.6 \times 0.8 = 4.48\) | M1 |
| (ii) | \(AB = \sqrt{7^2 + 10^2 - 2 \times 7 \times 10 \times \cos 0.8} = 7.17\) | M1 |
| (ii) | \(ABCD = 7.17 + (10 - 5.6) + 4.48 + (7 - 5.6) = 17.5\) cm | M1 |
| (ii) | A1 |
Total: [4]
(i) | $0.5 \times 10 \times 7 \times \sin 0.8 = 25.107$ | M1 | Attempt area of triangle
(i) | $0.5 \times r^2 \times 0.8 = 12.55$ | M1 | Equate area of sector to half of area of triangle and attempt to find $r$
(i) | $r = 5.6$ cm | A1 | Obtain correct value for $r$ (5.6 or better (5.60217…))
Total: [3]
(ii) | $CD = 5.6 \times 0.8 = 4.48$ | M1 | Attempt arc length using $r\theta$
(ii) | $AB = \sqrt{7^2 + 10^2 - 2 \times 7 \times 10 \times \cos 0.8} = 7.17$ | M1 | Attempt $AB$ using cosine rule
(ii) | $ABCD = 7.17 + (10 - 5.6) + 4.48 + (7 - 5.6) = 17.5$ cm | M1 | Attempt perimeter of $ABCD$
(ii) | | A1 | Obtain 17.5 cm (Allow 17.4 www)
Total: [4]4\\
\includegraphics[max width=\textwidth, alt={}, center]{e3942549-bfc0-432a-bf49-7d01d44af01a-5_487_789_251_639}
The diagram shows the triangle $A O B$, in which angle $A O B = 0.8$ radians, $O A = 7 \mathrm {~cm}$ and $O B = 10 \mathrm {~cm}$. $C D$ is the arc of a circle with centre $O$ and radius $O C$. The area of the triangle $A O B$ is twice the area of the sector COD\\
(i) Find the length $O C$.\\
(ii) Find the perimeter of the region $A B C D$.
\hfill \mbox{\textit{OCR H240/01 2018 Q4 [7]}}