| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2005 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Radians, Arc Length and Sector Area |
| Type | Simultaneous equations with arc/area |
| Difficulty | Standard +0.3 This is a straightforward application of standard arc length and sector area formulas (s=rθ, A=½r²θ) leading to simple simultaneous equations that solve cleanly. Part (iii) requires subtracting triangle area from sector area, which is a standard technique. Slightly above average difficulty due to the multi-step nature and need to recall multiple formulas, but all steps are routine for C2 level. |
| Spec | 1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(r\theta = 12\), \(\frac{1}{2}r^2\theta = 36\) | B1, B1 | For \(r\theta = 12\) stated correctly at any point; For \(\frac{1}{2}r^2\theta = 36\) stated correctly at any point |
| (ii) \(\frac{1}{2}r \times 12 = 36 \Rightarrow r = 6\) | B1 | For showing given value correctly |
| Hence \(\theta = 2\) | B1 | For correct value 2 (or 0.637π) |
| (iii) Segment area is \(36 - \frac{1}{2} \times 6^2 \times \sin 2 = 19.6 \text{ cm}^2\) | M1, M1dep*, A1 | For use of \(A = \frac{1}{2}r^2(\theta - \sin\theta)\) or equivalent; For attempt at \(36 - \Delta\); For correct value (rounding to) 19.6 |
**(i)** $r\theta = 12$, $\frac{1}{2}r^2\theta = 36$ | B1, B1 | For $r\theta = 12$ stated correctly at any point; For $\frac{1}{2}r^2\theta = 36$ stated correctly at any point
**(ii)** $\frac{1}{2}r \times 12 = 36 \Rightarrow r = 6$ | B1 | For showing given value correctly
Hence $\theta = 2$ | B1 | For correct value 2 (or 0.637π)
**(iii)** Segment area is $36 - \frac{1}{2} \times 6^2 \times \sin 2 = 19.6 \text{ cm}^2$ | M1, M1dep*, A1 | For use of $A = \frac{1}{2}r^2(\theta - \sin\theta)$ or equivalent; For attempt at $36 - \Delta$; For correct value (rounding to) 19.6
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\includegraphics[max width=\textwidth, alt={}, center]{387a37c4-0997-484c-8e28-954639169ebe-2_579_895_817_625}
A sector $O A B$ of a circle of radius $r \mathrm {~cm}$ has angle $\theta$ radians. The length of the arc of the sector is 12 cm and the area of the sector is $36 \mathrm {~cm} ^ { 2 }$ (see diagram).\\
(i) Write down two equations involving $r$ and $\theta$.\\
(ii) Hence show that $r = 6$, and state the value of $\theta$.\\
(iii) Find the area of the segment bounded by the arc $A B$ and the chord $A B$.
\hfill \mbox{\textit{OCR C2 2005 Q2 [7]}}