| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2012 |
| Session | January |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Radians, Arc Length and Sector Area |
| Type | Sector area calculation |
| Difficulty | Moderate -0.8 This is a straightforward application of standard sector formulas (arc length = rθ and area = ½r²θ) with all values given directly. It requires only substitution into memorized formulas with no problem-solving, making it easier than average but not trivial since students must recall the correct formulas and handle the reflex angle correctly. |
| Spec | 1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use \(s = 12\theta\) | M1* | Allow equiv method using fractions of a circle. If working in degrees, must use 180 and \(\pi\) (or 360 and \(2\pi\)) to find angle. M0 if \(12\theta\) used with \(\theta\) in degrees. M0 if \(4.2\pi\) used instead of 4.2. M1 if attempting arc of minor sector (\(12 \times 2.1\) or better) |
| Attempt perimeter of sector | M1d* | Add 24 to their attempt at \(12\theta\). M0 if using minor sector |
| Obtain 74.4 | A1 | Units not required. Allow a more accurate answer that rounds to 74.4, with no errors seen (possibly resulting from working in degrees) |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use \(A = \frac{1}{2}12^2\theta\) | M1 | Condone omission of \(\frac{1}{2}\), but no other error. Allow equiv method using fractions of a circle. M0 if \(\frac{1}{2}12^2\theta\) used with \(\theta\) in degrees. M0 if \(4.2\pi\) used instead of 4.2. M1 if attempting area of minor sector |
| area \(= \frac{1}{2} \times 12^2 \times 4.2 = 302.45\ \text{cm}^2\); Obtain 302, or better | A1 | Units not required. Allow 302 or a more accurate answer that rounds to 302.4, with no errors seen (could be slight inaccuracy if using fractions of a circle) |
| [2] |
## Question 1:
### Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use $s = 12\theta$ | M1* | Allow equiv method using fractions of a circle. If working in degrees, must use 180 and $\pi$ (or 360 and $2\pi$) to find angle. M0 if $12\theta$ used with $\theta$ in degrees. M0 if $4.2\pi$ used instead of 4.2. M1 if attempting arc of minor sector ($12 \times 2.1$ or better) |
| Attempt perimeter of sector | M1d* | Add 24 to their attempt at $12\theta$. M0 if using minor sector |
| Obtain 74.4 | A1 | Units not required. Allow a more accurate answer that rounds to 74.4, with no errors seen (possibly resulting from working in degrees) |
| **[3]** | | |
### Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use $A = \frac{1}{2}12^2\theta$ | M1 | Condone omission of $\frac{1}{2}$, but no other error. Allow equiv method using fractions of a circle. M0 if $\frac{1}{2}12^2\theta$ used with $\theta$ in degrees. M0 if $4.2\pi$ used instead of 4.2. M1 if attempting area of minor sector |
| area $= \frac{1}{2} \times 12^2 \times 4.2 = 302.45\ \text{cm}^2$; Obtain 302, or better | A1 | Units not required. Allow 302 or a more accurate answer that rounds to 302.4, with no errors seen (could be slight inaccuracy if using fractions of a circle) |
| **[2]** | | |
---1\\
\includegraphics[max width=\textwidth, alt={}, center]{ad3083ae-caa6-42d8-a1f2-e984150cb104-2_319_454_246_810}
The diagram shows a sector $A O B$ of a circle with centre $O$ and radius 12 cm . The reflex angle $A O B$ is 4.2 radians.\\
(i) Find the perimeter of the sector.\\
(ii) Find the area of the sector.
\hfill \mbox{\textit{OCR C2 2012 Q1 [5]}}