| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2016 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Radians, Arc Length and Sector Area |
| Type | Sector perimeter calculation |
| Difficulty | Moderate -0.8 This is a straightforward application of basic radian conversion and arc length formula. Part (i) is simple recall (54° = 54π/180 = 3π/10), and part (ii) requires setting up the equation 2r + rθ = 60 and solving for r. Both parts are routine exercises with no problem-solving insight required, making this easier than average. |
| Spec | 1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(54° \times \frac{\pi}{180} = \frac{3\pi}{10}\) | M1 | Attempt to use conversion factor of \(\frac{\pi}{180}\). Must use \(\frac{\pi}{180}\) or \(\frac{2\pi}{360}\) or equiv. Can also use 1 rad \(= 57.3°\) or \(1° = 0.0175\) rad. Must use fractions correct way up. \(0.942\) (or better) with no working will imply M1 |
| Obtain \(\frac{3\pi}{10}\) | A1 | Allow exact simplified equiv ie \(0.3\pi\). A0 if not fully simplified. No ISW if decimal equiv (0.942) given as final answer. However, if both decimal and exact answers seen, allow A1 if, and only if, the exact answer is indicated as their only intended final answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{3\pi}{10}r + 2r = 60\) | M1* | Attempt perimeter in terms of \(r\). Must be using \(r\theta\) as arc length, and also including \(2r\). Allow use of an incorrect \(\theta\) from (i). Only allow incorrect \(\theta\) if seen in (i), so \(0.3r + 2r\) is M0 unless 0.3 was their (i). Could be using decimal equiv for \(\theta\) (0.942). M0 if using \(54°\). M0 if using radians incorrectly eg \(0.942\pi\) |
| \(r = 20.4\) | M1d* | Equate to 60, and attempt to solve. Must be a valid solution attempt. M0 for \(2.3\pi r = 60\). Could be working exactly or in decimals |
| A1 | Obtain 20.4, or better. If \(> 3\)sf, allow answers in the range [20.39, 20.40] |
# Question 2:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $54° \times \frac{\pi}{180} = \frac{3\pi}{10}$ | M1 | Attempt to use conversion factor of $\frac{\pi}{180}$. Must use $\frac{\pi}{180}$ or $\frac{2\pi}{360}$ or equiv. Can also use 1 rad $= 57.3°$ or $1° = 0.0175$ rad. Must use fractions correct way up. $0.942$ (or better) with no working will imply M1 |
| Obtain $\frac{3\pi}{10}$ | A1 | Allow exact simplified equiv ie $0.3\pi$. A0 if not fully simplified. No ISW if decimal equiv (0.942) given as final answer. However, if both decimal and exact answers seen, allow A1 if, and only if, the exact answer is indicated as their only intended final answer |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{3\pi}{10}r + 2r = 60$ | M1* | Attempt perimeter in terms of $r$. Must be using $r\theta$ as arc length, and also including $2r$. Allow use of an incorrect $\theta$ from (i). Only allow incorrect $\theta$ if seen in (i), so $0.3r + 2r$ is M0 unless 0.3 was their (i). Could be using decimal equiv for $\theta$ (0.942). M0 if using $54°$. M0 if using radians incorrectly eg $0.942\pi$ |
| $r = 20.4$ | M1d* | Equate to 60, and attempt to solve. Must be a valid solution attempt. M0 for $2.3\pi r = 60$. Could be working exactly or in decimals |
| | A1 | Obtain 20.4, or better. If $> 3$sf, allow answers in the range [20.39, 20.40] |
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\includegraphics[max width=\textwidth, alt={}, center]{555f7205-5e2a-4471-901d-d8abc9dd4b4a-2_417_476_1030_790}
The diagram shows a sector $A O B$ of a circle with centre $O$ and radius $r \mathrm {~cm}$. The angle $A O B$ is $54 ^ { \circ }$. The perimeter of the sector is 60 cm .\\
(i) Express $54 ^ { \circ }$ exactly in radians, simplifying your answer.\\
(ii) Find the value of $r$, giving your answer correct to 3 significant figures.
\hfill \mbox{\textit{OCR C2 2016 Q2 [5]}}