| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2012 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Radians, Arc Length and Sector Area |
| Type | Segment area calculation |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question testing standard sector formulas. Part (i) is routine degree-to-radian conversion, part (ii) applies the sector area formula directly to find r, and part (iii) requires subtracting triangle area from sector area—all standard C2 techniques with no novel problem-solving required. Slightly easier than average due to the step-by-step scaffolding. |
| Spec | 1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(72° = 72 \times \frac{\pi}{180} = \frac{2\pi}{5}\) radians | B1 | State \(\frac{2\pi}{5}\), \(\frac{2}{5}\pi\) or \(0.4\pi\). Must be simplified. B0 for decimal equiv (1.26), but isw if exact simplified value seen but then given in decimals. |
| [1] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{1}{2} \times r^2 \times \frac{2\pi}{5} = 45\pi\), equate attempt at area using \(\frac{1}{2}r^2\theta\) to \(45\pi\) and attempt to solve for \(r\) | M1 | Condone omission of \(\frac{1}{2}\), but no other error. Must be equated to \(45\pi\). Allow M1 for using 0.4 or \(1.26\pi\). Allow if using incorrect angle from part (i) as long as clearly intended to be in radians. |
| \(r = 15\) cm | A1 | Must be exact working only – any use of decimals is A0 (but allow 15.0 if giving previously exact answer to 3sf). |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Area of triangle \(= \frac{1}{2} \times 15^2 \times \sin\!\left(\frac{2\pi}{5}\right) = 106.99\), attempt area of triangle using \(\frac{1}{2}r^2\sin\theta\) | M1* | Condone omission of \(\frac{1}{2}\), but no other error. Must be using their \(r\) and angle linked to their \(\theta\). Could be using degrees or radians. |
| Area of segment \(= 45\pi - 106.99\), attempt \(45\pi -\) area of triangle | M1d* | Must be using \(45\pi\) (not 45). M0 if area of triangle is greater than \(45\pi\). |
| \(= 34.4\) cm² | A1 | If \(>3\)sf then allow any values rounding to 34.4. |
| [3] |
## Question 3:
### Part (i): Convert $72°$ to radians
| Answer | Marks | Guidance |
|--------|-------|----------|
| $72° = 72 \times \frac{\pi}{180} = \frac{2\pi}{5}$ radians | B1 | State $\frac{2\pi}{5}$, $\frac{2}{5}\pi$ or $0.4\pi$. Must be simplified. B0 for decimal equiv (1.26), but isw if exact simplified value seen but then given in decimals. |
| | **[1]** | |
### Part (ii): Find radius $r$
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{1}{2} \times r^2 \times \frac{2\pi}{5} = 45\pi$, equate attempt at area using $\frac{1}{2}r^2\theta$ to $45\pi$ and attempt to solve for $r$ | M1 | Condone omission of $\frac{1}{2}$, but no other error. Must be equated to $45\pi$. Allow M1 for using 0.4 or $1.26\pi$. Allow if using incorrect angle from part (i) as long as clearly intended to be in radians. |
| $r = 15$ cm | A1 | Must be exact working only – any use of decimals is A0 (but allow 15.0 if giving previously exact answer to 3sf). |
| | **[2]** | |
### Part (iii): Find area of segment
| Answer | Marks | Guidance |
|--------|-------|----------|
| Area of triangle $= \frac{1}{2} \times 15^2 \times \sin\!\left(\frac{2\pi}{5}\right) = 106.99$, attempt area of triangle using $\frac{1}{2}r^2\sin\theta$ | M1* | Condone omission of $\frac{1}{2}$, but no other error. Must be using their $r$ and angle linked to their $\theta$. Could be using degrees or radians. |
| Area of segment $= 45\pi - 106.99$, attempt $45\pi -$ area of triangle | M1d* | Must be using $45\pi$ (not 45). M0 if area of triangle is greater than $45\pi$. |
| $= 34.4$ cm² | A1 | If $>3$sf then allow any values rounding to 34.4. |
| | **[3]** | |
---3\\
\includegraphics[max width=\textwidth, alt={}, center]{f25e2580-ba0b-42ce-bf86-63f2c2075223-2_436_526_762_767}
The diagram shows a sector $A O B$ of a circle, centre $O$ and radius $r \mathrm {~cm}$. Angle $A O B$ is $72 ^ { \circ }$.\\
(i) Express $72 ^ { \circ }$ exactly in radians, simplifying your answer.
The area of the sector $A O B$ is $45 \pi \mathrm {~cm} ^ { 2 }$.\\
(ii) Find the value of $r$.\\
(iii) Find the area of the segment bounded by the arc $A B$ and the chord $A B$, giving your answer correct to 3 significant figures.
\hfill \mbox{\textit{OCR C2 2012 Q3 [6]}}