| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2014 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Radians, Arc Length and Sector Area |
| Type | Segment area calculation |
| Difficulty | Moderate -0.8 This is a straightforward application of standard arc length and segment area formulas with a given radius and angle in radians. Part (i) requires direct substitution into s = rθ, and part (ii) requires finding sector area minus triangle area using basic formulas. The angle 2π/3 leads to exact values involving √3, but no problem-solving or geometric insight is needed—just routine formula application. |
| Spec | 1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\text{arc} = 12 \times \frac{2\pi}{3}\) | M1 | Attempt use of \(r\theta\). Allow M1 if using \(\theta\) as \(\frac{2}{3}\). M1 implied by sight of 25.1 or better. M0 if \(r\theta\) used with \(\theta\) in degrees. M1 for equiv method using fractions of a circle with \(\theta\) as \(120°\) |
| \(= 8\pi\) | A1 | Obtain \(8\pi\) only. Given as final answer — A0 if followed by 25.1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\text{sector} = \frac{1}{2} \times 12^2 \times \frac{2\pi}{3} = 48\pi\) | M1* | Obtain area of sector using \(\frac{1}{2}r^2\theta\). Must have \(r = 12\). Allow M1 if using \(\theta\) as \(\frac{2}{3}\). M0 if \(\frac{1}{2}r^2\theta\) used with \(\theta\) in degrees. M1 for equiv method using fractions of a circle with \(\theta\) as \(120°\). M1 implied by sight of 151 or better |
| \(\text{triangle} = \frac{1}{2} \times 12^2 \times \sin\frac{2\pi}{3} = 36\sqrt{3}\) | M1* | Attempt area of triangle using \(\frac{1}{2}r^2\sin\theta\). Must have \(r = 12\). Allow M1 if using \(\theta\) as \(\frac{2}{3}\). Allow even if evaluated in incorrect mode (2.63 or 41.8). If using \(\frac{1}{2} \times b \times h\), then must be valid use of trig to find \(b\) and \(h\). M1 implied by sight of 62.4 or better |
| \(\text{segment} = 48\pi - 36\sqrt{3}\) | M1d* | Correct method to find segment area. Area of sector \(-\) area of triangle. M0 if using \(\theta\) as \(\frac{2}{3}\). Could be exact or decimal values |
| A1 | Obtain either \(48\pi - 36\sqrt{3}\) or 88.4. Allow decimal answer in range [88.44, 88.45] if \(> 3\)sf | |
| A1 | Obtain \(48\pi - 36\sqrt{3}\) only. Given as final answer — A0 if followed by 88.4 |
# Question 3:
## Part (i)
| Answer | Mark | Guidance |
|--------|------|----------|
| $\text{arc} = 12 \times \frac{2\pi}{3}$ | M1 | Attempt use of $r\theta$. Allow M1 if using $\theta$ as $\frac{2}{3}$. M1 implied by sight of 25.1 or better. M0 if $r\theta$ used with $\theta$ in degrees. M1 for equiv method using fractions of a circle with $\theta$ as $120°$ |
| $= 8\pi$ | A1 | Obtain $8\pi$ only. Given as final answer — A0 if followed by 25.1 |
## Part (ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| $\text{sector} = \frac{1}{2} \times 12^2 \times \frac{2\pi}{3} = 48\pi$ | M1* | Obtain area of sector using $\frac{1}{2}r^2\theta$. Must have $r = 12$. Allow M1 if using $\theta$ as $\frac{2}{3}$. M0 if $\frac{1}{2}r^2\theta$ used with $\theta$ in degrees. M1 for equiv method using fractions of a circle with $\theta$ as $120°$. M1 implied by sight of 151 or better |
| $\text{triangle} = \frac{1}{2} \times 12^2 \times \sin\frac{2\pi}{3} = 36\sqrt{3}$ | M1* | Attempt area of triangle using $\frac{1}{2}r^2\sin\theta$. Must have $r = 12$. Allow M1 if using $\theta$ as $\frac{2}{3}$. Allow even if evaluated in incorrect mode (2.63 or 41.8). If using $\frac{1}{2} \times b \times h$, then must be valid use of trig to find $b$ and $h$. M1 implied by sight of 62.4 or better |
| $\text{segment} = 48\pi - 36\sqrt{3}$ | M1d* | Correct method to find segment area. Area of sector $-$ area of triangle. M0 if using $\theta$ as $\frac{2}{3}$. Could be exact or decimal values |
| | A1 | Obtain either $48\pi - 36\sqrt{3}$ or 88.4. Allow decimal answer in range [88.44, 88.45] if $> 3$sf |
| | A1 | Obtain $48\pi - 36\sqrt{3}$ only. Given as final answer — A0 if followed by 88.4 |
---3\\
\includegraphics[max width=\textwidth, alt={}, center]{9e95415c-00f5-4b52-a443-0b946602b3b4-2_350_597_1695_735}
The diagram shows a sector $O A B$ of a circle, centre $O$ and radius 12 cm . The angle $A O B$ is $\frac { 2 } { 3 } \pi$ radians.\\
(i) Find the exact length of the $\operatorname { arc } A B$.\\
(ii) Find the exact area of the shaded segment enclosed by the arc $A B$ and the chord $A B$.
\hfill \mbox{\textit{OCR C2 2014 Q3 [7]}}