| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2013 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Radians, Arc Length and Sector Area |
| Type | Multiple circles or sectors |
| Difficulty | Standard +0.3 This is a standard two-circle intersection problem requiring cosine rule to find angles, then arc length and sector area formulas. While it involves multiple steps (showing an angle, finding perimeter, finding area), each step follows routine procedures taught in C2 with no novel insight required. The 'show that' part provides the answer to check against, making it slightly easier than average. |
| Spec | 1.05b Sine and cosine rules: including ambiguous case1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\cos^{-1}\frac{6}{7} = 0.5411\) | M1 | Attempt correct method to find angle \(CAB\). Either use cosine rule or right-angled trigonometry. Allow M1 for \(\cos A = \frac{6}{7}\) or equiv from cosine rule |
| Obtain \(0.5411\) | A1 | Must be given to exactly 4sf. If angle found as \(31°\) then conversion to radians must be shown explicitly |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Arc length \(= 7 \times (2 \times 0.5411) = 7.575\), perimeter \(= 15.2\) | M1 | Attempt arc length using \(7\theta\). Must be using \(r = 7\). Allow if using \(\theta = 0.5411\) not \(1.0822\). M0 if using angle other than \(0.5411\) or \(1.0822\) |
| Obtain perimeter as \(15.2\), or better | A1 | Allow \(15.15\), or anything that rounds to this with no errors seen |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Sector area \(= \frac{1}{2} \times 7^2 \times (2 \times 0.5411) = 26.51\) | M1* | Attempt area of one sector using \(\frac{1}{2} \times 7^2 \times \theta\), or equiv. Allow if using \(\theta = 0.5411\) not \(1.0822\). Allow M1 if \(13.3\) or \(26.5\) seen with no method |
| Triangle area \(= \frac{1}{2} \times 7^2 \times \sin 1.082 = 21.63\) | M1* | Attempt area of relevant triangle or area of rhombus. Allow if using \(\theta = 0.5411\) not \(1.0822\) in \(\frac{1}{2} \times 7^2 \times \sin\theta\) |
| Area of segment \(= 4.88\) | A1 | Obtain \(4.88\), or better, either as final answer or soi in method. Allow values in range \([4.85, 4.9]\) |
| Shaded area \(= 9.76 \text{ cm}^2\) | M1d* | Attempt correct method to find required area. Must be full and valid method including attempted use of correct angle and subtraction in correct order |
| Obtain \(9.76\), or better | A1 | Allow answer rounding to \(9.76\), no errors seen |
## Question 7:
### Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\cos^{-1}\frac{6}{7} = 0.5411$ | M1 | Attempt correct method to find angle $CAB$. Either use cosine rule or right-angled trigonometry. Allow M1 for $\cos A = \frac{6}{7}$ or equiv from cosine rule |
| Obtain $0.5411$ | A1 | Must be given to exactly 4sf. If angle found as $31°$ then conversion to radians must be shown explicitly |
### Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Arc length $= 7 \times (2 \times 0.5411) = 7.575$, perimeter $= 15.2$ | M1 | Attempt arc length using $7\theta$. Must be using $r = 7$. Allow if using $\theta = 0.5411$ not $1.0822$. M0 if using angle other than $0.5411$ or $1.0822$ |
| Obtain perimeter as $15.2$, or better | A1 | Allow $15.15$, or anything that rounds to this with no errors seen |
### Part (iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Sector area $= \frac{1}{2} \times 7^2 \times (2 \times 0.5411) = 26.51$ | M1* | Attempt area of one sector using $\frac{1}{2} \times 7^2 \times \theta$, or equiv. Allow if using $\theta = 0.5411$ not $1.0822$. Allow M1 if $13.3$ or $26.5$ seen with no method |
| Triangle area $= \frac{1}{2} \times 7^2 \times \sin 1.082 = 21.63$ | M1* | Attempt area of relevant triangle or area of rhombus. Allow if using $\theta = 0.5411$ not $1.0822$ in $\frac{1}{2} \times 7^2 \times \sin\theta$ |
| Area of segment $= 4.88$ | A1 | Obtain $4.88$, or better, either as final answer or soi in method. Allow values in range $[4.85, 4.9]$ |
| Shaded area $= 9.76 \text{ cm}^2$ | M1d* | Attempt correct method to find required area. Must be full and valid method including attempted use of correct angle and subtraction in correct order |
| Obtain $9.76$, or better | A1 | Allow answer rounding to $9.76$, no errors seen |7\\
\includegraphics[max width=\textwidth, alt={}, center]{87012792-fa63-4003-875d-b8e7739037f1-3_412_707_751_680}
The diagram shows two circles of radius 7 cm with centres $A$ and $B$. The distance $A B$ is 12 cm and the point $C$ lies on both circles. The region common to both circles is shaded.\\
(i) Show that angle $C A B$ is 0.5411 radians, correct to 4 significant figures.\\
(ii) Find the perimeter of the shaded region.\\
(iii) Find the area of the shaded region.
\hfill \mbox{\textit{OCR C2 2013 Q7 [9]}}