| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2011 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Radians, Arc Length and Sector Area |
| Type | Segment area calculation |
| Difficulty | Standard +0.3 This is a standard C2 radians question requiring the formula for triangle area (½r²sinθ), then sector area minus triangle area for the segment, and arc length plus chord for perimeter. All formulas are given in the formula booklet and the question guides students through each step methodically. Slightly above average difficulty only because it requires careful application of multiple related formulas and working with an obtuse angle. |
| Spec | 1.05c Area of triangle: using 1/2 ab sin(C)1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{1}{2} \times 5^2 \times \sin\theta = 8\), \(\sin\theta = 0.64\) | M1* | Attempt to solve \(\frac{1}{2}r^2\sin\theta = 8\) to find value for \(\theta\). Allow M1 if using \(r^2\sin\theta = 8\). Need to get as far as attempting \(\theta\). |
| \(\theta = \pi - 0.694 = 2.45\) | M1d* | Attempt to find obtuse angle from principal value. i.e. \(\pi - \theta\) in radians, or \(180° - \theta\) in degrees. |
| \(\theta = 2.45\) | A1 | 3 marks Obtain \(\theta = 2.45\) or better. Must be in radians. A0 if acute angle then becomes \(2.45\pi\). |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{1}{2} \times 5^2 \times 2.447 = 30.6\) | M1* | Attempt area of sector using \(\frac{1}{2}r^2\theta\). \(\theta\) must be numerical and in radians. |
| Area \(= 30.6 - 8 = 22.6\) cm² | M1d* | Attempt area of segment. Subtract 8 from their sector area. |
| \(22.6\) cm² | A1 | 3 marks Obtain area of segment as 22.6. Allow more sig fig as long as rounds to 22.6 with no errors seen. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Arc \(= 5 \times 2.447 = 12.2\) | B1ft | State or imply arc length is \(5\theta\). \(\theta\) must be numerical and in radians. |
| Chord \(= 2 \times 5\sin 1.22 = 9.40\) or \(AB^2 = 5^2 + 5^2 - 2\times5\times5\times\cos 2.447\) | M1 | Attempt length of chord \(AB\). Any reasonable method. If using cosine rule must be correct formula. |
| \(AB = 9.40\) (allow 9.41) | A1 | Obtain 9.40. Allow any answer in range \(9.40 \leq AB \leq 9.41\). |
| Perimeter \(= 12.2 + 9.40 = 21.6\) cm | A1 | 4 marks Obtain perimeter as 21.6 (allow 21.7). Allow any answer in range \(21.6 \leq \text{perimeter} \leq 21.7\). |
# Question 8:
## Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{1}{2} \times 5^2 \times \sin\theta = 8$, $\sin\theta = 0.64$ | M1* | Attempt to solve $\frac{1}{2}r^2\sin\theta = 8$ to find value for $\theta$. Allow M1 if using $r^2\sin\theta = 8$. Need to get as far as attempting $\theta$. |
| $\theta = \pi - 0.694 = 2.45$ | M1d* | Attempt to find obtuse angle from principal value. i.e. $\pi - \theta$ in radians, or $180° - \theta$ in degrees. |
| $\theta = 2.45$ | A1 | **3 marks** Obtain $\theta = 2.45$ or better. Must be in radians. A0 if acute angle then becomes $2.45\pi$. |
## Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{1}{2} \times 5^2 \times 2.447 = 30.6$ | M1* | Attempt area of sector using $\frac{1}{2}r^2\theta$. $\theta$ must be numerical and in radians. |
| Area $= 30.6 - 8 = 22.6$ cm² | M1d* | Attempt area of segment. Subtract 8 from their sector area. |
| $22.6$ cm² | A1 | **3 marks** Obtain area of segment as 22.6. Allow more sig fig as long as rounds to 22.6 with no errors seen. |
## Part (iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Arc $= 5 \times 2.447 = 12.2$ | B1ft | State or imply arc length is $5\theta$. $\theta$ must be numerical and in radians. |
| Chord $= 2 \times 5\sin 1.22 = 9.40$ or $AB^2 = 5^2 + 5^2 - 2\times5\times5\times\cos 2.447$ | M1 | Attempt length of chord $AB$. Any reasonable method. If using cosine rule must be correct formula. |
| $AB = 9.40$ (allow 9.41) | A1 | Obtain 9.40. Allow any answer in range $9.40 \leq AB \leq 9.41$. |
| Perimeter $= 12.2 + 9.40 = 21.6$ cm | A1 | **4 marks** Obtain perimeter as 21.6 (allow 21.7). Allow any answer in range $21.6 \leq \text{perimeter} \leq 21.7$. |
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\includegraphics[max width=\textwidth, alt={}, center]{c52fe7e9-0442-4b3e-b924-2e5e4b3e98f5-03_420_729_1027_708}
The diagram shows a sector $A O B$ of a circle with centre $O$ and radius 5 cm . Angle $A O B$ is $\theta$ radians. The area of triangle $A O B$ is $8 \mathrm {~cm} ^ { 2 }$.\\
(i) Given that the angle $\theta$ is obtuse, find $\theta$.
The shaded segment in the diagram is bounded by the chord $A B$ and the arc $A B$.\\
(ii) Find the area of the segment, giving your answer correct to 3 significant figures.\\
(iii) Find the perimeter of the segment, giving your answer correct to 3 significant figures.
\hfill \mbox{\textit{OCR C2 2011 Q8 [10]}}