| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Session | Specimen |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sine and Cosine Rules |
| Type | Triangle with additional point or sector |
| Difficulty | Standard +0.3 This is a straightforward multi-part question testing standard C2 content: sine rule for finding an angle, triangle area formula (½ab sin C), and sector perimeter/area formulas. All parts require direct application of memorized formulas with no problem-solving insight needed. Slightly above average difficulty only due to the multi-step nature and radian mode requirement. |
| Spec | 1.05b Sine and cosine rules: including ambiguous case1.05c 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 |
|---|---|---|
| \(\frac{3}{\sin C} = \frac{5}{\sin 2.1} \Rightarrow \sin C = \frac{3}{5}\sin 2.1\) | M1 | For any correct initial statement of the sine rule, together with an attempt to find \(\sin C\) |
| A1 | 2 | For correct value |
| Hence \(C = 0.544\) |
| Answer | Marks | Guidance |
|---|---|---|
| Angle \(A\) is \(\pi - 2.1 - 0.5444 = 0.4972\) | M1 | For calculation of angle \(A\) |
| M1 | For any complete method for the area | |
| Area is \(\frac{1}{2} \times 5 \times 3 \times \sin 0.4972\) | ||
| i.e. \(3.58\) cm² | A1′ | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Sector perimeter is \(6 + 3 \times 0.4972\) i.e. \(7.49\) cm | M1 | For using \(r\theta\) with their \(A\) in radians |
| A1† | For correct value, following their \(A\) | |
| Sector area is \(\frac{1}{2} \times 3^2 \times 0.4972\) i.e. \(2.24\) cm² | M1 | For using \(\frac{1}{2}r^2\theta\) with their \(A\) in radians |
| A1′ | 4 | For correct value, following their \(A\) |
| Total: 9 |
## Part (i)
$\frac{3}{\sin C} = \frac{5}{\sin 2.1} \Rightarrow \sin C = \frac{3}{5}\sin 2.1$ | M1 | For any correct initial statement of the sine rule, together with an attempt to find $\sin C$
| A1 | 2 | For correct value
Hence $C = 0.544$ | |
## Part (ii)
Angle $A$ is $\pi - 2.1 - 0.5444 = 0.4972$ | M1 | For calculation of angle $A$
| M1 | For any complete method for the area
Area is $\frac{1}{2} \times 5 \times 3 \times \sin 0.4972$ | |
i.e. $3.58$ cm² | A1′ | 3 | For correct value, following their $C$
## Part (iii)
Sector perimeter is $6 + 3 \times 0.4972$ i.e. $7.49$ cm | M1 | For using $r\theta$ with their $A$ in radians
| A1† | | For correct value, following their $A$
Sector area is $\frac{1}{2} \times 3^2 \times 0.4972$ i.e. $2.24$ cm² | M1 | For using $\frac{1}{2}r^2\theta$ with their $A$ in radians
| A1′ | 4 | For correct value, following their $A$
| **Total: 9** | |\includegraphics{figure_6}
The diagram shows triangle $ABC$, in which $AB = 3$ cm, $AC = 5$ cm and angle $ABC = 2.1$ radians. Calculate
\begin{enumerate}[label=(\roman*)]
\item angle $ACB$, giving your answer in radians, [2]
\item the area of the triangle. [3]
\end{enumerate}
An arc of a circle with centre $A$ and radius 3 cm is drawn, cutting $AC$ at the point $D$.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{2}
\item Calculate the perimeter and the area of the sector $ABD$. [4]
\end{enumerate}
\hfill \mbox{\textit{OCR C2 Q6 [9]}}