| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Session | Specimen |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Find exact trigonometric values |
| Difficulty | Moderate -0.3 Part (i) requires applying addition formulae and collecting terms—a standard identity proof with clear structure. Part (ii) is a straightforward substitution (x=-15°) once part (i) is established. This is typical C3 material testing routine application of addition formulae rather than problem-solving insight, making it slightly easier than average. |
| Spec | 1.01a Proof: structure of mathematical proof and logical steps1.05g Exact trigonometric values: for standard angles1.05l Double angle formulae: and compound angle formulae |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\sin x(\frac{1}{3}\sqrt{3}) + \cos x(\frac{1}{3}) + (\sqrt{3})(\cos x(\frac{1}{3}\sqrt{3}) - \sin x(\frac{1}{3}))\) | M1 | For expanding both compound angles |
| \(= \frac{1}{3}\cos x + \frac{1}{3}\cos x = 2\cos x\), as required | A1, M1, A1 | For completely correct expansion; For using exact values of sin 30° and cos 30°; For showing given answer correctly |
| (ii) \(\sin 45° + (\sqrt{3})\cos 45° = 2\cos 15°\) Hence \(\cos 15° = \frac{1 + \sqrt{3}}{2\sqrt{2}}\) | M1, A1 | For letting \(x = 15°\) throughout; For any correct exact form |
**(i)** $\sin x(\frac{1}{3}\sqrt{3}) + \cos x(\frac{1}{3}) + (\sqrt{3})(\cos x(\frac{1}{3}\sqrt{3}) - \sin x(\frac{1}{3}))$ | M1 | For expanding both compound angles
$= \frac{1}{3}\cos x + \frac{1}{3}\cos x = 2\cos x$, as required | A1, M1, A1 | For completely correct expansion; For using exact values of sin 30° and cos 30°; For showing given answer correctly
**(ii)** $\sin 45° + (\sqrt{3})\cos 45° = 2\cos 15°$ Hence $\cos 15° = \frac{1 + \sqrt{3}}{2\sqrt{2}}$ | M1, A1 | For letting $x = 15°$ throughout; For any correct exact form
**Total for Question 2: 6 marks**
---2 (i) Prove the identity
$$\sin \left( x + 30 ^ { \circ } \right) + ( \sqrt { } 3 ) \cos \left( x + 30 ^ { \circ } \right) \equiv 2 \cos x$$
where $x$ is measured in degrees.\\
(ii) Hence express $\cos 15 ^ { \circ }$ in surd form.
\hfill \mbox{\textit{OCR C3 Q2 [6]}}