| Exam Board | OCR MEI |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2015 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Inverse trigonometric function equations |
| Difficulty | Moderate -0.8 Both parts are straightforward inverse trig equations requiring only basic rearrangement and knowledge of special angle values. Part (i) is simple algebra to isolate x, giving x=sin(π/6)=1/2. Part (ii) uses the identity that arcsin x = arccos x when x=√2/2, or can be solved by taking sin/cos of both sides. No multi-step problem-solving or novel insight required—this is below-average difficulty for C3. |
| Spec | 1.05i Inverse trig functions: arcsin, arccos, arctan domains and graphs |
| Answer | Marks |
|---|---|
| \(\arcsin x = \pi/6 \Rightarrow x = \sin \pi/6 = \frac{1}{2}\) | M1 |
| A1 | allow unsupported answers |
| Answer | Marks | Guidance |
|---|---|---|
| \(\sin \pi/4 = \cos \pi/4 = 1/\sqrt{2}\) | ||
| \(\Rightarrow \arcsin(1/\sqrt{2}) = \arccos(1/\sqrt{2})\) | ||
| \(\Rightarrow x = 1/\sqrt{2}\) | B2 | o.e. e.g. \(\sqrt{2}/2\), must be exact; SCB1 0.707… |
**(i)**
$\arcsin x = \pi/6 \Rightarrow x = \sin \pi/6 = \frac{1}{2}$ | M1 |
| A1 | allow unsupported answers
**Total: [2]**
**(ii)**
$\sin \pi/4 = \cos \pi/4 = 1/\sqrt{2}$ | |
$\Rightarrow \arcsin(1/\sqrt{2}) = \arccos(1/\sqrt{2})$ | |
$\Rightarrow x = 1/\sqrt{2}$ | B2 | o.e. e.g. $\sqrt{2}/2$, must be exact; SCB1 0.707…
**Total: [2]**
---6 Solve each of the following equations, giving your answers in exact form.\\
(i) $6 \arcsin x - \pi = 0$.\\
(ii) $\arcsin x = \arccos x$.
\hfill \mbox{\textit{OCR MEI C3 2015 Q6 [4]}}