| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2009 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trigonometric equations in context |
| Type | Reduce to quadratic in trig |
| Difficulty | Moderate -0.3 Part (i) is a straightforward double angle equation requiring basic knowledge of sin(2x) and reference angles. Part (ii) involves rearranging to a quadratic in sin(x) using the Pythagorean identity, then solving—a standard C2 technique but requires more steps and algebraic manipulation than typical routine questions. Overall slightly easier than average due to limited range and standard methods. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| \(2x = 30°, 150°\) | M1 | Attempt \(\sin^{-1} 0.5\), then divide or multiply by 2 |
| \(x = 15°, 75°\) | A1 | Obtain \(15°\) (allow \(\frac{\pi}{12}\) or \(0.262\)) |
| A1 (3) | Obtain \(75°\) (not radians), and no extra solutions in range |
| Answer | Marks | Guidance |
|---|---|---|
| \(2(1-\cos^2 x) = 2 - \sqrt{3}\cos x\) | M1 | Use \(\sin^2 x = 1 - \cos^2 x\) |
| \(2\cos^2 x - \sqrt{3}\cos x = 0\) | A1 | Obtain \(2\cos^2 x - \sqrt{3}\cos x = 0\) or equiv (no constant terms) |
| \(\cos x(2\cos x - \sqrt{3}) = 0\) | M1 | Attempt to solve quadratic in \(\cos x\) |
| \(\cos x = 0,\ \cos x = \frac{1}{2}\sqrt{3}\) | A1 | Obtain \(30°\) (allow \(\frac{\pi}{6}\) or \(0.524\)), and no extra solutions in range |
| \(x = 90°,\ x = 30°\) | B1 (5) | Obtain \(90°\) (allow \(\frac{\pi}{2}\) or \(1.57\)), from correct quadratic only |
| SR | Answer only: B1 one correct solution, B1 second correct solution and no others |
## Question 5:
### Part (i):
| $2x = 30°, 150°$ | M1 | Attempt $\sin^{-1} 0.5$, then divide or multiply by 2 |
|---|---|---|
| $x = 15°, 75°$ | A1 | Obtain $15°$ (allow $\frac{\pi}{12}$ or $0.262$) |
| | A1 **(3)** | Obtain $75°$ (not radians), and no extra solutions in range |
### Part (ii):
| $2(1-\cos^2 x) = 2 - \sqrt{3}\cos x$ | M1 | Use $\sin^2 x = 1 - \cos^2 x$ |
|---|---|---|
| $2\cos^2 x - \sqrt{3}\cos x = 0$ | A1 | Obtain $2\cos^2 x - \sqrt{3}\cos x = 0$ or equiv (no constant terms) |
| $\cos x(2\cos x - \sqrt{3}) = 0$ | M1 | Attempt to solve quadratic in $\cos x$ |
| $\cos x = 0,\ \cos x = \frac{1}{2}\sqrt{3}$ | A1 | Obtain $30°$ (allow $\frac{\pi}{6}$ or $0.524$), and no extra solutions in range |
| $x = 90°,\ x = 30°$ | B1 **(5)** | Obtain $90°$ (allow $\frac{\pi}{2}$ or $1.57$), from correct quadratic only |
| | SR | Answer only: B1 one correct solution, B1 second correct solution and no others |
---5 Solve each of the following equations for $0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }$.\\
(i) $\sin 2 x = 0.5$\\
(ii) $2 \sin ^ { 2 } x = 2 - \sqrt { 3 } \cos x$
\hfill \mbox{\textit{OCR C2 2009 Q5 [8]}}