| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2013 |
| Session | June |
| Marks | 8 |
| Topic | Harmonic Form |
| Type | Express and solve equation |
| Difficulty | Moderate -0.3 This is a standard harmonic form question requiring routine application of the R sin(θ + α) formula with straightforward coefficient matching (R=2, α=π/3) and solving a basic trigonometric equation. While it involves multiple steps, the techniques are well-practiced and require no novel insight, making it slightly easier than average. |
| Spec | 1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc |
**(i)** State or imply use of $R\sin\theta\cos\alpha + R\cos\theta\sin\alpha$ — M1
Obtain $R\cos\alpha = 1$ and $R\sin\alpha = \sqrt{3}$ — A1
Obtain $\alpha = \dfrac{\pi}{3}$ or $60°$ — A1
Obtain $R = 2$ — A1 **[4]**
**(ii)** State $2\sin(\theta + \pi/3) = 0.8$ — B1
Attempt to solve (correct order of operations) — M1
Obtain either: $-0.636$ or $1.68$ — A1
Obtain $1.68$ only — A1 **[4]** **[8]**9 (i) Show that $\sin \theta + \sqrt { 3 } \cos \theta$ can be expressed in the form $R \sin ( \theta + \alpha )$ where $R > 0$ and $0 < \alpha < \frac { 1 } { 2 } \pi$. State the values of $R$ and $\alpha$.\\
(ii) Hence find the value of $\theta$, where $0 < \theta < \pi$, such that $\sin \theta + \sqrt { 3 } \cos \theta = 0.8$.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2013 Q9 [8]}}