| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2007 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Express and solve equation |
| Difficulty | Standard +0.3 This is a standard harmonic form question requiring the routine technique of expressing acos(x) + bsin(x) as Rsin(x+α) using R=√(a²+b²) and tan(α)=a/b, followed by solving a straightforward trigonometric equation. While it requires multiple steps, both parts follow well-practiced procedures taught explicitly in C3 with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(R^2 = (\sqrt{3})^2 + 1^2 \Rightarrow R = 2\); \(\tan\alpha = \sqrt{3} \Rightarrow \alpha = \frac{\pi}{3}\); accept awrt 1.05 | M1, A1, M1, A1 | (4 marks) |
| (b) \(\sin(x + \text{their }\alpha) = \frac{1}{2}\); \(x + \text{their }\alpha = \frac{\pi}{6}\left(\frac{5\pi}{6}, \frac{13\pi}{6}\right)\); \(x = \frac{\pi}{2}, \frac{11\pi}{6}\); accept awrt 1.57, 5.76 | M1, A1, M1, A1 | (4 marks) |
(a) $R^2 = (\sqrt{3})^2 + 1^2 \Rightarrow R = 2$; $\tan\alpha = \sqrt{3} \Rightarrow \alpha = \frac{\pi}{3}$; accept awrt 1.05 | M1, A1, M1, A1 | (4 marks)
(b) $\sin(x + \text{their }\alpha) = \frac{1}{2}$; $x + \text{their }\alpha = \frac{\pi}{6}\left(\frac{5\pi}{6}, \frac{13\pi}{6}\right)$; $x = \frac{\pi}{2}, \frac{11\pi}{6}$; accept awrt 1.57, 5.76 | M1, A1, M1, A1 | (4 marks)
**Guidance:** The use of degrees loses only one mark in this question. Penalise the first time it occurs in an answer and then ignore.
---5.
\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\includegraphics[alt={},max width=\textwidth]{a4ad749b-181b-4680-8771-94d9b581125a-07_865_926_301_516}
\end{center}
\end{figure}
Figure 1 shows an oscilloscope screen.
The curve shown on the screen satisfies the equation
$$y = \sqrt { 3 } \cos x + \sin x$$
\begin{enumerate}[label=(\alph*)]
\item Express the equation of the curve in the form $y = R \sin ( x + \alpha )$, where $R$ and $\alpha$ are constants, $R > 0$ and $0 < \alpha < \frac { \pi } { 2 }$.
\item Find the values of $x , 0 \leqslant x < 2 \pi$, for which $y = 1$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 2007 Q5 [8]}}