| Exam Board | AQA |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2005 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Express and solve equation |
| Difficulty | Moderate -0.3 This is a standard C4 harmonic form question with routine application of the R sin(x+α) method. Part (a) requires finding R and α using standard formulas (R²=a²+b², tan α=b/a), and part (b) involves solving a straightforward equation once the harmonic form is established. While it requires multiple steps, the techniques are well-practiced and follow a predictable pattern, 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) \(R = \sqrt{5}\) (or \(\sqrt{1 + 2^2}\) or 2.23 or 2.24) | B1 | |
| \(\frac{\sin \alpha}{\cos \alpha} = \frac{1}{2}\), \(\alpha = 26.6°\) | M1A1 | CAO; SC 63.4° \(\frac{1}{2}\) |
| (b) \(\sin(x + 26.6) = \frac{1}{\sqrt{5}}\) | M1 | |
| \(x = \sin^{-1}\left(\frac{1}{\sqrt{5}}\right) - 26.6\) | m1 | |
| \(x = 0°\), \(x = 126.8°\) (126.9°, or 127° or 126° with working) | B1, A1 | B1 \(x = 0°\) lose if extra solution in range; SC calculator trace: 126.9 full marks; SC ft from 63.4° |
**(a)** $R = \sqrt{5}$ (or $\sqrt{1 + 2^2}$ or 2.23 or 2.24) | B1 |
$\frac{\sin \alpha}{\cos \alpha} = \frac{1}{2}$, $\alpha = 26.6°$ | M1A1 | CAO; SC 63.4° $\frac{1}{2}$
**(b)** $\sin(x + 26.6) = \frac{1}{\sqrt{5}}$ | M1 |
$x = \sin^{-1}\left(\frac{1}{\sqrt{5}}\right) - 26.6$ | m1 |
$x = 0°$, $x = 126.8°$ (126.9°, or 127° or 126° with working) | B1, A1 | B1 $x = 0°$ lose if extra solution in range; SC calculator trace: 126.9 full marks; SC ft from 63.4°
**Total: 7 marks**
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\begin{enumerate}[label=(\alph*)]
\item Express $2 \sin x + \cos x$ in the form $R \sin ( x + \alpha )$ where $R$ is a positive constant and $\alpha$ is an acute angle. Give your value of $\alpha$ to the nearest $0.1 ^ { \circ }$.
\item Solve the equation $2 \sin x + \cos x = 1$ for $0 ^ { \circ } \leqslant x < 360 ^ { \circ }$.
\end{enumerate}
\hfill \mbox{\textit{AQA C4 2005 Q1 [7]}}