| Exam Board | AQA |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2011 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Find value where max/min occurs |
| Difficulty | Moderate -0.3 This is a standard harmonic form question requiring routine application of the R sin(x+α) formula. Part (a) involves straightforward use of R=√(a²+b²) and tan α=b/a, while parts (b)(i) and (b)(ii) follow directly from understanding that the maximum of R sin(x+α) is R, occurring when sin(x+α)=1. This is a textbook exercise 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) etc |
| Answer | Marks | Guidance |
|---|---|---|
| \(R\sin(x+\alpha) = R\sin x\cos\alpha + R\cos x\sin\alpha\) | M1 | Expanding and comparing coefficients |
| \(R\cos\alpha = 2\), \(R\sin\alpha = 5\) | ||
| \(R = \sqrt{4+25} = \sqrt{29}\) | A1 | |
| \(\alpha = \arctan\left(\frac{5}{2}\right) = 68.2°\) | A1 | Accept 68.2° |
| Answer | Marks | Guidance |
|---|---|---|
| Maximum value \(= \sqrt{29}\) | B1 | ft their \(R\) |
| Answer | Marks | Guidance |
|---|---|---|
| Maximum when \(\sin(x + 68.2°) = 1\), so \(x + 68.2° = 90°\) | M1 | |
| \(x = 21.8°\) | A1 | ft their \(\alpha\) |
# Question 1:
## Part (a)
| $R\sin(x+\alpha) = R\sin x\cos\alpha + R\cos x\sin\alpha$ | M1 | Expanding and comparing coefficients |
|---|---|---|
| $R\cos\alpha = 2$, $R\sin\alpha = 5$ | | |
| $R = \sqrt{4+25} = \sqrt{29}$ | A1 | |
| $\alpha = \arctan\left(\frac{5}{2}\right) = 68.2°$ | A1 | Accept 68.2° |
## Part (b)(i)
| Maximum value $= \sqrt{29}$ | B1 | ft their $R$ |
|---|---|---|
## Part (b)(ii)
| Maximum when $\sin(x + 68.2°) = 1$, so $x + 68.2° = 90°$ | M1 | |
|---|---|---|
| $x = 21.8°$ | A1 | ft their $\alpha$ |
---1
\begin{enumerate}[label=(\alph*)]
\item Express $2 \sin x + 5 \cos x$ in the form $R \sin ( x + \alpha )$, where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$. Give your value of $\alpha$ to the nearest $0.1 ^ { \circ }$.
\item \begin{enumerate}[label=(\roman*)]
\item Write down the maximum value of $2 \sin x + 5 \cos x$.
\item Find the value of $x$ in the interval $0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }$ at which this maximum occurs, giving the value of $x$ to the nearest $0.1 ^ { \circ }$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C4 2011 Q1 [6]}}