| Exam Board | AQA |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2009 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Find value where max/min occurs |
| Difficulty | Standard +0.3 This is a standard harmonic form question requiring routine application of the R sin(x - α) formula, followed by straightforward reading of minimum value and solving a simple equation. The technique is well-practiced in C4, though it involves multiple steps and radian arithmetic, making it slightly above average difficulty. |
| 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 |
|---|---|---|
| \(R = \sqrt{10}\), \(\tan\alpha = 3\), \(\alpha = 1.25\) | B1, M1, A1 (3 marks) | Accept \(R = 3.16\) or better; OE (can be implied by \(71.57°\)); A0 if extra answers within given range; SC1 \(\tan\alpha = \frac{1}{3}\), \(\alpha = 0.32\) |
| Answer | Marks | Guidance |
|---|---|---|
| min value \(= -\sqrt{10}\) \(\left(\text{or} \geq \sqrt{-10}\right)\) | B1F (1 mark) | ft on \(R\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\sin(x - \alpha) = -1\), \(x = 5.96\) | M1, A1F (2 marks) | or \(\sin^{-1}\frac{3\pi}{2}\); ft on their \(\alpha\) (to 2 dp) \(+ \frac{3\pi}{2}\) |
## Question 2:
### Part (a)
| $R = \sqrt{10}$, $\tan\alpha = 3$, $\alpha = 1.25$ | B1, M1, A1 (3 marks) | Accept $R = 3.16$ or better; OE (can be implied by $71.57°$); A0 if extra answers within given range; SC1 $\tan\alpha = \frac{1}{3}$, $\alpha = 0.32$ |
### Part (b)(i)
| min value $= -\sqrt{10}$ $\left(\text{or} \geq \sqrt{-10}\right)$ | B1F (1 mark) | ft on $R$ |
### Part (b)(ii)
| $\sin(x - \alpha) = -1$, $x = 5.96$ | M1, A1F (2 marks) | or $\sin^{-1}\frac{3\pi}{2}$; ft on their $\alpha$ (to 2 dp) $+ \frac{3\pi}{2}$ |
---2
\begin{enumerate}[label=(\alph*)]
\item Express $\sin x - 3 \cos x$ in the form $R \sin ( x - \alpha )$, where $R > 0$ and $0 < \alpha < \frac { \pi } { 2 }$. Give your value of $\alpha$ in radians to two decimal places.
\item Hence:
\begin{enumerate}[label=(\roman*)]
\item write down the minimum value of $\sin x - 3 \cos x$;
\item find the value of $x$ in the interval $0 < x < 2 \pi$ at which this minimum value occurs, giving your value of $x$ in radians to two decimal places.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C4 2009 Q2 [6]}}