| Exam Board | AQA |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2006 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Find exact trigonometric values |
| Difficulty | Moderate -0.3 This is a standard R-formula (harmonic form) question requiring routine application of the compound angle method: expand R cos(θ+α), equate coefficients to find R=√13 and tan α=2/3, then read off the maximum. It's slightly easier than average because it's a textbook template question with clear signposting through parts (a), (b), (c), requiring no problem-solving insight beyond knowing the standard technique. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(R = \sqrt{13}\) or \(3.6\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{\sin\alpha}{\cos\alpha} = \tan\alpha = \frac{2}{3}\), \(\alpha \approx 33.7\) | M1A1 | Allow M1 for \(\tan\alpha = -\frac{2}{3}\) or \(\pm\frac{3}{2}\); AG convincingly obtained |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| maximum value \(= \sqrt{13}\) | B1F | |
| \(\cos(\theta + 33.7) = 1\), \((\theta = -33.7)\) | M1 | |
| \(\theta = 326.3\) | A1 | AWRT 326 |
## Question 3:
### Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $R = \sqrt{13}$ or $3.6$ | B1 | |
### Part (b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{\sin\alpha}{\cos\alpha} = \tan\alpha = \frac{2}{3}$, $\alpha \approx 33.7$ | M1A1 | Allow M1 for $\tan\alpha = -\frac{2}{3}$ or $\pm\frac{3}{2}$; AG convincingly obtained |
### Part (c)
| Answer | Marks | Guidance |
|--------|-------|----------|
| maximum value $= \sqrt{13}$ | B1F | |
| $\cos(\theta + 33.7) = 1$, $(\theta = -33.7)$ | M1 | |
| $\theta = 326.3$ | A1 | AWRT 326 |
---3 It is given that $3 \cos \theta - 2 \sin \theta = R \cos ( \theta + \alpha )$, where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$.
\begin{enumerate}[label=(\alph*)]
\item Find the value of $R$.
\item Show that $\alpha \approx 33.7 ^ { \circ }$.
\item Hence write down the maximum value of $3 \cos \theta - 2 \sin \theta$ and find a positive value of $\theta$ at which this maximum value occurs.
\end{enumerate}
\hfill \mbox{\textit{AQA C4 2006 Q3 [6]}}