| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2011 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard trigonometric equations |
| Type | General solution — then find specific solutions |
| Difficulty | Moderate -0.3 This is a straightforward Further Maths question requiring standard technique for general solutions of trigonometric equations. Part (a) involves recognizing cos θ = √3/2 gives θ = ±π/6 + 2nπ, then solving for x. Part (b) is simple substitution to find a specific value. While it's FP1, the method is routine and requires no problem-solving insight, making it slightly easier than average overall. |
| Spec | 1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\cos\frac{\pi}{6} = \frac{\sqrt{3}}{2}\) | B1 | OE stated or used; deg/dec penalised at 5th mark |
| \(\cos\left(-\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}\) | B1F | OE; ft wrong first value |
| Introduction of \(2n\pi\) | M1 | (or \(n\pi\)) at any stage |
| Going from \(3x - \frac{\pi}{6}\) to \(x\) | m1 | incl division of all terms by 3 |
| GS: \(x = \frac{\pi}{18} \pm \frac{\pi}{18} + \frac{2}{3}n\pi\) | A1F | ft wrong first value |
| Total | 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(n = 8\) will give the required solution | M1 | GS must include \(\frac{2}{3}n\pi\) for this |
| ... which is \(\frac{16}{3}\pi\) \((\approx 16.755)\) | A1 | from correct GS; allow \(\frac{48}{9}\pi\) or dec approx |
| Total | 2 |
## Question 5:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\cos\frac{\pi}{6} = \frac{\sqrt{3}}{2}$ | B1 | OE stated or used; deg/dec penalised at 5th mark |
| $\cos\left(-\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}$ | B1F | OE; ft wrong first value |
| Introduction of $2n\pi$ | M1 | (or $n\pi$) at any stage |
| Going from $3x - \frac{\pi}{6}$ to $x$ | m1 | incl division of all terms by 3 |
| GS: $x = \frac{\pi}{18} \pm \frac{\pi}{18} + \frac{2}{3}n\pi$ | A1F | ft wrong first value |
| **Total** | **5** | |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $n = 8$ will give the required solution | M1 | GS must include $\frac{2}{3}n\pi$ for this |
| ... which is $\frac{16}{3}\pi$ $(\approx 16.755)$ | A1 | from correct GS; allow $\frac{48}{9}\pi$ or dec approx |
| **Total** | **2** | |
---5
\begin{enumerate}[label=(\alph*)]
\item Find the general solution of the equation
$$\cos \left( 3 x - \frac { \pi } { 6 } \right) = \frac { \sqrt { 3 } } { 2 }$$
giving your answer in terms of $\pi$.
\item Use your general solution to find the smallest solution of this equation which is greater than $5 \pi$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2011 Q5 [7]}}