| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2009 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard trigonometric equations |
| Type | General solution — then find specific solutions |
| Difficulty | Standard +0.3 This is a straightforward Further Maths question on general solutions of trig equations. Part (a) requires standard technique (solving cos θ = 1/2 then back-substituting), and part (b) is routine inequality work to find specific solutions in a range. While it's FP1 content, the execution is mechanical with no conceptual challenges, making it slightly easier than average overall. |
| Spec | 1.05g Exact trigonometric values: for standard angles1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| \(\cos(3x - \pi) = \frac{1}{2}\), so \(3x - \pi = \pm\frac{\pi}{3} + 2n\pi\) | M1 A1 | |
| \(3x = \pi \pm \frac{\pi}{3} + 2n\pi\) | M1 | |
| \(x = \frac{\pi}{3} \pm \frac{\pi}{9} + \frac{2n\pi}{3}\) | A1 | |
| General solutions: \(x = \frac{4\pi}{9} + \frac{2n\pi}{3}\) and \(x = \frac{2\pi}{9} + \frac{2n\pi}{3}\) | A1 A1 | Both required |
| Answer | Marks |
|---|---|
| Testing integer values of \(n\) to find solutions between \(10\pi\) and \(11\pi\) | M1 |
| \(x = \frac{4\pi}{9} + \frac{2n\pi}{3}\): try \(n=15\): \(x = \frac{4\pi}{9} + 10\pi = \frac{94\pi}{9}\) ✓ | A1 |
| \(x = \frac{2\pi}{9} + \frac{2n\pi}{3}\): try \(n=15\): \(x = \frac{2\pi}{9} + 10\pi = \frac{92\pi}{9}\) ✓ | A1 |
# Question 5:
## Part (a)
| $\cos(3x - \pi) = \frac{1}{2}$, so $3x - \pi = \pm\frac{\pi}{3} + 2n\pi$ | M1 A1 | |
| $3x = \pi \pm \frac{\pi}{3} + 2n\pi$ | M1 | |
| $x = \frac{\pi}{3} \pm \frac{\pi}{9} + \frac{2n\pi}{3}$ | A1 | |
| General solutions: $x = \frac{4\pi}{9} + \frac{2n\pi}{3}$ and $x = \frac{2\pi}{9} + \frac{2n\pi}{3}$ | A1 A1 | Both required |
## Part (b)
| Testing integer values of $n$ to find solutions between $10\pi$ and $11\pi$ | M1 | |
| $x = \frac{4\pi}{9} + \frac{2n\pi}{3}$: try $n=15$: $x = \frac{4\pi}{9} + 10\pi = \frac{94\pi}{9}$ ✓ | A1 | |
| $x = \frac{2\pi}{9} + \frac{2n\pi}{3}$: try $n=15$: $x = \frac{2\pi}{9} + 10\pi = \frac{92\pi}{9}$ ✓ | A1 | |5
\begin{enumerate}[label=(\alph*)]
\item Find the general solution of the equation
$$\cos ( 3 x - \pi ) = \frac { 1 } { 2 }$$
giving your answer in terms of $\pi$.
\item From your general solution, find all the solutions of the equation which lie between $10 \pi$ and $11 \pi$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2009 Q5 [9]}}