| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2014 |
| 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 FP1 trigonometric equation requiring standard techniques: solving cos(θ) = √2/2, writing the general solution with ±2nπ, then identifying which solutions fall in a given interval and summing them. While it involves multiple steps and careful arithmetic with fractions of π, it follows a completely standard algorithm taught in FP1 with no novel insight required. Slightly above average difficulty due to the fractional coefficient and the summation part, but still routine for Further Maths students. |
| Spec | 1.05g Exact trigonometric values: for standard angles1.05o Trigonometric equations: solve in given intervals |
\begin{enumerate}[label=(\alph*)]
\item Find the general solution of the equation
$$\cos\left(\frac{5}{4}x - \frac{\pi}{3}\right) = \frac{\sqrt{2}}{2}$$
giving your answer for $x$ in terms of $\pi$. [5 marks]
\item Use your general solution to find the sum of all the solutions of the equation
$$\cos\left(\frac{5}{4}x - \frac{\pi}{3}\right) = \frac{\sqrt{2}}{2}$$
that lie in the interval $0 \leqslant x \leqslant 20\pi$. Give your answer in the form $k\pi$, stating the exact value of $k$. [4 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2014 Q8 [9]}}