| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2005 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trig Graphs & Exact Values |
| Type | Find general solution of trig equation |
| Difficulty | Standard +0.3 This is a straightforward Further Pure 1 question requiring standard technique: recognize 1/√2 = cos(π/4), solve the compound angle equation using the general solution formula cos(θ)=cos(α) gives θ=±α+2nπ, then count roots in an interval. While it's FP1, it follows a routine algorithmic approach with exact values that students practice extensively. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Attempt at \(\cos^{-1}\frac{1}{\sqrt{2}}\) | M1 | Allow degrees or decimals |
| \(\frac{\pi}{4}\) appearing in solution | A1 | Must be exact |
| Introduction of \(\pm\) | M1 | |
| Introduction of \(\ldots + 2n\pi\) | M1 | Or \(360n\) |
| Making \(x\) the subject | M1 | From \(2x + \frac{\pi}{6} = kn\pi + \alpha\) (or \(\pm\alpha\)) |
| \(x = -\frac{\pi}{12} \pm \frac{\pi}{8} + n\pi\) | A1 (6 marks) | OE |
| Answer | Marks | Guidance |
|---|---|---|
| Number of roots is 4 | M1A1ft (2 marks) | M1 e.g. for answer consistent with candidate's general solution |
## Question 6:
**Part (a)**
Attempt at $\cos^{-1}\frac{1}{\sqrt{2}}$ | M1 | Allow degrees or decimals
$\frac{\pi}{4}$ appearing in solution | A1 | Must be exact
Introduction of $\pm$ | M1 |
Introduction of $\ldots + 2n\pi$ | M1 | Or $360n$
Making $x$ the subject | M1 | From $2x + \frac{\pi}{6} = kn\pi + \alpha$ (or $\pm\alpha$)
$x = -\frac{\pi}{12} \pm \frac{\pi}{8} + n\pi$ | A1 (6 marks) | OE
**Part (b)**
Number of roots is 4 | M1A1ft (2 marks) | M1 e.g. for answer consistent with candidate's general solution
---6 The angle $x$ radians satisfies the equation
$$\cos \left( 2 x + \frac { \pi } { 6 } \right) = \frac { 1 } { \sqrt { 2 } }$$
\begin{enumerate}[label=(\alph*)]
\item Find the general solution of this equation, giving the roots as exact values in terms of $\pi$.
\item Find the number of roots of the equation which lie between 0 and $2 \pi$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2005 Q6 [8]}}