| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2023 |
| Session | June |
| Marks | 6 |
| Topic | Harmonic Form |
| Type | Express and solve equation |
| Difficulty | Moderate -0.3 This is a standard Further Maths harmonic form question with routine application. Part (i) uses the textbook method R cos(x-α) = R cos α cos x + R sin α sin x to find R = √(64+25) = √89 and tan α = 5/8. Part (ii) is straightforward solving R cos(x-α) = 6, giving two solutions in the range. While it requires multiple steps and careful arithmetic, it follows a well-practiced algorithm with no novel insight needed, making it slightly easier than average. |
| Spec | 1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc1.05o Trigonometric equations: solve in given intervals |
In this question you must show detailed reasoning.
\begin{enumerate}[label=(\roman*)]
\item Express $8\cos x + 5\sin x$ in the form $R\cos(x - \alpha)$, where $R$ and $\alpha$ are constants with $R > 0$ and $0 < \alpha < \frac{\pi}{2}$. [3]
\item Hence solve the equation $8\cos x + 5\sin x = 6$ for $0 \leqslant x < 2\pi$, giving your answers correct to 4 decimal places. [3]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2023 Q2 [6]}}