| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2023 |
| Session | October |
| Marks | 12 |
| Topic | Harmonic Form |
| Type | Solve equation directly given harmonic form |
| Difficulty | Moderate -0.3 This is a standard harmonic form (R cos(x + α)) question requiring the formula R² = a² + b², tan α = b/a, then solving a trigonometric equation and finding minimum values. While it has multiple parts (12 marks total), each step follows a well-established procedure taught in C3/C4 with no novel problem-solving required. Slightly easier than average due to its routine nature, though the multi-part structure and mark allocation suggest it's a substantial question. |
| Spec | 1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc1.05o Trigonometric equations: solve in given intervals |
$$f(x) = 12 \cos x - 4 \sin x.$$
Given that $f(x) = R \cos(x + \alpha)$, where $R \geq 0$ and $0 \leq \alpha \leq 90°$,
\begin{enumerate}[label=(\alph*)]
\item find the value of $R$ and the value of $\alpha$. [4]
\item Hence solve the equation
$$12 \cos x - 4 \sin x = 7$$
for $0 \leq x < 360°$, giving your answers to one decimal place. [5]
\item
\begin{enumerate}[label=(\roman*)]
\item Write down the minimum value of $12 \cos x - 4 \sin x$. [1]
\item Find, to 2 decimal places, the smallest positive value of $x$ for which this minimum value occurs. [2]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2023 Q4 [12]}}