| Exam Board | SPS |
|---|---|
| Module | SPS SM (SPS SM) |
| Year | 2021 |
| Session | November |
| Marks | 11 |
| Topic | Harmonic Form |
| Type | Range of rational function with harmonic denominator |
| Difficulty | Standard +0.3 This is a standard A-level trigonometry question covering the R-formula technique (harmonic form), which is a routine topic in C3/C4. Part (a) is textbook application of the R-formula, part (b)(i) is straightforward solving once in harmonic form, part (c)(i) requires identifying transformations (standard but slightly more thought needed), and part (c)(ii) applies range knowledge to find max/min of a reciprocal function. All parts follow predictable patterns with no novel problem-solving required, making it slightly easier than average overall. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc |
\begin{enumerate}[label=(\alph*)]
\item Express $2\sqrt{3} \cos 2x - 6 \sin 2x$ in the form $R\cos(2x + \alpha)$ where $R > 0$ and $0 < \alpha < \frac{\pi}{2}$ [3]
\item Hence
\begin{enumerate}[label=(\roman*)]
\item Solve the equation $2\sqrt{3} \cos 2x - 6 \sin 2x = 6$ for $0 \leq x \leq 2\pi$
Giving your answers in terms of $\pi$. [3]
\end{enumerate}
\item It can be shown that $y = 9 \sin 2x + 4 \cos 2x$ can be written as $y = \sqrt{97} \sin(2x + 24.0°)$
\begin{enumerate}[label=(\roman*)]
\item State the transformations in the order of occurrence which transform the curve $y = 9 \sin 2x + 4 \cos 2x$ to the curve $y = \sin x$ [3]
\item Find the exact maximum and minimum values of the function;
$$f(x) = \frac{1}{11 - 9 \sin 2x - 4 \cos 2x}$$ [2]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM 2021 Q8 [11]}}