| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2021 |
| Session | June |
| Marks | 9 |
| Topic | Harmonic Form |
| Type | Range of squared harmonic expression |
| Difficulty | Standard +0.3 Part (a) requires expressing 3cos θ + 3sin θ in the form R cos(θ - α), which is a standard A-level technique, then identifying the transformations (stretch and translation). Part (b) applies this result to find max/min values of a quadratic expression, which is straightforward once (a) is complete. This is a typical Further Maths question testing standard methods with clear structure, slightly above average difficulty due to the multi-step reasoning and justification required. |
| 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 Determine a sequence of transformations which maps the graph of $y = \cos \theta$ onto the graph of $y = 3\cos \theta + 3\sin \theta$
Fully justify your answer.
[6 marks]
\item Hence or otherwise find the least value and greatest value of
$$4 + (3\cos \theta + 3\sin \theta)^2$$
Fully justify your answer.
[3 marks]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2021 Q7 [9]}}