| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2021 |
| Session | June |
| Marks | 7 |
| Topic | Function Transformations |
| Type | Stationary points after transformation |
| Difficulty | Standard +0.3 This question involves routine trigonometric concepts: identifying minimum points of a cosine function (part a), applying horizontal stretches and translations (part b), and solving a trigonometric equation by converting to a single function (part c). While part (c) requires algebraic manipulation and calculator work across a specified range, all techniques are standard A-level fare with no novel problem-solving required. The 7 total marks and straightforward structure place this slightly easier than average. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.05o Trigonometric equations: solve in given intervals |
\includegraphics{figure_3}
Figure 3 shows part of the curve with equation $y = 3\cos x^2$.
The point $P(c, d)$ is a minimum point on the curve with $c$ being the smallest negative value of $x$ at which a minimum occurs.
\begin{enumerate}[label=(\alph*)]
\item State the value of $c$ and the value of $d$. [1]
\item State the coordinates of the point to which $P$ is mapped by the transformation which transforms the curve with equation $y = 3\cos x^2$ to the curve with equation
\begin{enumerate}[label=(\roman*)]
\item $y = 3\cos\left(\frac{x^2}{4}\right)$
\item $y = 3\cos(x - 36)^2$
\end{enumerate}
[2]
\item Solve, for $450° \leq \theta < 720°$,
$$3\cos\theta = 8\tan\theta$$
giving your solution to one decimal place. [4]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2021 Q9 [7]}}