| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2024 |
| Session | February |
| Marks | 6 |
| Topic | Trig Graphs & Exact Values |
| Type | Prove or show algebraic identity |
| Difficulty | Moderate -0.3 This question tests standard double-angle formula manipulation (sin²x = (1-cos2x)/2) and solving a basic trigonometric equation. Part (a) is routine algebraic rearrangement, part (b) requires recognizing the period from the transformed equation, and part (c) involves solving a simple linear equation in cos2x. While multi-part, each component uses well-practiced A-level techniques without requiring novel insight or complex problem-solving. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
5.
The diagram shows the graph of $y = 1.5 + \sin ^ { 2 } x$ for $0 \leqslant x \leqslant 2 \pi$.\\
\includegraphics[max width=\textwidth, alt={}, center]{ede204ac-09c3-486b-8877-df935e6ed015-10_513_1266_349_210}
\begin{enumerate}[label=(\alph*)]
\item Show that the equation of the graph can be written in the form $y = a - b \cos 2 x$ where $a$ and $b$ are constants to be determined.
\item Write down the period of the function $1.5 + \sin ^ { 2 } x$.
\item Determine the $x$-coordinates of the points of intersection of the graph of $y = 1.5 + \sin ^ { 2 } x$ with the graph of $y = 1 + \cos 2 x$ in the interval $0 \leqslant x \leqslant 2 \pi$.
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2024 Q5 [6]}}