It is given that $\mathrm{f}(x) = 2\sin\frac{x}{2}$ and $\mathrm{g}(x) = 3\cos x + 1$, wher

Question

It is given that $\mathrm{f}(x) = 2\sin\frac{x}{2}$ and $\mathrm{g}(x) = 3\cos x + 1$, where $0 \le x \le 2\pi$. State the period of $\mathrm{f}(x)$.[1] State the smallest value of $\mathrm{f}(x)$.[1] State the largest value of $\mathrm{g}(x)$.[1] Sketch on the same axes, the graphs of $y = \mathrm{f}(x)$ and $y = \mathrm{g}(x)$ for $0 \le x \le 2\pi$. Label your graphs clearly. [4] The solutions to the equation $\mathrm{f}(x) = \mathrm{g}(x)$ for $0 \le x \le 2\pi$ are $a$ and $b$, where $a < b$. State, in terms of $a$ and $b$, the range of values of $x$ for which $\mathrm{f}(x) > \mathrm{g}(x)$. [1]

Accepted Answer

(a) $4\pi$ (b) $0$ (c) $4$ (d) $y=\mathrm{f}(x):(0,0),(\pi,2),(2\pi,0)$, smooth half-sine arch above the $x$-axis; $y=\mathrm{g}(x):(0,4),(\frac{\pi}{2},1),(\pi,-2),(\frac{3\pi}{2},1),(2\pi,4)$, smooth cosine curve with midline $y=1$; both curves labelled (e) $a<x<b$

2025 PHS PRELIMS P1 Q10

2025 PHS PRELIMS P1 Q10

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