2025 Ngee Ann Secondary School Prelims A Math Paper 1
Prove that \(\dfrac{1-\cos x+\sin x}{1+\cos x+\sin x}=\tan \dfrac{x}{2}\).[4]
It is given that \(f(x)=1-2\sin 3x\).
State the least and greatest values of \(f(x)\).[2]
State the period of \(f(x)\).[1]
Solve \(1-2\sin 3x=0\) for \(0\le x\le \pi\).[3]
Sketch the graph of \(y=f(x)\) for \(0\le x\le \pi\).[3]
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Answer:\(\text{(a)}\hspace{0.5em}\frac{1-\cos x+\sin x}{1+\cos x+\sin x}=\tan\frac{x}{2}\text{, where the original expression is defined.}\) \(\text{(b)(i) Least value}=-1\text{ and greatest value}=3.\) \(\text{(b)(ii) Period}=\frac{2\pi}{3}.\) \(\text{(b)(iii)}\hspace{0.5em}x=\frac{\pi}{18},\frac{5\pi}{18},\frac{13\pi}{18},\frac{17\pi}{18}.\) \(\text{(b)(iv) Draw a smooth sinusoidal curve through}\hspace{0.5em}(0,1),(\frac{\pi}{6},-1),(\frac{\pi}{3},1),(\frac{\pi}{2},3),(\frac{2\pi}{3},1),(\frac{5\pi}{6},-1),(\pi,1)\text{, with midline}\hspace{0.5em}y=1.\)