Express $5 \sin^2 x - \cos^2 x - 1$ in the form $1 - k \cos 2x$.[2] State the amplitude and peri

Question

Express $5 \sin^2 x - \cos^2 x - 1$ in the form $1 - k \cos 2x$.[2] State the amplitude and period, in radians, of $5 \sin^2 x - \cos^2 x - 1$.[2] Sketch the graph of $y = 5 \sin^2 x - \cos^2 x - 1$ for $-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}$.[3] By drawing the line $y=1-\frac{3x}{2\pi}$ on the same axes, state the number of solutions to the equation $2\pi-3x=2\pi(5\sin^2x-\cos^2x-1)$ in the range of $-\frac{\pi}{2} \leq x\leq \frac{\pi}{2}$[2]

Accepted Answer

(i) $k=3$ (ii) amplitude $=3$, period $=\pi$ radians (iii) key points $\left(-\dfrac{\pi}{2},4\right),\left(-\dfrac{\pi}{4},1\right),(0,-2),\left(\dfrac{\pi}{4},1\right),\left(\dfrac{\pi}{2},4\right)$ (iv) $2$ solutions

(A Math) 2025 CCHY PRELIMS P1 Q9

(A Math) 2025 CCHY PRELIMS P1 Q9

Solution:

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