The answer
(a)(i) amplitude \(2\), period \(2\pi\)
(a)(ii) amplitude \(4\), period \(\pi\)
(b) sketch
O-Level A-Math 2022 Paper 1 Question 10 · Verified worked solution by the Genius Plus Academy teaching team
The question
(a) Find the amplitude and period of (i) \(2\sin x\) [1]; (ii) \(3 - 4\cos 2x\) [2].
(b) Sketch \(y = 2\sin x\) and \(y = 3 - 4\cos 2x\) on one diagram for \(0 \leqslant x \leqslant 2\pi\) radians. [3]
(a)(i) For \(2\sin x\) the amplitude is \(2\) and the period is \(\dfrac{2\pi}{1} = 2\pi\).
(ii) For \(3 - 4\cos 2x\) the amplitude is \(|{-4}| = 4\) and the period is \(\dfrac{2\pi}{2} = \pi\).
(b) Over \(0 \leqslant x \leqslant 2\pi\): \(y = 2\sin x\) is one full sine wave oscillating between \(-2\) and \(2\), zero at \(x = 0, \pi, 2\pi\), maximum \(2\) at \(x = \tfrac{\pi}{2}\), minimum \(-2\) at \(x = \tfrac{3\pi}{2}\). \(y = 3 - 4\cos 2x\) oscillates about the line \(y = 3\) with amplitude \(4\), so it runs between \(-1\) and \(7\); with period \(\pi\) it completes two cycles. Since it starts at \(3 - 4\cos 0 = -1\) (a minimum), it has minima \(-1\) at \(x = 0, \pi, 2\pi\) and maxima \(7\) at \(x = \tfrac{\pi}{2}, \tfrac{3\pi}{2}\).
Answer: (a)(i) amplitude \(2\), period \(2\pi\)
(a)(ii) amplitude \(4\), period \(\pi\)
(b) sketch
Same structure, different numbers
Swap the constants, dress a quadratic as a length, hide a derivative inside an integral, and a student sees a brand new problem. The structure underneath is the same, and so is the method. Once a student can name the structure, a whole row of questions that look different start to open the same way.
That is where marks really leak: in choosing the method, not in the algebra that follows. We call it Lock and Key, name the lock, then the key follows.
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Genius Plus Academy · O-Level & IP Mathematics
Our O-Level A-Math tuition trains the same recognise-the-structure method these worked solutions show, taught by a team that has marked these papers for years. It runs within our weekly Secondary Math programme, Sec 1 to 4 and IP.
It is a amplitude and period of a x and a bx question from Trig functions, identities & equations, worth 6 marks: (1 + 2) + 3.
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