The answer
(a) shown
(b) \(v = 20\pi\cos\tfrac{\pi}{3}t\), maximum speed \(20\pi\) cm/s
(c) \(\dfrac{20\pi^2}{3}\) cm/s\(^2\)
O-Level A-Math 2022 Paper 1 Question 12 · Verified worked solution by the Genius Plus Academy teaching team
What this question tests
This is Question 12 of the O-Level A-Math 2022 Paper 1. It tests amplitude and period of a sinusoidal model, in the Applications of differentiation (kinematics) area. It is worth 8 marks: 3 + 3 + 2. It is a worded / diagram-based question, so open your Ten-Year Series (TYS) or the official paper at this question, then follow our full worked solution below.
(a) \(x = a\sin nt\) measures displacement from the centre \(O\); the dot swings \(60\) cm each side of \(O\) (half of the \(120\) cm width), so the amplitude is \(a = \tfrac{120}{2} = 60\). One complete oscillation is the period of \(\sin nt\), namely \(\dfrac{2\pi}{n}\); setting this equal to \(6\) s gives \(\dfrac{2\pi}{n} = 6 \Rightarrow n = \dfrac{2\pi}{6} = \dfrac{\pi}{3}\). (shown)
(b) With \(x = 60\sin\tfrac{\pi}{3}t\), the velocity is \(v = \dfrac{dx}{dt} = 60 \cdot \dfrac{\pi}{3}\cos\tfrac{\pi}{3}t = 20\pi\cos\tfrac{\pi}{3}t\) cm/s. Since \(-1 \leqslant \cos\tfrac{\pi}{3}t \leqslant 1\), the speed \(|v|\) is greatest when \(\cos\tfrac{\pi}{3}t = \pm 1\), giving a maximum speed of \(20\pi\) cm/s.
(c) The acceleration is \(\dfrac{dv}{dt} = -20\pi \cdot \dfrac{\pi}{3}\sin\tfrac{\pi}{3}t = -\dfrac{20\pi^2}{3}\sin\tfrac{\pi}{3}t\) cm/s\(^2\). At \(A\) the dot is at the extreme of the motion, \(x = 60\), so \(60\sin\tfrac{\pi}{3}t = 60 \Rightarrow \sin\tfrac{\pi}{3}t = 1\). Hence the magnitude of the acceleration there is \(\dfrac{20\pi^2}{3}\) cm/s\(^2\).
Answer: (a) shown
(b) \(v = 20\pi\cos\tfrac{\pi}{3}t\), maximum speed \(20\pi\) cm/s
(c) \(\dfrac{20\pi^2}{3}\) cm/s\(^2\)
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 sinusoidal model question from Applications of differentiation (kinematics), worth 8 marks: 3 + 3 + 2.
Yes. IP (Integrated Programme) schools teach the same O-Level Mathematics content; they just sequence it differently and set their own internal exams, so these worked solutions apply to IP students too.
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