The answer
shown (\(4 - \dfrac{2\pi\sqrt3}{3}\) units\(^2\))
O-Level A-Math 2021 Paper 2 Question 10 · Verified worked solution by the Genius Plus Academy teaching team
What this question tests
This is Question 10 of the O-Level A-Math 2021 Paper 2. It tests stationary point of a trig curve, in the Applications of integration area. It is worth 12 marks. 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.
The maximum: \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = 2\cos\tfrac x2 - 1 = 0 \Rightarrow \cos\tfrac x2 = \tfrac12\). For \(0 \leqslant x \leqslant \pi\) (\(0 \leqslant \tfrac x2 \leqslant \tfrac\pi2\)), \(\tfrac x2 = \tfrac\pi3\), so \(x = \tfrac{2\pi}{3}\). Then \(y_M = 4\sin\tfrac\pi3 - \tfrac{2\pi}{3} = 2\sqrt3 - \tfrac{2\pi}{3}\), so \(M = \left(\tfrac{2\pi}{3},\ 2\sqrt3 - \tfrac{2\pi}{3}\right)\).
The shaded region is the area under the curve minus the area of triangle \(OM\) (with base \(\tfrac{2\pi}{3}\) along the \(x\)-axis), over \(0 \leqslant x \leqslant \tfrac{2\pi}{3}\).
Area under curve: \[\int_0^{2\pi/3}\!\left(4\sin\tfrac x2 - x\right)\mathrm{d}x = \left[-8\cos\tfrac x2 - \tfrac{x^2}{2}\right]_0^{2\pi/3} = \left(-8\cdot\tfrac12 - \tfrac{2\pi^2}{9}\right) - (-8) = 4 - \frac{2\pi^2}{9}.\] Area of triangle \(OM\): \(\tfrac12\cdot\tfrac{2\pi}{3}\cdot\!\left(2\sqrt3 - \tfrac{2\pi}{3}\right) = \tfrac{2\pi\sqrt3}{3} - \tfrac{2\pi^2}{9}\). Shaded area \(= \left(4 - \tfrac{2\pi^2}{9}\right) - \left(\tfrac{2\pi\sqrt3}{3} - \tfrac{2\pi^2}{9}\right) = 4 - \dfrac{2\pi\sqrt3}{3}\) units\(^2\). (shown)
Answer: shown (\(4 - \dfrac{2\pi\sqrt3}{3}\) units\(^2\))
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 stationary point of a trig curve question from Applications of integration, worth 12 marks.
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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