The answer
\(y = -3\cos x + \sin 2x + 4x + 9 - 2\pi\)
O-Level A-Math 2021 Specimen Paper 1 Question 6 · Verified worked solution by the Genius Plus Academy teaching team
The question
\(\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2} = 3\cos x - 4\sin 2x\); curve through \(P\left(\tfrac{\pi}{2}, 9\right)\), gradient \(5\) at \(P\). Find the equation. [6]
Integrate once: \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = 3\sin x + 2\cos 2x + c\) (since \(\int -4\sin 2x\,\mathrm{d}x = 2\cos 2x\)). At \(P\) the gradient is \(5\): \(3\sin\tfrac{\pi}{2} + 2\cos\pi + c = 3 + 2(-1) + c = 1 + c = 5 \Rightarrow c = 4\). Integrate again: \(y = -3\cos x + \sin 2x + 4x + k\). At \(P\left(\tfrac{\pi}{2}, 9\right)\): \(-3\cos\tfrac{\pi}{2} + \sin\pi + 4\cdot\tfrac{\pi}{2} + k = 0 + 0 + 2\pi + k = 9 \Rightarrow k = 9 - 2\pi\). So \(y = -3\cos x + \sin 2x + 4x + 9 - 2\pi\).
Answer: \(y = -3\cos x + \sin 2x + 4x + 9 - 2\pi\)
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 integrate x question from Integration, worth 6 marks.
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