The answer
(a) \(y = 6x - 12\) (i.e. \(y = 6(x - 2)\))
(b) \(4\) units\(^2\)
O-Level A-Math 2021 Specimen Paper 2 Question 3 · Verified worked solution by the Genius Plus Academy teaching team
What this question tests
This is Question 3 of the O-Level A-Math 2021 Specimen Paper 2. It tests tangent at a point, in the Applications of differentiation / Applications of integration area. It is worth 10 marks: 5 + 5. 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) \(P\) lies on both the \(x\)-axis (\(y = 0\)) and the curve: \(2x - \dfrac{16}{x^2} = 0 \Rightarrow 2x^3 = 16 \Rightarrow x^3 = 8 \Rightarrow x = 2\), so \(P(2, 0)\). Differentiating, \(\dfrac{dy}{dx} = 2 + \dfrac{32}{x^3}\); at \(x = 2\) this is \(2 + \dfrac{32}{8} = 6\). The tangent at \(P\) has gradient \(6\) and passes through \((2, 0)\): \[y - 0 = 6(x - 2) \Rightarrow y = 6x - 12.\]
(b) \(Q\) is on \(x = 4\) and on \(PQ\): \(y = 6(4 - 2) = 12\), so \(Q(4, 12)\). The shaded region is the area under the tangent (a triangle on base \(PQ'\) where the line \(x = 4\) closes it) minus the area under the curve, both from \(x = 2\) to \(x = 4\). The straight-line piece gives a right-angled triangle of base \((4 - 2) = 2\) and height \(12\): \[\text{Area} = \tfrac12(4 - 2)(12) - \int_2^4\!\left(2x - \frac{16}{x^2}\right)dx = 12 - \left[x^2 + \frac{16}{x}\right]_2^4 = 12 - \left[\left(16 + 4\right) - \left(4 + 8\right)\right] = 12 - 8 = 4 \text{ units}^2.\]
Answer: (a) \(y = 6x - 12\) (i.e. \(y = 6(x - 2)\))
(b) \(4\) 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 tangent at a point question from Applications of differentiation / Applications of integration, worth 10 marks: 5 + 5.
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