The answer
(a) \(Q\left(\tfrac{11}{2}, 0\right)\)
(b) \(\dfrac{139}{9} = 15\tfrac49 \approx 15.4\) units\(^2\)
O-Level A-Math 2021 Paper 1 Question 14 · Verified worked solution by the Genius Plus Academy teaching team
What this question tests
This is Question 14 of the O-Level A-Math 2021 Paper 1. It tests chain rule for (4+3x)^1/2, in the 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) \(\dfrac{dy}{dx} = \tfrac12(4 + 3x)^{-1/2}\cdot 3 = \dfrac{3}{2\sqrt{4 + 3x}}\). At \(P(4, 4)\): \(\dfrac{dy}{dx} = \dfrac{3}{2\sqrt{16}} = \dfrac{3}{8}\), so the normal gradient is \(-\dfrac{8}{3}\). The normal at \(P\) is \(y - 4 = -\tfrac83(x - 4)\). Setting \(y = 0\): \(-4 = -\tfrac83(x - 4) \Rightarrow x - 4 = \tfrac32 \Rightarrow x = \tfrac{11}{2}\). So \(Q\left(\tfrac{11}{2}, 0\right)\).
(b) The region is the area under the curve from \(x = 0\) to \(x = 4\) plus the triangle under the normal from \(P(4, 4)\) to \(Q(\tfrac{11}{2}, 0)\). \[\int_0^4 (4 + 3x)^{1/2}\,dx = \left[\frac{2}{9}(4 + 3x)^{3/2}\right]_0^4 = \frac{2}{9}\big(16^{3/2} - 4^{3/2}\big) = \frac{2}{9}(64 - 8) = \frac{112}{9}.\] The triangle has base \(\tfrac{11}{2} - 4 = \tfrac32\) and height \(4\), area \(\tfrac12\cdot\tfrac32\cdot 4 = 3\). Total area \(= \dfrac{112}{9} + 3 = \dfrac{139}{9} = 15\tfrac49 \approx 15.4\) units\(^2\).
Answer: (a) \(Q\left(\tfrac{11}{2}, 0\right)\)
(b) \(\dfrac{139}{9} = 15\tfrac49 \approx 15.4\) 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 chain rule for (4+3x)^1/2 question from Applications of integration, worth 10 marks: 5 + 5.
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