The answer
(a) \(AQ = \dfrac{80}{x} - 4\)
(c) \(x = 18 \mapsto y = 73.8\)
(e) \(x \approx 12.7\)
(f) minimum \(\approx 56 > 50\)
O-Level E-Math 2021 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 E-Math 2021 Paper 2. It tests forming an area expression, in the Functions & graphs / Mensuration area. It is worth 11 marks: (a) 2, (b) 3, (c) 1, (d) 3, (e) 1, (f) 1. 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) The rectangle is \(x\) by \(\dfrac{80}{x}\) (area 80). \(AB = \dfrac{80}{x}\) and \(QB = 4\), so \(AQ = AB - QB = \dfrac{80}{x} - 4\).
(b) Each removed triangle has area \(\dfrac12 (x - 4)\left(\dfrac{80}{x} - 4\right)\), and there are two, so removed area \(= (x - 4)\left(\dfrac{80}{x} - 4\right) = 80 - 4x - \dfrac{320}{x} + 16 = 96 - 4x - \dfrac{320}{x}\). Shaded \(y = 80 - \left(96 - 4x - \dfrac{320}{x}\right) = \dfrac{320}{x} + 4x - 16\). (shown)
(c) At \(x = 18\): \(y = \dfrac{320}{18} + 4(18) - 16 = 17.8 + 72 - 16 = 73.8\).
(d) Plot the nine points and join with a smooth curve (a minimum of about 56 near \(x = 9\)).
(e) Solving \(\dfrac{320}{x} + 4x - 16 = 60\) gives \(x^2 - 19x + 80 = 0\), \(x \approx 6.3\) or \(12.7\); the greatest value is \(x \approx 12.7\).
(f) The lowest point of the curve is about \(y = 56\), which is greater than 50, so the curve never reaches \(y = 50\); hence the shaded area cannot be 50 cm².
Answer: (a) \(AQ = \dfrac{80}{x} - 4\)
(c) \(x = 18 \mapsto y = 73.8\)
(e) \(x \approx 12.7\)
(f) minimum \(\approx 56 > 50\)
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 E-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 forming an area expression question from Functions & graphs / Mensuration, worth 11 marks: (a) 2, (b) 3, (c) 1, (d) 3, (e) 1, (f) 1.
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