The answer
(i) plot below
(ii) \(P \approx 3.32\,e^{0.123x}\)
(iii) 2030
O-Level A-Math 2020 Paper 2 Question 6 · Verified worked solution by the Genius Plus Academy teaching team
What this question tests
This is Question 6 of the O-Level A-Math 2020 Paper 2. It tests linearise y = aeᵏˣ as ln p = kx + ln a, in the Exponential & logarithmic (linear law) area. It is worth 9 marks: 2 + 4 + 3. 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.
(i) Compute \(\ln P\) against \(x\):
| \(x\) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| \(\ln P\) | 1.200 | 1.322 | 1.445 | 1.569 | 1.692 |
These points lie close to a straight line; draw the line of best fit.
(ii) Taking logs of \(P = Ae^{kx}\) gives \(\ln P = kx + \ln A\), so the gradient of the line is \(k\) and the intercept is \(\ln A\). \[k \approx \dfrac{1.692 - 1.200}{4 - 0} = \dfrac{0.492}{4} = 0.123, \qquad \ln A \approx 1.20 \Rightarrow A = e^{1.20} \approx 3.32.\] Hence \(P \approx 3.32\,e^{0.123x}\). (Values depend slightly on the line drawn.)
(iii) Require \(P > 8\): \(3.32\,e^{0.123x} > 8 \Rightarrow e^{0.123x} > 2.41 \Rightarrow 0.123x > 0.880 \Rightarrow x > 7.15\). Since \(x\) counts 5-year steps from 1995 (\(x = 7 \to 2030\), \(x = 8 \to 2035\)), the population first exceeds 8 million in the interval \(x = 7\) to \(x = 8\). The first year of that interval is 2030.
Answer: (i) plot below
(ii) \(P \approx 3.32\,e^{0.123x}\)
(iii) 2030
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 linearise y = aeᵏˣ as ln p = kx + ln a question from Exponential & logarithmic (linear law), worth 9 marks: 2 + 4 + 3.
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