The answer
(a) plot \(\tfrac{d}{v}\) against \(v\)
(b)(i) model reasonable
(b)(ii) \(\approx 74\) tigers
(b)(iii) extrapolation beyond the data range
O-Level A-Math 2021 Paper 2 Question 7 · Verified worked solution by the Genius Plus Academy teaching team
What this question tests
This is Question 7 of the O-Level A-Math 2021 Paper 2. It tests linearise d=av^2+bv via d/v, in the Exponential & logarithmic (linear law) area. It is worth 11 marks: 4 + (4 + 2 + 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) Dividing by \(v\), \(\dfrac{d}{v} = av + b\). Plotting \(\dfrac{d}{v}\) (vertical axis) against \(v\) (horizontal axis) gives a straight line of gradient \(a\) and vertical intercept \(b\); so \(a\) is the gradient and \(b\) is the vertical intercept.
(b) Taking logs of \(n = ab^t\): \(\ln n = (\ln b)t + \ln a\), a straight line of \(\ln n\) against \(t\). The values \(\ln n = 6.70, 6.11, 5.48, 4.91\) at \(t = 1, 2, 3, 4\).
(i) These four points lie close to a straight line, so the model \(n = ab^t\) is reasonable.
(ii) The decade 2000 to 2009 is \(t = 5\). Extending the line, \(\ln n \approx 4.3\) at \(t = 5\), so \(n = e^{4.3} \approx 74\) (nearest whole number).
(iii) \(t = 5\) lies beyond the collected data (1960 to 1999), so the estimate relies on extrapolation; the underlying trend (through conservation, habitat change, poaching, etc.) may shift, making the model unreliable for later decades.
Answer: (a) plot \(\tfrac{d}{v}\) against \(v\)
(b)(i) model reasonable
(b)(ii) \(\approx 74\) tigers
(b)(iii) extrapolation beyond the data range
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 d=av^2+bv via d/v question from Exponential & logarithmic (linear law), worth 11 marks: 4 + (4 + 2 + 1).
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