The answer
(b) \(\approx 20.1\) g
(c) \(\approx 2\) min \(48\) s
O-Level A-Math 2022 Paper 1 Question 2 · Verified worked solution by the Genius Plus Academy teaching team
What this question tests
This is Question 2 of the O-Level A-Math 2022 Paper 1. It tests linear law (linearise m = ae^-kt), in the Exponential & logarithmic area. It is worth 6 marks: 2 + 2 + 2. 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.
Taking natural logs linearises the model: \(\ln m = \ln A - kt\), a straight line with \(\ln m\) on the vertical axis against \(t\) on the horizontal axis, intercept \(\ln A\) and gradient \(-k\). The plotted points \((t, \ln m)\) are approximately \((1, 2.75)\), \((2, 2.49)\), \((3, 2.25)\), \((4, 2.00)\), \((5, 1.74)\), lying close to a straight line.
(b) "At the start" means \(t = 0\), i.e. the vertical intercept. Extending the line to \(t = 0\) gives \(\ln m \approx 3.0\), so the initial amount is \(A = e^{3.0} \approx 20.1\) g (3 s.f.).
(c) "\(50\%\) dissolved" means \(m\) is half the initial amount, \(m = \tfrac12(20.1) \approx 10.05\), so \(\ln m \approx \ln 10.05 \approx 2.3\). Reading across to the line and down, \(t \approx 2.8\) min. Converting the fractional part, \(0.8 \times 60 = 48\) s, so the time is about \(2\) minutes \(48\) seconds.
Answer: (b) \(\approx 20.1\) g
(c) \(\approx 2\) min \(48\) s
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 linear law (linearise m = ae^-kt) question from Exponential & logarithmic, worth 6 marks: 2 + 2 + 2.
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