The answer
(a) 50
(b) \(p = 70.7\)
(c) increasing curve through the table values
(d) \(\approx 3.3\) hours
(e)(i) \(\approx 140\) (accept 120 to 150)
(ii) the rate at which the number of bacteria is increasing
O-Level E-Math 2017 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 2017 Paper 2. It tests evaluate an exponential model, in the Functions & graphs (exponential growth) area. It is worth 9 marks: 1 + 1 + 3 + 1 + 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) At the start \(t = 0\): \(y = 50 \times 2^{0} = 50\). So 50 bacteria.
(b) \(p\) is the value at \(t = 0.5\): \(p = 50 \times 2^{0.5} = 50\sqrt{2} = 70.7\) (3 s.f.).
(c) Plot the eight tabulated points (and \((0, 50)\), \((0.5, 70.7)\)) and join with a smooth curve that rises ever more steeply; it doubles every hour (50, 100, 200, 400, 800).
(d) From the curve, \(y = 500\) at \(t \approx 3.3\) hours. *(Read from graph; checking algebraically, \(500 = 50 \times 2^t \Rightarrow 2^t = 10 \Rightarrow t = \log_2 10 = 3.32\) h, consistent.)*
(e)(i) Draw a tangent to the curve at \((2, 200)\), read two points that lie on it, and work out rise \(\div\) run. A carefully drawn tangent gives a gradient of about 140 bacteria per hour. (Read from a hand-drawn tangent; accept roughly 120 to 150.)
(ii) The gradient is the instantaneous rate of change of \(y\) with \(t\): it represents how fast the number of bacteria is increasing (bacteria per hour) at the moment \(t = 2\).
Answer: (a) 50
(b) \(p = 70.7\)
(c) increasing curve through the table values
(d) \(\approx 3.3\) hours
(e)(i) \(\approx 140\) (accept 120 to 150)
(ii) the rate at which the number of bacteria is increasing
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 evaluate an exponential model question from Functions & graphs (exponential growth), worth 9 marks: 1 + 1 + 3 + 1 + 2 + 1.
Yes. IP (Integrated Programme) schools teach the same O-Level Mathematics content; they just sequence it differently and set their own internal exams, so these worked solutions apply to IP students too.
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