The answer
(a) \(1.75\) m
(b) \(h = 3 - 5\left(t - \tfrac12\right)^2\)
(c) \(3\) m at \(t = 0.5\) s
(d) not thrown from ground level (asymmetric)
(e) \(\approx 0.894\) s
O-Level A-Math 2023 Paper 1 Question 12 · Verified worked solution by the Genius Plus Academy teaching team
What this question tests
This is Question 12 of the O-Level A-Math 2023 Paper 1. It tests completing the square (vertex), in the Quadratic functions area. It is worth 10 marks: 1 + 3 + 2 + 1 + 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.
(a) At \(t = 0\), \(h = 1.75\) m.
(b) \(h = -5(t^2 - t) + 1.75 = -5\left(t - \tfrac12\right)^2 + \tfrac54 + 1.75 = 3 - 5\left(t - \tfrac12\right)^2\) (so \(a = 3\), \(b = -5\), \(c = -\tfrac12\)).
(c) Maximum height \(3\) m, at \(t = \tfrac12 = 0.5\) s.
(d) The ball is thrown from \(1.75\) m above the ground, not from ground level, so its path is not symmetric about the highest point: it takes longer to fall from \(3\) m to the ground than to rise from \(1.75\) m to \(3\) m. Hence the total time is not twice the time to the maximum.
(e) \(h \geqslant 2\): \(1.75 + 5t - 5t^2 \geqslant 2 \Rightarrow 5t^2 - 5t + 0.25 \leqslant 0 \Rightarrow t^2 - t + 0.05 \leqslant 0\). Roots \(t = \dfrac{1 \pm \sqrt{0.8}}{2}\), so the interval has length \(\sqrt{0.8} = 0.894\) s (3 s.f.).
Answer: (a) \(1.75\) m
(b) \(h = 3 - 5\left(t - \tfrac12\right)^2\)
(c) \(3\) m at \(t = 0.5\) s
(d) not thrown from ground level (asymmetric)
(e) \(\approx 0.894\) 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 completing the square (vertex) question from Quadratic functions, worth 10 marks: 1 + 3 + 2 + 1 + 3.
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