The answer
(a) \((2, 3)\)
(b) \(k = 45\)
O-Level A-Math 2022 Paper 1 Question 1 · Verified worked solution by the Genius Plus Academy teaching team
The question
The equation of a curve is \(y = 2x^2 - 8x + 11\).
(a) By expressing \(2x^2 - 8x + 11\) in the form \(a(x + b)^2 + c\), find the coordinates of the stationary point. [2]
(b) The line \(y = 2x + 3\) intersects the curve at points \(A\) and \(B\). Find \(k\) for which the distance \(AB\) can be expressed as \(\sqrt{k}\). [4]
(a) Take out the factor \(2\) and complete the square: \(2x^2 - 8x + 11 = 2(x^2 - 4x) + 11 = 2\big[(x - 2)^2 - 4\big] + 11 = 2(x - 2)^2 + 3\). The vertex (stationary point of this upward parabola) is where \((x - 2)^2 = 0\), i.e. \(x = 2\), \(y = 3\), so the stationary point is \((2, 3)\).
(b) At the intersections, \(2x + 3 = 2x^2 - 8x + 11 \Rightarrow 2x^2 - 10x + 8 = 0 \Rightarrow x^2 - 5x + 4 = 0 \Rightarrow (x - 1)(x - 4) = 0\), so \(x = 1\) or \(x = 4\). The \(y\)-values on the line are \(y = 5\) and \(y = 11\), giving \(A(1, 5)\), \(B(4, 11)\). Then \(AB^2 = (4 - 1)^2 + (11 - 5)^2 = 9 + 36 = 45\), so \(AB = \sqrt{45}\) and \(k = 45\).
Answer: (a) \((2, 3)\)
(b) \(k = 45\)
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 question from Quadratic functions, worth 6 marks: 2 + 4.
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