The answer
(a) \(x = \dfrac{1}{2}\)
(b) \(x \approx 2.56\) or \(x \approx -1.56\)
(c)(ii) \(y = -\dfrac{3}{2}x + 2\)
O-Level E-Math 2018 Paper 1 Question 23 · Verified worked solution by the Genius Plus Academy teaching team
What this question tests
This is Question 23 of the O-Level E-Math 2018 Paper 1. It tests line of symmetry from the roots, in the Functions & graphs (quadratics, tangents) area. It is worth 6 marks: 1 + 2 + 1 + 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.
(a) The curve \(y = \dfrac12(x - x^2) = -\dfrac12 x^2 + \dfrac12 x\) is a parabola; its line of symmetry runs through the vertex. The roots of \(x - x^2 = 0\) are \(x = 0\) and \(x = 1\), so the axis of symmetry is midway between them: \(x = \dfrac{1}{2}\).
(b) \(x - x^2 = -4\) is the same as \(\dfrac12(x - x^2) = -2\), i.e. \(y = -2\) on this graph. Read the \(x\)-coordinates where the curve meets the line \(y = -2\). Algebraically \(x^2 - x - 4 = 0\) gives \(x = \dfrac{1 \pm \sqrt{17}}{2}\), so \(x \approx 2.56\) or \(x \approx -1.56\). *(Read from the graph; accept roughly \(x \approx 2.6\) and \(x \approx -1.6\).)*
(c)(i) Draw the straight line through \(P(-2, 5)\) that just touches the curve (it touches near \((2, -1)\)).
(ii) Read two points off the tangent you drew in (i). The tangent from \(P(-2, 5)\) touches the curve at about \((2, -1)\), so use the two points \(P(-2, 5)\) and the point of contact \((2, -1)\). Gradient \(= \dfrac{-1 - 5}{2 - (-2)} = \dfrac{-6}{4} = -\dfrac32\). The line through \(P(-2, 5)\) with gradient \(-\dfrac32\) is \(y - 5 = -\dfrac32(x + 2)\), so \(y = -\dfrac{3}{2}x + 2\). (The point of contact is read from the graph; any carefully drawn tangent from \(P\) should give a gradient close to \(-1.5\) and an equation close to \(y = -1.5x + 2\).)
Answer: (a) \(x = \dfrac{1}{2}\)
(b) \(x \approx 2.56\) or \(x \approx -1.56\)
(c)(ii) \(y = -\dfrac{3}{2}x + 2\)
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.
Want more questions like this, with worked solutions?
Join our mailing list and we will send practice sets and worked solutions. One email, no spam.
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 line of symmetry from the roots question from Functions & graphs (quadratics, tangents), worth 6 marks: 1 + 2 + 1 + 2.
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.
Yes. Every worked solution here is free to read, with no sign-up wall.
Browse E-Math and A-Math by year in our worked-solutions library at /resources/solutions/o-level/.
Book a free trial and diagnostic. We will look at a real paper and show you exactly where the marks are going.