The answer
(a) explained below
(b) \(2x^2 - x - 12 = 0\)
(c)(i) \(y = 3x - 4\)
(ii) \(x \approx -0.44\) or \(x \approx 3.44\)
O-Level E-Math 2017 Paper 1 Question 21 · Verified worked solution by the Genius Plus Academy teaching team
What this question tests
This is Question 21 of the O-Level E-Math 2017 Paper 1. It tests why no solution for some k, in the Functions & graphs (solving equations graphically) area. It is worth 6 marks: 1 + 1 + 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) The curve \(y = 2x^2 - 3x - 7\) has a lowest point (a minimum). The equation \(2x^2 - 3x - 7 = k\) asks where the curve meets the horizontal line \(y = k\). For values of \(k\) below the curve's minimum value, the line \(y = k\) lies entirely under the curve and never meets it, so there are no solutions.
(b) Intersections occur where \(2x^2 - 3x - 7 = 5 - 2x\). Bring all terms to one side: \(2x^2 - 3x - 7 - 5 + 2x = 0\), i.e. \(2x^2 - x - 12 = 0\).
(c)(i) We want a line \(y = mx + c\) so that \(\text{curve} - \text{line}\) gives \(2x^2 - 6x - 3\). Since \((2x^2 - 3x - 7) - (2x^2 - 6x - 3) = 3x - 4\), drawing the line \(y = 3x - 4\) and reading the intersections with the curve solves \(2x^2 - 6x - 3 = 0\). So the line is \(y = 3x - 4\).
(ii) The line \(y = 3x - 4\) meets the curve where \(2x^2 - 3x - 7 = 3x - 4\), i.e. \(2x^2 - 6x - 3 = 0\). Reading the two intersection points off the grid gives \(x \approx -0.44\) and \(x \approx 3.44\). *(Read from the drawn line; these match the algebraic roots \(x = \tfrac{6 \pm \sqrt{60}}{4} = -0.436\) and \(3.436\).)*
Answer: (a) explained below
(b) \(2x^2 - x - 12 = 0\)
(c)(i) \(y = 3x - 4\)
(ii) \(x \approx -0.44\) or \(x \approx 3.44\)
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 why no solution for some k question from Functions & graphs (solving equations graphically), worth 6 marks: 1 + 1 + 1 + 3.
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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