The answer
(a) \(x = -\dfrac{1}{3}\)
(b) \(y < -2\)
(c)(i) \(c = \dfrac{9}{4}\)
(ii) \(a = \dfrac{3bc - 3}{bc + b - 1}\)
(d) \(x = 0.56\) or \(x = 6.44\)
O-Level E-Math 2023 Paper 2 Question 2 · Verified worked solution by the Genius Plus Academy teaching team
What this question tests
This is Question 2 of the O-Level E-Math 2023 Paper 2. It tests linear equations, in the Equations & inequalities / Algebraic manipulation area. It is worth 12 marks: (a) 2, (b) 1, (c) 1 + 3, (d) 5. 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) \(6 - 7x = 5 - 10x \Rightarrow -7x + 10x = 5 - 6 \Rightarrow 3x = -1 \Rightarrow x = -\dfrac{1}{3}\).
(b) \(5 - y > 7 \Rightarrow -y > 2 \Rightarrow y < -2\) (dividing by \(-1\) reverses the inequality).
(c)(i) \(c = \dfrac{1}{4} + \dfrac{2}{3 - 2} = \dfrac{1}{4} + 2 = \dfrac{9}{4}\).
(ii) \(c - \dfrac{1}{b} = \dfrac{a}{3 - a}\). Cross-multiply: \(\left(c - \tfrac{1}{b}\right)(3 - a) = a \Rightarrow 3\left(c - \tfrac{1}{b}\right) = a + a\left(c - \tfrac{1}{b}\right) = a\left(1 + c - \tfrac{1}{b}\right)\). So \(a = \dfrac{3\left(c - \frac{1}{b}\right)}{1 + c - \frac{1}{b}}\); multiplying numerator and denominator by \(b\) gives \(a = \dfrac{3bc - 3}{bc + b - 1}\).
(d) Multiply by \((2x - 1)(4 - x)\): \(x(4 - x) - 6(2x - 1) = 3(2x - 1)(4 - x)\). LHS \(= 4x - x^2 - 12x + 6 = -x^2 - 8x + 6\). RHS \(= 3(-2x^2 + 9x - 4) = -6x^2 + 27x - 12\). \(0 = -6x^2 + 27x - 12 - (-x^2 - 8x + 6) = -5x^2 + 35x - 18\), i.e. \(5x^2 - 35x + 18 = 0\). \(x = \dfrac{35 \pm \sqrt{35^2 - 4(5)(18)}}{10} = \dfrac{35 \pm \sqrt{865}}{10}\), so \(x = 6.44\) or \(x = 0.56\) (2 d.p.).
Answer: (a) \(x = -\dfrac{1}{3}\)
(b) \(y < -2\)
(c)(i) \(c = \dfrac{9}{4}\)
(ii) \(a = \dfrac{3bc - 3}{bc + b - 1}\)
(d) \(x = 0.56\) or \(x = 6.44\)
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 linear equations question from Equations & inequalities / Algebraic manipulation, worth 12 marks: (a) 2, (b) 1, (c) 1 + 3, (d) 5.
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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