The answer
\(x = \sqrt{\dfrac{3 + ay}{y - 1}}\)
O-Level E-Math 2018 Paper 1 Question 17 · Verified worked solution by the Genius Plus Academy teaching team
The question
Rearrange the formula \(y = \dfrac{x^2 + 3}{x^2 - a}\) to make \(x\) the subject. [4]
Multiply both sides by \((x^2 - a)\): \(y(x^2 - a) = x^2 + 3\), i.e. \(yx^2 - ay = x^2 + 3\). Collect the \(x^2\) terms on one side: \(yx^2 - x^2 = 3 + ay\), so \(x^2(y - 1) = 3 + ay\). Then \(x^2 = \dfrac{3 + ay}{y - 1}\) and \(x = \pm\sqrt{\dfrac{3 + ay}{y - 1}}\) (take the positive root unless a sign is specified): \(x = \sqrt{\dfrac{3 + ay}{y - 1}}\).
Answer: \(x = \sqrt{\dfrac{3 + ay}{y - 1}}\)
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 clear the fraction question from Algebraic manipulation (subject of a formula), worth 4 marks.
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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