The answer
(a) shown
(b) \(x = 2\) or \(x = 3 \pm \sqrt{11}\)
O-Level A-Math 2022 Paper 1 Question 9 · Verified worked solution by the Genius Plus Academy teaching team
The question
The curve is \(y = x^3 - ax^2 + bx + 4\).
(a) Show that if \(y\) is always increasing then \(a^2 < 3b\). [3]
(b) For \(a = 8\), \(b = 10\), find the \(x\)-coordinates of the three points where the curve meets the \(x\)-axis. [4]
(a) \(\dfrac{dy}{dx} = 3x^2 - 2ax + b\). The curve is always increasing if \(\dfrac{dy}{dx} > 0\) for all real \(x\). For this upward quadratic to be strictly positive everywhere, its discriminant must be negative: \((-2a)^2 - 4(3)(b) < 0 \Rightarrow 4a^2 - 12b < 0 \Rightarrow a^2 < 3b\). (shown)
(b) With \(a = 8\), \(b = 10\): \(y = x^3 - 8x^2 + 10x + 4\). Let \(f(x) = x^3 - 8x^2 + 10x + 4\). Testing \(x = 2\): \(f(2) = 8 - 32 + 20 + 4 = 0\), so by the factor theorem \((x - 2)\) is a factor. Dividing, \(f(x) = (x - 2)(x^2 - 6x - 2)\). Setting \(f(x) = 0\): \(x = 2\), or \(x^2 - 6x - 2 = 0 \Rightarrow x = \dfrac{6 \pm \sqrt{36 + 8}}{2} = \dfrac{6 \pm \sqrt{44}}{2} = 3 \pm \sqrt{11}\). The three \(x\)-intercepts are \(x = 2\) and \(x = 3 \pm \sqrt{11}\).
Answer: (a) shown
(b) \(x = 2\) or \(x = 3 \pm \sqrt{11}\)
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 always-increasing condition (discriminant < 0) question from Polynomials & partial fractions (factor theorem) / Applications of differentiation, worth 7 marks: 3 + 4.
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