The answer
(a)(i) \(-2 \mapsto 8\), \(4 \mapsto -4\)
(iii) the line \(y=x\) meets the curve once
(iv) \(x \approx -0.9,\ 1.2,\ 3.7\) (from the graph)
(b) \(p = 405\)
O-Level E-Math 2024 Paper 2 Question 4 · Verified worked solution by the Genius Plus Academy teaching team
What this question tests
This is Question 4 of the O-Level E-Math 2024 Paper 2. It tests cubic/power-function graph, in the Functions & graphs area. It is worth 11 marks: (a) 1 + 3 + 1 + 3, (b) 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)(i) At \(x=-2\): \(2(4) - \frac{(-8)}{2} - 4 = 8 + 4 - 4 = 8\). At \(x=4\): \(2(16) - \frac{64}{2} - 4 = 32 - 32 - 4 = -4\).
(ii) Plot the seven points and join with a smooth curve (rises to a local maximum near \(x=3\), \(y\approx0.5\), then falls).
(iii) The solutions of \(2x^2 - \frac{x^3}{2} - 4 = x\) are the \(x\)-coordinates where the curve meets the line \(y = x\). On the grid that line cuts the curve at exactly one point (near \(x \approx -1\)), so the equation has only one solution.
(iv) Rearrange to use the drawn curve. \(4x^2 - x^3 - 4 = 0\); divide by 2: \(2x^2 - \frac{x^3}{2} - 2 = 0\), i.e. \(2x^2 - \frac{x^3}{2} - 4 = -2\). So draw the line \(y = -2\); its intersections with the curve give the roots: \(x \approx -0.9,\ 1.2,\ 3.7\) (read from the graph).
(b) \(A(0,5)\): \(5 = ka^0 \Rightarrow k = 5\). \(B(3,135)\): \(135 = 5a^3 \Rightarrow a^3 = 27 \Rightarrow a = 3\). So \(y = 5(3)^x\), and at \(C\), \(p = 5(3)^4 = 5 \times 81 = 405\).
Answer: (a)(i) \(-2 \mapsto 8\), \(4 \mapsto -4\)
(iii) the line \(y=x\) meets the curve once
(iv) \(x \approx -0.9,\ 1.2,\ 3.7\) (from the graph)
(b) \(p = 405\)
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 cubic/power-function graph question from Functions & graphs, worth 11 marks: (a) 1 + 3 + 1 + 3, (b) 3.
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