The answer
(a) \(p = -0.5\)
(c) the line \(y = 6\) cuts the curve once
(d) gradient \(\approx -3.5\) (read)
(e)(ii) \(x \approx 2.9\) (read)
(e)(iii) \(A = -4\), \(B = -12\)
O-Level E-Math 2016 Paper 2 Question 5 · Verified worked solution by the Genius Plus Academy teaching team
What this question tests
This is Question 5 of the O-Level E-Math 2016 Paper 2. It tests evaluate to complete a table, in the Functions & graphs area. It is worth 13 marks: (a) 1, (b) 3, (c) 2, (d) 2, (e) 2 + 1 + 2. 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) \(p = \dfrac{(-3)^3}{2} - 5(-3) - 2 = \dfrac{-27}{2} + 15 - 2 = -13.5 + 13 = -0.5\).
(b) Plot \((-3, -0.5)\), \((-2, 4)\), \((-1, 2.5)\), \((0, -2)\), \((1, -6.5)\), \((2, -8)\), \((3, -3.5)\), \((4, 10)\) on the stated scales and draw one smooth curve (a cubic dipping to a local minimum near \(x = 2\) and rising steeply after \(x = 3\)).
(c) Rearrange: \(\dfrac{x^3}{2} - 5x = 8\) means \(\dfrac{x^3}{2} - 5x - 2 = 6\), i.e. \(y = 6\). Drawing the horizontal line \(y = 6\) on the graph, it meets the curve at only one point within \(-3 \leqslant x \leqslant 4\), so the equation has only one solution.
(d) Draw a tangent to the curve at \((1, -6.5)\), read two points that lie on the tangent line, and work out its gradient as rise \(\div\) run. A carefully drawn tangent gives a gradient of about \(-3.5\). **(Read from the drawn tangent; accept roughly \(-3\) to \(-4\).)**
(e)(i) The line \(y = 4 - 3x\) passes through \((-1, 7)\) and \((4, -8)\); draw the straight segment between them.
(ii) The line meets the curve at one point in the range; reading the graph gives \(x \approx 2.9\). **(Read from graph; accept about \(2.8\) to \(3.0\).)**
(iii) Setting curve \(=\) line: \(\dfrac{x^3}{2} - 5x - 2 = 4 - 3x \Rightarrow \dfrac{x^3}{2} - 2x - 6 = 0\). Multiply by 2: \(x^3 - 4x - 12 = 0\). Comparing with \(x^3 + Ax + B = 0\) gives \(A = -4\) and \(B = -12\). (Its real root is \(x \approx 2.86\), consistent with the reading in (ii).)
Answer: (a) \(p = -0.5\)
(c) the line \(y = 6\) cuts the curve once
(d) gradient \(\approx -3.5\) (read)
(e)(ii) \(x \approx 2.9\) (read)
(e)(iii) \(A = -4\), \(B = -12\)
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 evaluate to complete a table question from Functions & graphs, worth 13 marks: (a) 1, (b) 3, (c) 2, (d) 2, (e) 2 + 1 + 2.
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