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WMA11/01 Oct 2023, Question 4: Sketching Cubics and Reciprocal Curves

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 1 (WMA11) · Calculator · 7 marks

The question, in short

For the curve y = 1/(x + 2) shown in Figure 1, find a vertical asymptote, factorise a cubic, sketch that cubic on the same axes with axis intercepts, and count real solutions of a related equation.

Summarised in our own words. Use the official paper for the exact wording and the diagram.

Part (a) · 1 mark

Give the equation of the asymptote of C parallel to the y-axis.

  1. 1The curve is undefined when x + 2 = 0
  2. 2Vertical asymptote: x = −2

Answerx = −2

Part (b) · 2 marks

Factorise x³ + 4x² + 4x completely.

  1. 1Take out x: x(x² + 4x + 4)
  2. 2x² + 4x + 4 = (x + 2)², so x(x + 2)²

Answerx(x + 2)²

Part (c) · 3 marks

On a copy of Figure 1, sketch y = x³ + 4x² + 4x and state the coordinates where it meets or cuts the axes.

  1. 1y = x(x + 2)²: positive x³ coefficient, so it rises from bottom-left to top-right
  2. 2Repeated root x = −2: the curve touches the x-axis at (−2, 0)
  3. 3Single root x = 0: the curve crosses through the origin (0, 0)
  4. 4Between −2 and 0, y < 0, so there is a minimum there

AnswerPositive cubic: comes from bottom-left, touches the x-axis at (−2, 0) (a maximum turning point there, on the asymptote), dips to a minimum between −2 and 0, then passes through the origin (0, 0) and rises to the top-right.

Part (d) · 1 mark

State how many real solutions (x + 2)(x³ + 4x² + 4x) = 1 has, with a reason.

  1. 1(x + 2)(x³ + 4x² + 4x) = 1 is the same as x³ + 4x² + 4x = 1/(x + 2)
  2. 2From the sketch the two curves meet twice (once in the first quadrant, once in the third)
  3. 3So there are 2 real solutions

Answer2 real solutions, because the curves y = x³ + 4x² + 4x and y = 1/(x + 2) intersect twice.

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Also searched as: WMA11 October 2023 · WMA11/01 · wma11 oct23 · IAS Maths Pure Mathematics 1 October 2023 · question 4.
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