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4PM1/02 Nov 2021, Question 8: Solving Equations by Graphical Methods

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2 · Calculator · 9 marks

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The question, in short

Complete a table for y = 2x + 3/x² − 3, plot the curve for 0.5 ≤ x ≤ 5, then use a straight line to estimate roots of 4x³ − 10x² + 3 = 0.

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

Part (a) · 2 marks

Fill in the missing y-values for y = 2x + 3/x² − 3 at the given x-values, rounding to 2 decimal places where needed.

  1. 1x = 0.75: 1.5 + 5.333… − 3 = 3.83
  2. 2x = 1.5: 3 + 1.333… − 3 = 1.33
  3. 3x = 2: 4 + 0.75 − 3 = 1.75
  4. 4x = 3: 6 + 0.333… − 3 = 3.33

Answerx = 0.75: y = 3.83; x = 1.5: y = 1.33; x = 2: y = 1.75; x = 3: y = 3.33

Part (b) · 2 marks

Draw the graph of y = 2x + 3/x² − 3 on the grid for x from 0.5 to 5.

  1. 1Plot the points from the table
  2. 2Join with a smooth curve

AnswerPoints plotted and joined with a smooth curve: steep fall from (0.5, 10) to a minimum near x ≈ 1.44 (y ≈ 1.33), then rising to (5, 7.12)

Part (c) · 5 marks

Draw a suitable straight line on the grid and use its intersections to estimate, to 1 decimal place, the roots of 4x³ − 10x² + 3 = 0 for 0.5 ≤ x ≤ 5.

  1. 1Divide 4x³ − 10x² + 3 = 0 by x²: 4x − 10 + 3/x² = 0
  2. 2So 2x + 3/x² − 3 = 7 − 2x
  3. 3Draw the line y = 7 − 2x (through (0, 7) and (3.5, 0))
  4. 4Read off the intersections: x ≈ 0.6 and x ≈ 2.4

AnswerDraw y = 7 − 2x; x = 0.6 and x = 2.4 (accept 0.7 and 2.3)

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