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4024/21 May/Jun 2021, Question 7: Solving Equations by Graphical Methods

Cambridge O Level Mathematics 4024 · Paper 21 (Calculator) · Calculator · 10 marks

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The question, in short

Match two sketched lines to their equations, find a and b from a quadratic sketch, and use a cubic graph to solve equations.

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

Part (a) · 2 marks

Choose which of the four given equations matches each of the two sketched lines.

  1. 1Left: positive gradient, y-intercept below O → y = 2x − 1
  2. 2Right: negative gradient, y-intercept above O → y = −2x + 1

AnswerLeft: y = 2x − 1 Right: y = −2x + 1

Part (b) · 2 marks

Use the x-intercepts at −3 and 2 to find the values of a and b in y = x² + ax + b.

  1. 1y = (x + 3)(x − 2)
  2. 2= x² + x − 6
  3. 3a = 1, b = −6

Answera = 1, b = −6

Part (ci) · 2 marks

Using the cubic graph, find the values of k for which the equation has exactly 2 solutions.

  1. 1Exactly 2 solutions when y = k touches a turning point
  2. 2Maximum ≈ 8.2 and minimum ≈ −4.1
  3. 3k ≈ 8.2 or k ≈ −4.1

Answer8 to 8.5, −4.3 to −3.8

Part (cii) · 4 marks

Draw a suitable line on the grid and use it to solve x³ − 2x² − 7x + 5 = 0.

  1. 1x³ − 2x² − 7x + 5 = 0 → x³ − 2x² − 5x + 6 = 2x + 1
  2. 2Draw the ruled line y = 2x + 1
  3. 3Read the x-coordinates where it crosses the curve
  4. 4x ≈ −2.2, x ≈ 0.6, x ≈ 3.6

AnswerRuled line y = 2x + 1 drawn; the three x values read from its intersections with the curve

Stuck on a step? Read the free lesson: Solving Equations by Graphical Methods

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