Lemmara logoLemmaraWorked solution

4024/22 Oct/Nov 2024, Question 6: Solving Equations by Graphical Methods

Cambridge O Level Mathematics 4024 · Paper 22 (Calculator) · Calculator · 10 marks

اقرأ هذا الحل مشروحاً بالعربي ←

The question, in short

Work with the cubic y = (x/4)(2x² − x − 10): fill a missing table value, plot its curve for −3 ≤ x ≤ 3, then use the graph to solve related equations.

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

Part (a) · 1 mark

Work out the value of y when x = −3 for the given cubic formula.

  1. 1x = −3: y = (−3/4)(2 × 9 + 3 − 10) = (−3/4) × 11
  2. 2= −8.25

Answer−8.25

Part (b) · 3 marks

Plot the cubic curve on the grid for x values from −3 to 3.

  1. 1Plot (−3, −8.25), (−2, 0), (−1, 1.75), (0, 0), (1, −2.25), (2, −2), (3, 3.75)
  2. 2Join with a smooth curve: maximum near x = −1.1, minimum near x = 1.5

AnswerCorrect smooth curve

Part (c) · 2 marks

Using your curve, state the k values for which the equation has exactly two solutions.

  1. 1y = k meets the curve exactly twice when it touches the maximum or the minimum
  2. 2Maximum ≈ 1.8, minimum ≈ −2.6
  3. 3k ≈ 1.8 or k ≈ −2.6

Answer1.7 to 1.9, −2.7 to −2.5

Part (d) · 4 marks

Add a suitable straight line to the grid and use it to solve 2x³ − x² − 10x = 2x − 4.

  1. 1Divide by 4: (x/4)(2x² − x − 10) = (2x − 4)/4 = x/2 − 1
  2. 2Draw y = x/2 − 1, e.g. through (−2, −2), (0, −1) and (2, 0)
  3. 3Read the x-coordinates where the line meets the curve
  4. 4x ≈ −2.4, x ≈ 0.3, x ≈ 2.5

Answer−2.6 to −2.3, 0.2 to 0.4, 2.3 to 2.7

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

Practise more solving equations by graphical methods questions

Real 4024 past-paper questions, marked on your working like the exam, with a worked solution for every part.

Practise free on Lemmara

More past-paper questions on solving equations by graphical methods

Also searched as: 4024/22/O/N/24 · 4024_w24_qp_22 · 4024 w24 22 · question 6.
Lemmara is an independent study tool, not affiliated with or endorsed by Cambridge International Education. The worked solutions are Lemmara’s own.