EDEXCEL GCSE MATHS · HIGHER
Exam Practice: Cubic equations
Cubic equations · Equations and graphs · Lesson 5 of 6 · 14 marks · 30 minutes
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Instructions
• Answer all the questions.
• Write your answers in the spaces provided.
• The marks for each question are shown in brackets - use this as a guide to how much to write.
• The answers are on separate pages at the back. Attempt every question before you look at them.
• Answer all questions. Show your working.
Question 1 COMPLETE THE TABLE (2 marks)
Complete the table of values for y = x³ − 4x for x = −3, −2, −1, 0, 1, 2, 3.
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(Total for Question 1 = 2 marks)
Question 2 USE THE GRAPH (3 marks)
The graph of y = x³ − 4x is drawn on the grid. Use the graph to find estimates for the solutions of x³ − 4x = 1.
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(Total for Question 2 = 3 marks)
Question 3 SOLVE (3 marks)
Factorise x³ − 4x and hence solve x³ − 4x = 0.
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(Total for Question 3 = 3 marks)
Question 4 EXPLAIN (2 marks)
Use the graph to explain why x³ − 4x = 5 has only one solution.
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(Total for Question 4 = 2 marks)
Question 5 WORK OUT (2 marks)
Work out the value of x³ − 4x when x = 1.2.
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(Total for Question 5 = 2 marks)
Question 6 DESCRIBE (2 marks)
Describe the shape of the graph of y = x³ and write down its root.
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(Total for Question 6 = 2 marks)
TOTAL FOR PAPER = 14 MARKS
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Answers and mark scheme
Check your answer only once you have written one.
Question 1 (2 marks)
y = −15, 0, 3, 0, −3, 0, 15
• At least 4 correct M1
• All correct A1
Question 2 (3 marks)
Draw y = 1. Solutions x ≈ −1.9, −0.3 and 2.1. Accept −2.0 to −1.8, −0.4 to −0.2 and 2.0 to 2.2.
• Draws y = 1 M1
• Two correct estimates A1
• All three A1
Question 3 (3 marks)
x(x − 2)(x + 2) = 0, so x = 0, x = 2 or x = −2.
• x(x² − 4) M1
• x(x − 2)(x + 2) A1
• Three solutions A1
Question 4 (2 marks)
The line y = 5 is above the local maximum of the curve (about 3.1), so it meets the curve only once, on the right.
• Line y = 5 is above the maximum M1
• Only one intersection C1
Question 5 (2 marks)
1.728 − 4.8 = −3.072
• 1.2³ = 1.728 M1
• −3.072 A1
Question 6 (2 marks)
A smooth S-shaped curve rising from bottom left to top right through the origin; the root is x = 0.
• S-shaped curve B1
• x = 0 B1