Question Simulator

IGCSE Additional Mathematics (0606)

Equations, inequalities and graphs Banker

13 questions · about 5% of the marks we indexed · examined in 9 of 9 sessions.

Questions by year

2026

  • 2026 May/June · Paper 22 · Q4 (a, b)7 marksModulus and cubic graphs · Modulus equations and inequalities

2025

  • 2025 Oct/Nov · Paper 12 · Q1 (a, b)5 marksCubic inequalities · Modulus and cubic graphs
  • 2025 Oct/Nov · Paper 12 · Q65 marksModulus equations and inequalities
  • 2025 May/June · Paper 12 · Q13 marksModulus and cubic graphs
  • 2025 May/June · Paper 12 · Q24 marksEquations reducible to quadratics
  • 2025 May/June · Paper 22 · Q14 marksModulus equations and inequalities

2024

  • 2024 Oct/Nov · Paper 12 · Q2 (b, c)5 marksModulus and cubic graphs
  • 2024 Oct/Nov · Paper 22 · Q45 marksModulus equations and inequalities
  • 2024 May/June · Paper 22 · Q1 (a, b(i), b(ii))6 marksCubic inequalities · Modulus and cubic graphs

2023

  • 2023 Oct/Nov · Paper 12 · Q1 (a, b)5 marksCubic inequalities · Modulus and cubic graphs
  • 2023 May/June · Paper 12 · Q3 (a, b)7 marksModulus and cubic graphs · Modulus equations and inequalities

2022

  • 2022 Oct/Nov · Paper 12 · Q2 (b)4 marksModulus and cubic graphs
  • 2022 May/June · Paper 22 · Q25 marksModulus equations and inequalities

References only, not question text. Open Official paper ↗ where Cambridge publishes the paper, or see the official Cambridge past-papers page; for other papers, ask your school. “Paper 42” means Paper 4, variant 2.

Practice questions

Original questions written by HomiLabs in the style of the exam, with answers and mark schemes. They are not Cambridge past paper questions and have not been reviewed by Cambridge. Try each one before opening the answer.

Q1Modulus and cubic graphs · Core · Paper 1 style · no calculator · 5 marks
  1. (a) Sketch the graph of y = |2x − 3| for −1 ≤ x ≤ 4, stating the coordinates of the points where the graph meets the axes. [3]
  2. (b) Hence find the values of x for which |2x − 3| = 2. [2]
Show answer
  1. (a) V-shaped graph with vertex on the x-axis at (1.5, 0); meets the y-axis at (0, 3)
    B1 V shape; B1 (1.5, 0); B1 (0, 3)
  2. (b) x = 0.5, x = 2.5
    2x − 3 = ±2 M1; x = 0.5, 2.5 A1
Q2Modulus and cubic graphs · Core · Paper 1 style · no calculator · 3 marks
  1. (a) Sketch the graph of y = (x + 1)(x − 2)(x − 4), stating the coordinates of the points where the curve meets the axes. [3]
Show answer
  1. (a) positive cubic shape (rising from bottom left to top right) crossing the x-axis at (−1, 0), (2, 0) and (4, 0); meets the y-axis at (0, 8)
    B1 correct cubic shape; B1 x-intercepts −1, 2, 4; B1 y-intercept 8
Q3Modulus and cubic graphs · Core · Paper 2 style · calculator · 3 marks
  1. (a) Sketch the graph of y = (2 − x)(x + 3)², stating the coordinates of the points where the curve meets the axes. [3]
Show answer
  1. (a) negative cubic shape (from top left to bottom right) touching the x-axis at (−3, 0) (a minimum there) and crossing it at (2, 0); meets the y-axis at (0, 18)
    B1 shape with negative x³ coefficient; B1 touches at −3, crosses at 2; B1 (0, 18)
Q4Modulus and cubic graphs · Core · Paper 1 style · no calculator · 3 marks
  1. (a) Sketch the graph of y = |x + 4| − 3, stating the coordinates of its vertex and of the points where it meets the axes. [3]
Show answer
  1. (a) V shape with vertex (−4, −3); meets the x-axis at (−7, 0) and (−1, 0) and the y-axis at (0, 1)
    B1 vertex; B1 x-intercepts; B1 y-intercept
Q5Modulus and cubic graphs · Standard · Paper 1 style · no calculator · 6 marks
  1. (a) Sketch the graph of y = |(x − 1)(x − 5)|, stating the coordinates of the points where it meets the axes and the coordinates of its local maximum point. [4]
  2. (b) Find the set of values of k for which |(x − 1)(x − 5)| = k has exactly 4 solutions. [2]
Show answer
  1. (a) W shape: meets the x-axis at (1, 0) and (5, 0) (cusps), meets the y-axis at (0, 5); the part between x = 1 and x = 5 is reflected, giving a maximum at (3, 4)
    B1 W shape with cusps on the x-axis; B1 (1, 0), (5, 0); B1 (0, 5); B1 (3, 4)
  2. (b) 0 < k < 4
    B1 for 4 as upper limit; B1 strict inequality 0 < k < 4
Q6Modulus and cubic graphs · Standard · Paper 1 style · no calculator · 4 marks
  1. (a) The graph of a cubic y = f(x) crosses the x-axis at (−2, 0), touches the x-axis at (3, 0) and meets the y-axis at (0, 36). Find f(x) in the form a(x + p)(x + q)². [3]
  2. (b) Write down the set of values of x for which f(x) < 0. [1]
Show answer
  1. (a) f(x) = 2(x + 2)(x − 3)²
    f(x) = a(x + 2)(x − 3)² M1; a × 2 × 9 = 36 M1; a = 2 A1
  2. (b) x < −2
    B1 (f(x) ≥ 0 at x = 3)
Q7Modulus and cubic graphs · Standard · Paper 2 style · calculator · 5 marks
  1. (a) Sketch the graph of y = |x³ − 4x| for −3 ≤ x ≤ 3, stating the coordinates of the points where it meets the x-axis. [3]
  2. (b) The local maximum points of the graph between x = −2 and x = 2 have y-coordinate 3.08, correct to 3 significant figures. Find the number of solutions of the equation |x³ − 4x| = 1. [2]
Show answer
  1. (a) graph lies on or above the x-axis, meeting it (with cusps) at (−2, 0), (0, 0) and (2, 0), with two humps between −2 and 2
    B1 factorised x(x − 2)(x + 2); B1 shape with negative parts reflected; B1 intercepts
  2. (b) 6
    line y = 1 is below both humps (1 < 3.08), so it cuts each hump twice M1; it also cuts the rising branches for x < −2 and x > 2 once each; 6 solutions A1
Q8Modulus and cubic graphs · Standard · Paper 1 style · no calculator · 6 marks
  1. (a) On the same axes, sketch the graphs of y = |3x + 6| and y = |x − 2|, stating where each graph meets the axes. [3]
  2. (b) Find the x-coordinates of the points where the graphs intersect. [3]
Show answer
  1. (a) y = |3x + 6|: vertex (−2, 0), y-intercept (0, 6); y = |x − 2|: vertex (2, 0), y-intercept (0, 2)
    B1 each V shape; B1 all intercepts
  2. (b) x = −4, x = −1
    3x + 6 = x − 2 gives x = −4 M1 A1; 3x + 6 = −(x − 2) gives x = −1 A1
Q9Modulus and cubic graphs · Standard · Paper 2 style · calculator · 5 marks
  1. (a) Show that x³ − 3x² − 9x + 27 = (x − 3)²(x + 3). [2]
  2. (b) Hence sketch y = x³ − 3x² − 9x + 27, stating the coordinates of the points where it meets the axes. [3]
Show answer
  1. (a) (x² − 6x + 9)(x + 3) = x³ − 3x² − 9x + 27
    M1 expansion; A1
  2. (b) positive cubic crossing the x-axis at (−3, 0), touching it at (3, 0) (a minimum), y-intercept (0, 27)
    B1 shape; B1 (−3, 0) crossing and (3, 0) touching; B1 (0, 27)
Q10Modulus and cubic graphs · Challenge · Paper 1 style · no calculator · 4 marks
  1. (a) The graph of y = |f(x)|, where f(x) is a cubic, touches the x-axis at x = −1, meets the x-axis at x = 4 and meets the y-axis at (0, 8). Find the two possible expressions for f(x). [4]
Show answer
  1. (a) f(x) = 2(x + 1)²(x − 4) or f(x) = −2(x + 1)²(x − 4)
    f(x) = k(x + 1)²(x − 4) M1; |k × 1 × (−4)| = 8 M1; k = ±2 A1; both expressions given A1
Q11Modulus and cubic graphs · Challenge · Paper 2 style · calculator · 6 marks
  1. (a) Sketch the graph of y = |x² − 2x − 8|, stating the coordinates of the points where it meets the axes and of its local maximum point. [4]
  2. (b) Write down the value of k for which |x² − 2x − 8| = k has exactly 3 solutions, and the set of values of k for which it has exactly 2 solutions. [2]
Show answer
  1. (a) W shape: meets the x-axis at (−2, 0) and (4, 0), the y-axis at (0, 8); maximum of the reflected part at (1, 9)
    B1 shape; B1 (−2, 0), (4, 0); B1 (0, 8); B1 (1, 9)
  2. (b) k = 9; k > 9 or k = 0
    B1 k = 9; B1 k > 9 or k = 0
Q12Modulus and cubic graphs · Challenge · Paper 1 style · no calculator · 6 marks
  1. (a) On the same axes, sketch y = |2x − 5| and y = 3 − x for 0 ≤ x ≤ 4. Find the x-coordinates of the points where the graphs meet. [4]
  2. (b) Hence solve the inequality |2x − 5| < 3 − x. [2]
Show answer
  1. (a) x = 2 and x = 8/3
    both graphs sketched B1; 2x − 5 = 3 − x gives x = 8/3 M1 A1; −(2x − 5) = 3 − x gives x = 2 A1
  2. (b) 2 < x < 8/3
    B1 B1
Q13Modulus equations and inequalities · Core · Paper 1 style · no calculator · 2 marks
  1. (a) Solve the equation |3x − 4| = 8. [2]
Show answer
  1. (a) x = 4, x = −4/3
    3x − 4 = 8 or 3x − 4 = −8 M1; x = 4, −4/3 A1
Q14Modulus equations and inequalities · Core · Paper 2 style · calculator · 2 marks
  1. (a) Solve the inequality |x + 2| < 5. [2]
Show answer
  1. (a) −7 < x < 3
    −5 < x + 2 < 5 M1; −7 < x < 3 A1
Q15Modulus equations and inequalities · Core · Paper 1 style · no calculator · 3 marks
  1. (a) Solve the inequality |2x + 1| ≥ 7. [3]
Show answer
  1. (a) x ≤ −4 or x ≥ 3
    critical values 2x + 1 = ±7 M1; −4 and 3 A1; x ≤ −4 or x ≥ 3 A1
Q16Modulus equations and inequalities · Standard · Paper 1 style · no calculator · 3 marks
  1. (a) Solve the equation |4x − 3| = |x + 6|. [3]
Show answer
  1. (a) x = 3, x = −3/5
    4x − 3 = x + 6 gives x = 3 M1 A1; 4x − 3 = −(x + 6) gives x = −3/5 A1
Q17Modulus equations and inequalities · Standard · Paper 1 style · no calculator · 4 marks
  1. (a) Solve the inequality |x − 3| > |2x + 1|. [4]
Show answer
  1. (a) −4 < x < 2/3
    square both sides: x² − 6x + 9 > 4x² + 4x + 1 M1; 3x² + 10x − 8 < 0 A1; (3x − 2)(x + 4) < 0 M1; −4 < x < 2/3 A1
Q18Modulus equations and inequalities · Standard · Paper 2 style · calculator · 3 marks
  1. (a) Solve the equation |x² − 5| = 4. [3]
Show answer
  1. (a) x = −3, −1, 1, 3
    x² − 5 = 4 gives x = ±3 M1 A1; x² − 5 = −4 gives x = ±1 A1
Q19Modulus equations and inequalities · Standard · Paper 1 style · no calculator · 4 marks
  1. (a) Solve the equation |x² − 3x| = 2x. [4]
Show answer
  1. (a) x = 0, 1, 5
    x² − 3x = 2x gives x = 0 or 5 M1 A1; x² − 3x = −2x gives x = 0 or 1 A1; all satisfy 2x ≥ 0, so x = 0, 1, 5 A1
Q20Modulus equations and inequalities · Standard · Paper 2 style · calculator · 3 marks
  1. (a) Solve the inequality |5 − 2x| < 9. [3]
Show answer
  1. (a) −2 < x < 7
    −9 < 5 − 2x < 9 M1; −14 < −2x < 4 A1; −2 < x < 7 A1
Q21Modulus equations and inequalities · Challenge · Paper 1 style · no calculator · 4 marks
  1. (a) Solve the inequality |2x − 1| ≤ |x + 4|. [4]
Show answer
  1. (a) −1 ≤ x ≤ 5
    4x² − 4x + 1 ≤ x² + 8x + 16 M1; 3x² − 12x − 15 ≤ 0 A1; (x − 5)(x + 1) ≤ 0 M1; −1 ≤ x ≤ 5 A1
Q22Modulus equations and inequalities · Challenge · Paper 2 style · calculator · 4 marks
  1. (a) Solve the equation |x² − 4x| = 3, giving any irrational answers in the form a ± √b. [4]
Show answer
  1. (a) x = 1, 3, 2 + √7, 2 − √7
    x² − 4x − 3 = 0 M1; x = 2 ± √7 A1; x² − 4x + 3 = 0 M1; x = 1, 3 A1
Q23Modulus equations and inequalities · Challenge · Paper 1 style · no calculator · 3 marks
  1. (a) Solve the equation |x − 4| = 2x + 1. [3]
Show answer
  1. (a) x = 1 only
    x − 4 = 2x + 1 gives x = −5, but then 2x + 1 = −9 < 0, so reject M1 A1; x − 4 = −(2x + 1) gives x = 1 (check |−3| = 3) A1
Q24Modulus equations and inequalities · Challenge · Paper 1 style · no calculator · 6 marks
  1. (a) Solve the equation |2ˣ − 5| = 3. [3]
  2. (b) Explain why the equation |2ˣ − 5| = 6 has only one solution, and find it correct to 3 significant figures. [3]
Show answer
  1. (a) x = 1, x = 3
    2ˣ − 5 = 3 gives 2ˣ = 8, x = 3 M1 A1; 2ˣ − 5 = −3 gives 2ˣ = 2, x = 1 A1
  2. (b) 2ˣ = −1 is impossible since 2ˣ > 0; x = log₂ 11 = 3.46
    2ˣ = 11 or 2ˣ = −1 M1; 2ˣ > 0 so reject −1 A1; x = lg 11/lg 2 = 3.46 A1
Q25Cubic inequalities · Standard · Paper 1 style · no calculator · 3 marks
  1. (a) Solve the inequality (x + 2)(x − 1)(x − 4) > 0. [3]
Show answer
  1. (a) −2 < x < 1 or x > 4
    critical values −2, 1, 4 B1; sketch or sign table M1; −2 < x < 1, x > 4 A1
Q26Cubic inequalities · Standard · Paper 2 style · calculator · 5 marks
  1. (a) Show that x = 1 is a root of x³ − 6x² + 11x − 6 = 0 and factorise x³ − 6x² + 11x − 6 completely. [3]
  2. (b) Hence solve x³ − 6x² + 11x − 6 ≤ 0. [2]
Show answer
  1. (a) (x − 1)(x − 2)(x − 3)
    f(1) = 0 B1; quadratic factor x² − 5x + 6 M1; (x − 1)(x − 2)(x − 3) A1
  2. (b) x ≤ 1 or 2 ≤ x ≤ 3
    B1 critical values used; B1 correct set with ≤
Q27Cubic inequalities · Challenge · Paper 1 style · no calculator · 6 marks
  1. (a) Show that f(x) = −x³ + 6x² − 3x − 10 can be written as (2 − x)(x + 1)(x − 5). [2]
  2. (b) Sketch y = f(x), stating the intercepts with the axes, and hence solve f(x) ≥ 0. [4]
Show answer
  1. (a) (2 − x)(x² − 4x − 5) = −x³ + 6x² − 3x − 10
    M1 expansion; A1
  2. (b) negative cubic through (−1, 0), (2, 0), (5, 0) and (0, −10); x ≤ −1 or 2 ≤ x ≤ 5
    B1 shape; B1 intercepts; M1 reading regions above the axis; A1 x ≤ −1 or 2 ≤ x ≤ 5
Q28Equations reducible to quadratics · Core · Paper 1 style · no calculator · 3 marks
  1. (a) Solve the equation x⁴ − 13x² + 36 = 0. [3]
Show answer
  1. (a) x = −3, −2, 2, 3
    let u = x²: u² − 13u + 36 = 0 M1; u = 4 or 9 A1; x = ±2, ±3 A1
Q29Equations reducible to quadratics · Standard · Paper 1 style · no calculator · 4 marks
  1. (a) Solve the equation 2x − 7√x + 3 = 0. [4]
Show answer
  1. (a) x = 9, x = 1/4
    let u = √x: 2u² − 7u + 3 = 0 M1; (2u − 1)(u − 3) = 0 M1; u = 1/2 or 3 A1; x = 1/4 or 9 A1
Q30Equations reducible to quadratics · Challenge · Paper 2 style · calculator · 4 marks
  1. (a) Solve the equation x^(2/3) − 5x^(1/3) + 4 = 0. [4]
Show answer
  1. (a) x = 1, x = 64
    let u = x^(1/3): u² − 5u + 4 = 0 M1; u = 1 or 4 A1; x = 1³ = 1 or 4³ = 64 M1 A1

Similar questions

Questions that test the same subtopic in different years, grouped so you can work on one skill at a time.

References only: find each question in its paper (official link where available, otherwise through your school).

Modulus and cubic graphs Banker8 questions · 8/9 sessions
Modulus equations and inequalities 6 questions · 6/9 sessions
Cubic inequalities 3 questions · 3/9 sessions

Subtopics

Our analysis of how often each subtopic was examined and how many marks it carried, 2022 to May/June 2026.

SubtopicMarks indexedSessions
Modulus and cubic graphs Banker288/9
Modulus equations and inequalities 276/9
Cubic inequalities 63/9
Equations reducible to quadratics Rare41/9

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