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About Pre-DP Review — Class 10 IB

Consolidate MYP skills and bridge to IB Diploma Programme mathematics with mixed-topic revision. This topic is part of the IB Class 10 mathematics syllabus (chapter: Unit 11). On this page you can practice 60 questions across three difficulty levels — 20 easy, 20 medium, and 20 hard — each with a visual step-by-step solution, plus a timed 35-question mastery test with a live leaderboard. Worked examples and sample questions from the topic are below.

Pre-DP Review — solved examples for Class 10 IB

Example 1easy

Which of the following statements correctly describes a mathematical function?
  1. A)A. A relation where each input has exactly one output.
  2. B)B. A relation where each input can have multiple outputs.
  3. C)C. A relation where each output has exactly one input.
  4. D)D. Any set of ordered pairs (x, y).

Step-by-step solution

  1. A function maps each element from its domain (inputs) to exactly one element in its codomain (outputs).
  2. This means for every x-value, there can only be one corresponding y-value. This is also known as the vertical line test for graphs.
  3. Options B, C, and D describe relations that are not necessarily functions.

Answer: A. A relation where each input has exactly one output.

Example 2medium

Simplify the expression: (x² - 4x + 4) / (x² - 4) ÷ (x - 2) / (x + 2)
  1. A)1
  2. B)x - 2
  3. C)(x - 2)² / (x² - 4)
  4. D)(x + 2)² / (x - 2)

Step-by-step solution

  1. Factorise the numerator and denominator of the first fraction: x² - 4x + 4 = (x - 2)² and x² - 4 = (x - 2)(x + 2).
  2. Rewrite the expression with factorised terms: [(x - 2)² / ((x - 2)(x + 2))] ÷ [(x - 2) / (x + 2)].
  3. Change division to multiplication by the reciprocal of the second fraction: [(x - 2)² / ((x - 2)(x + 2))] × [(x + 2) / (x - 2)].
  4. Cancel common factors: (x - 2) / (x + 2) × (x + 2) / (x - 2) = 1.

Answer: 1

Example 3hard

Given the functions f(x) = 2x + 1 and g(x) = x². Find the value(s) of x for which f(g(x)) = g(f(x)).
  1. A)A. x = -1, 0
  2. B)B. x = 0, 1
  3. C)C. x = 0, -2
  4. D)D. x = -1, 2

Step-by-step solution

  1. Calculate f(g(x)): f(x²) = 2(x²) + 1 = 2x² + 1.
  2. Calculate g(f(x)): g(2x + 1) = (2x + 1)² = 4x² + 4x + 1.
  3. Set f(g(x)) = g(f(x)): 2x² + 1 = 4x² + 4x + 1.
  4. Rearrange and solve the quadratic equation: 0 = 2x² + 4x => 0 = 2x(x + 2). Thus, x = 0 or x = -2.

Answer: C. x = 0, -2

Practice questions on Pre-DP Review

  1. Q1.easy

    To determine the nature of the roots of a quadratic equation ax² + bx + c = 0, which expression is most crucial to evaluate?
    1. A)A. b² - 4ac
    2. B)B. -b / 2a
    3. C)C. (b² - 4ac) / 2a
    4. D)D. c / a
    Show answer

    Answer: A. b² - 4ac

    Hint: The discriminant provides direct information about whether the roots are real, distinct, equal, or complex.

  2. Q2.easy

    A sequence starts with 5 and each subsequent term is found by adding 3 to the previous term. What type of sequence is this, and what is its 4th term?
    1. A)A. Arithmetic, 14
    2. B)B. Geometric, 13
    3. C)C. Arithmetic, 11
    4. D)D. Geometric, 14
    Show answer

    Answer: A. Arithmetic, 14

    Hint: Consider how each term is generated from the previous one. Is it by adding a constant or multiplying by a constant?

  3. Q3.easy

    In a right-angled triangle ABC, with the right angle at B, if side AB is 8 units and side BC is 6 units, which expression correctly represents tan(A)?
    1. A)A. 6/8
    2. B)B. 8/6
    3. C)C. 6/10
    4. D)D. 8/10
    Show answer

    Answer: A. 6/8

    Hint: Recall the definition of tangent (SOH CAH TOA) and identify the opposite and adjacent sides relative to angle A.

  4. Q4.medium

    A rectangular garden has an area of 108 m². If its length is 3 m more than twice its width, what is the perimeter of the garden?
    1. A)39 m
    2. B)42 m
    3. C)45 m
    4. D)48 m
    Show answer

    Answer: 42 m

    Hint: Set up a quadratic equation using the given area and the relationship between length and width, then solve for the dimensions.

  5. Q5.medium

    Consider the function f(x) = 1 / (x - 3). Which of the following statements about f(x) and its inverse f⁻¹(x) is true?
    1. A)The domain of f(x) is x ≠ 3, and the range of f(x) is y ≠ 3.
    2. B)The domain of f⁻¹(x) is x ≠ 3, and f⁻¹(x) = 1/x + 3.
    3. C)f⁻¹(x) = 1/x + 3, and its domain is x ≠ 0.
    4. D)f⁻¹(x) = 3 - 1/x, and its range is y ≠ 3.
    Show answer

    Answer: f⁻¹(x) = 1/x + 3, and its domain is x ≠ 0.

    Hint: To find the inverse, swap x and y and solve for y. Remember that the domain of f is the range of f⁻¹, and vice-versa.

  6. Q6.medium

    A cuboid has dimensions 12 cm by 4 cm by 3 cm. What is the exact length of the longest diagonal (space diagonal) of the cuboid?
    1. A)√153 cm
    2. B)√169 cm
    3. C)12 cm
    4. D)13 cm
    Show answer

    Answer: 13 cm

    Hint: The space diagonal can be found using the 3D diagonal formula: d = √(l² + w² + h²).

  7. Q7.hard

    The p-th, q-th, and r-th terms of an arithmetic progression are in geometric progression. If p, q, r are distinct positive integers, which of the following must be true?
    1. A)A. p, q, r are in AP
    2. B)B. p, q, r are in GP
    3. C)C. p, q, r are in HP
    4. D)D. (q-p)² = (r-q)²
    Show answer

    Answer: A. p, q, r are in AP

    Hint: Express the p-th, q-th, and r-th terms of the AP using the first term and common difference. Use the property of a geometric progression (the square of the middle term equals the product of the other two) and simplify the resulting equation, considering the non-trivial case.

  8. Q8.hard

    If logₐb = 2 and logₓa = 3, then logₓ(ab) equals:
    1. A)A. 5
    2. B)B. 6
    3. C)C. 7
    4. D)D. 9
    Show answer

    Answer: D. 9

    Hint: Use the properties of logarithms, specifically log(XY) = logX + logY and log(Xⁿ) = n logX. Convert the given logarithmic equations into exponential form if it helps.

  9. Q9.hard

    If the quadratic equation ax² + bx + c = 0 has no real roots, and a > 0, which of the following statements must always be true?
    1. A)A. b² - 4ac < 0 and c > 0
    2. B)B. b² - 4ac < 0 and the parabola opens downwards
    3. C)C. b² - 4ac > 0 and c < 0
    4. D)D. b² - 4ac < 0 and the parabola has a minimum point above the x-axis
    Show answer

    Answer: D. b² - 4ac < 0 and the parabola has a minimum point above the x-axis

    Hint: Recall the condition for no real roots in terms of the discriminant. Consider how the sign of the leading coefficient 'a' affects the parabola's shape and its position relative to the x-axis when there are no real roots.

These are 9 of the 60 questions available for Pre-DP Review. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.