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Chapter 6 Practice Paper: Formulae, functions and proof — question paper

QuizQuipPractice Series
QQ/GCSE-MATHS/CH6

GCSE (9–1) · GCSE Mathematics (Foundation with Higher stretch)

Chapter 6 Practice Paper

Formulae, functions and proof

Duration: 40 minutes
Total: 30 marks
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READ THESE INSTRUCTIONS FIRST

  • Answer all questions.
  • Section A contains ten multiple-choice questions. Choose one answer, A, B, C or D, for each.
  • Section B contains structured questions. Write your answers in the spaces provided.
  • Where a question asks for a proof, use algebra — not worked examples.
  • The number of marks is given in brackets [ ] at the end of each question or part question.
  • The total mark for this paper is 30.

Original practice material written for the MOE GCSE Mathematics syllabus. It is not an official MOE or school examination paper. Chapter 6: substitution into formulae including scientific formulas, changing the subject, equation-versus-identity arguments and algebraic proof (Higher), inverse and composite functions (Higher), and iteration (Higher).

For Examiner’s Use
Section A/ 10
Section B/ 20
Total/ 30

Section A: Multiple-choice questions[10 marks]

Choose one answer for each question. Every correct answer scores one mark.

  1. 1

    Given T = 5x − 2y with x = 4 and y = 3, T equals

    1. A14
    2. B26
    3. C7
    4. D6
    ( )
  2. 2

    With a = −3, the value of a² + 2a is

    1. A−3
    2. B−15
    3. C3
    4. D15
    ( )
  3. 3

    Rearrange d = 5t to make t the subject.

    1. At = 5d
    2. Bt = d + 5
    3. Ct = d⁄5
    4. Dt = d − 5
    ( )
  4. 4

    In the triangle-area formula A = bh⁄2, rearrange to make h the subject.

    1. Ah = A⁄2b
    2. Bh = 2b⁄A
    3. Ch = Ab⁄2
    4. Dh = 2A⁄b
    ( )
  5. 5

    From y = 3(x − 4), write x in terms of y.

    1. Ax = y − 12
    2. Bx = (y − 4)⁄3
    3. Cx = y⁄3 − 4
    4. Dx = y⁄3 + 4
    ( )
  6. 6

    The expansion (x + 2)(x + 5) ≡ x² + 7x + 10 is best justified by

    1. Atesting x = 1 and x = 2
    2. Bexpanding the brackets
    3. Cdrawing a table of values
    4. Dmeasuring a graph
    ( )
  7. 7

    The function f maps x to x + 6. The value of f(10) is

    1. A16
    2. B60
    3. C4
    4. D1.6
    ( )
  8. 8

    For the same function, the inverse f⁻¹(x) is (Higher)

    1. Ax + 6
    2. B6x
    3. Cx − 6
    4. Dx⁄6
    ( )
  9. 9

    In the composite fg the function g runs first. With f(x) = 2x and g(x) = x + 1, fg(3) is (Higher)

    1. A7
    2. B8
    3. C9
    4. D12
    ( )
  10. 10

    With x₀ = 1 and xₙ₊₁ = ∛(10 − 2xₙ), x₁ equals (Higher)

    1. A∛12 ≈ 2.29
    2. B∛8 = 2
    3. C3
    4. D∛10 ≈ 2.15
    ( )

Section B: Structured questions[20 marks]

Show all your working. Marks are given for method as well as for the final answer.

  1. 11

    Substitution.

    1. (a)

      The formula for kinetic energy is E = mv²⁄2. Find E when m = 6 and v = 4.

      [2]
    2. (b)

      In the formula s = ut + ½at², u = −2, a = 3 and t = 4. Find s.

      [2]
  2. 12

    Changing the subject.

    1. (a)

      The formula v² = u² + 2as connects four quantities. Write a in terms of the other three.

      [2]
    2. (b)

      Starting from c = √(a² + d²), isolate d.

      [2]
  3. 13

    Proof. (Higher)

    1. (a)

      Prove that adding four consecutive odd numbers always gives a multiple of 8.

      [3]
    2. (b)

      Show that (2x + 1)(x + 4) ≡ 2x² + 9x + 4.

      [2]
  4. 14

    Functions. (Higher)

    1. (a)

      f(x) = 6 − 2x. Find f⁻¹(x).

      [2]
    2. (b)

      With g(x) = x², find gf(3).

      [2]
  5. 15

    Approximate roots by iteration. (Higher)

    1. (a)

      The equation x³ = 20 − 5x is solved by the iteration xₙ₊₁ = ∛(20 − 5xₙ). Starting with x₀ = 2, find x₁, x₂ and x₃ to 4 decimal places.

      [3]

— End of paper —