Paper 1-style (Higher)
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Paper 1-style practice paper: Higher (non-calculator) — question paper

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QQ/GCSE-MATHS/P1HS

GCSE (9–1) · GCSE Mathematics (Higher)

Paper 1-style practice paper

Higher (non-calculator)

Duration: 90 minutes
Total: 80 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.
  • You may not use a calculator. Give exact answers: fractions in their simplest form, surds left in surd form.
  • The number of marks is given in brackets [ ] at the end of each question or part question.
  • The total mark for this paper is 80.

Original practice material written for the MOE GCSE Mathematics syllabus. It is not an official MOE or school examination paper. Higher content across all six topic areas, non-calculator: surds and indices, algebraic manipulation and functions, quadratics, circle theorems, set notation and Venn diagrams, exact trigonometry, sequences and Higher probability.

For Examiner’s Use
Section A/ 10
Section B/ 70
Total/ 80

Section A: Multiple-choice questions[10 marks]

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

  1. 1

    √50 simplifies to

    1. A2√5
    2. B25√2
    3. C10√5
    4. D5√2
    ( )
  2. 2

    (2x²)³ equals

    1. A2x⁶
    2. B8x⁶
    3. C6x⁶
    4. D8x⁵
    ( )
  3. 3

    The exact value of sin 30° is

    1. A√3⁄2
    2. B√2⁄2
    3. C½
    4. D1
    ( )
  4. 4

    Solve x² − 5x + 6 = 0.

    1. Ax = 2 or x = 3
    2. Bx = −2 or x = −3
    3. Cx = 1 or x = 6
    4. Dx = −1 or x = −6
    ( )
  5. 5

    The nth term of the sequence 5, 8, 11, 14, … is

    1. A3n + 2
    2. B5n
    3. C3n − 2
    4. D5n − 3
    ( )
  6. 6

    The interior angle of a regular hexagon is

    1. A60°
    2. B108°
    3. C135°
    4. D120°
    ( )
  7. 7

    f(x) = 2x − 3. The inverse function f⁻¹(x) is

    1. A(x − 3)⁄2
    2. B2x + 3
    3. C(x + 3)⁄2
    4. D3 − 2x
    ( )
  8. 8

    A and B are independent with P(A) = 0.3 and P(B) = 0.4. P(A and B) is

    1. A0.12
    2. B0.7
    3. C0.58
    4. D0.1
    ( )
  9. 9

    The angle at the centre of a circle is 140°. The angle at the circumference subtended by the same arc is

    1. A140°
    2. B70°
    3. C110°
    4. D40°
    ( )
  10. 10

    2⁄√8 written in the form a√b is

    1. A√2
    2. B2√2
    3. C√2⁄2
    4. D½
    ( )

Section B: Structured questions[70 marks]

Show all your working. You may not use a calculator. Give exact answers.

  1. 11

    Surds and indices.

    1. (a)

      Simplify √72.

      [2]
    2. (b)

      Expand and simplify (3 + √2)(3 − √2).

      [2]
    3. (c)

      Evaluate 27^(2⁄3).

      [2]
  2. 12

    Algebraic manipulation.

    1. (a)

      Factorise 2x² + 5x − 12.

      [2]
    2. (b)

      Expand and simplify (x + 5)(x − 5).

      [2]
    3. (c)

      Simplify fully (x² − 9)⁄(x + 3).

      [3]
  3. 13

    Functions.

    1. (a)

      f(x) = x² + 1 and g(x) = 2x. Find fg(3).

      [2]
    2. (b)

      Find f⁻¹(x).

      [2]
    3. (c)

      Solve fg(x) = 9.

      [2]
  4. 14

    Quadratics.

    1. (a)

      Solve x² = 3x.

      [3]
    2. (b)

      Write x² + 6x + 2 in the form (x + a)² + b.

      [3]
    3. (c)

      Hence state the coordinates of the minimum point of the graph of y = x² + 6x + 2.

      [2]
  5. 15

    Circle theorems.

    1. (a)

      Points A and B lie on a circle with centre O, and angle OAB = 28°. Explain why angle OBA is also 28°.

      [3]
    2. (b)

      Find angle AOB.

      [3]
    3. (c)

      Point C lies on the major arc AB. Find angle ACB, giving the theorem you use.

      [2]
  6. 16

    Sets and Venn diagrams.

    1. (a)

      Of 60 pupils, 35 study French (F), 28 study Spanish (S) and 15 study neither. Find n(F ∩ S).

      [3]
    2. (b)

      Find P(F ∩ S).

      [2]
    3. (c)

      Find n(F′).

      [2]
  7. 17

    Exact trigonometry and similarity.

    1. (a)

      Write down the exact value of tan 60° × tan 30°.

      [2]
    2. (b)

      A right-angled triangle has shorter sides 1 cm and 2 cm. Write the exact length of the hypotenuse.

      [2]
    3. (c)

      Two similar triangles have areas 9 cm² and 25 cm². A side of the smaller triangle is 6 cm. Find the corresponding side of the larger triangle.

      [4]
  8. 18

    Inverse proportion.

    1. (a)

      y is inversely proportional to x². When x = 3, y = 4. Find an equation for y in terms of x.

      [4]
    2. (b)

      Find the value of y when x = 2.

      [2]
    3. (c)

      Find the exact value of x when y = 6. Give your answer in surd form.

      [1]
  9. 19

    Quadratic sequences.

    1. (a)

      A sequence begins 4, 7, 12, 19, 28. Write down the 6th term, showing how you found it.

      [1]
    2. (b)

      Find an expression for the nth term of the sequence.

      [4]
  10. 20

    Probability without replacement.

    1. (a)

      A bag holds 4 red and 3 blue counters. Two are drawn without replacement. Copy and complete the tree diagram by writing the four second-stage probabilities.

      [2]
    2. (b)

      Find P(both counters are red).

      [2]
    3. (c)

      Find P(the two counters are the same colour).

      [2]
    4. (d)

      Find P(at least one red).

      [2]

— End of paper —