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GCSE Maths Higher Paper 1: printable

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GCSE Maths Higher Paper 1 (non-calculator)

Name: ______________________   Date: __________   Marks: ____ / 80

Time: 90 minutes. You must not use a calculator. Answer all questions. Show your working. The marks for each part are shown in brackets. steplyclass.com/gcse-maths/higher-paper-1/

  1. 1 Question 1

    Write 252 as a product of its prime factors. Give your answer in index form.

    [2 marks]

    Answer: ____________________

  2. 2 Question 2

    Work out 3 14 − 1 5/6. Give your answer as a mixed number.

    [3 marks]

    Answer: ____________________

  3. 3 Question 3

    Work out the value of 642/3.

    [2 marks]

    Answer: ____________________

  4. 5 Question 5

    Write the recurring decimal 0.2727... as a fraction in its simplest form. Show your working.

    [3 marks]

    Answer: ____________________

  5. 6(a) Question 6(a)

    Simplify √75 − √12.

    [2 marks]

    Answer: ____________________

  6. 6(b) Question 6(b)

    Rationalise the denominator and simplify 6/(√3 + 1).

    [3 marks]

    Answer: ____________________

  7. 7 Question 7

    Work out (8 × 10⁴) × (6 × 10⁻⁷). Give your answer in standard form.

    [2 marks]

    Answer: ____________________

  8. 8 Question 8

    Factorise fully 3x² − 12.

    [2 marks]

    Answer: ____________________

  9. 9 Question 9

    Solve 3x² − 10x − 8 = 0.

    [3 marks]

    Answer: ____________________

  10. 10 Question 10

    Write x² + 8x + 3 in the form (x + a)² + b.

    [3 marks]

    Answer: ____________________

  11. 11 Question 11

    Solve x² − x − 12 < 0.

    [3 marks]

    Answer: ____________________

  12. 12 Question 12

    Solve the simultaneous equations 4x + 3y = 17 and 3x − 2y = 0.

    [4 marks]

    Answer: ____________________

  13. 13 Question 13

    Simplify fully x² − 16x² − 2x − 8.

    [3 marks]

    Answer: ____________________

  14. 14 Question 14

    Write 3/(x − 1) − 2/(x + 2) as a single fraction.

    [3 marks]

    Answer: ____________________

  15. f(x) = 3x − 2 and g(x) = x² + 1.

    15(a) Question 15(a)

    Work out fg(2).

    [2 marks]

    Answer: ____________________

  16. 15(b) Question 15(b)

    Work out f⁻¹(x).

    [2 marks]

    Answer: ____________________

  17. 16 Question 16

    Work out an expression for the nth term of 6, 9, 14, 21, 30, ...

    [3 marks]

    Answer: ____________________

  18. 17 Question 17

    Work out the equation of the line through (4, 1) that is perpendicular to y = 2x − 5.

    [3 marks]

    Answer: ____________________

  19. 18 Question 18

    A circle has centre (0, 0) and radius 7. Write down its equation.

    [2 marks]

    Answer: ____________________

  20. 19 Question 19

    Describe the transformation that maps y = x² onto y = (x − 3)² + 1.

    [2 marks]

    Answer: ____________________

  21. 20 Question 20

    A, B and C lie on a circle with centre O. Angle AOB = 124°. Work out angle ACB. Give a reason.

    [2 marks]

    Answer: ____________________

  22. 22 Question 22

    A tangent from P touches a circle, centre O, at T. The radius is 6 cm and OP = 10 cm. Work out PT.

    [2 marks]

    Answer: ____________________

  23. 24 Question 24

    A bag has 4 red and 5 blue sweets. Two sweets are taken at random without replacement. Work out the probability that both are red.

    [3 marks]

    Answer: ____________________

  24. 25 Question 25

    A survey: men, 18 said yes and 6 said no; women, 14 said yes and 12 said no. A man is chosen at random. Work out the probability he said yes.

    [2 marks]

    Answer: ____________________

  25. 26 Question 26

    y is inversely proportional to x. When x = 3, y = 8. Work out y when x = 6.

    [3 marks]

    Answer: ____________________

  26. 27 Question 27

    Two similar shapes have lengths in the ratio 2 : 5. The small shape has area 12 cm². Work out the area of the large shape.

    [2 marks]

    Answer: ____________________

  27. 28 Question 28

    Make x the subject of 5x + a = bx − 1.

    [3 marks]

    Answer: ____________________

  28. 29 Question 29

    Expand and simplify (x + 2)(x − 1)(x + 4).

    [3 marks]

    Answer: ____________________

  29. 30 Question 30

    Solve 10x − 3 = 2.

    [3 marks]

    Answer: ____________________

  30. 32 Question 32

    Work out the equation of the perpendicular bisector of the line from (2, 0) to (6, 4).

    [3 marks]

    Answer: ____________________

  31. 33 Question 33

    Prove that the sum of any two odd numbers is even.

    [2 marks]

Answers

  1. 1 252 = 22 × 32 × 7
  2. 2 1 512
  3. 3 16
  4. 5 x = 311
  5. 6(a) 3√3
  6. 6(b) 3√3 − 3
  7. 7 4.8 × 10−2
  8. 8 3(x + 2)(x − 2)
  9. 9 x = −23 or x = 4
  10. 10 (x + 4)2 − 13
  11. 11 −3 < x < 4
  12. 12 (x, y) = (2, 3)
  13. 13 x + 4x + 2
  14. 14 x + 8(x − 1)(x + 2)
  15. 15(a) f(g(2)) = 13
  16. 15(b) f −1(x) = x + 23
  17. 16 an = n2 + 5
  18. 17 y = −12x + 3
  19. 18 x2 + y2 = 49
  20. 19 turning point (3, 1)
  21. 20 x = 62
  22. 22 b = 8
  23. 24 P(all red) = 16
  24. 25 P(yes | man) = 34
  25. 26 y = 4
  26. 27 large area = 75
  27. 28 x = −1 − a5 − b
  28. 29 x³ + 5x² + 2x − 8
  29. 30 x = 2
  30. 32 y = −x + 6
  31. 33 2(a + b + 1)

Full mark scheme: steplyclass.com/gcse-maths/higher-paper-1/mark-scheme/

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