MK Maths

Original practice

Questions · Answers · Reasoning

MK Maths resources · Set 01

GCSE Algebra Diagnostic

Twenty original questions arranged from core algebraic fluency to higher-tier reasoning. Work without notes where possible and show each stage clearly.

  • GCSE / IGCSE
  • Foundation to Higher
  • Approximately 35 minutes
  • Calculator optional
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Questions

The labels indicate the principal level and skill, although several questions connect more than one curriculum objective.

  1. Simplify 4a + 3a.

    FoundationSimplifying

  2. Simplify 7x − 2x + 5.

    FoundationSimplifying

  3. Expand 3(2x + 5).

    FoundationExpansion

  4. Find the value of 3x² + 4 when x = −2.

    FoundationSubstitution

  5. Solve 5x + 7 = 27.

    FoundationEquations

  6. Solve 3(x − 2) = 15.

    FoundationEquations

  7. The first four terms of a sequence are 5, 8, 11, 14. Find an expression for its nth term.

    FoundationSequences

  8. Expand and simplify (x + 4)(x + 2).

    HigherExpansion

  9. Factorise fully 6x + 18.

    FoundationFactorising

  10. Factorise x² + 7x + 12.

    HigherQuadratics

  11. Solve 2x + 5 = 3x − 4.

    FoundationEquations

  12. Make a the subject of v = u + at.

    HigherRearranging

  13. Simplify (3x²y)(2xy³).

    HigherIndices

  14. Solve x² − 5x + 6 = 0.

    HigherQuadratics

  15. Factorise fully 9x² − 25.

    HigherDifference of squares

  16. Solve simultaneously: x + y = 9 and 2x − y = 6.

    HigherSimultaneous equations

  17. Solve the inequality 3x − 4 < 11.

    FoundationInequalities

  18. The first four terms of a sequence are 2, 6, 12, 20. Find an expression for its nth term.

    HigherQuadratic sequences

  19. Simplify (x² − 9) / (x² + 2x − 3). State the values of x for which the original expression is undefined.

    HigherAlgebraic fractions

  20. A rectangle has sides x + 3 cm and x − 1 cm. Its area is 60 cm². Form and solve an equation to find the dimensions of the rectangle.

    HigherProblem solving

Answers

  1. 1. 7a
  2. 2. 5x + 5
  3. 3. 6x + 15
  4. 4. 16
  5. 5. x = 4
  6. 6. x = 7
  7. 7. 3n + 2
  8. 8. x² + 6x + 8
  9. 9. 6(x + 3)
  10. 10. (x + 3)(x + 4)
  11. 11. x = 9
  12. 12. a = (v − u) / t
  13. 13. 6x³y⁴
  14. 14. x = 2 or x = 3
  15. 15. (3x − 5)(3x + 5)
  16. 16. x = 5, y = 4
  17. 17. x < 5
  18. 18. n(n + 1)
  19. 19. (x − 3) / (x − 1), where x ≠ −3 and x ≠ 1
  20. 20. 10 cm by 6 cm

Selected worked solutions

Question 5 · Linear equation

Start with 5x + 7 = 27. Subtract 7 from both sides to obtain 5x = 20. Divide both sides by 5, giving x = 4.

Question 10 · Factorising a quadratic

We need two numbers with a product of 12 and a sum of 7. These numbers are 3 and 4, so x² + 7x + 12 = (x + 3)(x + 4).

Question 12 · Rearranging a formula

From v = u + at, subtract u from both sides: v − u = at. Divide both sides by t to isolate a: a = (v − u) / t, where t ≠ 0.

Question 16 · Simultaneous equations

Add x + y = 9 and 2x − y = 6. The y terms cancel, leaving 3x = 15, so x = 5. Substitute into x + y = 9 to obtain 5 + y = 9, hence y = 4.

Question 18 · Quadratic sequence

The first differences are 4, 6 and 8, so the second difference is constant. The terms can be recognised as 1 × 2, 2 × 3, 3 × 4 and 4 × 5. Therefore the nth term is n(n + 1).

Question 19 · Algebraic fraction

Factorise the numerator and denominator: x² − 9 = (x − 3)(x + 3), while x² + 2x − 3 = (x + 3)(x − 1). Cancel the common factor to obtain (x − 3) / (x − 1). The original denominator is zero when x = −3 or x = 1, so both values must be excluded.

Question 20 · Forming and solving an equation

The area gives (x + 3)(x − 1) = 60. Expanding and rearranging gives x² + 2x − 63 = 0, which factorises to (x + 9)(x − 7) = 0. The possible values are −9 and 7, but −9 would produce negative side lengths. Therefore x = 7 and the rectangle measures 10 cm by 6 cm.

This is an original MK Maths resource aligned with algebra skills commonly assessed by AQA and Pearson Edexcel GCSE Mathematics and Cambridge IGCSE Mathematics. It is not endorsed by any examination board.