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GCSE Maths · Edexcel 1MA1 · Grade Ladder · Algebra

Grade Ladder: Algebra

18 questions climbing from grade 1 to grade 9 across the algebra content of the Edexcel GCSE, two at each grade. Each question shows its grade, its specification code and whether it is non-calculator.

How to use this ladderFoundation students: work through Grades 1–5. Higher students: start at Grade 4 and push on to 9. Questions marked Non-calculator should be done without a calculator, as in Paper 1.
Grades 1–3 · Foundation
  1. (1 mark)
    Grade 1A4Non-calculator
    Simplify a + a + a + b.
  2. (1 mark)
    Grade 1A2Non-calculator
    Work out the value of 3x + 2 when x = 4.
  3. (2 marks)
    Grade 2A17Non-calculator
    Solve (a) x + 7 = 12 (b) 3x = 21.
  4. (1 mark)
    Grade 2A23Non-calculator
    Write down the next two terms of 3, 7, 11, 15, …
  5. (1 mark)
    Grade 3A4Non-calculator
    Expand 4(2x − 3).
  6. (2 marks)
    Grade 3A17Non-calculator
    Solve 5x − 4 = 2x + 11.
Grades 4–5 · Foundation and Higher
  1. (2 marks)
    Grade 4A4Non-calculator
    Factorise x² + 7x + 12.
  2. (2 marks)
    Grade 4A25Non-calculator
    Find the nth term of 5, 8, 11, 14, …
  3. (3 marks)
    Grade 5A19Non-calculator
    Solve the simultaneous equations 2x + y = 11 and x − y = 1.
  4. (2 marks)
    Grade 5A5Non-calculator
    Make r the subject of A = πr².
Grades 6–7 · Higher
  1. (3 marks)
    Grade 6A18Non-calculator
    Solve x² − 5x − 6 = 0.
  2. (2 marks)
    Grade 6A22Non-calculator
    Solve 3(2x − 1) > 4x + 5.
  3. (3 marks)
    Grade 7A18
    Solve 2x² + 3x − 7 = 0. Give your answers to 2 decimal places.
  4. (3 marks)
    Grade 7A7Non-calculator
    f(x) = 3x − 2 and g(x) = x². (a) Find fg(2). (b) Find f⁻¹(x).
Grades 8–9 · Higher
  1. (3 marks)
    Grade 8A25Non-calculator
    Find the nth term of the quadratic sequence 4, 9, 16, 25, 36, …
  2. (3 marks)
    Grade 8A4Non-calculator
    Simplify fully x² − 9x² + x − 6.
  3. (4 marks)
    Grade 9A18
    Solve 3x + 1 + 2x − 2 = 1. Give your answers in surd form.
  4. (3 marks)
    Grade 9A6Non-calculator
    Prove that (2n + 1)² − (2n − 1)² is a multiple of 8 for all integers n.
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Worked solutions

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GCSE Maths · Edexcel 1MA1 · Grade Ladder · Algebra

Grade Ladder: Algebra — Worked Solutions

Full worked solutions to all 18 questions (41 marks).

Grades 1–3 · Foundation
  1. (1 mark)
    Grade 1A4Non-calculator
    Simplify a + a + a + b.
    Solution
    1. Collect like terms.
    3a + b
  2. (1 mark)
    Grade 1A2Non-calculator
    Work out the value of 3x + 2 when x = 4.
    Solution
    1. 3 × 4 + 2.
    14
  3. (2 marks)
    Grade 2A17Non-calculator
    Solve (a) x + 7 = 12 (b) 3x = 21.
    Solution
    1. (a) Subtract 7.
    2. (b) Divide by 3.
    (a) 5 (b) 7
  4. (1 mark)
    Grade 2A23Non-calculator
    Write down the next two terms of 3, 7, 11, 15, …
    Solution
    1. The terms go up by 4.
    19, 23
  5. (1 mark)
    Grade 3A4Non-calculator
    Expand 4(2x − 3).
    Solution
    1. 4 × 2x − 4 × 3.
    8x − 12
  6. (2 marks)
    Grade 3A17Non-calculator
    Solve 5x − 4 = 2x + 11.
    Solution
    1. 3x − 4 = 11 → 3x = 15.
    x = 5
Grades 4–5 · Foundation and Higher
  1. (2 marks)
    Grade 4A4Non-calculator
    Factorise x² + 7x + 12.
    Solution
    1. Two numbers that multiply to 12 and add to 7: 3 and 4.
    (x + 3)(x + 4)
  2. (2 marks)
    Grade 4A25Non-calculator
    Find the nth term of 5, 8, 11, 14, …
    Solution
    1. Difference 3 → 3n. 5 − 3 = 2.
    3n + 2
  3. (3 marks)
    Grade 5A19Non-calculator
    Solve the simultaneous equations 2x + y = 11 and x − y = 1.
    Solution
    1. Add: 3x = 12 → x = 4.
    2. 4 − y = 1 → y = 3.
    x = 4, y = 3
  4. (2 marks)
    Grade 5A5Non-calculator
    Make r the subject of A = πr².
    Solution
    1. r² = A ÷ π, then square-root.
    r = √(A/π)
Grades 6–7 · Higher
  1. (3 marks)
    Grade 6A18Non-calculator
    Solve x² − 5x − 6 = 0.
    Solution
    1. (x − 6)(x + 1) = 0.
    x = 6 or x = −1
  2. (2 marks)
    Grade 6A22Non-calculator
    Solve 3(2x − 1) > 4x + 5.
    Solution
    1. 6x − 3 > 4x + 5 → 2x > 8.
    x > 4
  3. (3 marks)
    Grade 7A18
    Solve 2x² + 3x − 7 = 0. Give your answers to 2 decimal places.
    Solution
    1. x = −3 ± √(9 + 56)4 = −3 ± √654.
    x = 1.27 or x = −2.77
  4. (3 marks)
    Grade 7A7Non-calculator
    f(x) = 3x − 2 and g(x) = x². (a) Find fg(2). (b) Find f⁻¹(x).
    Solution
    1. (a) g(2) = 4, then f(4) = 10.
    2. (b) y = 3x − 2 → x = (y + 2)/3.
    (a) 10 (b) (x + 2)/3
Grades 8–9 · Higher
  1. (3 marks)
    Grade 8A25Non-calculator
    Find the nth term of the quadratic sequence 4, 9, 16, 25, 36, …
    Solution
    1. Second difference 2 → n². Subtract: 4 − 1, 9 − 4, 16 − 9 = 3, 5, 7 → 2n + 1.
    2. nth term = n² + 2n + 1.
    n² + 2n + 1 = (n + 1)²
  2. (3 marks)
    Grade 8A4Non-calculator
    Simplify fully x² − 9x² + x − 6.
    Solution
    1. Numerator (x − 3)(x + 3). Denominator (x + 3)(x − 2).
    2. Cancel (x + 3).
    x − 3x − 2
  3. (4 marks)
    Grade 9A18
    Solve 3x + 1 + 2x − 2 = 1. Give your answers in surd form.
    Solution
    1. Multiply by (x + 1)(x − 2): 3(x − 2) + 2(x + 1) = (x + 1)(x − 2).
    2. 5x − 4 = x² − x − 2 → x² − 6x + 2 = 0.
    3. x = 6 ± √282 = 3 ± √7.
    x = 3 ± √7
  4. (3 marks)
    Grade 9A6Non-calculator
    Prove that (2n + 1)² − (2n − 1)² is a multiple of 8 for all integers n.
    Solution
    1. (4n² + 4n + 1) − (4n² − 4n + 1) = 8n.
    2. 8n = 8 × n is a multiple of 8.
    8n
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