Question Simulator

IGCSE Additional Mathematics (0606)

Indices and surds

11 questions · about 3% of the marks we indexed · examined in 6 of 9 sessions.

Questions by year

2024

  • 2024 Oct/Nov · Paper 22 · Q13 marksIndices and equations with indices
  • 2024 Oct/Nov · Paper 22 · Q65 marksSurds
  • 2024 May/June · Paper 12 · Q5 (a, b, c)9 marksSurds
  • 2024 May/June · Paper 22 · Q3 (b)4 marksIndices and equations with indices

2023

  • 2023 Oct/Nov · Paper 12 · Q94 marksIndices and equations with indices
  • 2023 Oct/Nov · Paper 22 · Q34 marksSurds
  • 2023 May/June · Paper 12 · Q25 marksSurds
  • 2023 May/June · Paper 12 · Q5 (a)4 marksIndices and equations with indices

2022

  • 2022 Oct/Nov · Paper 12 · Q34 marksIndices and equations with indices
  • 2022 May/June · Paper 22 · Q13 marksSurds
  • 2022 May/June · Paper 22 · Q5 (a)3 marksIndices and equations with indices

References only, not question text. Open Official paper ↗ where Cambridge publishes the paper, or see the official Cambridge past-papers page; for other papers, ask your school. “Paper 42” means Paper 4, variant 2.

Practice questions

Original questions written by HomiLabs in the style of the exam, with answers and mark schemes. They are not Cambridge past paper questions and have not been reviewed by Cambridge. Try each one before opening the answer.

Q1Surds · Core · Paper 1 style · no calculator · 2 marks
  1. (a) Simplify √75 + √48 − √27, giving your answer in the form k√3. [2]
Show answer
  1. (a) 6√3
    M1 5√3 + 4√3 − 3√3; A1
Q2Surds · Core · Paper 1 style · no calculator · 2 marks
  1. (a) Expand and simplify (3 + √5)(2 − √5). [2]
Show answer
  1. (a) 1 − √5
    M1 6 − 3√5 + 2√5 − 5; A1
Q3Surds · Core · Paper 1 style · no calculator · 3 marks
  1. (a) Write 14/(3 − √2) in the form a + b√2, where a and b are integers. [3]
Show answer
  1. (a) 6 + 2√2
    M1 multiply by (3 + √2)/(3 + √2); B1 denominator 7; A1
Q4Surds · Core · Paper 1 style · no calculator · 2 marks
  1. (a) Expand (2√3 − 1)², giving your answer in the form a + b√3. [2]
Show answer
  1. (a) 13 − 4√3
    M1 12 − 4√3 + 1; A1
Q5Surds · Core · Paper 1 style · no calculator · 3 marks
  1. (a) Simplify (√18 + √8)/√2. [2]
  2. (b) Write √12 × √6 in the form k√2. [1]
Show answer
  1. (a) 5
    M1 3√2 + 2√2 = 5√2; A1
  2. (b) 6√2
    B1 √72 = 6√2
Q6Surds · Core · Paper 1 style · no calculator · 2 marks
  1. (a) Write 6/√3 + √27 in the form k√3. [2]
Show answer
  1. (a) 5√3
    M1 6/√3 = 2√3 and √27 = 3√3; A1
Q7Surds · Standard · Paper 1 style · no calculator · 3 marks
  1. (a) Write (5 + 2√3)/(2 − √3) in the form a + b√3, where a and b are integers. [3]
Show answer
  1. (a) 16 + 9√3
    M1 × (2 + √3); M1 numerator 10 + 5√3 + 4√3 + 6, denominator 1; A1
Q8Surds · Standard · Paper 2 style · calculator · 4 marks
  1. (a) A rectangular solar panel has area (11 + 6√2) m² and width (1 + √2) m. Find the length of the panel in the form (a + b√2) m. Do not use a calculator. [3]
  2. (b) Find the perimeter of the panel in the form (c + d√2) m. [1]
Show answer
  1. (a) (1 + 5√2) m
    M1 divide and multiply by (√2 − 1); M1 11√2 − 11 + 12 − 6√2; A1
  2. (b) (4 + 12√2) m
    B1
Q9Surds · Standard · Paper 1 style · no calculator · 3 marks
  1. (a) Solve the equation x(√5 − 1) = 8, giving x in the form a + b√5. [3]
Show answer
  1. (a) x = 2 + 2√5
    M1 x = 8/(√5 − 1); M1 × (√5 + 1), denominator 4; A1
Q10Surds · Standard · Paper 2 style · calculator · 4 marks
  1. (a) A right-angled triangle has shorter sides of length (2 + √3) cm and (2 − √3) cm. Find the exact length of the hypotenuse. Do not use a calculator. [3]
  2. (b) Find the exact area of the triangle. [1]
Show answer
  1. (a) √14 cm
    M1 (7 + 4√3) + (7 − 4√3); A1 14; A1 √14
  2. (b) ½ cm²
    B1 ½(4 − 3)
Q11Surds · Standard · Paper 1 style · no calculator · 4 marks
  1. (a) Given that (a + b√2)² = 27 − 10√2, where a and b are integers and a > 0, find a and b. Hence write √(27 − 10√2) in the form p + q√2. [4]
Show answer
  1. (a) a = 5, b = −1; √(27 − 10√2) = 5 − √2
    M1 a² + 2b² = 27 and 2ab = −10; M1 solve; A1 a = 5, b = −1; A1 5 − √2 (positive since 5 > √2)
Q12Surds · Standard · Paper 1 style · no calculator · 3 marks
  1. (a) Show that 1/(√7 − √5) − 1/(√7 + √5) simplifies to √5. [3]
Show answer
  1. (a) √5
    M1 common denominator 7 − 5 = 2; M1 (√7 + √5) − (√7 − √5) = 2√5; A1
Q13Surds · Challenge · Paper 1 style · no calculator · 4 marks
  1. (a) Solve the equation x² − 2√3x − 9 = 0, giving your answers in the form k√3. [4]
Show answer
  1. (a) x = 3√3 or x = −√3
    M1 formula or completing the square; M1 discriminant 12 + 36 = 48 = (4√3)²; A1 A1
Q14Surds · Challenge · Paper 2 style · calculator · 6 marks
  1. (a) In triangle ABC, AB = (3 + √2) cm, AC = (3 − √2) cm and angle BAC = 60°. Without using a calculator, find the exact length of BC. [4]
  2. (b) Find the exact area of triangle ABC. [2]
Show answer
  1. (a) √15 cm
    M1 cosine rule; M1 AB² + AC² = 22; M1 2 × AB × AC × ½ = 7; A1 √15
  2. (b) (7√3)/4 cm²
    M1 ½ × 7 × sin 60°; A1
Q15Surds · Challenge · Paper 1 style · no calculator · 5 marks
  1. (a) Given that tan θ = (√3 + 1)/(√3 − 1), write tan θ in the form a + √b, where a and b are integers. [3]
  2. (b) Hence find tan² θ in the form c + d√3. [2]
Show answer
  1. (a) 2 + √3
    M1 × (√3 + 1); M1 (4 + 2√3)/2; A1
  2. (b) 7 + 4√3
    M1 (2 + √3)²; A1
Q16Surds · Challenge · Paper 2 style · calculator · 5 marks
  1. (a) A cuboid-shaped storage box has a square base of side (√3 + 1) cm and volume (10 + 6√3) cm³. Without using a calculator, find the area of the base in the form a + b√3. [2]
  2. (b) Find the height of the box in the form (p + q√3) cm. [3]
Show answer
  1. (a) 4 + 2√3
    M1 (√3 + 1)²; A1
  2. (b) (1 + √3) cm
    M1 (10 + 6√3)/(4 + 2√3) = (5 + 3√3)/(2 + √3); M1 × (2 − √3); A1
Q17Indices and equations with indices · Core · Paper 1 style · no calculator · 3 marks
  1. (a) Find the value of 27^(2/3). [1]
  2. (b) Find the value of 16^(−3/4). [1]
  3. (c) Find the value of (8/125)^(−1/3). [1]
Show answer
  1. (a) 9
    B1
  2. (b) 1/8
    B1
  3. (c) 5/2
    B1
Q18Indices and equations with indices · Core · Paper 1 style · no calculator · 3 marks
  1. (a) Simplify (2x³y⁻¹)³ ÷ (4x⁵y⁻⁴). [3]
Show answer
  1. (a) 2x⁴y
    B1 8x⁹y⁻³; M1 subtract powers; A1
Q19Indices and equations with indices · Core · Paper 1 style · no calculator · 3 marks
  1. (a) Solve 2^(x + 3) = 32. [1]
  2. (b) Solve 9^x = 27. [2]
Show answer
  1. (a) x = 2
    B1
  2. (b) x = 3/2
    M1 3^(2x) = 3³; A1
Q20Indices and equations with indices · Core · Paper 1 style · no calculator · 2 marks
  1. (a) Write (√x)³ × x^(−1/2) ÷ ∛x as a single power of x. [2]
Show answer
  1. (a) x^(2/3)
    M1 3/2 − 1/2 − 1/3; A1
Q21Indices and equations with indices · Core · Paper 2 style · calculator · 3 marks
  1. (a) Solve the equation 5^(2x − 1) = 125^(x − 2). [3]
Show answer
  1. (a) x = 5
    M1 125 = 5³; M1 2x − 1 = 3x − 6; A1
Q22Indices and equations with indices · Standard · Paper 1 style · no calculator · 4 marks
  1. (a) Using the substitution y = 2^x, solve 4^x − 6(2^x) + 8 = 0. [4]
Show answer
  1. (a) x = 1, x = 2
    M1 y² − 6y + 8 = 0; A1 y = 2, 4; M1 2^x = y; A1
Q23Indices and equations with indices · Standard · Paper 1 style · no calculator · 4 marks
  1. (a) Solve 3^(2x + 1) − 10(3^x) + 3 = 0. [4]
Show answer
  1. (a) x = 1, x = −1
    M1 3y² − 10y + 3 = 0 with y = 3^x; A1 y = 3, ⅓; M1; A1
Q24Indices and equations with indices · Standard · Paper 1 style · no calculator · 3 marks
  1. (a) Simplify (a^(1/2) b^(−2/3))⁶ ÷ (a² b⁻³). [3]
Show answer
  1. (a) a/b (= ab⁻¹)
    B1 a³b⁻⁴; M1 subtract powers; A1
Q25Indices and equations with indices · Standard · Paper 1 style · no calculator · 5 marks
  1. (a) Write (9^x × 27^(x + 1)) ÷ 3^(2x − 1) in the form 3^(px + q). [3]
  2. (b) Hence solve (9^x × 27^(x + 1)) ÷ 3^(2x − 1) = 81^x. [2]
Show answer
  1. (a) 3^(3x + 4)
    M1 3^(2x) × 3^(3x + 3); M1 ÷ 3^(2x − 1); A1
  2. (b) x = 4
    M1 3x + 4 = 4x; A1
Q26Indices and equations with indices · Standard · Paper 2 style · calculator · 5 marks
  1. (a) Solve the simultaneous equations 2^x × 4^y = 64 and 3^x ÷ 9^y = 1/3. [5]
Show answer
  1. (a) x = 5/2, y = 7/4
    M1 x + 2y = 6; M1 x − 2y = −1; M1 solve; A1 x = 5/2; A1 y = 7/4
Q27Indices and equations with indices · Challenge · Paper 1 style · no calculator · 5 marks
  1. (a) Solve 2^(2x + 1) + 2^(x + 3) = 10, explaining why one solution of your quadratic is rejected. [5]
Show answer
  1. (a) x = 0 (2^x = −5 is rejected because 2^x > 0 for all x)
    M1 2y² + 8y − 10 = 0, y = 2^x; A1 y = 1 or y = −5; B1 reject −5 with reason; A1 x = 0
Q28Indices and equations with indices · Challenge · Paper 1 style · no calculator · 4 marks
  1. (a) Given that (16x⁴y⁻²)^(3/4) ÷ (2x⁻¹y)² = 2^a x^p y^q, find a, p and q. [4]
Show answer
  1. (a) a = 1, p = 5, q = −7/2
    M1 (16x⁴y⁻²)^(3/4) = 8x³y^(−3/2); M1 (2x⁻¹y)² = 4x⁻²y²; A1 a = 1, p = 5; A1 q = −7/2
Q29Indices and equations with indices · Challenge · Paper 1 style · no calculator · 4 marks
  1. (a) Solve 5^x + 5^(1 − x) = 6. [4]
Show answer
  1. (a) x = 0, x = 1
    M1 y + 5/y = 6 with y = 5^x; M1 y² − 6y + 5 = 0; A1 y = 1, 5; A1
Q30Indices and equations with indices · Challenge · Paper 2 style · calculator · 4 marks
  1. (a) Using a suitable substitution, solve x^(2/3) − 5x^(1/3) + 4 = 0. [4]
Show answer
  1. (a) x = 1, x = 64
    M1 u = x^(1/3): u² − 5u + 4 = 0; A1 u = 1, 4; M1 x = u³; A1

Similar questions

Questions that test the same subtopic in different years, grouped so you can work on one skill at a time.

References only: find each question in its paper (official link where available, otherwise through your school).

Surds 5 questions · 5/9 sessions
Indices and equations with indices 6 questions · 6/9 sessions

Subtopics

Our analysis of how often each subtopic was examined and how many marks it carried, 2022 to May/June 2026.

SubtopicMarks indexedSessions
Surds 265/9
Indices and equations with indices 226/9

← All Additional Mathematics topics