IGCSE Maths 0580 Surds Past Paper Questions and Worked Solutions

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IGCSE Maths (0580) · Number · 1.18

Every surds question from 9 Cambridge IGCSE Mathematics (0580) past papers (May/June 2025 to May/June 2026), syllabus section 1.18. 9 Extended-tier questions. Each question is labelled with its question paper code (for example 0580/21/M/J/26) and every final answer matches the official Cambridge mark scheme. Try each question first, then click “Show solution” to check your working. This page is part of our IGCSE Maths 0580 Number past paper questions.

0580 Surds past paper questions: Extended (Papers 2 and 4)

17 marksExtended

May/June 2026 · Paper 2 · 0580/21/M/J/26 · Q26

(a) Expand and simplify (5 + √2)(3 − √8).

(b) √(29⁄20) + 6⁄√5 = k√5. Find the value of k.

Show solution
Part (a)
  1. Simplify √8 first: √8 = √4 × √2 = 2√2.
  2. Expand (5 + √2)(3 − 2√2): 5 × 3 = 15, 5 × (−2√2) = −10√2, √2 × 3 = 3√2, √2 × (−2√2) = −2 × 2 = −4.
  3. Collect: numbers 15 − 4 = 11 and surds −10√2 + 3√2 = −7√2.
  4. 11 − 7√2
Answer 11 − 7√2
Part (b)
  1. Change the mixed number: 29⁄20 = 49⁄20, so √(29⁄20) = 7⁄√20.
  2. √20 = 2√5, so 7⁄√20 = 7⁄2√5. Rationalise: 7√5⁄10.
  3. Rationalise the second term: 6⁄√5 = 6√5⁄5 = 12√5⁄10.
  4. Add: 7√5⁄10 + 12√5⁄10 = 19√5⁄10.
  5. So k√5 = 1.9√5 and k = 1.9
Answer k = 1.9
25 marksExtended

May/June 2026 · Paper 2 · 0580/22/M/J/26 · Q17

(a) Expand and simplify (4 − √7)(3 + 2√7).

(b) Rationalise the denominator and simplify 2⁄3 − √5.

Show solution
Part (a)
  1. Multiply each term in the first bracket by each term in the second.
  2. 4 × 3 = 12, 4 × 2√7 = 8√7, −√7 × 3 = −3√7, −√7 × 2√7 = −2 × 7 = −14.
  3. Collect: numbers 12 − 14 = −2 and surds 8√7 − 3√7 = 5√7.
  4. −2 + 5√7
Answer −2 + 5√7
Part (b)
  1. The bottom is 3 − √5. Multiply top and bottom by 3 + √5 (change the sign in the middle).
  2. Bottom: (3 − √5)(3 + √5) = 9 − 5 = 4 (the surds cancel).
  3. Top: 2(3 + √5).
  4. 2(3 + √5)⁄4 = 3 + √5⁄2
Answer 3 + √5⁄2
34 marksExtended

May/June 2026 · Paper 2 · 0580/23/M/J/26 · Q21

(a) Simplify (√0.1)². (b) Rationalise the denominator of 4⁄2 − √2. Give your answer in its simplest form.

Show solution
Part (a)
  1. Squaring a square root gives back the number inside: (√0.1)² = 0.1
Answer 0.1
Part (b)
  1. The bottom is 2 − √2. Multiply top and bottom by 2 + √2 (change the sign in the middle).
  2. Bottom: (2 − √2)(2 + √2) = 4 − 2 = 2.
  3. Top: 4(2 + √2) = 8 + 4√2.
  4. 8 + 4√2⁄2 = 4 + 2√2
Answer 4 + 2√2
42 marksExtended

Feb/Mar 2026 · Paper 2 · 0580/22/F/M/26 · Q15

Simplify √40 + √160.

Show solution
  1. Find the largest square number that divides into each number.
  2. √40 = √(4 × 10) = √4 × √10 = 2√10.
  3. √160 = √(16 × 10) = √16 × √10 = 4√10.
  4. Add like surds: 2√10 + 4√10 = 6√10
Answer 6√10
54 marksExtended

Oct/Nov 2025 · Paper 2 · 0580/21/O/N/25 · Q21

(a) (3 − √5)(2 + 3√5) = a + b√5. Find the value of a and the value of b.

(b) Rationalise the denominator of 6⁄√2. Write your answer in its simplest form.

Show solution
Part (a)
  1. Multiply each term in the first bracket by each term in the second.
  2. 3 × 2 = 6, 3 × 3√5 = 9√5, −√5 × 2 = −2√5, −√5 × 3√5 = −3 × 5 = −15.
  3. Collect: numbers 6 − 15 = −9 and surds 9√5 − 2√5 = 7√5.
  4. −9 + 7√5, so a = −9, b = 7
Answer a = −9, b = 7
Part (b)
  1. Multiply the top and bottom by √2: 6 × √2⁄√2 × √2 = 6√2⁄2.
  2. Simplify: 3√2
Answer 3√2
63 marksExtended

Oct/Nov 2025 · Paper 2 · 0580/22/O/N/25 · Q18

Rationalise the denominator and simplify 20⁄4 + √6.

Show solution
  1. The bottom is 4 + √6. Multiply top and bottom by 4 − √6 (change the sign in the middle).
  2. Bottom: (4 + √6)(4 − √6) = 16 − 6 = 10.
  3. Top: 20(4 − √6).
  4. 20(4 − √6)⁄10 = 2(4 − √6) = 8 − 2√6
Answer 8 − 2√6
76 marksExtended

Oct/Nov 2025 · Paper 2 · 0580/23/O/N/25 · Q24

(a) Simplify √125 − √20. (b) A rectangle measures 2√2 cm by 3√8 cm. Its area is k cm². Work out the value of k. (c) Rationalise the denominator of 1⁄√7 + 2.

Show solution
Part (a)
  1. Simplify each surd using square-number factors.
  2. √125 = √(25 × 5) = 5√5 and √20 = √(4 × 5) = 2√5.
  3. 5√5 − 2√5 = 3√5
Answer 3√5
Part (b)
  1. Area = 2√2 × 3√8. Multiply the numbers and the surds separately: 2 × 3 = 6 and √2 × √8 = √16 = 4.
  2. Area = 6 × 4 = 24, so k = 24
Answer 24
Part (c)
  1. The bottom is √7 + 2. Multiply top and bottom by √7 − 2 (change the sign in the middle).
  2. Bottom: (√7 + 2)(√7 − 2) = 7 − 4 = 3. Top: 1 × (√7 − 2) = √7 − 2.
  3. √7 − 2⁄3
Answer √7 − 2⁄3
84 marksExtended

May/June 2025 · Paper 2 · 0580/21/M/J/25 · Q15

(a) Expand and simplify. (2 − √5)(1 − 3√5)

(b) Rationalise the denominator. Give your answer in its simplest form. 6⁄√10

Show solution
Part (a)
  1. Multiply each term in the first bracket by each term in the second.
  2. 2 × 1 = 2, 2 × (−3√5) = −6√5, −√5 × 1 = −√5, −√5 × (−3√5) = 3 × 5 = 15.
  3. Collect: numbers 2 + 15 = 17 and surds −6√5 − √5 = −7√5.
  4. 17 − 7√5
Answer 17 − 7√5
Part (b)
  1. Multiply the top and bottom by √10 to remove the surd from the bottom: 6√10⁄10.
  2. Divide top and bottom by 2: 3√10⁄5
Answer 3√10⁄5
94 marksExtended

May/June 2025 · Paper 2 · 0580/22/M/J/25 · Q18

(a) Rationalise the denominator. Give your answer in its simplest form. 9⁄√3

(b) (5 − √2)(1 + 3√2) = c + k√2. Find the value of c and the value of k.

Show solution
Part (a)
  1. Multiply the top and bottom by √3: 9 × √3⁄√3 × √3 = 9√3⁄3.
  2. Simplify: 9 ÷ 3 = 3, so the answer is 3√3
Answer 3√3
Part (b)
  1. Multiply each term in the first bracket by each term in the second.
  2. 5 × 1 = 5, 5 × 3√2 = 15√2, −√2 × 1 = −√2, −√2 × 3√2 = −3 × 2 = −6.
  3. Collect: numbers 5 − 6 = −1 and surds 15√2 − √2 = 14√2.
  4. −1 + 14√2, so c = −1, k = 14
Answer c = −1, k = 14

Question papers used on this page

All surds questions above, listed by 0580 question paper code.

QP codeFile namePaperQuestions
0580/21/M/J/260580_s26_qp_21May/June 2026 · Paper 2 (Extended)Q26
0580/22/M/J/260580_s26_qp_22May/June 2026 · Paper 2 (Extended)Q17
0580/23/M/J/260580_s26_qp_23May/June 2026 · Paper 2 (Extended)Q21
0580/22/F/M/260580_m26_qp_22Feb/Mar 2026 · Paper 2 (Extended)Q15
0580/21/O/N/250580_w25_qp_21Oct/Nov 2025 · Paper 2 (Extended)Q21
0580/22/O/N/250580_w25_qp_22Oct/Nov 2025 · Paper 2 (Extended)Q18
0580/23/O/N/250580_w25_qp_23Oct/Nov 2025 · Paper 2 (Extended)Q24
0580/21/M/J/250580_s25_qp_21May/June 2025 · Paper 2 (Extended)Q15
0580/22/M/J/250580_s25_qp_22May/June 2025 · Paper 2 (Extended)Q18

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