Question 12 marks
Question
Simplify. 7c - 5d + c + 3d …… [2]
Mark scheme
1 8c – 2d final answer 2 B1 for answer 8c – kd or kc – 2d or for correct answer seen and spoilt
Mathematics 0580/21 May/June 2025: question paper and mark scheme
Cambridge IGCSE Mathematics (0580), Paper 2 (Extended), May/June 2025. Read both free below, or download them. Done the paper? Get it marked by AI against this mark scheme.
0580_s25_qp_210580_s25_ms_210580/21/M/J/25
- Paper
- Paper 2 (Extended)
- Total marks
- 100
- Time
- 2 hours
- Code on the paper
- 0580/21/M/J/25
0580/21 mark scheme, question by question
Each question as it's printed, and the mark scheme's points for it, as text. Diagrams, tables and graphs are in the PDFs above.
Question 24 marks
Question
x° NOT TO SCALE w° y° 76° 158° The diagram shows two parallel lines intersecting two straight lines. Find the values of w, x and y. w = …… x = …… y = …… [4] * 0000800000004 * DFD 4
Mark scheme
2 [w =] 158 4 B1 for w correct [x =] 76 B1 for x correct [y =] 82 B2FT for y = 158 – their x correctly evaluated or B1 for 22 (identified) or 82 in position vertically opposite to y or for y = 158 – their x
Question 33 marks
Question
Sally invests $1500 at 3% per year simple interest. Work out the total value of her investment at the end of 6 years. $ …… [3]
Mark scheme
3 1770 3 B2 for 270 1500 × 3× 6 or M2 for 1500 + 100 1500 × 3 [×6] or M1 for 100
Question 43 marks
Question
Work out. 5 2 3 - # 6 3 8 …… [3]
Mark scheme
4 7 3 6 oe M1 for oe 12 24 M1 for correct use of common denominator in 5 6 20 6 subtraction – their , e.g. and oe 6 24 24 24 1 If 0 scored, SC1 for answer oe 16
Question 52 marks
Question
The interior angle of a regular polygon is 150°. Find the number of sides of this polygon. …… [2] * 0000800000005 * DFD 5
Mark scheme
5 12 2 360 180 ( n − 2 ) M1 for or for = 150 180 − 150 n
Question 63 marks
Question
y 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 x The line x + y = 7 is drawn on the grid. (a) On the grid, draw the line y = 2x + 1. [2] (b) Use your graph to solve these simultaneous equations. x+y = 7 y = 2x + 1 x = …… y = …… [1]
Mark scheme
6(a) Correct line drawn 2 M1 for a line with gradient 2 or for a line with positive gradient and intercept at y = 1 6(b) [x =] 2, [y =] 5 1 FT intersection of their (a) with the given line
Question 73 marks
Question
Write the recurring decimal 0.26o as a fraction. Give your answer in its simplest form. …… [3] * 0000800000006 * DFD 6 11 8
Mark scheme
7 4 3 24 B2 for oe 15 90 or M1 for 26.66… – 2.66… oe or for 90x = 24 oe 2 6 or for + oe 10 90
Question 84 marks
Question
m =e o n =e o 5 -3 (a) Find 2m - n. f p [2] 5 (b) The vector e o has a magnitude of 7. y Find the value of y. y = …… [2]
Mark scheme
8(a) 14 2 14 k 22 B1 for or for or for 13 k 13 10 8(b) 24 2 M1 for y + 52 = 72 or better
Question 93 marks
Question
The table shows some information about the marks scored by a group of students in a test. Test mark 4 5 8 Frequency 2 4 n The mean mark is 6. Work out the value of n. n = …… [3] * 0000800000007 * DFD 7
Mark scheme
9 4 nfww 3 M2 for 8 + 20 + 8n = 6(2 + 4 + n) or better 2 × 4 + 4 × 5 + n ×8 or M1 for [= 6] oe 2+4+ n or 8 + 20 + 8n or 6(2 + 4 + n)
Question 104 marks
Question
A D E NOT TO SCALE O 35° 40° C B A, B and C are three points on a circle, centre O. DE is a tangent to the circle at A. Angle ACO = 35° and angle BCO = 40° . Find (a) angle AOC Angle AOC = …… [1] (b) angle ABC Angle ABC = …… [1] (c) angle DAC Angle DAC = …… [1] (d) angle OAB. Angle OAB = …… [1] * 0000800000008 * DFD 8
Mark scheme
10(a) 110 1 10(b) 55 1 FT their 110 ÷ 2 10(c) 55 1 FT their (b) Provided their (b) < 90 10(d) 15 1 FT 70 – their (b) Provided their (b) < 70
Question 115 marks
Question
The diagram shows the graph of y = f (x) and the point P (- 2, 11) . y 15 P 10 5 –2 –1 0 1 2 3 x –5 – 10 The tangent from P touches the graph of y = f (x) at the point (a, b). The values of a and b are integers. (a) By drawing this tangent, find the value of a and the value of b. a = …… , b = …… [2] (b) Find the equation of the tangent. y = …… [3] * 0000800000009 * DFD 9
Mark scheme
11(a) For correct ruled tangent and 2 B1 for correct ruled tangent or both values [a =] 1, [b =] 2 correct without a correct tangent 11(b) [y =] 5 – 3x 3 3FT their (a) provided m < 0, c ≠ 0 B1 for (their –3)x + c rise or M1 for correct for their line run B1 for mx + c where c is the correct intercept for their graph, m ≠ 0
Question 124 marks
Question
The time spent on the internet by each of 120 adults is recorded for one day. The cumulative frequency diagram shows this information. 120 100 80 Cumulative frequency 60 40 20 0 0 2 4 6 8 10 Time (hours) (a) Use the cumulative frequency diagram to find an estimate of the interquartile range. …… h [2] (b) 70% of the adults spent less than k hours on the internet. Use the cumulative frequency diagram to find an estimate of the value of k. k = …… [2] * 0000800000010 * DFD 10
Mark scheme
12(a) 1.4 2 B1 for [UQ =] 5.6 or for [LQ =] 4.2 or SC1 for 0.7 12(b) 5.4 2 B1 for 84 seen
Question 138 marks
Question
4 cm 10 cm NOT TO h cm SCALE 6 cm Solid A Solid B The diagram shows solid A and solid B. Solid A is made from a hemisphere and a cone each with radius 6 cm. The cone has sloping edge 10 cm. Solid B is a cylinder with radius 4 cm and height h cm. The total surface area of solid A is equal to the total surface area of solid B. (a) Work out the value of h. h = …… [5] (b) Work out the height of solid A. …… cm [3] * 0000800000011 * DFD 11
Mark scheme
13(a) 12.5 oe 5 M4 for 8[π]h = 100[π] OR M3 for π × 6 ×10 + 2 × π × 62 = 2 × π × 42 + 2 × π × 4 × h oe OR M1 for π × 6 ×10 + 2 × π × 62 oe M1 2 × π × 42 + 2 × π × 4 × h oe OR SC2 for answer 16.5 13(b) 14 3 M2 for 102 − 62 or M1 for 62 + x2 = 102
Question 148 marks
Question
f (x) = 3x - 4 g (x) = 4x + 1 (a) Find f (- 2) . …… [1] (b) Find f -1 (x) . f -1 (x) = …… [2] (c) fg (x) = ax + b Find the value of a, and the value of b. a = …… b = …… [2] (d) Simplify. 2 5 - f (x) g (x) Give your answer as a single fraction in terms of x. …… [3] * 0000800000012 * DFD 12
Mark scheme
14(a) –10 1 14(b) x+4 2 M1 for correct first step oe 3 y + 4 = 3x or x = 3y – 4 y 4 or = x − 3 3 14(c) a = 12, b = –1 2 B1 for either a or b correct or M1 for 3(4x + 1) – 4 14(d) 22 − 7 x 22 − 7 x 3 B1 for 2(4x + 1) – 5(3x – 4) oe or better isw or ( 3x − 4 )( 4 x + 1) 12 x 2 − 13x − 4 B1 for common denominator (3x – 4)(4x + 1) final answer oe isw
Question 154 marks
Question
(a) Expand and simplify. ( 2 - 5 ) ( 1 - 3 5) …… [2] (b) Rationalise the denominator. Give your answer in its simplest form. 6 10 …… [2]
Mark scheme
15(a) 17 – 7 5 final answer 2 B1 for 3 correct terms from 2–6 5 – 5 +3× 5 × 5 oe 15(b) 3 10 2 6 10 cao M1 for × oe 5 10 10
Question 163 marks
Question
Expand and simplify. (x + 4) (x - 3) (3x + 2) …… [3] * 0000800000013 * DFD 13
Mark scheme
16 3x3 + 5x2 – 34x – 24 final answer 3 B2 for correct expansion unsimplified or simplified four-term expression of correct form with 3 terms correct or B1 for one pair of brackets expanded with at least 3 terms out of 4 correct
Question 177 marks
Question
(a) A bag contains 6 red marbles, 3 green marbles and 1 blue marble. Two marbles are picked at random from the bag with replacement. Find the probability that both marbles are green. …… [2] (b) Another bag contains 4 red counters and 2 yellow counters. Two counters are picked at random from this bag without replacement. (i) Complete the tree diagram. First Second counter counter 3 Red 5 Red 4 6 …… Yellow Red 2 6 Yellow …… Yellow [2] (ii) Find the probability that one of the two counters is yellow. …… [3] * 0000800000014 * DFD 14
Mark scheme
17(a) 9 2 3 3 oe M1 for × 100 10 10 17(b)(i) 2 2 2 B1 for 5 5 4 5 1 5 17(b)(ii) 16 3 3FT their tree diagram dep on probabilities < 1 oe 30 4 2 2 4 M2FT for × + × 6 5 6 5 4 2 2 4 or M1FT for × or for × 6 5 6 5
Question 1810 marks
Question
One day, Anya runs 12 km at a speed of x km/h. The next day she walks 10 km at a speed of (x - 4) km/h. (a) Write down an expression, in terms of x, for the time she spends running. …… h [1] (b) Write down an expression, in terms of x, for the time she spends walking. …… h [1] (c) The time Anya spends walking is 1 hour more than the time she spends running. Write an equation in terms of x and show that it simplifies to x 2 - 2x - 48 = 0 . [4] (d) Use factorisation to solve the equation x 2 - 2x - 48 = 0 . x = …… or x = …… [3] (e) Find the time Anya spends running. …… h [1] * 0000800000015 * DFD 15 2
Mark scheme
18(a) 12 1 x 18(b) 10 1 x−4 18(c) 10 12 M1 their – their = 1 oe x−4 x 10x – 12x + 48 = x2 – 4x M2 Correctly multiplying their brackets and clearing algebraic fractions 12 e.g. ( x − 4 ) + 1 = 10 x 48 leading to 12 − + x − 4 = 10 and then x 2 12 x − 48 + x − 4 x = 10 x or M1 for correctly clearing, or correctly collecting into a single fraction, two fractions both with different algebraic denominators e.g. 10x – 12(x – 4) = x(x – 4) or 10 x − 12 ( x − 4 ) [= 1] x ( x − 4) Leading to 0 = x2 – 2x – 48 A1 With no errors or omissions seen, dep on M3 18(d) (x + 6)(x – 8) M2 M1 for x(x – 8) + 6(x – 8) or x(x + 6) – 8(x + 6) or (x + a)(x + b) where ab = –48 or a + b = –2 –6, 8 B1 18(e) 1 1 12 1.5 or 1 FT 2 their 8
Question 192 marks
Question
Find the value of 27 - 3 . …… [2]
Mark scheme
19 1 2 1 1 M1 for 9–1 or for 2 or 3 9 3 729
Question 204 marks
Question
NOT TO 6 cm SCALE 30° x cm Find the exact value of x. x = …… [4] * 0000800000016 * DFD 16
Mark scheme
20 6 3 oe 4 1 3 B1 for tan30 = or 3 3 3 1 or for sin 60 = and sin 30 = 2 2 6 6sin ( 60 ) M2 for or tan 30 sin ( 30 ) 6 sin 60 sin 30 or M1 for = tan 30 or = x x 6
Question 215 marks
Question
A B n P NOT TO 2m SCALE O C OABC is a rhombus and O is the origin. The diagonals of the rhombus intersect at P. OP = 2 m and AP = n . (a) Find, in terms of m and n, in its simplest form (i) OA OA = …… [1] (ii) OC . OC = …… [1] (b) D is the point such that AD = 10m - 3n . Show that OADC is a trapezium. [3] * 0000800000017 * DFD 17
Mark scheme
21(a)(i) 2m – n 1 21(a)(ii) 2m + n 1 21(b) CD = 10m – 5n M2 Allow M2 for equivalents CD = –2n + 10m – 3n or CD = − ( 2m + n ) + 2m − n + 10m − 3n For M2, FT their (a) e.g. CD = their CO + their OA +10m − 3n or M1 for correct route for CD using the lines of the diagram with AD e.g. CD = CA + AD oe CD = 5 OA A1 Dependent on M2 leading to CD is parallel to OA [ OACD is a trapezium]
Question 226 marks
Question
A curve has equation y = x n + qx 2 + 9x . dy = 3x 2 - 12x + 9 dx (a) Find the value of n, and the value of q. n = …… q = …… [2] (b) Work out the coordinates of the turning points of the curve. (…… , ……) and (…… , ……) [4] * 0000800000018 * DFD 18
Mark scheme
22(a) [n =] 3, [q =] – 6 2 B1 for each correct value 22(b) (1, 4) and (3, 0) 4 B3 for (1, 4) or (3, 0) or for two correct values of x or M2 for [3](x – 1)(x – 3) [ = 0] oe − ( −12 ) ± ( −12)2 − 4 × 3 × 9 or x = oe 2×3 dy or M1 for writing =0 dx or for 3x 2 − 12 x + 9 = 0
Question 233 marks
Question
Simplify. 2x 2 + 10x x 2 - 25 …… [3] DFD 19 * 0000800000020 * DFD 20
Mark scheme
23 2x 3 B1 for 2x(x + 5) final answer x −5 B1 for (x – 5)(x + 5)
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On this page, free: the mark scheme for IGCSE Mathematics 0580/21, May/June 2025 (0580_s25_ms_21), and the question paper (0580_s25_qp_21), to read or download, with each question's marking points written out underneath.
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Mathematics 0580/21 for May/June 2025 is out of 100 marks, with 2 hours to answer it.
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