Cambridge IGCSE Mathematics 0580 — 2025 Oct/Nov Paper 2 · Variant 1

1 Divide $90 in the ratio 2 : 3. $ .................... , $ .................... [2]

Mark scheme: Question Answer Marks Partial Marks 1 36, 54 2 90 M1 for k where k = 1, 2 or 3 2 + 3

More questions on Ratio and proportion

2 148° NOT TO 82° B SCALE C A In the diagram, AC = BC. Work out angle CAB. Angle CAB = ................................................ [3]

Mark scheme: 2 25 3 M1 for 360 – 148 – 82 M1 for (180 – their 130) ÷ 2

More questions on Angles

3 Find the interior angle of a regular 20-sided polygon. ................................................. [2]

Mark scheme: 3 162 2 360 180 ( 20 − 2 ) M1 for 180 – or for 20 20

More questions on Angles

4 The area of a triangle is 12 cm 2. The length of the base of the triangle is 8 cm. Work out the height of the triangle. ............................................ cm [2]

Mark scheme: 4 3 2 1 M1 for 8 h = 12 oe 2

More questions on Area and perimeter

5 y 4 3 P 2 T 1 – 4 – 3 – 2 – 1 0 1 2 3 4 x – 1 – 2 (a) Describe fully the single transformation that maps triangle T onto triangle P. ..................................................................................................................................................... ..................................................................................................................................................... [2] (b) Draw the image of triangle T after an enlargement of scale factor 2, centre (3, 3). [2]

Mark scheme: 5(a) Translation 2 B1 for each  −3     1  5(b) Triangle at (3, 1) , (3, –1) (–1, –1) 2 B1 for correct size and orientation but wrong position

More questions on Transformations

6 Find the value of (a) 5 -5 # 5 5 ................................................. [1] 3 (b) 125 2. ................................................. [2]

Mark scheme: 6(a) 1 cao 1 6(b) 25 cao 2 2 2 3 3 125 5 3 M1 for or 3 1252 or ( ) ( ) or B1 for 3 125 = 5

More questions on Powers and roots

7 Simplify. p 2 (a) ' t t .................................................. [2] 3x x - 1 (b) - 4 2 .................................................. [2]

Mark scheme: 7(a) p 2 p t final answer M1 for  or for cross cancelling of t 2 t 2 7(b) x + 2 x 1 2 3 x − 2 ( x − 1) or + final answer M1 for oe 4 4 2 4 x − 2 If 0 scored, SC1 for final answer 4 oe

More questions on The four operations

8 The cost of one orange is t cents. The cost of one apple is w cents. The total cost of 3 oranges and 1 apple is 51 cents. The total cost of 6 oranges and 5 apples is 129 cents. Use simultaneous equations to find the value of t and the value of w. You must show all your working. t = ................................................ w = ................................................ [5]

Mark scheme: 8 3t + w = 51 2 B1 for each 6t + 5w = 129 Correctly eliminating one variable from M1 e.g. 6t + 2w = 102 and 6t + 5w = 129 their equations leading to 3w = 27 or w = 51 − 3t and 6t + 5(51 − 3t ) = 129 [t =] 14 A2 A1 for [t =] 14 [w =] 9 A1 for [w =] 9 If M1A0A0 scored, M1 SC1 for two values satisfying one of their original equations or if M0 scored, SC1 for 2 correct answers

More questions on Equations

9 Nina walks at an average speed of 5 km/h, correct to the nearest km/h. She walks for exactly 2 hours. Work out the lower bound for the distance Nina walks. ............................................ km [2]

Mark scheme: 9 9 2 B1 for 4.5 seen

More questions on Limits of accuracy

10 % = {n: n is an integer and 1 G n G 8 } A = {factors of 12} B = {odd numbers} Find (a) A + B A + B = {................................} [1] (b) n ( Al , B ) . ................................................. [1]

Mark scheme: 10(a) 1, 3 1 10(b) 5 1

More questions on Sets

o o11 Write .024 as a fraction in its simplest form. ................................................. [2]

Mark scheme: 11 8 2 24 cao B1 for oe 33 99 or M1 for 24.24… – 0.24… oe

More questions on Fractions, decimals and percentages

12 The diagram shows a sector of a circle with centre O and radius 9 cm. 9 cm NOT TO SCALE O 9 cm The perimeter of the sector is ( 18 + 2 r ) cm. Find the area of the sector. Give your answer in terms of r. .......................................... cm2 [4]

Mark scheme: 12 9π 4 M3 for a fully correct method 2 π 2 e.g. π 9 oe 2 π  9 OR 1 B2 for 40 or oe 9 or M1 for 18 + k =2 π 9 18 + 2π oe M1 for ( their k )  π  9 2 oe OR SC2 for answer 81 + 9π oe

More questions on Circles, arcs and sectors

13 These are Rahul’s 10 test scores. 9 8 9 10 7 x 9 9 x 7 The mean of these scores is 8. Find the interquartile range. You must show all your working. ................................................. [4]

Mark scheme: 13 68 + 2 x M1 = 8 or better 10 x = 6 A1 Lower quartile = 6.5 or 6.75 or 7 B1 or upper quartile = 9 2 or 2.25 or 2.5 nfww A1

More questions on Averages and measures of spread

14 C 55° NOT TO B SCALE X 88° A D E A, B, C, D and E lie on the circle. AC and BD intersect at X. Angle ACD = 55° and angle CXD = 88°. (a) Complete the statements, giving a geometrical reason in each part. Angle CDB = ......................... because ...................................................................................... ..................................................................................................................................................... Angle ABD = ......................... because ...................................................................................... ..................................................................................................................................................... Angle AED = ......................... because ...................................................................................... ..................................................................................................................................................... [6] (b) Triangle CXD is mathematically similar to triangle BXA. DX = 8.0 cm, BX = 2.7 cm and AX = 4.0 cm. (i) Work out the length of CX. CX = ........................................... cm [2] (ii) Complete the statement. Area of triangle CXD : area of triangle BXA = ....................... : ....................... [1]

Mark scheme: 14(a) 37 2 B1 for each Angle sum of triangle = 180 55 2 B1 for each Angles in the same segment are equal 125 2 B1 for each Opposite angles of cyclic quadrilateral sum to 180 14(b)(i) 5.4 2 4 2.7 M1 for = oe 8 CX 14(b)(ii) 4 : 1 oe 1

More questions on Angles

15 (a) Write 66 000 in standard form. ................................................. [1] (b) Work out `3.7 # 10 8j + `3. 7 # 10 7j. Give your answer in standard form. ................................................. [2]

Mark scheme: 15(a) 6.6 × 104 1 15(b) 4.07 × 108 2 M1 for figs 407 or for 0.37 × 108 or 37 × 107 or 107(3.7 × 10 + 3.7) or 108(3.7+ 3.7 ÷ 10) oe

More questions on Standard form

16 y 6 5 4 R 3 2 1 0 1 2 3 4 x Write down all the inequalities that define the region R. .................................................. .................................................. .................................................. .................................................. [4]

Mark scheme: 16 x ⩽ 2.5 4 B3 for answer y > 3 x < 2.5 y ⩾ 3 y < 4 y < 2x y ⩽ 4 y ⩽ 2x OR B1 for each If 0 or 1 scored, instead award SC2 for recognition of x = 2.5, y = 3, y = 4, y = 2x If 0 scored, SC1 for recognition of y = 2x

More questions on Inequalities

17 I = M ( k2 + c 2 ) (a) Find the value of I when M = 7, k = 3 and c = 2. I = ................................................ [2] (b) Rearrange the formula to write k in terms of I, M and c. k = ................................................ [3]

Mark scheme: 17(a) 91 2 M1 for 7(32 + 22) oe 17(b) 2 3 M1 for correctly dividing by M I 2 I − Mc  − c or  M1 for correctly isolating term in k2 M M M1 for correctly taking square root final answer Maximum M2 if answer is incorrect

More questions on Algebraic manipulation

18 f ( )x = 2x + 5 f ( x) f ( x) - ff ( x) = ax 2 + bc + c Find the value of a, the value of b and the value of c. a = ................................................ b = ................................................ c = ................................................ [4]

Mark scheme: 18 [a =] 4 4 B3 for two correct nfww [b =] 16 [c =] 10 OR B2 for 4 x 2 + 20 x + 25 or B1 for three terms correct from 4 x 2 + 10 x + 10 x + 25 M1 for 2(2x + 5) + 5 soi

More questions on Algebraic manipulation

19 Solve. x 1 x + 4 e o = 9 3 x = ................................................ [3]

Mark scheme: 19 8 2 3 M2 for –x = 2(x + 4) oe − or −2 oe 2 ( x + 4 ) 3 3 or M1 for (3–1)x, 3−x , (32)x+4 , 3 or −2( x + 4 ) 1  oe 3

More questions on Equations

20 Bag A Bag B Bag A contains 5 white balls and 3 black balls. Bag B contains 3 white balls and 1 black ball. (a) Two balls are picked at random from bag B without replacement. Find the probability that both balls are black. ................................................. [1] (b) The balls are replaced into bag B. Kyle picks a ball at random from each bag. (i) Complete the tree diagram. Bag A Bag B white ........ white 5 8 black ........ white ........ ........ black black ........ [2] (ii) Find the probability that the two balls are the same colour. ................................................. [3] (c) The balls are replaced into their bags. Jo picks a ball at random from bag A and places it into bag B. She then picks a ball at random from bag B. Find the probability that she picks a black ball from bag B. ................................................. [3]

Mark scheme: 20(a) 0 1 20(b)(i) 2 3 B1 for 8 3 1 or for and correctly placed on the 4 4 same pair of branches 20(b)(ii) 18 3 5 3 3 1 oe M2 for  their + their  their oe 32 8 4 8 4 or M1 for one correct branch e.g. 5 3 3 1  their or their  their oe 8 4 8 4 20(c) 11 3 5 1 3 2 oe M2 for  +  oe 40 8 5 8 5 or M1 for one product of the form 5 k 3 k  or  oe (k = 1, 2, 3 or 4) 8 5 8 5

More questions on Probability of combined events

21 (a) `3 - 5`j2 + 3 5j = a + b 5 Find the value of a and the value of b. a = ................................................ b = ................................................ [2] (b) Rationalise the denominator. Write your answer in its simplest form. 6 2 ................................................. [2]

Mark scheme: 21(a) [a =] –9 2 B1 for each [b =] 7 or for three correct terms from 6 + 9 5 − 2 5 − 3 5 5 oe 21(b) 3 2 final answer 2 2 M1 for  oe 2

More questions on Surds

22 Solve. 2 x = x - 1 x + 2 x = .................. or x = .................. [5]

Mark scheme: 22 –1 and 4 5 B2 for x2 – 3x – 4 [= 0] or M1 for 2(x + 2) = x(x – 1) or better M2 for a correct method to solve their three-term quadratic in the numerator or M1 for (x + a)(x + b) where ab = –4 or a + b = –3 or x ( x − 4) + [1]( x − 4) or x ( x + 1) − 4( x + 1) or M1 for ( −3)2 −4 [1] −4 or better p + q p − q or if in the form or then r r M1 for p = −(−3) and r = 2(1) or better

More questions on Equations

23 Find the coordinates of the turning point on the graph of y = 7 - 2x - x 2 . ( ...................... , ...................... ) [4]

Mark scheme: 23 (–1, 8) 4 B3 for x = –1 OR M1 for –2 – 2x M1 for their derivative = 0 OR −−( 2 ) M2 for [x =] or better 2 ( −1) −b or B1 for 2 a OR M2 for – ((x + 1)2 – 8) [= 0] oe or M1 for [±](x + 1)2 + k

More questions on Graphs of functions

24 A NOT TO a SCALE C M O B D b In the diagram, OBD and ACD are straight lines. O is the origin, the position vector of A is a and the position vector of B is b. 1 BC = OA 3 M is the midpoint of CD. Find the position vector of M. Give your answer in terms of a and b, in its simplest form. ................................................. [4] Questions 25 and 26 are printed on the next page.

Mark scheme: 24 1 5 4   1 1 a + b final answer B3 for DM or MC = − b + a 6 4 4 6   1 1 or MD or CM = b – a 4 6  1  3 or B2 for BD = b or OD = b 2 2  or M1 for a correct route for OM

More questions on Vectors in two dimensions

25 Simplify. 10 ax + 6bx - 25a - 15 b 4 x 2 - 25 ................................................. [4]

Mark scheme: 25 5a + 3b 4 B2 for (5a + 3b )(2 x − 5) final answer 2 x + 5 or B1 for 5a (2 x − 5) + 3b(2 x − 5) or for 2 x (5a + 3b ) − 5(5a + 3b ) B1 for (2 x + 5)(2 x − 5)

More questions on Algebraic manipulation

1 26 Solve tanx =- for 0° G x G 360° . 3 x = .................... , x = .................... [3]

Mark scheme: 26 150 and 330 3 B2 for one correct answer or B1 for [tan–1…] = 30 or –30 If 0 scored, SC1 for two answers in range with a difference of 180

More questions on Trigonometric functions