Cambridge IGCSE Mathematics 0580 — 2025 May/June Paper 2 · Variant 2

1 (a) Shade one more small square so that the diagram has one line of symmetry. [1] (b) Shade one more small square so that the diagram has rotational symmetry of order 2. [1]

Mark scheme: Question Answer Marks Partial Marks 1(a) or 1 1(b) 1

More questions on Symmetry

2 The scale drawing shows the positions of two villages, P and Q. The scale is 1 cm represents 0.5 km. North P North Q (a) Find the actual distance between village P and village Q. ............................................ km [2] (b) Measure the bearing of village Q from village P. ................................................. [1]

Mark scheme: 2(a) 4.4 to 4.6 2 B1 for 8.8 [cm] to 9.2 [cm] or M1 for 0.5 × their written measurement where their measurement is in the range 8 to 10 2(b) 108 to 112 1

More questions on Scale drawings

3 45° 65° NOT TO SCALE x° y° The diagram shows two straight lines intersecting two parallel lines. Find the value of x and the value of y. x = ................................................ y = ................................................ [3]

Mark scheme: 3 [x =] 70 3 B2 for either correct and or B1 for 45 and 65 correctly placed on [y =] 65 diagram or M1 for 180 – 45 – 65 oe

More questions on Angles

4 1 2 3 4 5 6 7 Samira picks one of these cards at random and replaces it. (a) Find the probability that she picks an odd number. ................................................. [1] (b) Samira repeats this 35 times. Calculate the number of times Samira is expected to pick an odd number. ................................................. [1]

Mark scheme: 4(a) 4 1 oe 7 4(b) 20 1 FT their (a) × 35 provided 0 < their (a) < 1

More questions on Relative and expected frequencies

6 Solve. (a) 8x + 7 = 39 x = ................................................ [2] (b) 2 ( 5y - 1 ) = 24 y = ................................................ [3]

Mark scheme: 6(a) 4 2 M1 for 8x = 39 – 7 or better 6(b) 13 3 M1 for correct first step e.g. 2.6 or oe 5 24 5y – 1 = or 10y – 2 = 24 or better 2 M1 for correctly isolating terms in y FT their first step e.g. 5y = 12 + 1 or 10y = 24 + 2

More questions on Equations

7 These are the first 4 terms of a sequence. 11 8 5 2 (a) Find the next term of this sequence. ................................................. [1] (b) Find the nth term of this sequence. ................................................. [2]

Mark scheme: 7(a) –1 1 7(b) 14 – 3n oe final answer 2 B1 for c – 3n or 14 – kn (k ≠ 0) or 14 – 3n seen then spoilt

More questions on Sequences

8 Find the highest common factor (HCF) of 36 and 54. ................................................. [2]

Mark scheme: 8 18 2 B1 for answer 2, 3, 6 or 9 or M1 for answer 2 × 32 oe or for 2 × 2 × 3 × 3 and 2 × 3 × 3 × 3 or 2 correct factor trees or tables

More questions on Types of number

9 A is the point ( 3 , - 1 ) . 2 AB = e o - 4 (a) AC = 2 AB Find the coordinates of the point C. ( .................. , .................. ) [2] (b) The length of AB is k 5. Find the value of k. k = ................................................ [2] (c) P is a point on AB. AP | PB = 1 | 3 Find the position vector of P. f p [2]

Mark scheme: 9(a) (7, –9) 2  7  B1 for (7, k) or (k, –9) or    −9   4  or for   seen  −8   3   2  or M1 for   + 2    −1   −4  9(b) 2 2 M1 for 22 + ([–]4)2 oe or better 9(c)  3.5  2  3.5   k    oe B1 for answer   or    −2   k   −2   1   0.5    or seen 2 or for        −1  1  −   3  1  2  or M1 for   +   oe  −1  4  −4 

More questions on Vectors in two dimensions

10 18 cm NOT TO SCALE 45° O The diagram shows a sector of a circle, centre O. The length of the arc is nr cm . Find the value of n. n = ................................................ [2]

Mark scheme: 10 9 2 45 oe M1 for 2 π  18 oe 2 360

More questions on Circles, arcs and sectors

11 (a) Write 0.007 08 in standard form. ................................................. [1] (b) Work out ( 3. 8 # 10 22 ) + ( 3. 8 # 10 23 ) . Give your answer in standard form. ................................................. [2]

Mark scheme: 11(a) 7.08 × 10–3 cao 1 11(b) 4.18 × 1023 cao 2 B1 for figs 418 or M1 for 0.38 × 1023 or 38 × 1022 or 1022(3.8 + 3.8 × 10) or 1023(3.8 ÷ 10 + 3.8) oe

More questions on Standard form

12 NOT TO R SCALE P 74° Q P, Q and R lie on a circle. QR is a diameter. Find angle PRQ. Give geometrical reasons for your answer. Angle PRQ = ................. because ...................................................................................................... ............................................................................................................................................................. [2]

Mark scheme: 12 16 2 B1 for 16 or angle in a semicircle = 90 and angle in a semicircle = 90 and angle sum of a triangle = 180

More questions on Circle theorems I

13 (a) 100 students solve a puzzle. The table shows information about the time taken by each student to solve the puzzle. Time (t seconds) 20 1 t G 40 40 1 t G 60 60 1 t G 100 Frequency 30 40 30 (i) Work out an estimate of the mean. ................................................s [4] (ii) Complete the histogram to show the information in the table. 4 3 Frequency 2 density 1 0 t 20 30 40 50 60 70 80 90 100 Time (seconds) [2] (b) 80 adults solve the same puzzle as the students. The cumulative frequency table shows information about the time taken by each adult to solve the puzzle. Time (t seconds) t G 20 t G 40 t G 60 t G 80 t G 100 t G 120 Cumulative frequency 0 12 36 60 74 80 (i) On the grid, draw a cumulative frequency diagram. 80 70 60 50 Cumulative 40 frequency 30 20 10 0 t 20 30 40 50 60 70 80 90 100 110 120 Time (seconds) [3] (ii) Use your cumulative frequency diagram to find an estimate for (a) the median ............................................... s [1] (b) the lower quartile. ............................................... s [1]

Mark scheme: 13(a)(i) 53 4 M1 for correct midpoints soi M1 for fx where x is in correct interval including boundaries M1 for fx ÷ 100 dep on second M1 13(a)(ii) Two correct bars with correct widths 2 B1 for one correct bar and heights 2 and 0.75 or M1 for 40/20 oe and 30/40 oe soi 13(b)(i) Correct diagram 3 B1 for correct horizontal placement for 6 plots B1 for correct vertical placement for 6 plots B1FT dep on at least B1 for reasonable increasing curve or polygon through their 6 points If 0 scored, SC1 for 5 out of 6 points correctly plotted 13(b)(ii)(a) 62 to 64 1 FT their increasing curve or polygon reading at 40 13(b)(ii)(b) 46 to 48 1 FT their increasing curve or polygon reading at 20

More questions on Statistical charts and diagrams

14 Write .025o as a fraction. ................................................. [2]

Mark scheme: 14 23 2 M1 for 25.55… – 2.55… oe oe fraction 90 or for 90x = 23 oe 2 5 or for + oe 10 90

More questions on Fractions, decimals and percentages

15 y 5 4 3 2 1 – 8 – 7 – 6 – 5 – 4 – 3 – 2 – 1 0 1 2 3 4 5 6 7 8 x – 1 – 2 – 3 – 4 – 5 2 The diagram shows the graph of y = - 1. x (a) Write down the coordinates of the point where the graph crosses the x-axis. ( .................. , .................. ) [1] (b) Write down the equation of each asymptote. ................................................. ................................................. [2] 2 (c) By drawing a suitable straight line on the grid, solve - x - 1 = 0 . x x = .................. or x = .................. [3]

Mark scheme: 15(a) (2, 0) 1 15(b) x = 0, y = –1 2 B1 for each 15(c) y = x ruled B1 x = –2 and x = 1 B2 B1 for one correct or for two correct answers FT from their line

More questions on Graphs of functions

16 NOT TO SCALE 6 cm 5 cm The diagram shows a solid made by joining a hemisphere to a cylinder. The radius of both the hemisphere and the cylinder is 6 cm. The height of the cylinder is 5 cm. Find the total surface area of the solid. Give your answer in terms of r. ......................................... cm2 [4]

Mark scheme: 16 168 π 4 B3 for answer 168 OR M3 for 2 1 2 π  6 +  4π  6 + 2π 6 5 oe 2 OR To a maximum of 2 marks ignoring extra areas added or subtracted M1 for π × 62 1 2 M1 for 4 π 6 oe 2 M1 for 2 × π × 6 × 5

More questions on Compound shapes and parts of shapes

17 Find the value of 2 (a) 125 3 ................................................. [2] - 25 (b) 4 . ................................................. [2]

Mark scheme: 17(a) 25 2 2 2 3 3 125 5 3 M1 for or 3 1252 or ( ) ( ) or B1 for 3 125 = 5 17(b) 1 2 1 1 [] M1 for or 32–1 or 32 2 5 1024

More questions on Indices II

9 18 (a) 3 Rationalise the denominator. Give your answer in its simplest form. ................................................. [2] (b) ( 5 - 2 )( 1 + 3 2 ) = c + k 2 Find the value of c and the value of k. c = ................................................ k = ................................................ [2]

Mark scheme: 18(a) 3 3 cao 2 9 3 M1 for × oe 3 3 18(b) [c =] –1 2 B1 for each [k =] 14 or for 3 correct terms from 5 + 15 2 − 2 − 3 2 2 oe

More questions on Powers and roots

19 Write as a single fraction in its simplest form. 5a 3 b (a) # 6 a ................................................. [2] p 3t (b) + 2 4 ................................................. [2] 2 3 (c) - x - 2 x + 1 ................................................. [3]

Mark scheme: 19(a) 5b 2 15 ab cao final answer B1 for or better seen 2 6 a 19(b) 2 p + 3t 2 M1 for adding two correct fractions with a cao final answer 4 p 6t 4 common denominator e.g. + 8 8 19(c) 8 − x 8 − x 3 B1 for 2(x + 1) – 3(x – 2) or better isw or 2 −−x 2 ( x − 2 )( x + 1) x B1 for common denominator (x – 2)(x + 1) oe isw cao final answer

More questions on Algebraic fractions

1 20 y \ x (a) When x = 9, y = 2 . Find the value of y when x = 36 . y = .................................................. [3] (b) When x is increased by a factor of 4, the value of y changes by a factor of p. Find the value of p. p = ................................................ [1]

Mark scheme: 20(a) [] 1 3 k M1 for 2 = oe 9 their k M1 for 36 OR M2 for 2  9 = y  36 20(b) 1 1 oe 2

More questions on Powers and roots

21 y Q NOT TO A O B x SCALE P The diagram shows the graph of y = 3 x - x 3 . The graph crosses the x-axis at A, at O and at B. The turning points of the graph are at P and at Q. (a) Find the x-coordinate of A and the x-coordinate of B. Give your answers as exact values. x-coordinate of A ................................................ x-coordinate of B ................................................ [3] (b) (i) Differentiate 3x - x 3 . ................................................. [2] (ii) Find the coordinates of P and Q. P ( ..................... , ..................... ) Q ( ..................... , ..................... ) [4]

Mark scheme: 21(a) [A =] − 3 oe 3 B2 for – 3 oe or 3 oe [B =] 3 oe or M1 for x 3 − x 2 = 0  or better ( ) 0  0 2 −−4 1 3 or oe 2 −1 21(b)(i) 3 – 3x2 final answer 2 B1 for 3 or – 3x2 correct in an expression or for correct answer spoilt 21(b)(ii) [P =] (–1, –2) 4 B3 for (–1, –2) or (1, 2) and or for two correct values of x [Q =] (1, 2) or M2 for x2 = 1 or for [3](1 – x)(1 + x) [= 0] oe factorised 0  0 2 −−4 3 3 or oe 2 − 3 OR M1 for [3](1 – x2) [= 0] or their (b)(i) = 0 d y or for stating = 0 d x M1 for correct method to solve their quadratic

More questions on Differentiation

22 (a) Write down the exact value of tan 60°. ................................................. [1] (b) Solve 2 sinx - 1 = 0 for 0 ° G x G 360° . x = ............... or x = ............... [3]

Mark scheme: 22(a) 3 1 22(b) 30, 150 3 B2 for 30 or 150 1 or M1 for sin x = 2 If 0 or M1 scored, SC1 for one acute angle and one obtuse angle with a sum of 180

More questions on Trigonometric functions

23 B C NOT TO SCALE b M O a A In the diagram, OA is parallel to BC. BC = 3OA M is the midpoint of AC. The position vector of A is a and the position vector of B is b. Find the position vector of M. Give your answer in terms of a and b, in its simplest form. ................................................. [3]

Mark scheme: 23 1 3 B2 for a correct route in terms of a and b 2a + b final answer not in its simplest form 2 1 1 or for AM (or AC ) = a + b oe 2 2 or B1 for AC = –a + b + 3a oe or M1 for correct route for OM using the lines of the diagram

More questions on Vector geometry

24 The line y = 7x + 3 intersects the curve y = x 2 + 5x - 12 at the points A and B. Find the coordinates of A and B. A ( ..................... , ..................... ) B ( ..................... , ..................... ) [5]

Mark scheme: 24 (5, 38) and (–3, –18) 5 B4 for one correct coordinate or for x = 5 and x = –3 OR M2 for x2 – 2x – 15 [= 0] or y2 – 20y – 684 [= 0] or M1 for 7x + 3 = x2 + 5x – 12 oe  y − 3  2  y − 3  or y = + 5 − 12      7   7  M1 for correct method to solve their three- term quadratic (x – 5)(x + 3) −−( 2 )  ( −2 ) 2 −−4 1 15 oe 2  1 If B0 scored and at least 2 method marks scored, SC1 for correct substitution of both of their x values or their y values into y = 7x + 3 or y = x2 + 5x – 12

More questions on Equations