Cambridge IGCSE Mathematics 0580 — 2021 Feb/March Paper 4 · Variant 2

1 Painter Plumber Electrician $35 per hour Fixed charge $40 $48 per hour for the first 2 hours plus then $26.50 per hour $32 per hour These are the rates charged by a painter, a plumber and an electrician who do some work for Mr Sharma. (a) The painter works for 7 hours. Calculate the amount Mr Sharma pays the painter. $ ................................................. [1] (b) Mr Sharma pays the plumber $252. Calculate how many hours the plumber works. ........................................ hours [2] (c) Mr Sharma pays the electrician $224. Calculate how many hours the electrician works. ........................................ hours [2] (d) Write down the ratio of the amount Mr Sharma pays to the painter, the plumber and the electrician. Give your answer in its lowest terms. painter : plumber : electrician = .................. : .................. : .................. [2]

Mark scheme: Question Answer Marks Partial Marks 1(a) 245 1 1(b) 8 2 M1 for 40 + 26.5x = 252 oe or B1 for 212 seen 1(c) 6 2 M1 for (224 − 2 × 48 ) ÷ 32 oe or 2 × 48 + 32 (x – 2) = 224 soi 1(d) 35 : 36 : 32 final answer 2 B1 for their (a) : 252 : 224 or equivalent ratio

More questions on Rates

2 y 14 12 10 B 8 A 6 4 2 – 14 – 12 – 10 – 8 – 6 – 4 – 2 0 2 4 6 8 10 x – 2 – 4 C – 6 – 8 – 10 (a) Describe fully the single transformation that maps (i) triangle A onto triangle B, ............................................................................................................................................. ............................................................................................................................................. [3] (ii) triangle A onto triangle C. ............................................................................................................................................. ............................................................................................................................................. [3] - 5 (b) Draw the image of triangle A after a translation by the vector [2] e- 10o. (c) Draw the image of triangle A after a reflection in the line y = 4 . [2]

Mark scheme: 2(a)(i) rotation 3 B1 for each 90 anticlockwise oe (–3, 2) 2(a)(ii) enlargement 3 B1 for each 1 − 2 ( −2, − 1) 2(b) Image at (– 3, – 5) (1, − 5) (1, 3) 2  − 5   k  B1 for translation by   or    k   −10  2(c) Image at (2, 3) (6, 3) (6, − 5) 2 B1 for reflection in y = k or x = 4

More questions on Transformations

3 (a) a° NOT TO 126° SCALE c° b° 63° The diagram shows two straight lines intersecting two parallel lines. Find the values of a, b and c. a = ................................................ b = ................................................ c = ................................................ [3] (b) Q NOT TO SCALE S R 58° x° P Points R and S lie on a circle with diameter PQ. RQ is parallel to PS. Angle RPQ = 58° . Find the value of x, giving a geometrical reason for each stage of your working. ..................................................................................................................................................... ..................................................................................................................................................... ..................................................................................................................................................... x = ................................................ [3] (c) NOT TO O SCALE 142° C A y° B Points A, B and C lie on a circle, centre O. Angle AOC = 142° . Find the value of y. y = ................................................ [2]

Mark scheme: 3(a) 126 3 B1 for each 54 117 3(b) angle [in a] semicircle is 90 B1 Do not accept triangle for angle Allied, co-interior [add to 180] B1 or Angles in triangle [ = 180] and alternate oe 32 B1 3(c) 109 2 B1 for 218 or 71 in correct places or correctly labelled

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4 (a) A shop gives each of 1000 people a voucher. 28 people use their voucher. The shop now gives each of 16 500 people a voucher. Calculate how many of these 16 500 people are expected to use their voucher. ................................................. [1] (b) In a class activity, all the 15 students wear hats. 7 students wear red hats, 6 students wear green hats and 2 students wear white hats. (i) One of these students is picked at random. Find the probability that this student wears a red hat. ................................................. [1] (ii) Two of the 15 students are picked at random. 37 Show that the probability that these two students wear hats of the same colour is . 105 [3] (iii) Three of the 15 students are picked at random. Find the probability that at least two of these three students wear red hats. ................................................. [4]

Mark scheme: 4(a) 462 1 4(b)(i) 7 1 oe 15 4(b)(ii) 7 6 6 5 2 1 3 M2 for addition of two of × + × + × 7 6 6 5 2 1 15 14 15 14 15 14 × + × + × 37 15 14 15 14 15 14 = or M1 for one of the products seen 105 4(b)(iii) 29 4 M3 for oe 7 6 5 7 6 6 7 6 2 65 × × + 3 × × × + 3 × × × oe 15 14 13 15 14 13 15 14 13  8 7 7   8 7 6  or 1 − 3  × ×  −  × ×  oe  15 14 13   15 14 13  or M2 for the sum of at least two of 7 6 5 7 6 6 7 6 2 × × , N × × × , N × × × 15 14 13 15 14 13 15 14 13 seen 7 6 13 or for × × 15 14 13 7 6 7 6 k or × + N × × × seen 15 14 15 14 13 or M1 for 7 6 5 7 6 6 7 6 2 × × or N × × × or N × × × 15 14 13 15 14 13 15 14 13 seen 1519 If 0 scored SC1 for oe 3375

More questions on Introduction to probability

5 C B 65° NOT TO SCALE 4.4 cm 9.7 cm A 8.6 cm 42° D (a) Calculate angle ADB. Angle ADB = ................................................ [3] (b) Calculate DC. DC = ........................................... cm [4] (c) Calculate the shortest distance from C to BD. ............................................ cm [3]

Mark scheme: 5(a) 27[.0] or 26.97… nfww 3 8.6 2 + 9.7 2 − 4.4 2 M2 for [cos = ] 2 × 8.6 × 9.7 or M1 for implicit form 5(b) 9.19 or 9.192 to 9.193 4 B1 for [angle BCD =] 73 seen 9.7 × sin65 M2 for oe sin (180 − 65 − 42) sin(180 − 65 − 42) sin65 or M1 for = oe 9.7 DC 5(c) 6.15 or 6.149 to 6.151… 3 d M2 for = sin42 oe their 9.19 or M1 for right angle between line from C to BD and BD soi

More questions on Non-right-angled triangles

6 y 5 4 3 2 1 x – 2 – 1 0 1 2 – 1 – 2 (a) The grid shows the graph of y = a + bx2 . The graph passes through the points with coordinates (0, 4) and (1, 1). (i) Find the value of a and the value of b. a = ................................................ b = ................................................ [2] (ii) Write down the equation of the tangent to the graph at (0, 4). ................................................. [1] (iii) The equation of the tangent to the graph at x =- 1 is y = 6x + 7 . Find the equation of the tangent to the graph at x = 1. ................................................. [2] 5(b) The table shows some values for y = 1 + for - 2 G x G 1.5 . 3 - x x – 2 – 1.5 – 1 – 0.5 0 0.5 1 1.5 y 2 2.11 2.43 3 4.33 (i) Complete the table. [3] 5 (ii) On the grid, draw the graph of y = 1 + for - 2 G x G 1.5 . [4] 3 - x (c) (i) Write down the values of x where the two graphs intersect. x = ................... or x = .................. [2] (ii) The answers to part(c)(i) are two solutions of a cubic equation in terms of x. Find this equation in the form ax 3 + bx 2 + cx + d = 0 , where a, b, c and d are integers. ............................................................................... [4]

Mark scheme: 6(a)(i) [a = ] 4 2 B1 for [a = ] 4 B1 for [b = ] – 3 nfww [b = ] – 3 nfww 6(a)(ii) y = 4 oe 1 6(a)(iii) y = − 6x + 7 oe final answer 2 B1 for answer − 6x + 7 or answers y = − 6x + c or y = kx + 7 (k < 0) 6(b)(i) 2.25 2.67 3.5 3 B1 for each 6(b)(ii) correct curve 4 B3 FT for 7 or 8 points or B2 FT for 5 or 6 points or B1 FT for 3 or 4 points 6(c)(i) –0.78 to –0.72 and 0.55 to 0.59 2 B1 for each 6(c)(ii) 3 x 3 − 9 x 2 − 3 x + 4 [ = 0] final answer 4 B3FT for 3 out of 4 correct terms or for bx3 – 3bx2 + (a – 1)x + 8 – 3a [ = 0] oe or B2FT for 2 out of 4 correct terms or for 3 out of 4 terms from bx3 – 3bx2 + (a – 1)x + 8 – 3a [ = 0] 5 2 or M1 for 1 + = their 4 + (their ( −3)) x oe 3 − x

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7 (a) The box-and-whisker plot shows information about the marks scored by some students in a test. 0 10 20 30 40 50 60 70 80 90 100 Mark (i) Write down the median mark. ................................................. [1] (ii) Work out the range. ................................................. [1] (iii) Jais scored a mark in the test that was higher than the marks scored by 75% of the students. Write down a possible mark for Jais. ................................................. [1] (iv) This box-and-whisker plot shows information about the marks scored by the same students in a second test. 0 10 20 30 40 50 60 70 80 90 100 Mark Make one comparison between the distributions of marks in the two tests. ............................................................................................................................................. ............................................................................................................................................. [1] (b) The table shows information about the height, h cm, of each of 50 plants. Height (h cm) 0 1 h G 20 20 1 h G 30 30 1 h G 34 34 1 h G 40 40 1 h G 60 Frequency 4 9 20 15 2 Calculate an estimate of the mean. ............................................ cm [4] (c) Some apples are weighed and the mass, m grams, of each apple is recorded. The table shows the results. Mass (m grams) 100 1 m G 110 110 1 m G 115 115 1 m G 125 125 1 m G 140 Frequency 50 x 44 51 The histogram shows some of the information from the table. 8 6 Frequency density 4 2 0 m 100 110 120 130 140 Mass (grams) (i) Work out the value of x. x = ................................................ [1] (ii) Complete the histogram. [2]

Mark scheme: 7(a)(i) 70 1 7(a)(ii) 78 1 7(a)(iii) Value in range 86 <V ≤ 90 1 7(a)(iv) One general comment interpreting 1 the median comparison nfww e.g. Students did better on second test oe OR One general comment interpreting IQR/range comparison nfww e.g. Students marks were more consistent on the 2nd test oe 7(b) 31.2 4 M1 for mid-values soi M1 for Σfm where m is any value in interval including boundaries M1 (dep on second M1) for their Σfm ÷ 50 7(c)(i) 38 1 7(c)(ii) Blocks of heights 4.4 and 3.4 with 2 B1 for each correct block correct widths If B0 scored, SC1 for both correct frequency densities soi

More questions on Statistical charts and diagrams

8 (a) O 53° NOT TO 9.5 cm A B SCALE X Y The diagram shows a sector OXY of a circle with centre O and radius 9.5 cm. The sector angle is 53°. A lies on OX, B lies on OY and OA = OB . (i) Show that the area of the sector is 41.7 cm 2, correct to 1 decimal place. [2] 1 (ii) The area of triangle OAB is of the area of sector OXY. 3 Calculate OA. OA = ........................................... cm [4] (b) O 60° NOT TO 24 cm SCALE P Q The diagram shows a sector OPQ of a circle with centre O and radius 24 cm. The sector angle is 60°. A cone is made from this sector by joining OP to OQ. O NOT TO SCALE P Q Calculate the volume of the cone. 1 2 [The volume, V, of a cone with radius r and height h is V = rr h .] 3 ......................................... cm3 [6]

Mark scheme: 8(a)(i) 53 2 M1 × π× 9.5 360 41.74 to 41.75 A1 8(a)(ii) 5.9[0] or 5.899 to 5.903.. 4 2 13 × 41.7 M3 for  OA =  oe   1 2 sin53 1 2 1 M2 for × OA × sin53 = × 41.7 oe 2 3 1 1 M1 for × OA × OB × sin53 = × 41.7 seen or 2 3 better 8(b) 396 or 397 or 396.4 to 396.6 6  60  M2 for [ r = ]  ×2× π×24  ÷ 2π oe or better  360  60 or M1 for 2πr = × 2 × π× 24 oe 360 M2 for 24 2 −a 2 or M1 for h2 + a2 = 242 1 2 M1 for 3π× their r × their h

More questions on Circles, arcs and sectors

9 (a) Factorise. (i) 5am + 10ap - bm - 2bp ................................................. [2] (ii) 15 ( k + g) 2 - 20 ( k + g) ................................................. [2] (iii) 4x 2 - y 4 ................................................. [2] (b) Expand and simplify. ( x - 3)( x + 1)( 3x - 4) ................................................. [3] (c) ( x + a) 2 = x 2 + 22x + b Find the value of a and the value of b. a = ................................................ b = ................................................ [2]

Mark scheme: 9(a)(i) ( 5a − b )( m + 2 p ) final answer 2 M1 for 5 a ( m + 2 p ) − b ( m + 2 p ) or m ( 5 a − b ) + 2 p ( 5 a − b ) or B1 for correct answer seen 9(a)(ii) 5 ( k + g )( 3k + 3 g − 4 ) final answer 2 M1 for correct partial factorisation by 5 or (k + g) isw eg 5 ( 3k 2 + 6 kg + 3 g 2 − 4 k − 4 g ) or 5(3(k + g)2 – 4(k+ g)) or ( k + g ) (15 ( k + g ) − 20 ) or (5k + 5g) (3k + 3g – 4) or B1 for correct answer seen 9(a)(iii) 2 2 2 M1 for answer in form (a + b) (a – b) 2 x − y 2 x + y final answer ( )( ) or B1 for correct answer seen 9(b) 3 x 3 − 10 x 2 − x + 12 final answer 3 B2 for correct unsimplified expansion or simplified expression with 3 terms correct in a 4-term expression of required form or B1 for correct expansion of two of the brackets with at least 3 terms correct 9(c) [a =] 11 2 B1 for each [b =] 121

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10 (a) A box is a cuboid with length 45 cm, width 30 cm and height 42 cm. The box is completely filled with 90.72 kg of sand. Calculate the density of this sand in kg/m3. [Density = mass ÷ volume] ....................................... kg/m3 [3] (b) A bag contains 15000 cm 3 of sand. Some of this sand is used to completely fill a hole in the shape of a cylinder. The hole is 30 cm deep and has radius 10 cm. Calculate the percentage of the sand from the bag that is used. ..............................................% [3] (c) Sand costs $98.90 per tonne. This cost includes a tax of 15%. Calculate the amount of tax paid per tonne of sand. $ ................................................. [3] (d) Raj buys some sand for 3540 rupees. Calculate the cost in dollars when the exchange rate is $1 = 70.8 rupees. $ ................................................. [2]

Mark scheme: 10(a) 1600 3 B2 for answer figs 16 or M2 for 90.72 ÷ (figs45 × figs3 × figs42) or M1 for volume = figs 45 × figs 3 × figs 42 isw 10(b) 62.8 or 62.83 to 62.84 3 π× 10 2 × 30 M2 for × 100 15000 or M1 for π × 10 2 × 30 10(c) 12.9[0] 3 B2 for 86 OR 98.9 98.9 M2 for × 0.15 oe or 98.9 − oe 15 15 1 + 1 + 100 100  15  or M1 for  1 +  a = 98.9 oe isw  100  10(d) 50 2 M1 for 3540 ÷ 70.8

More questions on Ratio and proportion

11 Gaya spends $48 to buy books that cost $x each. (a) Write down an expression, in terms of x, for the number of books Gaya buys. ................................................. [1] (b) Myra spends $60 to buy books that cost $( x + 2) each. Gaya buys 4 more books than Myra. Show that x 2 + 5x - 24 = 0 . [4] (c) Solve by factorisation. x 2 + 5x - 24 = 0 x = .................. or x = ................... [3] (d) Find the number of books Myra buys. ................................................. [1]

Mark scheme: 11(a) 48 1 Accept 48 ÷ x final answer x 11(b) 60 M1 FT their (a) provided expression in x their ( a ) − = 4 oe x + 2 48 ( x + 2 ) − 60 x = 4 x ( x + 2 ) oe M2 FT their 3 term eqn with algebraic denominators, x and x + 2, for M2 or M1 M1 for common denominator x ( x + 2 ) oe seen or any two terms in a 3 term equation from ± 48 (x + 2) , ± 60x , ± 4x(x + 2) oe seen 48x + 96 – 60x = 4x2 + 8x oe A1 With brackets expanded and no errors or omissions leading to x 2 + 5 x − 24 = 0 seen 11(c) ( x − 3 )( x + 8 ) B2 B1 for x(x + 8) – 3(x + 8) or x(x – 3) + 8(x – 3) or (x + a)(x + b) [= 0] where ab = – 24 or a + b = 5 [a, b integers] 3 and − 8 B1 11(d) 12 1

More questions on Equations

12 (a) Find the gradient of the curve y = 2x 3 - 7x + 4 when x =- 2 . ................................................. [3] (b) A is the point (7, 2) and B is the point (−5, 8). (i) Calculate the length of AB. ................................................. [3] (ii) Find the equation of the line that is perpendicular to AB and that passes through the point (−1, 3). Give your answer in the form y = mx + c . y = ................................................ [4] (iii) AB is one side of the parallelogram ABCD and - a • BC = where a 2 0 and b 2 0 e- bo • the gradient of BC is 1 • BC = 8 . Find the coordinates of D. ( ....................... , ......................) [4]

Mark scheme: 12(a) 17 3 M2 for 3 × 2 x 2 − 7 or better isw or M1 for 3 × 2 x 2 oe or kx2 – 7 seen 12(b)(i) 13.4 or 13.41 to 13.42 3 2 2 M2 for ( −−5 7 ) + ( 8 − 2 ) oe 2 2 or M1 for ( −−5 7 ) + ( 8 − 2 ) oe 12(b)(ii) [ y = ] 2 x + 5 final answer 4 8 − 2 M1 for [gradient of AB =] oe −−5 7 1 M1dep for gradient p = −÷1 their − oe 2 M1dep on previous M1 for substituting (−1, 3) into y = their px + c oe where their p ≠ 0 12(b)(iii) ( 5, 0) 4   − 2   2 B3 for AD =   or DA =  − 2  2   12  or coordinates of C (−7, 6) and  CD =      oe  − 6  seen or B2 for a = b = 2 soi or coordinates of C (−7, 6) 2 8 oe or M1 for a = b oe soi or for a 2 + b 2 = ( ) a or cos 45 = oe 8   −12    12  =  DC =  or for      or  CD    seen  6   − 6  y − 8 y − 2 or = 1 oe or = 1 x −−5 x − 7

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