Cambridge IGCSE Mathematics 0580 — 2022 Feb/March Paper 4 · Variant 2

1 A company employed 300 workers when it started and now employs 852 workers. (a) Calculate the percentage increase in the number of workers. ............................................. % [2] (b) Of the 852 workers, the ratio part-time workers : full-time workers = 5 : 7. Calculate the number of full-time workers. ................................................. [2] (c) The company makes 40 600 headphones in one year. Write this number (i) in words, ........................................................................................ [1] (ii) in standard form. ................................................. [1] (d) In one month, the company sells 3 000 headphones. 3 Of these, 48% are exported, are sold to shops and the rest are sold online. 8 Calculate the number of headphones that are sold online. ................................................. [3] (e) One year, sales increased by 15%. The following year sales increased by 18%. Calculate the overall percentage increase in sales. ............................................. % [3]

Mark scheme: Question Answer Marks Partial Marks 1(a) 184 2 852 − 300 M1 for [× 100 ] oe 300 852 or for × 100 [ −100 ] oe 300 1(b) 497 2 852 M1 for × k oe where k = 1, 5 or 7 5 + 7 1(c)(i) Forty thousand six hundred 1 1(c)(ii) 4.06 × 10 4 1 1(d) 435 3  48 3  M2 for 3000 ×  1 − −  oe  100 8  or B2 for 2565, or 1440 and 1125 or 1875 and 1440 or 1560 and 1125 48 3  48 3  or M1 for 1 − − or 3000 ×  +  oe 100 8  100 8  or B1 for 1440 or 1125 or 1560 or 1875 If 0 scored SC1 for answer 975 1(e) 35.7 3 100 + 15 100 + 18 M2 for × [–1] oe or better 100 100 100 + 15 100 + 18 or M1 for k × × oe 100 100

More questions on Types of number

2 12 The table shows some values for y = x - , x ! 0 . 3x The y-values are rounded to 1 decimal place. x -2 -1.5 -1 -0.75 -0.5 -0.25 -0.1 y 4.2 2.5 1.3 1.4 3.3 (a) Complete the table. [2] 2 1 (b) On the grid, draw the graph of y = x - for - 2 G x G - 0 .1 . 3x 2 1 The graph of y = x - for x 2 0 has been drawn for you. 3x y 4 3 2 1 – 2 – 1 0 1 2 x – 1 – 2 [4] 2 1 (c) By drawing a suitable line on the grid, solve the equation x - + 1 = 0 . 3x x = ................................................ [2]

Mark scheme: 2(a) 1[.0] 0.9 2 B1 for each 2(b) correct curve 4 B3 FT for 6 or 7 points B2 FT for 4 or 5 points B1 FT for 2 or 3 points 2(c) ruled line at y = − 1 B1 0.3 to 0.32 B1

More questions on Graphs of functions

3 f ( )x = 1 + 4 x g ( )x = x 2 (a) Find (i) gf(3), ................................................. [2] (ii) fg(x), ................................................. [1] (iii) f - 1 f ( )x . ................................................. [1] (b) Find the value of x when f ( )x = 15 . x = ................................................ [2]

Mark scheme: 3(a)(i) 169 2 2 M1 for g(13) or (1 + 4 x ) or better 3(a)(ii) 1 + 4 x 2 final answer 1 3(a)(iii) x 1 3(b) 7 2 M1 for 1 + 4 x = 15 3.5 or 2

More questions on Functions

4 (a) C NOT TO SCALE 29.5 cm 35.3 cm 51.6° A B 45 cm In triangle ABC, AB = 45 cm, AC = 29.5 cm, BC = 35.3 cm and angle CAB = 51.6°. (i) Calculate angle ABC. Angle ABC = ................................................ [3] (ii) Calculate the area of triangle ABC. ......................................... cm2 [2] (b) R NOT TO SCALE S 32 cm 47 cm 56° P 60° 32 cm Q The diagram shows a quadrilateral PQRS formed from two triangles, PQS and QRS. Triangle PQS is isosceles, with PQ = PS = 32 cm and angle SPQ = 56°. QR = 47 cm and angle SQR = 60°. (i) Calculate SR. SR = ........................................... cm [4] (ii) Calculate the shortest distance from P to SQ. ............................................ cm [3]

Mark scheme: 4(a)(i) 40.9 or 40.91… 3 29.5sin51.6 M2 for [sin ABC= ] oe 35.3 35.32 + 452 − 29.52 or for [cos ABC = ] 2 × 35.3 × 45 29.5 35.3 or M1 for = oe sin ABC sin51.6 or for correct implicit cosine rule 4(a)(ii) 520 or 520.0 to 520.2… 2 FT their (a)(i) if used provided working shown M1 for 0.5 × 29.5 × 45 × sin51.6 oe or for 0.5 × 35.3 × 45 × sin(their (a)(i)) or for 0.5 × 35.3 × 29.5sin (180–51.6–their(a)(i)) 4(b)(i) 41.2 or 41.21 to 41.23 4  1  M1 for SQ = 2 × 32 × sin  × 56  oe  2  or 32 2 + 32 2 − 2 × 32 × 32 × cos56 oe 32sin56 or oe sin((180 -56) ÷ 2) M2 for − 2 × 47 × their SQ × cos60 SR 2 = 47 2 + ( their SQ 2 ) or M1 for implicit form 4(b)(ii) 28.3 or 28.25 to 28.29… 3 M2 for 32 × sin62 oe or M1 for recognition that line from P is perpendicular to SQ

More questions on Non-right-angled triangles

5 The table shows information about the mass, m grams, of each of 120 letters. Mass (m grams) 0 1 m G 50 50 1 m G 100 100 1 m G 200 200 1 m G 500 Frequency 43 31 25 21 (a) Calculate an estimate of the mean mass. .............................................. g [4] (b) Iraj draws a histogram to show this information. He makes the height of the first bar 17.2 cm. Calculate the height of each of the remaining bars. height of bar for 50 1 m G 100 .......................................... cm height of bar for 100 1 m G 200 .......................................... cm height of bar for 200 1 m G 500 ........................................... cm [3] (c) Complete the cumulative frequency table. Mass (m grams) m G 50 m G 100 m G 200 m G 500 Cumulative frequency [2] (d) Draw a cumulative frequency diagram. 120 100 80 Cumulative 60 frequency 40 20 0 m 0 100 200 300 400 500 Mass (g) [3] (e) Use the cumulative frequency diagram to find an estimate for (i) the median, .............................................. g [1] (ii) the upper quartile, .............................................. g [1] (iii) the 40th percentile, .............................................. g [2] (iv) the number of letters with a mass m where 250 1 m G 400 . ................................................. [2]

Mark scheme: 5(a) 5 4 M1 for midpoints soi 121 or 120.8… or 120 6 M1 for use of ∑fx with x in correct interval including both boundaries but not if x is 50, 50, 100 and 300 M1 (dep on 2nd M1) for ∑fx ÷ 120 5(b) 12.4 5 1.4 3 B1 for each If 0 scored SC1 for fd’s [0.86,] 0.62, 0.25 and 0.07 oe 5(c) 43 74 99 120 2 B1 for 2 or 3 correct 5(d) Correct diagram 3 B1 for correct horizontal placement for 4 plots B1FT for correct vertical placement for 4 plots B1FT dep on at least B1 for reasonable increasing curve or polygon through their 4 points If 0 scored SC1 FT for 3 out of 4 points correctly plotted 5(e)(i) Strict FT their median reading 1 5(e)(ii) Strict FT their UQ reading 1 5(e)(iii) Strict FT their reading at 40th 2 B1 for 48 written or mark at cf = 48 on graph percentile 5(e)(iv) Strict FT their reading at 400 2 B1 for either correct reading at 250 or 400 – their reading at 250

More questions on Histograms

6 (a) The interior angle of a regular polygon is 156°. Calculate the number of sides of this polygon. ................................................. [2] (b) NOT TO SCALE C O 52° A B A, B and C lie on a circle, centre O. Angle OBA = 52°. Calculate angle ACB. Angle ACB = ................................................ [2] (c) S R W 112° NOT TO SCALE Q T P P, Q, R, S and T lie on a circle. WSR is a straight line and angle WSP = 112°. Calculate angle PTR. Angle PTR = ................................................ [2] (d) K NOT TO SCALE O M F G H G, K and M lie on a circle, centre O. FGH is a tangent to the circle at G and MG is parallel to OH. Show that triangle GKM is mathematically similar to triangle OHG. Give a geometrical reason for each statement you make. ..................................................................................................................................................... ..................................................................................................................................................... ..................................................................................................................................................... ..................................................................................................................................................... ..................................................................................................................................................... [4]

Mark scheme: 6(a) 15 2 360 180 ( n − 2 ) M1 for or for = 156 oe 180 − 156 n 6(b) 38 2 B1 for AOB = 76 6(c) 68 2 B1 for RSP = 68 or RQP = 112 6(d) Two pairs of equal angles identified M3 M2 for one pair of equal angles identified with fully with fully correct reasons correct reasons KMG = 90 angle in semicircle and OGH = 90 angle between tangent and radius OR KMG = OGH alternate segment OR GOH = MGK alternate angles OR Angle FGM = angle GHO corresponding and angle FGM = GKM alternate segment and angle H = angle K or M1 for KMG = 90, angle in semicircle or OGH = 90, angle between tangent and radius Two or three pairs of angles equal A1 Dep on M3 with no incorrect work seen [so similar] oe

More questions on Geometrical terms

7 Two rectangular picture frames are mathematically similar. (a) The areas of the frames are 350 cm2 and 1134 cm2. The width of the smaller frame is 17.5 cm. Calculate the width of the larger frame. ............................................ cm [3] (b) A picture in the smaller frame has length 15 cm and width 10.5 cm, both correct to the nearest 5 mm. Calculate the upper bound for the area of this picture. ......................................... cm2 [2] (c) In a sale, the price of a large frame is reduced by 18%. Parthi pays $166.05 for 5 large frames in the sale. Calculate the original price of one large frame. $ ................................................ [2] (d) Parthi advertises a large frame for a price of $57 or 48.20 euros. The exchange rate is $1= 0.88 euros. Calculate the difference between these prices, in dollars and cents, correct to the nearest cent. $ ................................................ [3]

Mark scheme: 7(a) 31.5 3 1134 M2 for 17.5 × oe 350 1134 350 or M1 for oe isw or oe isw 350 1134 1134  x  2 or for =   oe 350  17.5  7(b) 15 2 B1 for 15 + 0.25 or 10.5 + 0.25 or better seen 163.937 5 or 16316 final answer 7(c) 40.5[0] 2  18  166.05 M1 for x ×  1 −  = oe  100  [ 5 ] 7(d) $2.23 final answer 3 B2 for 2.227… or 2.23 seen OR 48.2 M2 for 57 – oe 0.88 48.2 or M1 for oe 0.88 If 0 scored SC1 for 57 × 0.88 oe seen

More questions on Limits of accuracy

8 Darpan runs a distance of 12 km and then cycles a distance of 26 km. His running speed is x km / h and his cycling speed is 10 km / h faster than his running speed. He takes a total time of 2 hours 48 minutes. 12 (a) An expression for the time, in hours, Darpan takes to run the 12 km is . x Write an equation, in terms of x, for the total time he takes in hours. ................................................. [3] (b) Show that this equation simplifies to 7x 2 - 25x - 300 = 0 . [4] (c) Use the quadratic formula to solve 7x 2 - 25x - 300 = 0 . You must show all your working. x = ................... or x = .................. [4] (d) Calculate the number of minutes Darpan takes to run the 12 km. .......................................... min [2]

Mark scheme: 8(a) 12 26 3 12 26 + = 2.8 oe isw B2 for + oe isw x x + 10 x x + 10 OR 26 B1 for seen x + 10 168 48 B1 for time = 2.8 or or 2 oe 60 60 8(b) 12 ( x + 10 ) + 26 x = 2.8 x ( x + 10 ) or M2 FT their time, provided 2 algebraic fractions one in x and other in ± x ± 10 better M1 for 12 ( x + 10 ) + 26 x seen or better 12 x + 120 + 26 x = 2.8 x 2 + 28 x M1 FT their equation dep on M2 2.8 x 2 − 10 x − 120 = 0 oe A1 or 30x + 300 + 65x = 7x2 + 70x or better 2 with no errors or omissions leading to 7 x − 25 x − 300 = 0 8(c) 2 B2 2 [ −− ]25 ± ( [ − ]25 ) − 4 × 7 × −300 B1 for ( [ − ]25 ) − 4(7)( −300) or better 2 × 7 [ −− ]25 + q [ −− ]25 − q oe or for or 2 × 7 2 × 7 − 5 and 8.57 or 8.571… B2 B1 for each or SC1 for final answers 5 and –8.57 8(d) 84 to 84.01… 2 720 FT to 3 sf or better their positive answer 12 M1 for [× 60 ] oe their positive answer

More questions on Equations

9 (a) O O NOT TO SCALE x° A B 2.4 cm AB The volume of a paper cone of radius 2.4 cm is 95.4 cm3. The paper is cut along the slant height from O to AB. The cone is opened to form a sector OAB of a circle with centre O. Calculate the sector angle x°. 1 2 [The volume, V, of a cone with radius r and height h is V = rr h .] 3 ................................................. [6] (b) An empty fuel tank is filled using a cylindrical pipe with diameter 8 cm. Fuel flows along this pipe at a rate of 2 metres per second. It takes 24 minutes to fill the tank. Calculate the capacity of the tank. Give your answer in litres. ........................................ litres [4]

Mark scheme: 9(a) 54[.0] or 53.99 to 54.03… 6 M2 for [h = ] 95.4 × 3 ÷ ( π × 2.42) oe 1 2 or M1 for 95.4 = × π × 2.4 × h 3 M2 for [slant ht , l = ] ( their h ) 2 + 2.4 2 or M1 for (their h)2 + 2.42 x M1 for × 2 × π × theirl = 2 × π × 2.4 oe 360 x 2 or × π × ( theirl ) = π × 2.4 × theirl 360 9(b) 14500 or 14470 to 14480 4 M3 for 200 × 60 × 24 × π × 42 [÷1000] or 2 × 60 ×24 ×π × 0.042 [×1000] or M2 for 200 × π × 42 or for 2 × π × 0.042 or M1 for π × 42 oe or π × 0.042 seen oe isw or 1000 cm3 = 1 litre soi or 1 m3 = 1000 litres soi or for 24 × 60 seen oe

More questions on Circles, arcs and sectors

10 (a) Expand and simplify. ( x + 1)( x - 2)( x + 3) ................................................. [3] (b) Make g the subject of the formula. 2fg M = g - c g = ................................................ [4] (c) Simplify. 4x 2 - 16 x x 2 - 16 ................................................. [3]

Mark scheme: 10(a) x 3 + 2 x 2 − 5 x − 6 final answer 3 B2 for correct expansion of three brackets unsimplified or for simplified expression of correct form with 3 out of 4 terms correct or B1 for correct expansion of 2 of the 3 given brackets with at least 3 terms out of four correct 10(b) Mc − Mc 4 M1 for clearing g – c from denominator or final answer e.g. M(g – c) = 2fg M − 2 f 2 f − M M1 for correctly isolating terms in g in numerator on one side M1 for correctly factorising or simplifying, to single term in g in an equation M1 for correctly dividing by bracket to final answer 10(c) 4 x 3 B1 for 4x(x – 4) final answer B1 for (x + 4) (x – 4) x + 4

More questions on Equations

11 (a) The probability that Shalini is late for school on any day is 1. 6 (i) Complete the tree diagram for Monday and Tuesday. Monday Tuesday Late ........ Late ........ ........ Not late Late ........ ........ Not late ........ Not late [2] (ii) Calculate the probability that Shalini is late on Monday but is not late on Tuesday. ................................................. [2] (b) The Venn diagram shows the number of students in a group of 50 students who wear glasses (G), who wear trainers (T ) and who have a mobile phone (M ). G T 0 2 3 2 14 19 1 9 M (i) Use set notation to describe the region that contains only one student. .................................................. [1] (ii) Find n T l + G , M b ` jl. ................................................. [1] (iii) One student is picked at random from the 50 students. Find the probability that this student wears trainers but does not wear glasses. ................................................. [1] (iv) Two students are picked at random from those wearing trainers. Find the probability that both students have mobile phones. ................................................. [3]

Mark scheme: 11(a)(i) 1 2 1 5 oe on all late branches B1 for one correct vertical pair oe and oe 6 6 6 5 oe on all not late branches 6 11(a)(ii) 5 2 FT their tree oe 1 5 36 M1 for their × their 6 6 11(b)(i) 1 ( G ∪ T ∪ M )′ oe 11(b)(ii) 28 1 11(b)(iii) 17 1 oe 50 11(b)(iv) 4 3 16 15 oe M2 for × 7 21 20 n n − 1 16 15 or M1 for × or for and seen 21 20 21 20 256 If 0 scored SC1 for answer oe 441

More questions on Probability of combined events

12 (a) Solve the equation tan x = 11.43 for 0° G x G 360 ° . x = ................... or x = .................. [2] (b) Sketch the curve y = x 3 - 4x . y x O [3] (c) A curve has equation y = x 3 + ax + b . The stationary points of the curve have coordinates (2, k) and (-2, 10 - k). Work out the value of a, the value of b and the value of k. a = ................................ , b = ................................ , k = ................................ [6]

Mark scheme: 12(a) 85[.0], 265[.0] and no others 2 B1 for each If 0 scored SC1 for two values in the range with a difference of 180 but not multiples of 90 12(b) correct shape and passes through 3 B1 for any positive cubic shape origin B1 for sketch with one max and one min and with 3 roots including zero If 0 scored, SC1 for x(x + 2)(x – 2) soi 12(c) a = –12 6 B5 for 2 correct b = 5 OR k = –11 B2 for 3x2 + a or B1 for 3x2 isw d y M1dep on at least B1 for their = 0 d x M1dep on at least B1M1 for x = 2 or x = – 2 d y substituted in their = 0 equation d x M1 for k = 23 + 2 × their a + b and 3 10 − k = ( −2 ) + ( −2 ) × their a + b

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