Cambridge IGCSE Mathematics (US) 0444 — 2020 May/June Paper 4 · Variant 1

1 y 6 5 4 3 2 T 1 0 x - 6 - 5 - 4 - 3 - 2 - 1 1 2 3 4 5 6 - 1 - 2 - 3 - 4 A - 5 - 6 (a) Draw the image of triangle T after a reflection in the line y =- 1. [2] (b) Draw the image of triangle T after a rotation through 90° clockwise about (0, 0). [2] (c) Describe fully the single transformation that maps triangle T onto triangle A. ..................................................................................................................................................... ..................................................................................................................................................... [2]

Mark scheme: Question Answer Marks Partial Marks 1(a) Triangle at (–4, –4) (–1, –3) 2 B1 for correct points not joined or for reflection in (–4, –3) any y = k or for reflection in x = –1 1(b) Triangle at (1, 1) (1, 4) (2, 4) 2 B1 for correct points not joined or rotation 90° clockwise around any point or rotation 90° anticlockwise around (0, 0) 1(c)  5  2 B1 for translation or correct vector oe Translation    − 6 

2 (a) In 2018, Gretal earned $32 000. (i) She paid tax of 24% on these earnings. Work out the amount she paid in tax in 2018. $ ................................................ [2] (ii) In 2019, Gretal’s earnings increased by 7%. Work out her earnings in 2019. $ ................................................ [2] (b) Gretal invests $5000 at a rate of 2% per year compound interest. Calculate the value of her investment at the end of 3 years. $ ................................................ [2] (c) One month, Gretal spent a total of $360 on presents. She spent 15 of this total on presents for her parents. She spent 23 of the remaining money on presents for her friends. She spent the rest of the money on presents for her sisters. Calculate the percentage of the $360 that she spent on presents for her sisters. ............................................. % [4] (d) Arjun earned $36 515 in 2019. This was an increase of 9% on his earnings in 2018. Work out his earnings in 2018. $ ................................................ [2] (e) Arjun and Gretal each pay rent. In 2018, the ratio of the amount each paid in rent was Arjun : Gretal = 5 : 7. In 2019, the ratio of the amount each paid in rent was Arjun : Gretal = 9 : 13. Arjun paid the same amount of rent in both 2018 and 2019. Gretal paid $290 more rent in 2019 than she did in 2018. Work out the amount Arjun paid in rent in 2019. $ ................................................ [4]

Mark scheme: 2(a)(i) 7680 2 M1 for 0.24 × 32 000 oe 2(a)(ii) 34 240 2 100 + 7 M1 for 32 000 × oe 100 2(b) 5306.04 2 3  100 + 2  M1 for 5000 ×   oe  100  2(c) 26.7 or 26.66... to 26.67 4 96 B3 for 96 or oe 360 OR 1 2 M3 for (1 − ) × (1 − ) × 100 oe 5 3 1 2 or M2 for (1 − ) and (1 − ) oe 5 3 OR M1 for 360 ÷ 5 [ × 4] oe M1 for their 288 ÷ 3 [×2] 2(d) 33 500 2 100 + 9 M1 for 36 515 ÷ oe 100 2(e) 6525 4  65 63  M3 for  −  [ A ] = 290 oe  45 45   13 7  or M2 for  −  [ A ] = 290 oe  9 5  or M1 for correct attempt to convert to a common ratio value for Rohsin 13 7 or for − oe 9 5

3 The heights, h meters, of the 120 boys in an athletics club are recorded. The table shows information about the heights of the boys. Height (h meters) 1.3 1 h G 1.4 1.4 1 h G 1.5 1.5 1 h G 1.6 1.6 1 h G 1 .7 1.7 1 h G 1.8 1.8 1 h G 1.9 Frequency 7 18 30 24 27 14 (a) (i) Write down the modal class. .................. 1 h G .................. [1] (ii) Calculate an estimate of the mean height. ............................................. m [4] (b) (i) One boy is chosen at random from the club. Find the probability that this boy has a height greater than 1.8 m. ................................................. [1] (ii) Three boys are chosen at random from the club. Calculate the probability that one of the boys has a height greater than 1.8 m and the other two boys each have a height of 1.4 m or less. ................................................. [4] (c) (i) Use the frequency table on page 6 to complete the cumulative frequency table. Height h G 1.4 h G 1.5 h G 1.6 h G 1.7 h G 1.8 h G 1.9(h meters) Cumulative 7 25frequency [2] (ii) On the grid, draw a cumulative frequency diagram to show this information. 120 110 100 90 80 Cumulative 70 frequency 60 50 40 30 20 10 0 1.3 1.4 1.5 1.6 1.7 1.8 1.9 h Height (m) [3] (d) Use your diagram to find an estimate for (i) the median height, ............................................. m [1] (ii) the 40th percentile. ............................................. m [2]

Mark scheme: 3(a)(i) 1.5 < h ⩽ 1.6 1 3(a)(ii) 1.62 or 1.623… nfww 4 M1 for 1.35, 1.45, 1.55, 1.65, 1.75 1.85 soi M1 for Σfx M1 dep for their Σfx ÷ 120 3(b)(i) 14 1 oe 120 3(b)(ii) 21 4  14 7 6  oe M3 for 3  × ×  20060  120 119 118  14 7 6 or M2 for × × isw 120 119 118 14 7 6 or M1 for , , 120 119 118 343 343 If 0 scored, SC1 for answer or 864000 288000 oe 3(c)(i) 55, 79, 106, 120 2 B1 for 2 or 3 correct 3(c)(ii) Correct diagram 3 B1 for correct horizontal plots B1FT for correct vertical 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 3(d)(i) 1.62 to 1.63 1 3(d)(ii) 1.57 to 1.58 2 B1 for 48 soi

4 (a) s = ut + 12 at 2 Find the value of s when u = 5.2 , t = 7 and a = 1.6 . s = ................................................ [2] (b) Simplify. (i) 3a - 5b - a + 2b ................................................. [2] 5 9x (ii) # 3x 20 ................................................. [2] (c) Solve. 15 (i) =- 3 x x = ................................................ [1] (ii) 4 ( 5 - 3)x = 23 x = ................................................ [3] (d) Simplify. 2 3 ( 27x 9 ) ................................................. [2] (e) Expand and simplify. (3x - 5y)(2x + y) ................................................. [2]

Mark scheme: 4(a) 75.6 2 1 M1 for 5.2 × 7 + × 1.6 × 72 2 4(b)(i) 2a – 3b final answer 2 B1 for answer 2a + kb or ka – 3b or for 2a – 3b seen in working 4(b)(ii) 3 2 45 x B1 for oe single fraction 4 60 x 4(c)(i) −5 1 4(c)(ii) 1 3 23 −0.25 or – M1 for 20 – 12x = 23 or for 5 – 3x = 4 4 M1 for correct completion to ax = b FT their first step 4(d) 9x6 2 B1 for 9xk or kx6 4(e) 6x2 – 7xy – 5y2 2 M1 for 3 terms out of 4 from 6x2 – 10xy + 3xy – 5y2

5 North B 80 m NOT TO A SCALE 72° 115 m C The diagram shows the positions of three points A, B, and C in a field. (a) Show that BC is 118.1 m, correct to 1 decimal place. [3] (b) Calculate angle ABC. Angle ABC = ................................................ [3] (c) The bearing of C from A is 147°. Find the bearing of (i) A from B, ................................................. [3] (ii) B from C. ................................................. [2] (d) Mitchell takes 35 seconds to run from A to C. Calculate his average running speed in kilometers per hour. ........................................ km/h [3] (e) Calculate the shortest distance from point B to AC. ............................................. m [3]

Mark scheme: 5(a) [BC2 = ] 802 + 1152 – 2 × 80 × M1 115cos 72oe 118.06… A2 A1 for 13939. …. 5(b) 67.8 or 67.9 or 67.83 to 67.88 3 115 × sin72 M2 for [sin B =] oe 118.1 115 118.1 or M1 for = oe sin B sin72 5(c)(i) 255 3 B1 for bearing of B from A is 75 soi M1 for 180 + 75 oe 5(c)(ii) [00]7.2 2 M1 for their (c)(i) – their (b) – 180 5(d) 11.8 or 11.82 to 11.83 3 M1 for 115 ÷ 35 oe M1 for their speed in m/s × 60 × 60 ÷ 1000 5(e) 76.1 or 76.08 to 76.09 3 distance M2 for = sin72 oe 80 or M1 distance required is perpendicular to AC soi

6 (a) (i) On the axes, sketch the graph of y = sin x for 0° G x G 360° . y 1 0 90° 180° 270° 360° x – 1 [2] (ii) Describe fully the symmetry of the graph of y = sin x for 0° G x G 360° . ............................................................................................................................................. ............................................................................................................................................. [2] 1 (iii) On the same diagram, sketch the line y = . [1] 2 1 (iv) Find the two exact values of x when sinx = for 0° G x G 360° . 2 x = ........................ or x = ........................ [2] (b) (i) Write x 2 + 10x + 14 in the form ( x + a) 2 + b . ................................................. [2] (ii) Write down the coordinates of the minimum point on the graph of y = x 2 + 10x + 14 . ( ........................ , ........................ ) [2] (iii) On the axes, sketch the graph of y = x 2 + 10x + 14 . y O x [1]

Mark scheme: 6(a)(i) Correct sketch labelled at 1 2 M1 for correct shape but inaccurate and –1 6(a)(ii) Rotational [symmetry] order 2 2 M1 for rotational [symmetry] [centre] (180, 0) 6(a)(iii) Correct line sketched 1 6(a)(iv) 30, 150 2 B1 for each 6(b)(i) (x + 5)2 – 11 2 M1 for (x + 5)2 + k or (x + their 5)2 + 14 – (their 5)2 or a = 5 6(b)(ii) (–5, –11) 2 B1 for each 6(b)(iii) Sketch of U-shaped parabola 1 with a minimum in third quadrant

7 NOT TO SCALE A 8 cm 165° B O The diagram shows a sector of a circle with center O, radius 8 cm and sector angle 165°. (a) Calculate the total perimeter of the sector. ............................................ cm [3] (b) The surface area of a sphere is the same as the area of the sector. Calculate the radius of the sphere. ............................................ cm [4] (c) O NOT TO h SCALE r A B A cone is made from the sector by joining OA to OB. (i) Calculate the radius, r, of the cone. r = ........................................... cm [2] (ii) Calculate the volume of the cone. ......................................... cm3 [4]

Mark scheme: 7(a) 39[.0] or 39.03 to 39.04.. 3 165 M2 for × 2 × π × 8 + 16 360 165 or M1 for × 2 × π × 8 360 7(b) 2.71 or 2.708.. 4 165 2 [× π ] × 8 360 M3 for oe 4 [× π ] 165 2 [× π ] × 8 360 or M2 for r2 = oe 4 [× π ] 165 2 or M1 for × π × 8 oe seen 360 7(c)(i) 3.67 or 3.666 to 3.667 2 165 M1 for ×2[×π]×8 = 2[×π]× r or better 360 or for 165[×π]×82 =[π×]r ×8 or better 360 7(c)(ii) 100 or 100.0 to 100.1… final 4 1 2 2 M3 for π × (their (c)(i))2 × 8 – their r a dius answer 3 or M2 for 8 2 – their r a d ius 2 or M1 for (their (c)(i))2 + h2 = 82

8 A rhombus ABCD has a diagonal AC where A is the point (-3, 10) and C is the point (4, -4). (a) Calculate the length AC. ................................................. [3] (b) Show that the equation of the line AC is y =- 2x + 4 . [2] (c) Find the equation of the line BD. ................................................. [4]

Mark scheme: 8(a) 15.7 or 15.65... 3 2 2 M2 for ( 4 – 10) + (4 – –3) oe or M1 for (–4 –10)2 + (4 – – 3)2 oe 8(b) –10 – 4 M1 [= –2] oe 4 – –3 10 = –2(–3) + c A1 or –4 = –2(4) + c and correct completion to y = –2x + 4 8(c) 1 11 4 1 y = x + oe M1 for grad = soi 2 4 2 1 M1 for [midpoint =] ( , 3) 2 1 M1 for substitution of ( , 3) into their y = mx + c 2 oe

9 (a) The cost of one apple is a cents and the cost of one pear is p cents. 5 apples and 1 pear cost a total of $2.21 . 3 apples and 2 pears cost a total of $1.97 . Find the value of a and the value of p. a = ................................................. p = ................................................ [5] (b) NOT TO SCALE x + 1 3x + 1 6x - 1 2x The area of the rectangle is double the area of the triangle. Find the value of x. x = ................................................ [4]

Mark scheme: 9(a) a = 35, p = 46 5 B1 for 5a + p = 221 (or 2.21) B1 for 3a + 2p = 197 (or 1.97) M1 for correctly eliminating one variable A1 for one correct answer If zero scored, SC1 for correct substitution and evaluation to find other variable 9(b) 1 4  1  oe M2 for (6 x − 1)( x + 1) = 2 × (2 x )(3 x + 1)   3  2  1 or M1 for (6 x − 1)( x + 1) or (2 x )(3 x + 1) 2 M1 for correct linear equation from correctly expanding two quadratic expressions

10 y 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 x 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 (a) On the grid, draw the lines y = 4 , y = 10 , y = x and 3x + 2y = 30 . [5] (b) Label the region R where x H 0 , y H 4 , y G 10 , y H x and 3x + 2y G 30 . [1] (c) For points in the region R, find (i) the smallest value of x + y , ................................................. [1] (ii) the largest value of x + y when x and y are integers. ................................................. [1] (d) Label the region S where y H 10 , y H x and 3x + 2y H 30 . [1]

Mark scheme: 10(a) y = 4 ruled correctly 1 y = 10 ruled correctly 1 y = x ruled correctly 1 3x + 2y = 30 ruled correctly 2 B1 for correct freehand or ruled line through (10, 0) or (0, 15) when extended 10(b) Correct region labelled cao 1 10(c)(i) 4 1 10(c)(ii) 13 1 10(d) Correct region labelled 1

11 (a) y x O On the diagram, sketch the graph of y = 2x . [1] (b) y x O On the diagram, sketch the graph of y = 0.9x . [1] x 1 (c) Find the exact value of x when 2 = . 4 2 x = ................................................ [2] (d) (i) f ( x) = 3 ( 1.04 ) x f ( x) is an exponential function representing a rate of increase of r%. Find the value of r. r = ................................................. [1] (ii) g ( x) is an exponential function representing a rate of decrease of 2%. g ( 0) = 7 Find g ( x) , giving your answer in its simplest form. g ( x) = ................................................ [2]

Mark scheme: 11(a) Correct sketch3333 1 2222 1111 3333 -2-2-2-2 -1-1-1-1 0000 0000 1111 2222 3333 -1-1-1-1 -2-2-2-2 -3-3-3-3 11(b) Correct sketch3333 1 2222 1111 3333 -2-2-2-2 -1-1-1-1 0000 0000 1111 2222 3333 -1-1-1-1 -2-2-2-2 -3-3-3-3 11(c) –2.5 oe 2 1 M1 for 2. 5 2 11(d)(i) 4 1 11(d)(ii) 7(0.98)x 2 M1 for (1 − 0.02) x or better seen