Cambridge IGCSE Mathematics (US) 0444 — 2021 Oct/Nov Paper 4 · Variant 3

1 The diagram shows three triangles, T, A, and B, drawn on a 1 cm2 grid. y 8 7 B 6 5 4 A 3 T 2 1 0 1 2 3 4 5 6 7 8 x (a) Describe fully the single transformation that maps triangle T onto triangle A. ..................................................................................................................................................... ..................................................................................................................................................... [3] (b) (i) Describe fully the single transformation that maps triangle T onto triangle B. ..................................................................................................................................................... ..................................................................................................................................................... [2] (ii) Calculate the distance that each point of triangle T moves when it is mapped onto triangle B. ............................................ cm [2]

Mark scheme: Question Answer Marks Partial Marks 1(a) Rotation 3 B1 for each 90° clockwise oe [centre] (5, 2) 1(b)(i) Translation 2 B1 for each  −1     4  1(b)(ii) 4.12 or 4.123... 2 M1 for (their (–1))2 + (their 4)2

2 (a) A NOT TO SCALE O 38° B P A, B, and P are points on a circle, center O and angle OBA = 38° . Find angle APB. Angle APB = ................................................. [3] (b) F 80° E NOT TO SCALE D C CDEF is a cyclic quadrilateral and FC = FE. Angle CFE = 80°. Find (i) angle CDE, Angle CDE = ................................................ [1] (ii) angle CDF. Angle CDF = ................................................ [2]

Mark scheme: 2(a) 52° 3 M1 for 180 – 2 × 38, implied by 104 M1 for their AOB ÷ 2 2(b)(i) 100° 1 2(b)(ii) 50° 2 M1 for angle FEC or angle FCE = 50

3 (a) $500 is invested at a rate of 3% per year. Calculate the total interest earned at the end of 7 years when (i) simple interest is paid, $ ................................................ [2] (ii) compound interest is paid. $ ................................................ [3] (b) Each year the value of a car decreases by 10% of its value at the beginning of the year. The value now is $6269.40 . Calculate the value of the car 3 years ago. $ ................................................ [3]

Mark scheme: 3(a)(i) 105 2 3 M1 for × 500 [× 7 ] 100 3(a)(ii) 115 or 114.9... 3 7  3  M2 for 500 ×  1 +  [ −500 ]  100   3  k or M1 for 500 ×  1 +  , k integer ⩾ 2  100  3(b) 8600 3 6269.4 M2 for oe  10  3  1 −   100   10  3 or M1 for C ×  1 −  = 6269.4 oe  100 

4 (a) 5 cm NOT TO SCALE 4 cm C D A B 10 cm The diagram shows a prism. The cross-section of the prism is a trapezoid with CD parallel to AB and AC = BD. AB = 10 cm, CD = 4 cm and the height of the trapezoid is 5 cm. The volume of the prism is 525 cm3. (i) The prism is made of iron. 1 cm3 of iron has a mass of 7.8 g. Calculate the mass of the prism. Give your answer in kilograms. ............................................ kg [2] (ii) Calculate the length of the prism. ............................................ cm [3] (iii) Calculate the total surface area of the prism. ......................................... cm2 [6] (iv) In a mathematically similar prism, the height of the trapezoid is 10 cm. Calculate the volume of this prism. ......................................... cm3 [3] (b) A new town is built with a boundary that is a circle of radius R miles. The population of the town is 50 000. The population density is 3500 persons per square mile. Calculate the value of R. R = ................................................ [3]

Mark scheme: 4(a)(i) 4.095 2 B1 for figs 4095 525 × 7.8 or M1 for 1000 4(a)(ii) 15 3 B2 for 35 OR 1 M2 for (10 + 4) × 5 × L = 525 oe 2 1 M1 for (10 + 4) × 5 oe 2 4(a)(iii) 455 or 454.9... 6 2 2 M3 for their [BD =] 3 + 5 × (their 15) [× 2] or B2 for 34 or 5.83 or 5.830 to 5.831 2  1  2 or M1 for 5 +  (10 − 4 )   2  and M1 for their 35 × 2 M1 for (their 15) × 10 and (their 15) × 4 4(a)(iv) 4200 3 3  10  M2 for 525 –   oe  5   10  3  5  3 or M1 for   or   oe  5   10  4(b) 2.13 or 2.132 to 2.133 3 M2 for 50000 ÷ 3500 ÷ π or M1 for 50000 ÷ 3500

5 (a) Solve the system of linear equations. You must show all your work. 5p - 3q = 18 3p + 2q = 7 p = ................................................... q = .................................................. [4] (b) Solve the equation. x 2x + = 1 4 3 x = ................................................ [2] (c) - 8 1 3x - 2 G 7 (i) Solve the inequality. ................................................. [3] (ii) Find the integer values of x that satisfy the inequality. ................................................. [1] (d) Factorize completely. 16a - 4 a 2 ................................................. [2] (e) Write each of the following as a single fraction, in its simplest form. 1 3 (i) ' 2a 4b ................................................. [2] x (ii) 2 - x - 1 ................................................. [2]

Mark scheme: 5(a) Correctly equate one set of coefficients M1 or for expressing one variable in terms of the other Correctly eliminate one variable M1 or for correct substitution p = 3 A2 A1 for each q = –1 If M0, SC1 for 2 values satisfying one of original equations If 0 scored SC1 for correct answers with no working 5(b) 1 12 2 3 x 8 x 1 or 1.09 or 1.090 to 1.091 M1 for + = 1 or better 11 11 12 12 5(c)(i) –2 < x ⩽ 3 3 B2 for –2 < x or x ⩽ 3 or M1 for –8 + 2 < 3x or 3x ⩽ 7 + 2 5(c)(ii) –1, 0, 1, 2, 3 1 FT dep on –ve and +ve values in their (c)(i) 5(d) 4 a (4 − a ) final answer 2 B1 for any correct partial factorisation 5(e)(i) 2b 2 1 4b final answer M1 for × or better 3a 2 a 3 5(e)(ii) x − 2 2 B1 for 2(x – 1) – x oe seen. final answer nfww x − 1

6 D 100° NOT TO SCALE 50° C 12 cm A 8 cm 11 cm B (a) Calculate AD. AD = ........................................... cm [3] (b) Calculate angle BAC and show that it rounds to 40.42°, correct to 2 decimal places. [4] (c) Calculate the area of the quadrilateral ABCD. .......................................... cm2 [3] (d) Calculate the shortest distance from B to AC. ............................................ cm [3]

Mark scheme: 6(a) 9.33 or 9.334... 3 12sin50 M2 for sin100 sin100 sin50 or M1 for = oe 12 AD 6(b) 112 + 12 2 − 8 2 M2 M1 for [cos =] 2 2 2 2 × 11 × 12 8 = 11 + 12 − 2 × 11 × 12cos( BAC ) 40.415... A2 201 67 A1 for 0.761... or or 264 88 6(c) 70.8 or 70.77 to 70.79... 3 M1 for 1 × 12 × their (a) × sin(180 − 100 − 50) 2 1 M1 for × 12 × 11 × sin(40.42) 2 6(d) 7.13 or 7.131 to 7.132... 3 dist M2 for = sin(40.42) 11 or M1 for recognition that shortest distance is perpendicular to AC

7 (a) Amir buys 3 cakes that cost c cents each and 2 loaves of bread that cost (2c - 11) cents each. He spends a total of $5.87 . Find the value of c. c = ................................................ [3] (b) A bottle of water costs $w. A bottle of juice costs $(w + 1). Alex spends $22 on bottles of water and $42 on bottles of juice. The number of bottles of water is equal to the number of bottles of juice. Find the value of w. w = ................................................ [3]

Mark scheme: 7(a) 87 3 M2 for 3c + 4c = 587 + 22 or better or M1 for 3c + 2(2c – 11) [= 587 or 5.87] 7(b) 1.1[0] 3 M2 for 22w + 22 = 42w or better 22 42 or M1 for = oe w w + 1 OR B2 for number of bottles = 20 or M1 for Nw = 22 and N(w+1) = 42

8 (a) Jean asks 600 people to choose their favorite sport. The pie chart shows some of this information. Football 105° 60° 27° Tennis Golf (i) Show that 100 people choose tennis. [1] (ii) Work out how many people choose golf. ................................................. [2] (iii) 125 people choose baseball and the rest choose swimming. Complete the pie chart to show this information. [2] (b) The heights of some plants are measured: • smallest height = 0.6 cm • range = 8.1 cm • median = 5.2 cm • lower quartile = 3.4 cm • interquartile range = 4.1 cm. On the grid, draw a box-and-whisker plot to show this information. 0 1 2 3 4 5 6 7 8 9 10 Height (cm) [3] (c) A dice is rolled 100 times. The frequency table shows the results. Score 1 2 3 4 5 6 Frequency 16 25 17 19 8 15 Find (i) the range, ................................................. [1] (ii) the mode, ................................................. [1] (iii) the median. ................................................. [1] (d) 50 students answer a mathematics question. The table shows the time, t seconds, taken by each student to answer the question. Time (t seconds) 10 1 t G 20 2 0 1 t G 25 25 1 t G 30 30 1 t G 50 50 1 t G 80 Frequency 2 8 12 16 12 Calculate an estimate of the mean. ............................................... s [4]

Mark scheme: 8(a)(i) 60 1 × 600 oe 360 8(a)(ii) 45 2 27 M1 for × 600 oe 360 8(a)(iii) Correct straight line on the pie chart 2 B1 for 75 8(b) Correct diagram 3 B1 for any three of 0.6, 3.4, 5.2, 7.5, 8.7 correctly placed B1 for 7.5 and 8.7 seen 0.6 3.4 5.2 7.5 8.7 8(c)(i) 5 1 8(c)(ii) 2 1 8(c)(iii) 3 1 8(d) 39.2 4 M1 for mid-values soi M1 for Σfx with x in correct interval including boundaries Σfx M1 dep for dep on second M1 50

9 (a) f ( x) = x ( x - 1)( x - 2) (i) Find the coordinates of the points where the graph of y = f ( x) crosses the x-axis. ( ............... , ............... ) ( ............... , ............... ) ( ............... , ............... ) [2] (ii) y y O x O x A B y y O x O x C D Which of the sketches shows the graph of y = f ( x) ? ................................................. [1] (b) The diagram shows a sketch of the graph of y = g ( x) . y (5, 1) 0 x Find g ( x) . g ( x) = ................................................ [2] (c) y 2 0 x 180 360 540 720 – 2 The graph shows the function h ( x) = a cos ( bx) for 0° G x G 720 ° . (i) Complete the range of h ( x) . .................... G h ( x) G .................... [1] (ii) Find the value of a and the value of b. a = ................................................ b = ................................................ [2] (d) Describe fully the single transformation that maps the graph of y = j ( x) onto the graph of (i) y = j ( x - 5) , ............................................................................................................................................. ............................................................................................................................................. [2] (ii) y = 5 j ( x) . ............................................................................................................................................. ............................................................................................................................................. [3]

Mark scheme: 9(a)(i) (0, 0), (1, 0), (2, 0) 2 B1 for any two correct If 0 scored, SC1 for all three x values clearly identified 9(a)(ii) C 1 9(b) 5 2 k [g(x) =] M1 for y = x x 9(c)(i) –2 ... 2 1 9(c)(ii) [a =] 2 2 B1 for each 1 [b =] 2 9(d)(i) Translation 2 B1 for each 5  0 9(d)(ii) Stretch 3 B1 for each [factor] 5 x-axis oe invariant

10 (a) Sarah spins a fair four-sided spinner numbered 0, 1, 1 and 3. (i) What number is the spinner most likely to land on? ................................................. [1] (ii) Sarah spins the spinner twice. Find the probability that it lands on the number 1 both times. ................................................. [2] (iii) Sarah spins the spinner until it lands on the number 3. 729 The probability that this happens on the nth spin is . 16384 Find the value of n. n = ................................................ [2] (b) Scott takes an examination. The examination is in two parts, a theory test and a practical test. Both parts must be passed to pass the examination. The probability that Scott passes the theory test is 0.9 . The probability that Scott passes the practical test is 0.8 . Find the probability that (i) Scott passes the examination, ................................................. [2] (ii) Scott passes the theory test or the practical test but not both. ................................................. [3]

Mark scheme: 10(a)(i) 1 1 10(a)(ii) 1 2 2 2 oe nfww M1 for × oe 4 4 4 10(a)(iii) 7 2 k M1 for trials with  3 × 1 soi    4  4 10(b)(i) 0.72 oe 2 M1 for 0.9 × 0.8 10(b)(ii) 0.26 oe 3 M2 for 0.9 × 0.2 + 0.1 × 0.8 or 1 – their (b)(i) – 0.1 × 0.2 or M1 for 0.9 × 0.2 or 0.1 × 0.8 or 1 – their (b)(i) or 1 – 0.1 × 0.2

11 f ( x) = 2 x - 1 g ( x) = x 2 + 2x h ( x) = 4x j ( x) = 2x (a) Find the value of (i) h(3), ................................................. [1] (ii) f(h(3)). ................................................. [1] (b) Solve the equation g ( f ( x)) = 0 . x = .................. or x = .................. [4] (c) p -1 ( x) = f ( x) Find p(x). ................................................. [2] 1(d) h ( x) j ( x) = 2 Find the value of x. x = ................................................ [3] Question 12 is printed on the next page.

Mark scheme: 11(a)(i) 64 1 11(a)(ii) 127 1 FT 2 × their (a)(i) – 1 11(b) 1 4 2 M1 for ( 2 x − 1) + 2(2 x − 1) ± oe nfww 2 2 B1 for 4 x − 2 x − 2 x + 1 or ( 2 x − 1)( 2 x −+1 2 ) B1 for 4 x 2 − 1 [= 0] or ( 2 x − 1)( 2 x + 1) [= 0] OR M1 for x(x + 2) = 0 (solving g(x) = 0) A1 for x = 0 or –2 B1 for 2x – 1 = 0 or 2x – 1 = –2 11(c) x + 1 2 M1 for oe final answer y 1 2 y + 1 = 2 x or = x − or x = 2 y − 1 2 2 11(d) 1 3 1 − oe nfww B2 for 3 x = − oe 6 2 OR 1 x M1 for 2 2 x × 2 x oe or 4 2 × 4 x oe or 8x oe 1 1 1 − − − M1 for 2 2 or 4 4 or 8 6 soi

12 Alicia walks a distance of 9 km at a speed of x km/h. She then runs a distance of 5 km at a speed of (2x + 1) km/h. The total time Alicia takes is 2.5 hours. (a) Show that 10x 2 - 41x - 18 = 0 . [4] (b) Work out Alicia’s running speed. You must show all your work. ........................................ km/h [4]

Mark scheme: 12(a) 9 5 M2 9 5 + = 2.5 oe M1 for or x 2 x + 1 x 2 x + 1 9(2 x + 1) + 5 x = 2.5 x (2 x + 1) oe M1 Correctly clearing fractions, or correctly collecting into a single fraction FT their expression dep on two fractions 9(2 x + 1) + 5 x or [= 2.5 oe] both with algebraic denominators x (2 x + 1) All brackets expanded leading to A1 10 x 2 − 41 x − 18 = 0 with no errors or omissions 12(b) (2 x − 9)(5 x + 2) M2 or B1 for ( ax + b )( cx + d ) 2 with ac = 10 and bd = –18 or ad + bc = −−( 41) ± ( − 41) − 4(10)( − 18) or –41 2(10) or ( − 41) 2 − 4(10)( − 18) −−( 41) or correct if formula in correct 2(10) form 10 A2 9 A1 for [x =] oe 2 or M1 for 2 × their positive root + 1