Mathematics 0580/42 May/June 2022: question paper and mark scheme
Cambridge IGCSE Mathematics (0580), Paper 4 (Extended), May/June 2022. Read both free below, or download them. Done the paper? Get it marked by AI against this mark scheme.
(a) Find the lowest common multiple (LCM) of 30 and 75.
…… [2]
(b) Share $608 in the ratio 4 : 5 : 7.
$ ……
$ ……
$ …… [3]
6.39 # 10 4
(c) Work out .
2.45 # 10 6
Give your answer in standard form.
…… [2]
o o as a fraction.
(d) Write 0.27
…… [1]
(e) A stone has volume 45 cm 3 and mass 126 g.
Find the density of the stone, giving the units of your answer.
[Density = mass ' volume]
…… …… [2]
Mark scheme
1(a) 150 2 B1 for answer 150k
or M1 for prime factors of 30 or 75 seen
or a list of multiples of both 30 and 75 with at
least 3 of each
30 × 75
or for oe
15
or for answer 2 × 3 × 52
1(b) 152 3 Accept in any order
B2 for two correct answers
190 608
or M1 for × k oe where k =1, 4, 5, 7
4+5+7
266
1(c) 2.61 × 10–2 2.61 × 10 −2 or 2 B1 for figs 2608 or 261 seen
2.608… × 10–2
If 0 scored, SC1 for answer 2.6[0] × 10–2 without
more accurate value in standard form seen
1(d) 27 1
oe fraction
99
1(e) 2.8 1
g/cm3 or g cm–3 1
(a)
R
Q NOT TO
S SCALE
29°
P
The points P, Q, R and S lie on a circle with diameter PR.
Work out the size of angle PSQ, giving a geometrical reason for each step of your working.
…… [3]
(b)
A
NOT TO
SCALE
B
98°
C T S
The points A, B and T lie on a circle and CTS is a tangent to the circle at T.
ABC is a straight line and AB = BT.
Angle ATS = 98°.
Work out the size of angle ACT.
Angle ACT = …… [4]
Mark scheme
2(a) PQR = 90 angle in semi-circle B1
PRQ = 61 angle sum of triangle B1
[= 180]
PSQ = 61 angle in same segment B1 If 0 scored SC1 for PSQ = PRQ [= 61] soi
2(b) 57 4 B1 for ABT = 98
B1 for TAB or ATB = 41
B1 for BTC = 41 or TBC = 82 or ATC =82 soi
A line, l, joins point F (3, 2) and point G (- 5, 4).
(a) Calculate the length of line l.
…… [3]
(b) Find the equation of the perpendicular bisector of line l in the form y = mx + c .
y = …… [5]
(c) A point H lies on the y-axis such that the distance GH = 13 units.
Find the coordinates of the two possible positions of H.
( …… , …… ) and ( …… , …… ) [4]
Mark scheme
3(a) 8.25 or 8.246… 3 M2 for ( 3 − −5)2 + ( 2 − 4 )2 oe or better
or M1 for ( 3 − −5 ) and ( 2 − 4 ) oe seen
3(b) [ y =] 4 x + 7 5 B1 for [midpoint] (− 1, 3) soi
4−2
M1 for [gradient of l =] oe
−5 − 3
1
M1 for gradient −1 / their −
4
M1dep on at least M1 for their (− 1, 3)
substituted into y = their m × x + c oe
3(c) (0, − 8) and (0, 16) 4 B3 for (0, −8) or (0, 16) or for –8 and 16
OR
B2 for distance = [±]12 soi
or M1 for 132 – (5[–0])2 oe
B1 for both answers (0, k), k ≠ 0 or 4
ALT METHOD
B3 for (0, −8) or (0 , 16) or for – 8 and 16
OR
M2 for y2 – 8y – 128 [= 0]
or for (y – 4)2 = 144 or better
or M1 for 132 = (–5 – 0)2 + (4 – y)2 oe
B1 for both answers (0, k), k ≠ 0 or 4
6.4 cm
D C
38° NOT TO
SCALE
10.9 cm
45°
A B
ABCD is a trapezium with DC parallel to AB.
DC = 6.4 cm, DB = 10.9 cm, angle CDB = 38° and angle DAB = 45°.
(a) Find CB.
CB = …… cm [3]
(b) (i) Find angle ADB.
Angle ADB = …… [1]
(ii) Find AB.
AB = …… cm [3]
(c) Calculate the area of the trapezium.
…… cm 2 [3]
Mark scheme
4(a) 7.06 or 7.058… or 7.059 3 M2 for 6.42 + 10.92 − 2 × 6.4 × 10.9 × cos38 oe
OR
M1 for 6.42 + 10.92 – 2 × 6.4 × 10.9 × cos 38 oe
A1= 49.8...
4(b)(i) 97 1
4(b)(ii) 15.3[0…] 3 10.9 × sin their 97
M2 for [AB =]
sin 45
sin their 97 sin 45
or M1 for = oe
AB 10.9
4(c) 72.8 to 72.81… 3 M2 for
1 1
( 6.4 ) × 10.9 × sin 38 + their 15.3 ×10.9 × sin 38
2 2
oe
or M1 for 12 × 6.4 × 10.9 × sin 38 oe
or 1
2
× their15.3 × 10.9 × sin 38 oe
or M1 for height =10.9 × sin38 oe
(a) Draw the lines of symmetry of the rectangle.
[2]
(b) y
8
7
6
5
4
3
B
2
1
–6 –5 –4 –3 –2 –1 0 x
1 2 3 4 5 6 7 8
–1
A
–2
–3
C
–4
–5
–6
–7
(i) Describe fully the single transformation that maps
(a) triangle A onto triangle B,
…… [2]
(b) triangle A onto triangle C.
…… [3]
(ii) (a) Draw the image of triangle A after reflection in y = 2 . [2]
(b) Draw the image of triangle A after enlargement by scale factor - 2, centre (- 1, 1). [2]
Mark scheme
5(a) Correct lines drawn 2 B1 for one correct with no incorrect lines
5(b)(i)(a) Translation or translate 2 B1 for each
−1
oe
4
5(b)(i)(b) Rotation or rotate 3 B1 for each
90 [anticlockwise] oe
[centre] (2, 1)
5(b)(ii)(a) Triangle at (– 5, 6) (– 2, 6) (– 2, 5) 2 B1 for reflection in y = k
5(b)(ii)(b) Triangle at (1, 5) (1, 7) (7, 7) 2 B1 for correct size and orientation, wrong
position
(a) At a festival, 380 people out of 500 people questioned say that they are camping.
There are 55 300 people at the festival.
Calculate an estimate of the total number of people camping at the festival.
…… [2]
(b) 12 friends travel to the festival.
5 travel by car, 4 travel by bus and 3 travel by train.
Two people are chosen at random from the 12 friends.
Calculate the probability that they travel by different types of transport.
…… [4]
(c) Arno buys a student ticket for $43.68 .
This is a saving of 16% on the full price of a ticket.
Calculate the full price of a ticket.
$ …… [2]
(d) At a football match, there are 29 800 people, correct to the nearest 100.
(i) At the end of the football match, the people leave at a rate of 400 people per minute, correct to the nearest 50 people.
Calculate the lower bound for the number of minutes it takes for all the people to leave.
…… min [3]
(ii) At a cricket match there are 27 500 people, correct to the nearest 100.
Calculate the upper bound for the difference between the number of people at the football match and at the cricket match.
…… [2]
Mark scheme
6(a) 42 028 2 380
M1 for oe soi isw
500
6(b) 47 4 0.712[1…]
oe
66 5 4 4 3 5 3
M3 for 2 × + 2 × + 2 ×
12 11 12 11 12 11
oe
5 4 4 3 3 2
or 1 – × + × + × oe
12 11 12 11 12 11
or M2 for sum of 3 or more correct product pairs
and no incorrect pairs
5 4 4 3 3 2
or for × + × + × and no other
12 11 12 11 12 11
pairs
k j
or M1 for × seen
12 11
94
If 0 scored SC1 for answer oe
144
6(c) 52 2 100 − 16
M1 for x × = 43.68 oe or better
100
6(d)(i) 70 or 70.16[5…] or 70.17 or 70.2 3 29750 to 29800 29750 to 29800
M2 for or or
400 + 25 400 + 24
29800 − 50
400 to 425
or B1 for 29 750 or 29 850 or 29 849 or 375 or
425 or 424 seen
6(d)(ii) 2399 2 B1 for 27 450 or 27 550 or 27 549 or 29 850 or
or 2400 nfww 29 849 seen
Information about the mass, m kg, of each of 150 children is recorded in the frequency table.
Mass (m kg) 0 1 m G 10 10 1 m G 20 20 1 m G 25 25 1 m G 40 40 1 m G 50
Frequency 12 38 32 50 18
(a) Calculate an estimate of the mean mass.
…… kg [4]
(b) Draw a histogram to show the information in the table.
8
7
6
5
Frequency density 4
3
2
1
0 m
0 10 20 30 40 50
Mass (kg)
[4]
(c) (i) Use the frequency table to complete this cumulative frequency table.
Mass (m kg) m G 10 m G 20 m G 25 m G 40 m G 50
Cumulative frequency
[2]
(ii) Calculate the percentage of children with a mass greater than 10 kg.
…… % [2]
Mark scheme
7(a) 25.2 or 25.23… 4 M1 for midpoints soi
M1 for use of ∑fx with x in correct interval
including both boundaries
M1 (dep on 2nd M1) for ∑fx ÷ 150
7(b) 5 correct blocks 4 B3 for 4 correct blocks
or B2 for 3 correct blocks
or B1 for 2 correct blocks
or block widths 10, 10, 5, 15, 10
If 0 scored SC1 for 4 correct frequency densities
from 1.2, 3.8, 6.4, 3.33[3…] and 1.8 oe soi
7(c)(i) 12, 50, 82, 132, 150 2 B1 for 3 or 4 correct
7(c)(ii) 92 2 M1 for 150 −12 oe seen
If 0 scored, SC1 for answer 8[%]
(a) Solve.
10 - 3p = 3 + 11p
p = …… [2]
(b) Make m the subject of the formula. mc 2 - 2k = mg
m = …… [3]
(c) Solve.
1 4
+ =1 x - 3 2x + 3
x = …… or x = …… [5]
(d) Solve the simultaneous equations.
You must show all your working. x + 2y = 12
5x + y 2 = 39
(e) Expand and simplify.
(2x - 3) (x + 6) (x - 4)
…… [3]
Mark scheme
8(a) 1 2 M1 for 10 − 3 = 11 p + 3 p oe or better
or 0.5 oe
2
8(b) 2k 3 M1 for correctly isolating m terms
[ m =] 2 oe final answer M1 for correctly factorising
c −g
M1 for dividing by a bracket with two terms to
the final answer
Maximum mark M2 if final answer incorrect
8(c) 0 4.5 oe 5 B4 for 2 x 2 − 9 x [ = 0] or 9x – 2x2 [= 0] or better
OR
M2 for ( 2 x + 3) + 4 ( x − 3) = ( x − 3)( 2 x + 3) or
better
or M1 for ( 2 x + 3) + 4 ( x − 3) seen oe
or common denominator ( x − 3)( 2 x + 3) oe
B1 for 2 x 2 − 6 x + 3 x − 9 or better seen
8(d) y 2 − 10 y + 21[= 0] or M2 M1 for y 2 + 5 (12 − 2 y ) = 39 oe
x 2 − 4 x − 12[= 0]
or 5 x +
(12 − x )2 = 39 seen oe
22
(y – 3)(y – 7) [= 0] M1 or for correct factors for their 3– term quadratic
or (x + 2)(x – 6) [= 0] equation
or for correct substitution into quadratic formula
or correctly completing the square for their 3–
term quadratic equation
x=−2 y=7 B2 B1 for x = − 2, x = 6
or for y = 7, y = 3
x=6 y=3 or for one correct pair of x and y values
8(e) 2 x 3 + x 2 − 54 x + 72 final answer 3 B2 correct expansion of three brackets
unsimplified
or for final answer of correct form with 3 out of
4 terms correct
or B1 correct expansion of two brackets with at
least three terms out of four correct
(a)
P
NOT TO
S SCALE
Q M R
In triangle PQR, M lies on QR and S lies on PR.
Explain, giving reasons, why triangle PMR is similar to triangle MSR.
…… [3]
(b)
C R
8.1 cm NOT TO
4.5 cm SCALE x cm 9.9 cm
A B
P Q
Triangle ABC is similar to triangle PQR.
(i) Find the value of x.
x = …… [2]
(ii) The area of triangle PQR is 25 cm 2 .
Calculate the area of triangle ABC.
…… cm 2 [2]
Mark scheme
9(a) PMR = MSR = right angle[s] or 90° B1
PRM = MRS same angle B1
AAA oe B1 Dep on B1B1 and no errors seen
OR
MPR = SMR 3rd angle of triangle
9(b)(i) 5.5 2 x 9.9
M1 for = oe
4.5 8.1
9(b)(ii) 16.7 or 16.73 to 16.74 2 8.1
2
M1 for 25 × oe
9.9
2
4.5
or 25 × oe
their 5.5
(a) Find all the positive integers which satisfy the inequality.
3n - 8 2 5n - 15
…… [2]
(b) y x=4
9
8
7
6
5
4 10y + 8x = 80
2y = x - 4
3
R
2
1
0 x
0 1 2 3 4 5 6 7 8 9 10 11
–1
–2
–3
The region marked R is defined by three inequalities.
(i) Find these three inequalities.
…… [3]
(ii) Write down the largest value of 3x + y in the region R for integers x and y.
…… [2]
Mark scheme
10(a) 1, 2, 3 2 M1 for 15 − 8 > 5n − 3n oe
If 0 scored, B1 for 2 correct answers and no
others
or 3 correct answers with one extra value
10(b)(i) 10y + 8x ⩽ 80 oe final answer 3 B1 for each
x > 4 oe final answer
2y > x – 4 oe final answer If 0 scored, SC1 for
10y + 8x < 80 oe final answer
and
x ⩾ 4 oe final answer
and
2y ⩾ x – 4 oe final answer
10(b)(ii) 23 final answer 2 M1 for 7 and 2 selected soi
(a)
A 28 cm D AD
NOT TO
SCALE
20 cm
N BC
B C
A rectangular sheet of paper ABCD is made into an open cylinder with the edge AB meeting the edge DC.
AD = 28 cm and AB = 20 cm.
(i) Show that the radius of the cylinder is 4.46 cm, correct to 3 significant figures.
[2]
(ii) Calculate the volume of the cylinder.
…… cm 3 [2]
(iii) N is a point on the base of the cylinder, such that BN is a diameter.
Calculate the angle between AN and the base of the cylinder.
…… [3]
(b) The volume of a solid cone is 310 cm 3 .
The height of the cone is twice the radius of its base.
Calculate the slant height of the cone.
1
[The volume, V, of a cone with radius r and height h is V = rr 2 h .]
3
…… cm [5]
Mark scheme
11(a)(i) 4.455 to 4.456… [= 4.46] 2 28
M1 for [r =] oe
2π
11(a)(ii) 1250 or 1247 to 1249.9… 2 M1 for 20 × π × 4.46 2 oe
11(a)(iii) 66[.0] or 65.95 to 66.02 3 20
M2 for [tan] = oe
2 × 4.46
or B1 for identifying angle ANB on cylinder not
on rectangle
11(b) 11.8 or 11.82 to 11.83 5 310 × 3
M2 for [r =] 3 oe
2π
310 × 3 × 4
or [h =] 3 oe
π
or M1 for 310 = 13 π × r 2 × 2r
2
1 h
or 310 = π h
3 2
(their r ) 2 + ( 2 × their r )
2
M2 for oe
or M1 for [l 2 =] ( their r ) + ( 2 × their r )
2 2
oe
A curve has equation y = x 3 - kx 2 + 1.
When x = 2 , the gradient of the curve is 6.
(a) Show that k = 1.5 .
[5]
(b) Find the coordinates of the two stationary points of y = x 3 - 1.5x 2 + 1.
You must show all your working.
( …… , …… ) and ( …… , …… ) [4]
(c) Sketch the curve y = x 3 - 1.5x 2 + 1.
y
O x
[2]
Mark scheme
12(a) 3 x 2 − 2 kx M2 M1 for 3x 2 or −2kx
dy M1 Dep on at least M1 for derivative
their =6
dx
dy M1 Dep on at least M1 for derivative
x = 2 substituted in their
dx
Correct working leading to 1.5 oe A1 A0 if any errors in working leading to 1.5
12(b) ( 0, 1) ( 1, 0.5) 4 B3 for x = 0 and x = 1 or for (1, 0.5)
OR
dy
M1 for their =0
dx
B1 for 3 x 2 − 3 x oe or better
12(c) correct sketch 2 with max on positive y-axis and min in 1st
quadrant
B1 for positive cubic
or for graph with one max which is on pos y-axis
and one min which is in 1st quadrant
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On this page, free: the mark scheme for IGCSE Mathematics 0580/42, May/June 2022 (0580_s22_ms_42), and the question paper (0580_s22_qp_42), to read or download, with each question's marking points written out underneath.
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Mathematics 0580/42 for May/June 2022 is out of 130 marks, with 2 hours 30 minutes to answer it.
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