Mathematics 0580/42 May/June 2020: question paper and mark scheme
Cambridge IGCSE Mathematics (0580), Paper 4 (Extended), May/June 2020. Read both free below, or download them. Done the paper? Get it marked by AI against this mark scheme.
(a) (i) Divide $24 in the ratio 7 : 5.
$ …… , $ …… [2]
(ii) Write $24.60 as a fraction of $2870.
Give your answer in its lowest terms.
…… [2]
(iii) Write $1.92 as a percentage of $1.60 .
…… % [1]
(b) In a sale the original prices are reduced by 15%.
(i) Calculate the sale price of a book that has an original price of $12.
$ …… [2]
(ii) Calculate the original price of a jacket that has a sale price of $38.25 .
$ …… [2]
(c) (i) Dean invests $500 for 10 years at a rate of 1.7% per year simple interest.
Calculate the total interest earned during the 10 years.
$ …… [2]
(ii) Ollie invests $200 at a rate of 0.0035% per day compound interest.
Calculate the value of Ollie’s investment at the end of 1 year.
[1 year = 365 days.]
$ …… [2]
(iii) Edna invests $500 at a rate of r % per year compound interest.
At the end of 6 years, the value of Edna’s investment is $559.78 .
Find the value of r.
r = …… [3]
4 -2
Mark scheme
1(a)(i) 14, 10 2 M1 for 24 ÷ (7 + 5)
1(a)(ii) 3 2 B1 for correct fraction not in lowest terms
350
1(a)(iii) 120 1
1(b)(i) 10.2[0] 2 15
M1 for ×12 oe or better
100
1(b)(ii) 45 2 38.25
M1 for oe
15
1−
100
1(c)(i) 85 2 500 ×1.7 ×10
M1 for oe
100
1(c)(ii) 203 or 202.5 to 202.6 2 365
0.0035
M1 for 200 × 1 +
100
1(c)(iii) 1.9 3 559.78
M2 for 6
500
6
r
or M1 for 500 1 + = 559.78
100
(a) p =e o q =e o
5 7
(i) Find 2p + q .
f p [2]
(ii) Find p .
…… [2]
-3
(b) A is the point (4, 1) and AB = e o.
1
Find the coordinates of B.
( …… , …… ) [1]
(c) The line y = 3x - 2 crosses the y-axis at G.
Write down the coordinates of G.
( …… , …… ) [1]
(d)
D
NOT TO
T
SCALE
M
O C
In the diagram, O is the origin, OT = 2TD and M is the midpoint of TC.
OC = c and OD = d .
Find the position vector of M.
Give your answer in terms of c and d in its simplest form.
…… [3]
Mark scheme
2(a)(i) 6 2 B1 for each
17
2(a)(ii) 6.4[0] or 6.403... 2 M1 for 42 + 52
2(b) (1, 2) 1
2(c) (0, –2) 1
2(d) 1 1 3 B2 for correct unsimplified answer
c+ d ρ 2
2 3 or M1 for C T = – c + d oe
3
ρ 2
or TC = c – d oe
3
or for correct route
The speed, v km/h, of each of 200 cars passing a building is measured.
The table shows the results.
Speed (v km/h) 0 1 v G 20 20 1 v G 40 40 1 v G 45 45 1 v G 50 50 1 v G 60 60 1 v G 80
Frequency 16 34 62 58 26 4
(a) Calculate an estimate of the mean.
…… km/h [4]
(b) (i) Use the frequency table to complete the cumulative frequency table.
Speed (v km/h) v G 20 v G 40 v G 45 v G 50 v G 60 v G 80
Cumulative frequency 16 50 196 200
[1]
(ii) On the grid, draw a cumulative frequency diagram.
200
180
160
140
120
Cumulative frequency 100
80
60
40
20
0
0 10 20 30 40 50 60 70 80 v
Speed (km/h)
[3]
(iii) Use your diagram to find an estimate of
(a) the upper quartile,
…… km/h [1]
(b) the number of cars with a speed greater than 35 km/h.
…… [2]
(c) Two of the 200 cars are chosen at random.
Find the probability that they both have a speed greater than 50 km/h.
…… [2]
(d) A new frequency table is made by combining intervals.
Speed (v km/h) 0 1 v G 40 40 1 v G 50 50 1 v G 80
Frequency 50 120 30
On the grid, draw a histogram to show the information in this table.
15
10
Frequency density
5
0
0 10 20 30 40 50 60 70 80 v
Speed (km/h)
[3]
Mark scheme
3(a) 41.4 4 M1 for 10, 30, 42.5, 47.5, 55, 70
M1 for Σ fx where x lies in or on the boundary of
each interval.
Σfx
M1 dep for dep on second M1
200
3(b)(i) 112, 170 1
3(b)(ii) Correct diagram 3 B1 for correct horizontal plot
B1FT for correct vertical plots
B1 FT dep on at least B1 earned for reasonable
increasing curve or polygon through their 6
points
If 0 scored SC1FT for 5 out of 6 points plotted
correctly
3(b)(iii)(a) 48 1
3(b)(iii)(b) 160 2 M1 for 40 seen
3(c) 87 2 30 29
oe M1 for × oe
3980 200 199
3(d) Correct histogram 3 B1 for each column
If 0 scored SC1 for correct frequency densities
soi 1.25, 12, 1
S
NOT TO
SCALE
55°
P 150 m
25° 45°
R
120 m
Q
The diagram shows two triangles.
(a) Calculate QR.
QR = …… m [3]
(b) Calculate RS.
RS = …… m [4]
(c) Calculate the total area of the two triangles.
…… m2 [3]
Mark scheme
4(a) 65.4 or 65.36 to 65.37 3 M1 for 1502 + 1202 – 2 × 150 × 120 cos 25
A1 for 4270 or 4272 to 4273
4(b) 125 or 124.7 to 124.8 4 B1 for [angle S =] 80
150sin55
M2 for
sin their80
sin their80 sin55
or M1 for = oe
150 RS
4(c) 10 400 or 10 410 to 10 440 nfww 3 1
M1 for ×120 ×150sin 25 oe
2
1
M1 for ×150 × their (b) sin 45 oe
2
North
D NOT TO
SCALE
A 140°
450 m
400 m
B 350 m C
The diagram shows a field ABCD.
The bearing of B from A is 140°.
C is due east of B and D is due north of C.
AB = 400 m, BC = 350 m and CD = 450 m.
(a) Find the bearing of D from B.
…… [2]
(b) Calculate the distance from D to A.
…… m [6]
(c) Jono runs around the field from A to B, B to C, C to D and D to A.
He runs at a speed of 3 m/s.
Calculate the total time Jono takes to run around the field.
Give your answer in minutes and seconds, correct to the nearest second.
…… min …… s [4]
Mark scheme
5(a) [0]38 or [0]37.9 or [0]37.87... 2 350
M1 for tan = oe
450
If 0 scored, SC1 for answer [0]52 or [0]52.1
or [0]52.12 to [0]52.13
5(b) 624 or 623.8 to 623.9 6 M2 for 450 – 400 sin 50
...
or M1 for sin 50 =
400
M2 for 350 + 400 cos 50
...
or M1 for cos 50 =
400
M1 for (their (450 – 400 sin 50))2 + (their (350 +
400 cos 50))2
5(c) 10 min 8 s 4 B3 for 10.1 or 10.13…
or
M2 for (400 + 350 + 450 + their DA) ÷ 3 [÷ 60]
oe
or M1 for any distance ÷ 3
M1 for rounding their minutes into minutes and
seconds to nearest second if clearly seen
f (x) = 3x + 2 g (x) = x 2 + 1 h (x) = 4 x
(a) Find h(4).
…… [1]
(b) Find fg(1).
…… [2]
(c) Find gf(x) in the form ax 2 + bx + c .
…… [3]
(d) Find x when f (x) = g (7) .
x = …… [2]
(e) Find f -1 (x) .
f -1 (x) = …… [2]
g (x)
(f) Find + x. f (x)
Give your answer as a single fraction, in terms of x, in its simplest form.
…… [3]
(g) Find x when h -1 (x) = 2 .
x = …… [1]
Mark scheme
6(a) 256 1
6(b) 8 2 M1 for 3(x2 + 1) + 2 or for 3(2) + 2
6(c) 9 x 2 + 12 x + 5 3 M1 for (3x + 2)2 + 1
B1 for [(3x + 2)2 =] 9 x 2 + 6 x + 6 x + 4 oe
6(d) 16 2 M1 for 3x + 2 = 72 + 1 or better
6(e) x−2 2 M1 for x = 3y + 2 or for y – 2 = 3x or for
oe final answer y 2
3 = x+
3 3
6(f) 4 x2 + 2 x + 1 3 B1 for x2 + 1 + x (3x + 2) or better seen
final answer M1 for common denominator 3x + 2
3x + 2
6(g) 16 1
Tanya plants some seeds.
The probability that a seed will produce flowers is 0.8 .
When a seed produces flowers, the probability that the flowers are red is 0.6 and the probability that the flowers are yellow is 0.3 .
(a) Tanya has a seed that produces flowers.
Find the probability that the flowers are not red and not yellow.
…… [1]
(b) (i) Complete the tree diagram.
Produces Colour flowers
Red
Yes Yellow
0.8
…… Other colours
…… No
[2]
(ii) Find the probability that a seed chosen at random produces red flowers.
…… [2]
(iii) Tanya chooses a seed at random.
Find the probability that this seed does not produce red flowers and does not produce yellow flowers.
…… [3]
(c) Two of the seeds are chosen at random.
Find the probability that one produces flowers and one does not produce flowers.
…… [3]
Mark scheme
7(a) 0.1 1
7(b)(i) 0.2 oe 2 B1 for 0.2
0.6, 0.3, 0.1 oe B1 for 0.6, 0.3, 0.1
7(b)(ii) 0.48 oe 2 FT their 0.6 from tree diagram
M1 for 0.8 × their 0.6
7(b)(iii) 0.28 oe 3 M2 for 0.2 + 0.8 × 0.1 oe
or M1 for
0.2 or 0.8 × 0.1 or 0.8 × (0.6 + 0.3)
7(c) 0.32 oe 3 M2 for 0.8 × 0.2 + 0.2 × 0.8 oe
M1 for one of these products
(a)
C R
NOT TO
SCALE
A 8 cm B
P Q
12 cm
Triangle ABC is mathematically similar to triangle PQR.
The area of triangle ABC is 16 cm2.
(i) Calculate the area of triangle PQR.
…… cm2 [2]
(ii) The triangles are the cross-sections of prisms which are also mathematically similar.
The volume of the smaller prism is 320 cm3.
Calculate the length of the larger prism.
…… cm [3]
(b) A cylinder with radius 6 cm and height h cm has the same volume as a sphere with radius 4.5 cm.
Find the value of h.
4
[The volume, V, of a sphere with radius r is V = rr 3 .]
3
h = …… [3]
(c) A solid metal cube of side 20 cm is melted down and made into 40 solid spheres, each of radius r cm.
Find the value of r.
4
[The volume, V, of a sphere with radius r is V = rr 3 .]
3
r = …… [3]
7x
(d) A solid cylinder has radius x cm and height cm.
2
The surface area of a sphere with radius R cm is equal to the total surface area of the cylinder.
Find an expression for R in terms of x.
[The surface area, A, of a sphere with radius r is A = 4rr 2 .]
R = …… [3]
Mark scheme
8(a)(i) 36 2 2 2
8 12
M1 for or oe
12 8
8(a)(ii) 30 3 12
M2 for 320 ÷ 16 × oe
8
or M1 for 3 2 0 ÷ 1 6
8(b) 3.375 cao 3 4
π × 4.53
M2 for 3 or better
π × 62
2 4 3
or M1 for π × 6 × h = × π × 4.5
3
8(c) 3.63 or 3.627 to 3.628 3 203
M2 for
4
40 × π
3
4 3 3
or M1 for 40 × × π × r = 20
3
8(d) 3x 1 3 B2 for 4 R 2 = 9 x 2 oe or better
or 1.5x or 1 x
2 2 2 2 7x
or M1 for 4πR = 2πx + π × 2 x ×
2
(a) (i) Write x 2 + 8x - 9 in the form (x + k) 2 + h .
…… [2]
(ii) Use your answer to part (a)(i) to solve the equation x 2 + 8x - 9 = 0 .
x = …… or x = …… [2]
- 7 + 61 - 7 - 61
(b) The solutions of the equation x 2 + bx + c = 0 are and .
2 2
Find the value of b and the value of c.
b = ……
c = …… [3]
(c) (i) y
O x
On the diagram,
(a) sketch the graph of y = (x - 1) 2 , [2]
1
(b) sketch the graph of y = x + 1. [2]
2
1
(ii) The graphs of y = (x - 1) 2 and y = x + 1 intersect at A and B.
2
Find the length of AB.
AB = …… [7]
Question 10 is printed on the next page.
Mark scheme
9(a)(i) ( x + 4)2 − 25 2 B1 for ( x + k )2 − 9 − (their k )2 or ( x + 4)2 − h
or k = 4
9(a)(ii) x + 4 = [± ] 5 M1 FT their (a)(i)
–9 and 1 A1
9(b) [b =] 7 3 B1 for [b = ] 7
[c =] –3 M1 for b2 – 4c = 61
9(c)(i)(a) Correct sketch 2 B2 for correct quadratic curve with min touching
x-axis
8 or B1 for parabola vertex downwards
6
4
2
0
-2 0 2 4
9(c)(i)(b) Correct sketch 2 B2 for correct straight line intersecting curve on
6
y-axis
5
or B1 for straight line with positive gradient and
4
positive y-intercept
3
2
1
0
4 -3 -2 -1 0 1 2
-1
9(c)(ii) 2.8[0] or 2.795... 7 2 5
B3 for x − x = 0 oe
2
2 1
or M1 for ( x −1) = x + 1
2
B1 for [(x – 1)2 =] x2 – x – x + 1
AND
B2 for (0, 1) and , oe
5 9
2 4
5
or B1 [x =] 0 and oe
2
AND
M1 for (difference in x )2 + (difference in y)2
(a) y = x 4 - 4x 3
(i) Find the value of y when x =- 1.
y = …… [2]
(ii) Find the two stationary points on the graph of y = x 4 - 4x 3 .
( …… , …… )
( …… , …… ) [6]
(b) y = x p + 2x q dy dy
= 11x 10 + 10x 4 , where is the derived function. dx dx
Find the value of p and the value of q.
p = ……
q = …… [2]
Mark scheme
10(a)(i) 5 2 M1 for (–1)4 – 4(–1)3
10(a)(ii) (0, 0) and (3, –27) 6 B2 for 4x3 – 12x2 [ = 0]
or B1 for 4x3 or 12x2
AND
M1 for derivative = 0 or their derivative = 0
M1 for 4x2(x – 3)[= 0]
B1 for
[x =] 0 and [ x =] 3 or [y =] 0 and [y =] –27
or for one correct coordinate pair
10(b) [p =] 11 2 B1 for each
[q =] 5 dy
or M1 for = px p −1 + 2qxq −1
dx
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Mathematics 0580/42 for May/June 2020 is out of 130 marks, with 2 hours 30 minutes to answer it.
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