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MATHEMATICS
0580/41
Paper 4 (Extended)
October/November 2024
2 hours 30 minutes
You must answer on the question paper.
You will need:
Geometrical instruments
INSTRUCTIONS
●
Answer all questions.
●
Use a black or dark blue pen. You may use an HB pencil for any diagrams or graphs.
●
Write your name, centre number and candidate number in the boxes at the top of the page.
●
Write your answer to each question in the space provided.
●
Do not use an erasable pen or correction fluid.
●
Do not write on any bar codes.
●
You should use a calculator where appropriate.
●
You may use tracing paper.
●
You must show all necessary working clearly.
●
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in
degrees, unless a different level of accuracy is specified in the question.
●
For r, use either your calculator value or 3.142.
INFORMATION
●
The total mark for this paper is 130.
●
The number of marks for each question or part question is shown in brackets [ ].
This document has 20 pages. Any blank pages are indicated.
DC (DE/FC) 336592/3
© UCLES 2024
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2
(a) (i) Write 70 as a product of its prime factors.
................................................. [2]
................................................. [2]
(iii) Find the lowest common multiple (LCM) of 70x 4 y 2 and 112x 3 y 5 .
................................................. [2]
(b) Simplify.
(i)
a 12 ' a 4
................................................. [1]
(ii)
5 bc
#
2b 20
................................................. [2]
(c) Solve.
4 + 2x = 15
x = ................................................ [2]
© UCLES 2024
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Find the highest common factor (HCF) of 70 and 112.
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(ii)
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1
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3
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(d) Solve.
34 + 2x
= 4-x
5
x = ................................................ [3]
3
P = d + m2
(e)
(i) Find P when d = 7 and m = -8.
P = ................................................ [2]
(ii)
Rearrange the formula to make m the subject.
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m = ................................................ [3]
© UCLES 2024
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0580/41/O/N/24
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4
,
,
2
y
6
5
B
4
3
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2
1
-6 -5 -4 -3 -2 -1 0
-1
1
-2
-3
2
3
4
5
6
x
A
-4
(a) On the grid, draw
(i) the image of triangle A after a reflection in the line x = 1
(ii) the image of triangle A after an enlargement by scale factor
[2]
1
with centre (5, 1).
2
[2]
(b) Describe fully the single transformation that maps triangle A onto triangle B.
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-5
(c) The point (a, b) is reflected in the line y = k where k is an integer and b 1 k .
Write the coordinates of the image of point (a, b) in terms of a, b and k.
(.................. , ..................) [2]
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5
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3
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(a) The table shows the waiting times for 120 patients at a medical centre.
Waiting time
(t minutes)
0 1 t G 10
10 1 t G 20
20 1 t G 40
40 1 t G 50
50 1 t G 80
Frequency
2
46
33
26
13
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Calculate an estimate of the mean waiting time.
.......................................... min [4]
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(b) The histogram shows some information about the waiting times at a different medical centre.
2
1.5
Frequency
density
1
0.5
0
0
10
20
30
40
50
Waiting time (minutes)
60
80 t
70
The total number of patients is 90 and no patient waits for more than 80 minutes.
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Complete the histogram for the patients that have a waiting time between 10 and 30 minutes.
[4]
© UCLES 2024
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0580/41/O/N/24
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4
,
(a) Enzo, Rashid and Blessy each swim as many lengths of a swimming pool as they can in 15 minutes.
The results are shown in the table.
Name
Number of lengths
Enzo
11.25
Rashid
18.75
Blessy
20
(i) Find the number of lengths Enzo swims as a percentage of the total number of lengths all
three people swim.
.............................................. % [2]
(ii) Write the ratio of the number of lengths each person swims in the form
Enzo : Rashid : Blessy.
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© UCLES 2024
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............... : ............... : ............... [2]
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Give your answer in its simplest form.
7
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(iii) Each length of the pool is 25 m.
(a) Work out Blessy’s average swimming speed for the 15 minutes.
Give your answer in metres per second.
........................................... m/s [3]
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(b) Rashid continues to swim at the same rate.
Calculate the time it takes Rashid to swim a total distance of 5 km.
Give your answer in hours and minutes.
..................... h ........................ min [4]
(iv) Blessy swims for one hour.
The number of lengths she swims decreases by 5% every 15 minutes.
Calculate the number of lengths she swims in the final 15 minutes.
................................................. [3]
(b) Another swimmer, Adam, swims 450 m, correct to the nearest 25 metres.
This takes 10 minutes, correct to the nearest minute.
Calculate the minimum distance Adam swims in one hour at this rate.
.............................................. m [3]
© UCLES 2024
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8
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5
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A box contains 3 blue pens and 5 red pens.
(a) Mia picks a pen from the box at random.
Find the probability that she picks a red pen.
................................................. [1]
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(b) Mia puts the pen back into the box.
She then picks a pen at random and replaces it.
She then picks a second pen at random.
(i) Complete the tree diagram.
Second Pen
Blue
..........
..........
..........
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..........
Blue
Red
Blue
Red
..........
Red
[2]
(ii) Find the probability that Mia picks two pens that have the same colour.
................................................. [3]
© UCLES 2024
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0580/41/O/N/24
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First Pen
9
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(c) Mia now picks 3 of the 8 pens in the box at random without replacement.
Find the probability that she picks 2 blue pens and 1 red pen.
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................................................. [3]
© UCLES 2024
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10
The diagram shows a field ABCD.
A straight path AC goes across the field.
830 m
NOT TO
SCALE
C
420 m
A
62°
1150 m
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106°
B
(a) Show that AC = 1028 m, correct to the nearest metre.
[3]
(b) Angle ACB is obtuse.
Calculate angle ACB.
Angle ACB = ................................................ [4]
© UCLES 2024
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D
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11
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(c) Part of the field, triangle ACD, is sold for $41 500.
Calculate the cost of 1 hectare of this part of the field.
Give your answer correct to the nearest dollar.
[1 hectare = 10 000 m 2 ]
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$ ................................................. [4]
© UCLES 2024
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12
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7
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A company makes scientific calculators and graphic calculators.
Each day they make x scientific calculators and y graphic calculators.
These inequalities describe the number of scientific and graphic calculators they make each day.
x 1 180
y G 90
x + y G 240
(a) Complete these two statements.
[2]
(b) Scientific calculators cost $12 to make.
Graphic calculators cost $18 to make.
Each day the company spends at least $2700 making calculators.
Show that 2x + 3y H 450 .
© UCLES 2024
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[1]
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The company can make a maximum of ...................... calculators each day.
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The company makes fewer than ...................... scientific calculators each day.
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13
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(c) The region R satisfies these four inequalities.
x 1 180
y G 90
x + y G 240
2x + 3y H 450
By drawing four suitable lines and shading unwanted regions, find and label the region R.
y
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250
200
150
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100
50
0
50
100
150
200
250 x
[7]
(d) Scientific calculators are sold for a profit of $10.
Graphic calculators are sold for a profit of $30.
Calculate the maximum profit made by the company in one day.
$ ................................................. [2]
© UCLES 2024
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0580/41/O/N/24
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g (x) = x 2 - 16
(a) f (x) = 7 - 3x
(i) Find the values of x when g (x) = 20 .
x = .................... or x = .................... [2]
(ii) Find f -1 (x) .
f -1 (x) = ................................................ [2]
(iii) Find gf (x) + 1, giving your answer in its simplest form.
................................................. [3]
(iv) On the axes, sketch the graph of y = g (x) .
On your sketch, indicate the values where the graph crosses the axes.
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x
[4]
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O
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y
15
,
,
(v) Find the equation of the tangent to the graph of
Give your answer in the form y = mx + c .
y = g (x) when x = -3.
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y = ................................................ [5]
h (x) = 3 x
(b)
(i) On the axes, sketch the graph of y = h (x) .
y
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O
x
[2]
(ii) Write down the equation of the asymptote to the graph of
y = h (x) .
................................................. [1]
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16
,
9
H
G
F
E
17 cm
D
A
10 cm
B
NOT TO
SCALE
C
8 cm
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ABCDEFGH is a solid cuboid.
AB = 10 cm, BC = 8 cm and CG = 17 cm.
(a) Work out the volume of the cuboid.
(b) Work out the total surface area of the cuboid.
.......................................... cm 2 [3]
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(c) Calculate the angle between GA and the base ABCD.
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.......................................... cm 3 [1]
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................................................. [4]
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17
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(d) A straight rod PQ is placed inside the cuboid.
One end of the rod, P, is placed at the midpoint of AB.
The other end of the rod, Q, rests on GH.
HQ : QG = 4 : 1 .
Q
H
F
E
17 cm
D
A
P
10 cm
B
NOT TO
SCALE
C
8 cm
Calculate the length of the rod PQ.
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G
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............................................ cm [4]
© UCLES 2024
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18
,
,
10 (a)
(x + 1)cm
NOT TO
SCALE
(2x + 3)cm
(ii) Work out the perimeter of the rectangle.
............................................ cm [2]
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[4]
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(i) By forming and solving an equation, show that x = 8.5 .
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This rectangle has area 190 cm 2 .
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19
,
(b)
A
r cm
O
NOT TO
SCALE
50°
B
The diagram shows a sector OAB of a circle, with centre O, and a chord AB.
The shaded segment has area 30 cm 2 .
(i) Show that r = 23.7 cm, correct to 1 decimal place.
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,
[4]
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(ii) Calculate the perimeter of the shaded segment.
............................................ cm [4]
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20
,
,
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every
reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the
publisher will be pleased to make amends at the earliest possible opportunity.
To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge
Assessment International Education Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download
at www.cambridgeinternational.org after the live examination series.
Cambridge Assessment International Education is part of Cambridge Assessment. Cambridge Assessment is the brand name of the University of Cambridge
Local Examinations Syndicate (UCLES), which is a department of the University of Cambridge.
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