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6 In this question write any probability as a fraction.
Navpreet has 15 cards with a shape drawn on each card.
5 cards have a square, 6 cards have a triangle and 4 cards have a circle drawn on them.
(a) Navpreet selects a card at random.
Write down the probability that the card has a circle drawn on it.
Answer(a) ................................................ [1]
(b) Navpreet selects a card at random and replaces it.
She does this 300 times.
Calculate the number of times she expects to select a card with a circle drawn on it.
Answer(b) ................................................ [1]
(c) Navpreet selects a card at random, replaces it and then selects another card.
Calculate the probability that
(i) one card has a square drawn on it and the other has a circle drawn on it,
Answer(c)(i) ................................................ [3]
(ii) neither card has a circle drawn on it.
Answer(c)(ii) ................................................ [3]
(d) Navpreet selects two cards at random, without replacement.
Calculate the probability that
(i) only one card has a triangle drawn on it,
Answer(d)(i) ................................................ [3]
(ii) the two cards have different shapes drawn on them.
Answer(d)(ii) ................................................ [4]
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(c) Solve f(x) = 0.
Answer(c) x = ................................................. [1]
(d) By drawing a suitable line on the grid, solve the equation f(x) = 1 – x.
Answer(d) x = ................................................. [3]
(e) By drawing a tangent at the point (–3, –0.6), estimate the gradient of the graph of y = f(x) when x = –3.
Answer(e) ................................................. [3]
© UCLES 2015 0580/43/M/J/15 [Turn over
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A A A A B B C
(a) One of these 7 cards is chosen at random.
Write down the probability that the card
(i) shows the letter A,
Answer(a)(i) ................................................. [1]
(ii) shows the letter A or B,
Answer(a)(ii) ................................................. [1]
(iii) does not show the letter B.
Answer(a)(iii) ................................................. [1]
(b) Two of the cards are chosen at random, without replacement.
Find the probability that
(i) both show the letter A,
Answer(b)(i) ................................................. [2]
(ii) the two letters are different.
Answer(b)(ii) ................................................. [3]
(c) Three of the cards are chosen at random, without replacement.
Find the probability that the cards do not show the letter C.
Answer(c) ................................................. [2]
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© UCLES 2015 0580/43/M/J/15 [Turn over
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17 (a)
S R
NOT TO
b SCALE
P 2a Q
PQRS is a trapezium with PQ = 2SR.
= 2a and = b.
Find in terms of a and b in its simplest form.
Answer(a) = ................................................ [2]
(b)
X
M
x NOT TO
SCALE
O y Y
= x and = y.
M is a point on XY such that XM : MY = 3 : 5.
Find in terms of x and y in its simplest form.
Answer(b) = ................................................ [2]
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© UCLES 2015 0580/22/F/M/15
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3 On the firt par t of a jour ney , Alan drove a distance of x km and his car used 6 litres of fuel.
600
The rate of fuel used by his car was litres per 100 km.
x
(a) Alan then drove another (x + 20) km and his car used another 6 litres of fuel.
(i) Write down an expression, in terms of x, for the rate of fuel used by his car on this part
of the journey.
Give your answer in litres per 100 km.
Answer(a)(i) .............................. litres per 100 km [1]
(ii) On this part of the journey the rate of fuel used by the car decreased by 1.5 litres per 100 km.
Show that x2 + 20x – 8000 = 0.
Answer(a)(ii)
[4]
(b) Solve the equation x2 + 20x – 8000 = 0.
Answer(b) x = ............................... or x = ............................... [3]
(c) Find the rate of fuel used by Alan’s car for the complete journey.
Give your answer in litres per 100 km.
Answer(c) .............................. litres per 100 km [2]
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s
© UCLES 2015 0580/42/M/J/15 [Turn over
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d (c) By drawing a suitable tangent, fin an est ima t e of the gr a dient of the graph at the point (1, 11).
Answer(c) ................................................
d [3]
12
(d) The equation x2 – 2x + = k has exactly two distinct solutions.
x
Use the graph to fin
(i) the value of k,
Answer(d)(i) k = ................................................ [1]
12
(ii) the solutions of x2 – 2x + = k.
x
Answer(d)(ii) x = ........................... or x = ........................... [2]
(e) The equation x3 + ax2 + bx + c = 0 can be solved by drawing the line y = 3x + 1 on the grid.
Find the value of a, the value of b and the value of c.
Answer(e) a = ................................................
b = ................................................
c = ................................................ [3]
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© UCLES 2015 0580/42/M/J/15 [Turn over
20
11 Gareth has 8 sweets in v
a bag.
4 sweets are orange flaour ed, 3 ar e lemo n flavur ed and 1 is strawberry flaour ed.
(a) He chooses two of the sweets at random.
Find the probability
v that the two sweets have different flaour s.
Answer(a) ................................................ [4]
(b) Gareth now chooses a third sweet.v
Find the probability that none of the three sweets is lemon flaour ed.
Answer(b) ................................................ [2]
__________________________________________________________________________________________
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 International
Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after
the live examination series.
Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local
Examinations Syndicate (UCLES), which is itselfo a department of the University of Cambridge.
© UCLES 2015 0580/42/M/J/15
6
14
Q b R NOT TO
SCALE
P M S
PQRS is a quadrilateral and M is the midpoint of PS.
PQ = a, QR = b and SQ = a – 2b.
(a) Show that PS = 2b.
Answer(a)
[1]
(b) Write down the mathematical name for the quadrilateral PQRM, giving reasons for your answer.
Answer(b) .............................................................. because ...............................................................
............................................................................................................................................................. [2]
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© UCLES 2015 0580/21/M/J/15
6
3 The diagram shows a horizontal water trough in the shape of a prism.
NOT TO
35 cm
SCALE
12 cm
6 cm
120 cm
25 cm
The cross section of this prism is a trapezium.
The trapezium has parallel sides of lengths 35 cm and 25 cm and a perpendicular height of 12 cm.
The length of the prism is 120 cm.
(a) Calculate the volume of the trough.
Answer(a) ......................................... cm3 [3]
(b) The trough contains water to a depth of 6 cm.
(i) Show that the volume of water is 19 800 cm3.
Answer (b)(i)
[2]
(ii) Calculate the percentage of the trough that contains water.
Answer(b)(ii) ............................................ % [1]
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(c) The water is drained from the trough at a rate of 12 litres per hour.
Calculate the time it takes to empty the trough.
Give your answer in hours and minutes.
Answer(c) ................. h .................
l min [4]
(d) The water from the trough just fils a cyl inder of radi us r cm and height 3r cm.
Calculate the value of r.
Answer(d) r = ................................................ [3]
(e) The cylinder has a mass of 1.2 kg.
1 cm3 of water has a mass of 1 g.
Calculate the total mass of the cylinder and the water.
Give your answer in kilograms.
Answer(e) ........................................... kg [2]
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© UCLES 2015 0580/43/O/N/15 [Turn over
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(d) There is only one negative integer value, k, for which f(x) = k has only one solution for all real x.
Write down this value of k.
Answer(d) k = ................................................ [1]
1
(e) The equation 2x – 2 – 2 = 0 can be solved using the graph of y = f(x) and a straight line graph.
2x
(i) Find the equation of this straight line.
Answer(e)(i) y = ................................................ [1]
1
(ii) On the grid, draw this straight line and solve the equation 2x – – 2 = 0.
2x2
Answer(e)(ii) x = ................................................ [3]
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(c) An apple is chosen at random from the 160 apples.
Find the probability that its mass is more than 120 g.
Answer(c) ................................................ [1]
(d) Two apples are chosen at random from the 160 apples, without replacement.
Find the probability that
(i) they both have a mass of more than 120 g,
Answer(d)(i) ................................................ [2]
(ii) one has a mass of more than 120 g and one has a mass of 80 g or less.
Answer(d)(ii) ................................................ [3]
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© UCLES 2015 0580/43/O/N/15 [Turn over
4
2 (a) Calculate 20.7.
Answer(a) ................................................ [1]
(b) Find the value of x in each of the following.
(i) 2x = 128
Answer(b)(i) x = ................................................ [1]
x
(ii) 2 × 29 = 213
Answer(b)(ii) x = ................................................ [1]
(iii) 29 ÷ 2x = 4
Answer(b)(iii) x = ................................................ [1]
(iv) 2x = 3
2
Answer(b)(iv) x = ................................................ [1]
(c) (i) Complete this table of values for y = 2x.
x –3 –2 –1 0 1 2 3
y 0.125 0.5 2 4 8
[2]
© UCLES 2015 0580/41/O/N/15
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10
C
NOT TO
b
SCALE
M
a
A
BC = a and AC = b.
(a) Find AB in terms of a and b.
Answer(a) AB = ................................................ [1]
(b) M is the midpoint of BC.
X divides AB in the ratio 1 : 4.
Find XM in terms of a and b.
Show all your working and write your answer in its simplest form.
Answer(b) XM = ................................................ [4]
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 International
Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after
the live examination series.
Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local
Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.
© UCLES 2015 0580/41/O/N/15
9
23
B
NOT TO
SCALE
b
C
O a A
In the diagram, O is the origin, OA = a and OB = b.
C is on the line AB so that AC : CB = 1 : 2.
Find, in terms of a and b, in its simplest form,
(a) AC ,
Answer(a) AC = ................................................ [2]
(b) the position vector of C.
Answer(b) ................................................ [2]
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23 A box contains 6 red pencils and 8 blue pencils.
A pencil is chosen at random and not replaced.
A second pencil is then chosen at random.
(a) Complete the tree diagram.
First pencil Second pencil
Red
.......
Red
6
14 8
13 Blue
Red
.......
....... Blue
....... Blue
[2]
(b) Calculate the probability that
(i) both pencils are red,
Answer(b)(i) ................................................. [2]
(ii) at least one of the pencils is red.
Answer(b)(ii) ................................................. [3]
© UCLES 2015 0580/22/O/N/15
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20 The table shows the probability that a person has blue, brown or green eyes.
Eye colour Blue Brown Green
Probability 0.4 0.5 0.1
Use the table to work out the probability that two people, chosen at random,
(a) have blue eyes,
Answer(a) ................................................ [2]
(b) have different coloured eyes.
Answer(b) ................................................ [4]
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© UCLES 2015 0580/21/O/N/15
