w
w
ap
eP
m
e
tr
.X
w
om
.c
s
er
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
General Certificate of Education
Advanced Subsidiary Level and Advanced Level
9709/61
MATHEMATICS
Paper 6 Probability & Statistics 1 (S1)
May/June 2011
1 hour 15 minutes
*9986337980*
Additional Materials:
Answer Booklet/Paper
Graph Paper
List of Formulae (MF9)
READ THESE INSTRUCTIONS FIRST
If you have been given an Answer Booklet, follow the instructions on the front cover of the Booklet.
Write your Centre number, candidate number and name on all the work you hand in.
Write in dark blue or black pen.
You may use a soft pencil for any diagrams or graphs.
Do not use staples, paper clips, highlighters, glue or correction fluid.
Answer all the questions.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in
degrees, unless a different level of accuracy is specified in the question.
The use of an electronic calculator is expected, where appropriate.
You are reminded of the need for clear presentation in your answers.
At the end of the examination, fasten all your work securely together.
The number of marks is given in brackets [ ] at the end of each question or part question.
The total number of marks for this paper is 50.
Questions carrying smaller numbers of marks are printed earlier in the paper, and questions carrying larger
numbers of marks later in the paper.
This document consists of 3 printed pages and 1 blank page.
JC11 06_9709_61/RP
© UCLES 2011
[Turn over
2
1
Biscuits are sold in packets of 18. There is a constant probability that any biscuit is broken,
independently of other biscuits. The mean number of broken biscuits in a packet has been found to
be 2.7. Find the probability that a packet contains between 2 and 4 (inclusive) broken biscuits. [4]
2
When Ted is looking for his pen, the probability that it is in his pencil case is 0.7. If his pen is in
his pencil case he always finds it. If his pen is somewhere else, the probability that he finds it is 0.2.
Given that Ted finds his pen when he is looking for it, find the probability that it was in his pencil
case.
[4]
3
The possible values of the random variable X are the 8 integers in the set {−2, −1, 0, 1, 2, 3, 4, 5}.
1 . The probabilities for all the other values of X are equal. Calculate
The probability of X being 0 is 10
(i) P(X < 2),
[2]
(ii) the variance of X ,
[3]
(iii) the value of a for which P(−a ≤ X ≤ 2a) =
4
17
.
35
[1]
A cricket team of 11 players is to be chosen from 21 players consisting of 10 batsmen, 9 bowlers
and 2 wicketkeepers. The team must include at least 5 batsmen, at least 4 bowlers and at least
1 wicketkeeper.
(i) Find the number of different ways in which the team can be chosen.
[4]
Each player in the team is given a present. The presents consist of 5 identical pens, 4 identical diaries
and 2 identical notebooks.
(ii) Find the number of different arrangements of the presents if they are all displayed in a row. [1]
(iii) 10 of these 11 presents are chosen and arranged in a row.
arrangements that are possible.
5
Find the number of different
[3]
(a) The random variable X is normally distributed with mean µ and standard deviation σ . It is given
that 3µ = 7σ 2 and that P(X > 2µ ) = 0.1016. Find µ and σ .
[4]
(b) It is given that Y ∼ N(33, 21). Find the value of a given that P(33 − a < Y < 33 + a) = 0.5.
© UCLES 2011
9709/61/M/J/11
[4]
3
6
There are 5000 schools in a certain country. The cumulative frequency table shows the number of
pupils in a school and the corresponding number of schools.
Number of pupils in a school
Cumulative frequency
≤ 100
≤ 150
≤ 200
≤ 250
≤ 350
≤ 450
≤ 600
200
800
1600
2100
4100
4700
5000
(i) Draw a cumulative frequency graph with a scale of 2 cm to 100 pupils on the horizontal axis
and a scale of 2 cm to 1000 schools on the vertical axis. Use your graph to estimate the median
number of pupils in a school.
[3]
(ii) 80% of the schools have more than n pupils. Estimate the value of n correct to the nearest ten.
[2]
7
(iii) Find how many schools have between 201 and 250 (inclusive) pupils.
[1]
(iv) Calculate an estimate of the mean number of pupils per school.
[4]
(a)
(i) Find the probability of getting at least one 3 when 9 fair dice are thrown.
[2]
(ii) When n fair dice are thrown, the probability of getting at least one 3 is greater than 0.9.
Find the smallest possible value of n.
[4]
(b) A bag contains 5 green balls and 3 yellow balls. Ronnie and Julie play a game in which they
take turns to draw a ball from the bag at random without replacement. The winner of the game is
the first person to draw a yellow ball. Julie draws the first ball. Find the probability that Ronnie
wins the game.
[4]
© UCLES 2011
9709/61/M/J/11
4
BLANK PAGE
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.
University of 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.
9709/61/M/J/11
w
w
ap
eP
m
e
tr
.X
w
om
.c
s
er
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
General Certificate of Education
Advanced Subsidiary Level and Advanced Level
9709/12
MATHEMATICS
Paper 1 Pure Mathematics 1 (P1)
May/June 2011
1 hour 45 minutes
*4457114786*
Additional Materials:
Answer Booklet/Paper
Graph Paper
List of Formulae (MF9)
READ THESE INSTRUCTIONS FIRST
If you have been given an Answer Booklet, follow the instructions on the front cover of the Booklet.
Write your Centre number, candidate number and name on all the work you hand in.
Write in dark blue or black pen.
You may use a soft pencil for any diagrams or graphs.
Do not use staples, paper clips, highlighters, glue or correction fluid.
Answer all the questions.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in
degrees, unless a different level of accuracy is specified in the question.
The use of an electronic calculator is expected, where appropriate.
You are reminded of the need for clear presentation in your answers.
At the end of the examination, fasten all your work securely together.
The number of marks is given in brackets [ ] at the end of each question or part question.
The total number of marks for this paper is 75.
Questions carrying smaller numbers of marks are printed earlier in the paper, and questions carrying larger
numbers of marks later in the paper.
This document consists of 3 printed pages and 1 blank page.
JC11 06_9709_12/2R
© UCLES 2011
[Turn over
2
1
2
Find ä x3 +
1
dx.
x3
[3]
(i) Find the terms in x2 and x3 in the expansion of 1 − 32 x .
6
[3]
(ii) Given that there is no term in x3 in the expansion of (k + 2x) 1 − 32 x , find the value of the
constant k.
[2]
6
3
The equation x2 + px + q = 0, where p and q are constants, has roots −3 and 5.
(i) Find the values of p and q.
[2]
(ii) Using these values of p and q, find the value of the constant r for which the equation
x2 + px + q + r = 0 has equal roots.
[3]
4
5
A curve has equation y =
4
and P (2, 2) is a point on the curve.
3x − 4
(i) Find the equation of the tangent to the curve at P.
[4]
(ii) Find the angle that this tangent makes with the x-axis.
[2]
(i) Prove the identity
cos θ
1
≡1+
.
tan θ (1 − sin θ )
sin θ
(ii) Hence solve the equation
6
cos θ
= 4, for 0◦ ≤ θ ≤ 360◦ .
tan θ (1 − sin θ )
The function f is defined by f : x →
(i) Show that ff (x) = x.
[3]
x+3
, x ∈ >, x ≠ 12 .
2x − 1
[3]
(ii) Hence, or otherwise, obtain an expression for f −1 (x).
7
[3]
[2]
The line L1 passes through the points A (2, 5) and B (10, 9). The line L2 is parallel to L1 and passes
through the origin. The point C lies on L2 such that AC is perpendicular to L2 . Find
(i) the coordinates of C,
[5]
(ii) the distance AC.
[2]
© UCLES 2011
9709/12/M/J/11
3
8
Relative to the origin O, the position vectors of the points A, B and C are given by
−−→
OA =
2
3!,
5
−−→
OB =
4
2!
3
and
−−→
OC =
10
0!.
6
(i) Find angle ABC .
[6]
The point D is such that ABCD is a parallelogram.
(ii) Find the position vector of D.
9
[2]
The function f is such that f (x) = 3 − 4 cosk x, for 0 ≤ x ≤ π , where k is a constant.
(i) In the case where k = 2,
(a) find the range of f,
[2]
(b) find the exact solutions of the equation f (x) = 1.
[3]
(a) sketch the graph of y = f (x),
[2]
(ii) In the case where k = 1,
(b) state, with a reason, whether f has an inverse.
10
[1]
(a) A circle is divided into 6 sectors in such a way that the angles of the sectors are in arithmetic
progression. The angle of the largest sector is 4 times the angle of the smallest sector. Given
that the radius of the circle is 5 cm, find the perimeter of the smallest sector.
[6]
(b) The first, second and third terms of a geometric progression are 2k + 3, k + 6 and k, respectively.
Given that all the terms of the geometric progression are positive, calculate
11
(i) the value of the constant k,
[3]
(ii) the sum to infinity of the progression.
[2]
y
M
y = 4 Öx – x
O
A
x
√
The diagram shows part of the curve y = 4 x − x. The curve has a maximum point at M and meets
the x-axis at O and A.
(i) Find the coordinates of A and M .
[5]
(ii) Find the volume obtained when the shaded region is rotated through 360◦ about the x-axis, giving
[6]
your answer in terms of π .
© UCLES 2011
9709/12/M/J/11
4
BLANK PAGE
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.
University of 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.
9709/12/M/J/11
w
w
ap
eP
m
e
tr
.X
w
om
.c
s
er
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
General Certificate of Education
Advanced Subsidiary Level and Advanced Level
9709/11
MATHEMATICS
Paper 1 Pure Mathematics 1 (P1)
May/June 2011
1 hour 45 minutes
*0212815821*
Additional Materials:
Answer Booklet/Paper
Graph Paper
List of Formulae (MF9)
READ THESE INSTRUCTIONS FIRST
If you have been given an Answer Booklet, follow the instructions on the front cover of the Booklet.
Write your Centre number, candidate number and name on all the work you hand in.
Write in dark blue or black pen.
You may use a soft pencil for any diagrams or graphs.
Do not use staples, paper clips, highlighters, glue or correction fluid.
Answer all the questions.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in
degrees, unless a different level of accuracy is specified in the question.
The use of an electronic calculator is expected, where appropriate.
You are reminded of the need for clear presentation in your answers.
At the end of the examination, fasten all your work securely together.
The number of marks is given in brackets [ ] at the end of each question or part question.
The total number of marks for this paper is 75.
Questions carrying smaller numbers of marks are printed earlier in the paper, and questions carrying larger
numbers of marks later in the paper.
This document consists of 4 printed pages.
JC11 06_9709_11/2R
© UCLES 2011
[Turn over
2
1
2
3
Find the coefficient of x in the expansion of x +
7
2
.
x2
[3]
The volume of a spherical balloon is increasing at a constant rate of 50 cm3 per second. Find the rate
of increase of the radius when the radius is 10 cm. [Volume of a sphere = 43 π r3 .]
[4]
(i) Sketch the curve y = (x − 2)2 .
[1]
(ii) The region enclosed by the curve, the x-axis and the y-axis is rotated through 360◦ about the
x-axis. Find the volume obtained, giving your answer in terms of π .
[4]
4
Q
B
R
S
P
5 cm
C
6 cm
k
2 cm
j
A
i
6 cm
D
The diagram shows a prism ABCDPQRS with a horizontal square base APSD with sides of length
6 cm. The cross-section ABCD is a trapezium and is such that the vertical edges AB and DC are of
lengths 5 cm and 2 cm respectively. Unit vectors i, j and k are parallel to AD, AP and AB respectively.
−−→
−−→
(i) Express each of the vectors CP and CQ in terms of i, j and k.
(ii) Use a scalar product to calculate angle PCQ.
5
[2]
[4]
(i) Show that the equation 2 tan2 θ sin2 θ = 1 can be written in the form
2 sin4 θ + sin2 θ − 1 = 0.
(ii) Hence solve the equation 2 tan2 θ sin2 θ = 1 for 0◦ ≤ θ ≤ 360◦ .
© UCLES 2011
9709/11/M/J/11
[2]
[4]
3
6
The variables x, y and ß can take only positive values and are such that
ß = 3x + 2y
(i) Show that ß = 3x +
and
xy = 600.
1200
.
x
[1]
(ii) Find the stationary value of ß and determine its nature.
7
8
A curve is such that
[6]
3
dy
and the point (1, 21 ) lies on the curve.
=
dx (1 + 2x)2
(i) Find the equation of the curve.
[4]
(ii) Find the set of values of x for which the gradient of the curve is less than 31 .
[3]
A television quiz show takes place every day. On day 1 the prize money is $1000. If this is not won
the prize money is increased for day 2. The prize money is increased in a similar way every day until
it is won. The television company considered the following two different models for increasing the
prize money.
Model 1:
Increase the prize money by $1000 each day.
Model 2:
Increase the prize money by 10% each day.
On each day that the prize money is not won the television company makes a donation to charity. The
amount donated is 5% of the value of the prize on that day. After 40 days the prize money has still
not been won. Calculate the total amount donated to charity
(i) if Model 1 is used,
[4]
(ii) if Model 2 is used.
[3]
9
S
A
B
r
P
T
2q
O
In the diagram, OAB is an isosceles triangle with OA = OB and angle AOB = 2θ radians. Arc PST
has centre O and radius r, and the line ASB is a tangent to the arc PST at S.
(i) Find the total area of the shaded regions in terms of r and θ .
[4]
(ii) In the case where θ = 13 π and r = 6, find the total perimeter of the shaded regions, leaving your
√
answer in terms of 3 and π .
[5]
[Questions 10 and 11 are printed on the next page.]
© UCLES 2011
9709/11/M/J/11
[Turn over
4
10
(i) Express 2x2 − 4x + 1 in the form a(x + b)2 + c and hence state the coordinates of the minimum
point, A, on the curve y = 2x2 − 4x + 1.
[4]
The line x − y + 4 = 0 intersects the curve y = 2x2 − 4x + 1 at points P and Q. It is given that the
coordinates of P are (3, 7).
11
(ii) Find the coordinates of Q.
[3]
(iii) Find the equation of the line joining Q to the mid-point of AP.
[3]
Functions f and g are defined for x ∈ > by
f : x → 2x + 1,
g : x → x2 − 2.
(i) Find and simplify expressions for fg(x) and gf (x).
[2]
(ii) Hence find the value of a for which fg(a) = gf (a).
[3]
(iii) Find the value of b (b ≠ a) for which g(b) = b.
[2]
(iv) Find and simplify an expression for f −1 g(x).
[2]
The function h is defined by
h : x → x2 − 2,
for x ≤ 0.
(v) Find an expression for h−1 (x).
[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.
University of 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 2011
9709/11/M/J/11
w
w
ap
eP
m
e
tr
.X
w
om
.c
s
er
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
General Certificate of Education
Advanced Subsidiary Level and Advanced Level
9709/62
MATHEMATICS
Paper 6 Probability & Statistics 1 (S1)
May/June 2010
1 hour 15 minutes
*8471957091*
Additional Materials:
Answer Booklet/Paper
Graph Paper
List of Formulae (MF9)
READ THESE INSTRUCTIONS FIRST
If you have been given an Answer Booklet, follow the instructions on the front cover of the Booklet.
Write your Centre number, candidate number and name on all the work you hand in.
Write in dark blue or black pen.
You may use a soft pencil for any diagrams or graphs.
Do not use staples, paper clips, highlighters, glue or correction fluid.
Answer all the questions.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in
degrees, unless a different level of accuracy is specified in the question.
The use of an electronic calculator is expected, where appropriate.
You are reminded of the need for clear presentation in your answers.
At the end of the examination, fasten all your work securely together.
The number of marks is given in brackets [ ] at the end of each question or part question.
The total number of marks for this paper is 50.
Questions carrying smaller numbers of marks are printed earlier in the paper, and questions carrying larger
numbers of marks later in the paper.
This document consists of 3 printed pages and 1 blank page.
© UCLES 2010
[Turn over
2
1
The times in minutes for seven students to become proficient at a new computer game were measured.
The results are shown below.
15
10
48
10
19
14
16
(i) Find the mean and standard deviation of these times.
[2]
(ii) State which of the mean, median or mode you consider would be most appropriate to use as a
measure of central tendency to represent the data in this case.
[1]
(iii) For each of the two measures of average you did not choose in part (ii), give a reason why you
consider it inappropriate.
[2]
2
The lengths of new pencils are normally distributed with mean 11 cm and standard deviation 0.095 cm.
(i) Find the probability that a pencil chosen at random has a length greater than 10.9 cm.
[2]
(ii) Find the probability that, in a random sample of 6 pencils, at least two have lengths less than
10.9 cm.
[3]
3
Cumulative
frequency
1000
900
800
Country A
700
Country B
600
500
400
300
200
100
0
0
1
2
3
4
5
6
Weight (kg)
The birth weights of random samples of 900 babies born in country A and 900 babies born in country B
are illustrated in the cumulative frequency graphs. Use suitable data from these graphs to compare
the central tendency and spread of the birth weights of the two sets of babies.
[6]
© UCLES 2010
9709/62/M/J/10
3
4
The random variable X is normally distributed with mean µ and standard deviation σ .
(i) Given that 5σ = 3µ , find P(X < 2µ ).
[3]
(ii) With a different relationship between µ and σ , it is given that P(X < 13 µ ) = 0.8524. Express µ in
terms of σ .
[3]
5
Two fair twelve-sided dice with sides marked 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 are thrown, and the
numbers on the sides which land face down are noted. Events Q and R are defined as follows.
Q : the product of the two numbers is 24.
R : both of the numbers are greater than 8.
6
(i) Find P(Q).
[2]
(ii) Find P(R).
[2]
(iii) Are events Q and R exclusive? Justify your answer.
[2]
(iv) Are events Q and R independent? Justify your answer.
[2]
A small farm has 5 ducks and 2 geese. Four of these birds are to be chosen at random. The random
variable X represents the number of geese chosen.
(i) Draw up the probability distribution of X .
[3]
8
7
[3]
(ii) Show that E(X ) =
and calculate Var(X ).
(iii) When the farmer’s dog is let loose, it chases either the ducks with probability 35 or the geese with
1
probability 52 . If the dog chases the ducks there is a probability of 10
that they will attack the dog.
3
If the dog chases the geese there is a probability of 4 that they will attack the dog. Given that the
dog is not attacked, find the probability that it was chasing the geese.
[4]
7
Nine cards, each of a different colour, are to be arranged in a line.
(i) How many different arrangements of the 9 cards are possible?
[1]
The 9 cards include a pink card and a green card.
(ii) How many different arrangements do not have the pink card next to the green card?
[3]
Consider all possible choices of 3 cards from the 9 cards with the 3 cards being arranged in a line.
(iii) How many different arrangements in total of 3 cards are possible?
[2]
(iv) How many of the arrangements of 3 cards in part (iii) contain the pink card?
[2]
(v) How many of the arrangements of 3 cards in part (iii) do not have the pink card next to the green
card?
[2]
© UCLES 2010
9709/62/M/J/10
4
BLANK PAGE
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.
University of 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.
9709/62/M/J/10
w
w
ap
eP
m
e
tr
.X
w
om
.c
s
er
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
General Certificate of Education
Advanced Subsidiary Level and Advanced Level
9709/61
MATHEMATICS
Paper 6 Probability & Statistics 1 (S1)
May/June 2010
1 hour 15 minutes
*1115486555*
Additional Materials:
Answer Booklet/Paper
Graph Paper
List of Formulae (MF9)
READ THESE INSTRUCTIONS FIRST
If you have been given an Answer Booklet, follow the instructions on the front cover of the Booklet.
Write your Centre number, candidate number and name on all the work you hand in.
Write in dark blue or black pen.
You may use a soft pencil for any diagrams or graphs.
Do not use staples, paper clips, highlighters, glue or correction fluid.
Answer all the questions.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in
degrees, unless a different level of accuracy is specified in the question.
The use of an electronic calculator is expected, where appropriate.
You are reminded of the need for clear presentation in your answers.
At the end of the examination, fasten all your work securely together.
The number of marks is given in brackets [ ] at the end of each question or part question.
The total number of marks for this paper is 50.
Questions carrying smaller numbers of marks are printed earlier in the paper, and questions carrying larger
numbers of marks later in the paper.
This document consists of 3 printed pages and 1 blank page.
© UCLES 2010
[Turn over
2
1
The probability distribution of the discrete random variable X is shown in the table below.
x
−3
−1
0
4
P(X = x)
a
b
0.15
0.4
Given that E(X ) = 0.75, find the values of a and b.
2
[4]
The numbers of people travelling on a certain bus at different times of the day are as follows.
17
22
6
5
14
23
2
25
19
23
35
21
16
17
23
31
27
8
8
12
26
(i) Draw a stem-and-leaf diagram to illustrate the information given above.
[3]
(ii) Find the median, the lower quartile, the upper quartile and the interquartile range.
[3]
(iii) State, in this case, which of the median and mode is preferable as a measure of central tendency,
and why.
[1]
3
The random variable X is the length of time in minutes that Jannon takes to mend a bicycle puncture.
X has a normal distribution with mean µ and variance σ 2 . It is given that P(X > 30.0) = 0.1480 and
P(X > 20.9) = 0.6228. Find µ and σ .
[5]
4
The numbers of rides taken by two students, Fei and Graeme, at a fairground are shown in the
following table.
Roller
coaster
Water
slide
Revolving
drum
Fei
4
2
0
Graeme
1
3
6
(i) The mean cost of Fei’s rides is $2.50 and the standard deviation of the costs of Fei’s rides is $0.
Explain how you can tell that the roller coaster and the water slide each cost $2.50 per ride. [2]
(ii) The mean cost of Graeme’s rides is $3.76. Find the standard deviation of the costs of Graeme’s
rides.
[5]
© UCLES 2010
9709/61/M/J/10
3
5
In the holidays Martin spends 25% of the day playing computer games. Martin’s friend phones him
once a day at a randomly chosen time.
(i) Find the probability that, in one holiday period of 8 days, there are exactly 2 days on which
Martin is playing computer games when his friend phones.
[2]
(ii) Another holiday period lasts for 12 days. State with a reason whether it is appropriate to use a
normal approximation to find the probability that there are fewer than 7 days on which Martin is
playing computer games when his friend phones.
[1]
(iii) Find the probability that there are at least 13 days of a 40-day holiday period on which Martin is
playing computer games when his friend phones.
[5]
6
(i) Find the number of different ways that a set of 10 different mugs can be shared between Lucy
and Monica if each receives an odd number of mugs.
[3]
(ii) Another set consists of 6 plastic mugs each of a different design and 3 china mugs each of a
different design. Find in how many ways these 9 mugs can be arranged in a row if the china
mugs are all separated from each other.
[3]
(iii) Another set consists of 3 identical red mugs, 4 identical blue mugs and 7 identical yellow mugs.
These 14 mugs are placed in a row. Find how many different arrangements of the colours are
possible if the red mugs are kept together.
[3]
7
In a television quiz show Peter answers questions one after another, stopping as soon as a question is
answered wrongly.
• The probability that Peter gives the correct answer himself to any question is 0.7.
• The probability that Peter gives a wrong answer himself to any question is 0.1.
• The probability that Peter decides to ask for help for any question is 0.2.
On the first occasion that Peter decides to ask for help he asks the audience. The probability that
the audience gives the correct answer to any question is 0.95. This information is shown in the tree
diagram below.
Peter answers correctly
0.7
0.1
Peter answers wrongly
0.95
0.2
Audience answers correctly
Peter asks for help
0.05
Audience answers wrongly
(i) Show that the probability that the first question is answered correctly is 0.89.
[1]
On the second occasion that Peter decides to ask for help he phones a friend. The probability that his
friend gives the correct answer to any question is 0.65.
(ii) Find the probability that the first two questions are both answered correctly.
[6]
(iii) Given that the first two questions were both answered correctly, find the probability that Peter
asked the audience.
[3]
© UCLES 2010
9709/61/M/J/10
4
BLANK PAGE
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.
University of 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.
9709/61/M/J/10
w
w
ap
eP
m
e
tr
.X
w
om
.c
s
er
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
General Certificate of Education
Advanced Subsidiary Level and Advanced Level
9709/12
MATHEMATICS
Paper 1 Pure Mathematics 1 (P1)
May/June 2010
1 hour 45 minutes
*0432216967*
Additional Materials:
Answer Booklet/Paper
Graph Paper
List of Formulae (MF9)
READ THESE INSTRUCTIONS FIRST
If you have been given an Answer Booklet, follow the instructions on the front cover of the Booklet.
Write your Centre number, candidate number and name on all the work you hand in.
Write in dark blue or black pen.
You may use a soft pencil for any diagrams or graphs.
Do not use staples, paper clips, highlighters, glue or correction fluid.
Answer all the questions.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in
degrees, unless a different level of accuracy is specified in the question.
The use of an electronic calculator is expected, where appropriate.
You are reminded of the need for clear presentation in your answers.
At the end of the examination, fasten all your work securely together.
The number of marks is given in brackets [ ] at the end of each question or part question.
The total number of marks for this paper is 75.
Questions carrying smaller numbers of marks are printed earlier in the paper, and questions carrying larger
numbers of marks later in the paper.
This document consists of 4 printed pages.
© UCLES 2010
[Turn over
2
1
(i) Show that the equation
3(2 sin x − cos x) = 2(sin x − 3 cos x)
can be written in the form tan x = − 34 .
[2]
(ii) Solve the equation 3(2 sin x − cos x) = 2(sin x − 3 cos x), for 0◦ ≤ x ≤ 360◦ .
2
[2]
y
a
y= x
O
3
1
x
a
, where a is a positive constant. Given that the volume
x
obtained when the shaded region is rotated through 360◦ about the x-axis is 24π , find the value of a.
[4]
The diagram shows part of the curve y =
3
The functions f and g are defined for x ∈ > by
f : x → 4x − 2x2 ,
g : x → 5x + 3.
(i) Find the range of f.
[2]
(ii) Find the value of the constant k for which the equation gf (x) = k has equal roots.
[3]
4
y
L1
C
(–1, 3)
A
L2
B (3, 1)
x
O
In the diagram, A is the point (−1, 3) and B is the point (3, 1). The line L1 passes through A and is
parallel to OB. The line L2 passes through B and is perpendicular to AB. The lines L1 and L2 meet at
C . Find the coordinates of C .
[6]
© UCLES 2010
9709/12/M/J/10
3
5
Relative to an origin O, the position vectors of the points A and B are given by
−−→
OA =
−2
3!
1
and
−−→
OB =
4
1!.
p
−−→
−−→
(i) Find the value of p for which OA is perpendicular to OB.
−−→
(ii) Find the values of p for which the magnitude of AB is 7.
6
(i) Find the first 3 terms in the expansion of (1 + ax)5 in ascending powers of x.
[2]
[4]
[2]
(ii) Given that there is no term in x in the expansion of (1 − 2x)(1 + ax)5 , find the value of the
constant a.
[2]
7
(iii) For this value of a, find the coefficient of x2 in the expansion of (1 − 2x)(1 + ax)5 .
[3]
(a) Find the sum of all the multiples of 5 between 100 and 300 inclusive.
[3]
(b) A geometric progression has a common ratio of − 23 and the sum of the first 3 terms is 35. Find
8
(i) the first term of the progression,
[3]
(ii) the sum to infinity.
[2]
A solid rectangular block has a square base of side x cm. The height of the block is h cm and the total
surface area of the block is 96 cm2 .
(i) Express h in terms of x and show that the volume, V cm3 , of the block is given by
V = 24x − 12 x3 .
[3]
Given that x can vary,
(ii) find the stationary value of V ,
[3]
(iii) determine whether this stationary value is a maximum or a minimum.
[2]
[Questions 9, 10 and 11 are printed on the next page.]
© UCLES 2010
9709/12/M/J/10
[Turn over
4
9
y
2
y = (x – 2)
A
y + 2x = 7
B
x
O
The diagram shows the curve y = (x − 2)2 and the line y + 2x = 7, which intersect at points A and B.
Find the area of the shaded region.
[8]
10
The equation of a curve is y = 16 (2x − 3)3 − 4x.
(i) Find
dy
.
dx
[3]
(ii) Find the equation of the tangent to the curve at the point where the curve intersects the y-axis.
[3]
(iii) Find the set of values of x for which 16 (2x − 3)3 − 4x is an increasing function of x.
11
[3]
The function f : x → 4 − 3 sin x is defined for the domain 0 ≤ x ≤ 2π .
(i) Solve the equation f (x) = 2.
[3]
(ii) Sketch the graph of y = f (x).
[2]
(iii) Find the set of values of k for which the equation f (x) = k has no solution.
[2]
The function g : x → 4 − 3 sin x is defined for the domain 12 π ≤ x ≤ A.
(iv) State the largest value of A for which g has an inverse.
[1]
(v) For this value of A, find the value of g−1 (3).
[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.
University of 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 2010
9709/12/M/J/10
w
w
ap
eP
m
e
tr
.X
w
om
.c
s
er
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
General Certificate of Education
Advanced Subsidiary Level and Advanced Level
9709/11
MATHEMATICS
Paper 1 Pure Mathematics 1 (P1)
May/June 2010
1 hour 45 minutes
*9210562645*
Additional Materials:
Answer Booklet/Paper
Graph Paper
List of Formulae (MF9)
READ THESE INSTRUCTIONS FIRST
If you have been given an Answer Booklet, follow the instructions on the front cover of the Booklet.
Write your Centre number, candidate number and name on all the work you hand in.
Write in dark blue or black pen.
You may use a soft pencil for any diagrams or graphs.
Do not use staples, paper clips, highlighters, glue or correction fluid.
Answer all the questions.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in
degrees, unless a different level of accuracy is specified in the question.
The use of an electronic calculator is expected, where appropriate.
You are reminded of the need for clear presentation in your answers.
At the end of the examination, fasten all your work securely together.
The number of marks is given in brackets [ ] at the end of each question or part question.
The total number of marks for this paper is 75.
Questions carrying smaller numbers of marks are printed earlier in the paper, and questions carrying larger
numbers of marks later in the paper.
This document consists of 4 printed pages.
© UCLES 2010
[Turn over
2
1
The acute angle x radians is such that tan x = k, where k is a positive constant. Express, in terms of k,
(i) tan(π − x),
[1]
(ii) tan( 12 π − x),
[1]
(iii) sin x.
[2]
5
2
3
(i) Find the first 3 terms in the expansion of 2x − in descending powers of x.
x
(ii) Hence find the coefficient of x in the expansion of 1 +
3
[3]
5
2
3
2x − .
2
x
x
[2]
The ninth term of an arithmetic progression is 22 and the sum of the first 4 terms is 49.
(i) Find the first term of the progression and the common difference.
[4]
The nth term of the progression is 46.
(ii) Find the value of n.
[2]
4
y
y = 6x – x 2
y=5
x
O
The diagram shows the curve y = 6x − x2 and the line y = 5. Find the area of the shaded region.
5
6
[6]
The function f is such that f (x) = 2 sin2 x − 3 cos2 x for 0 ≤ x ≤ π .
(i) Express f (x) in the form a + b cos2 x, stating the values of a and b.
[2]
(ii) State the greatest and least values of f (x).
[2]
(iii) Solve the equation f (x) + 1 = 0.
[3]
A curve is such that
1
dy
= 3x 2 − 6 and the point (9, 2) lies on the curve.
dx
(i) Find the equation of the curve.
[4]
(ii) Find the x-coordinate of the stationary point on the curve and determine the nature of the
stationary point.
[3]
© UCLES 2010
9709/11/M/J/10
3
7
y
C
y=2–
A
O
18
2x + 3
x
B
18
, which crosses the x-axis at A and the y-axis at B.
2x + 3
The normal to the curve at A crosses the y-axis at C .
The diagram shows part of the curve y = 2 −
(i) Show that the equation of the line AC is 9x + 4y = 27.
[6]
(ii) Find the length of BC .
[2]
8
y
B (15, 22)
C
x
O
A (3, –2)
The diagram shows a triangle ABC in which A is (3, −2) and B is (15, 22). The gradients of AB, AC
and BC are 2m, −2m and m respectively, where m is a positive constant.
(i) Find the gradient of AB and deduce the value of m.
[2]
(ii) Find the coordinates of C.
[4]
The perpendicular bisector of AB meets BC at D.
(iii) Find the coordinates of D.
© UCLES 2010
[4]
9709/11/M/J/10
[Turn over
4
9
The function f is defined by f : x → 2x2 − 12x + 7 for x ∈ >.
(i) Express f (x) in the form a(x − b)2 − c.
[3]
(ii) State the range of f.
[1]
(iii) Find the set of values of x for which f (x) < 21.
[3]
The function g is defined by g : x → 2x + k for x ∈ >.
(iv) Find the value of the constant k for which the equation gf (x) = 0 has two equal roots.
10
[4]
A
B
O
C
−−→
−−→
The diagram shows the parallelogram OABC . Given that OA = i + 3j + 3k and OC = 3i − j + k, find
−−→
(i) the unit vector in the direction of OB,
[3]
(ii) the acute angle between the diagonals of the parallelogram,
[5]
(iii) the perimeter of the parallelogram, correct to 1 decimal place.
[3]
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.
University of 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 2010
9709/11/M/J/10
w
w
ap
eP
m
e
tr
.X
w
om
.c
s
er
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
General Certificate of Education
Advanced Subsidiary Level and Advanced Level
9709/62
MATHEMATICS
Paper 6 Probability & Statistics 1 (S1)
May/June 2011
1 hour 15 minutes
*4236760161*
Additional Materials:
Answer Booklet/Paper
Graph Paper
List of Formulae (MF9)
READ THESE INSTRUCTIONS FIRST
If you have been given an Answer Booklet, follow the instructions on the front cover of the Booklet.
Write your Centre number, candidate number and name on all the work you hand in.
Write in dark blue or black pen.
You may use a soft pencil for any diagrams or graphs.
Do not use staples, paper clips, highlighters, glue or correction fluid.
Answer all the questions.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in
degrees, unless a different level of accuracy is specified in the question.
The use of an electronic calculator is expected, where appropriate.
You are reminded of the need for clear presentation in your answers.
At the end of the examination, fasten all your work securely together.
The number of marks is given in brackets [ ] at the end of each question or part question.
The total number of marks for this paper is 50.
Questions carrying smaller numbers of marks are printed earlier in the paper, and questions carrying larger
numbers of marks later in the paper.
This document consists of 3 printed pages and 1 blank page.
JC11 06_9709_62/RP
© UCLES 2011
[Turn over
2
1
A biased die was thrown 20 times and the number of 5s was noted. This experiment was repeated
many times and the average number of 5s was found to be 4.8. Find the probability that in the next
20 throws the number of 5s will be less than three.
[4]
2
In Scotland, in November, on average 80% of days are cloudy. Assume that the weather on any one
day is independent of the weather on other days.
(i) Use a normal approximation to find the probability of there being fewer than 25 cloudy days in
Scotland in November (30 days).
[4]
(ii) Give a reason why the use of a normal approximation is justified.
3
[1]
A sample of 36 data values, x, gave Σ(x − 45) = −148 and Σ(x − 45)2 = 3089.
(i) Find the mean and standard deviation of the 36 values.
[3]
(ii) One extra data value of 29 was added to the sample. Find the standard deviation of all 37 values.
[4]
4
(i) Find the number of different ways that the 9 letters of the word HAPPINESS can be arranged in
a line.
[1]
(ii) The 9 letters of the word HAPPINESS are arranged in random order in a line. Find the probability
that the 3 vowels (A, E, I) are not all next to each other.
[4]
(iii) Find the number of different selections of 4 letters from the 9 letters of the word HAPPINESS
which contain no Ps and either one or two Ss.
[3]
5
A hotel has 90 rooms. The table summarises information about the number of rooms occupied each
day for a period of 200 days.
Number of rooms occupied
Frequency
6
1 − 20
21 − 40
41 − 50
51 − 60
61 − 70
71 − 90
10
32
62
50
28
18
(i) Draw a cumulative frequency graph on graph paper to illustrate this information.
[4]
(ii) Estimate the number of days when over 30 rooms were occupied.
[2]
(iii) On 75% of the days at most n rooms were occupied. Estimate the value of n.
[2]
The lengths, in centimetres, of drinking straws produced in a factory have a normal distribution with
mean µ and variance 0.64. It is given that 10% of the straws are shorter than 20 cm.
(i) Find the value of µ .
[3]
(ii) Find the probability that, of 4 straws chosen at random, fewer than 2 will have a length between
21.5 cm and 22.5 cm.
[6]
© UCLES 2011
9709/62/M/J/11
3
7
Judy and Steve play a game using five cards numbered 3, 4, 5, 8, 9. Judy chooses a card at random,
looks at the number on it and replaces the card. Then Steve chooses a card at random, looks at the
number on it and replaces the card. If their two numbers are equal the score is 0. Otherwise, the
smaller number is subtracted from the larger number to give the score.
(i) Show that the probability that the score is 6 is 0.08.
[1]
(ii) Draw up a probability distribution table for the score.
[2]
(iii) Calculate the mean score.
[1]
If the score is 0 they play again. If the score is 4 or more Judy wins. Otherwise Steve wins. They
continue playing until one of the players wins.
(iv) Find the probability that Judy wins with the second choice of cards.
[3]
(v) Find an expression for the probability that Judy wins with the nth choice of cards.
[2]
© UCLES 2011
9709/62/M/J/11
4
BLANK PAGE
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.
University of 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.
9709/62/M/J/11