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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
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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
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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.
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[4]
9709/11/M/J/10
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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
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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
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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
3
1
O
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
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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.]
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9709/12/M/J/10
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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
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UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
General Certificate of Education
Advanced Subsidiary Level and Advanced Level
9709/13
MATHEMATICS
Paper 1 Pure Mathematics 1 (P1)
May/June 2010
1 hour 45 minutes
*2725209489*
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
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2
1
2
3
4
5
The first term of a geometric progression is 12 and the second term is −6. Find
(i) the tenth term of the progression,
[3]
(ii) the sum to infinity.
[2]
2 6
(i) Find the first three terms, in descending powers of x, in the expansion of x − .
x
[3]
2 6
(ii) Find the coefficient of x4 in the expansion of (1 + x2 )x − .
x
[2]
The function f : x → a + b cos x is defined for 0 ≤ x ≤ 2π . Given that f (0) = 10 and that f 23 π = 1, find
(i) the values of a and b,
[2]
(ii) the range of f,
[1]
(iii) the exact value of f 56 π .
[2]
(i) Show that the equation 2 sin x tan x + 3 = 0 can be expressed as 2 cos2 x − 3 cos x − 2 = 0.
[2]
(ii) Solve the equation 2 sin x tan x + 3 = 0 for 0◦ ≤ x ≤ 360◦ .
[3]
The equation of a curve is such that
dy
6
= √
. Given that the curve passes through the point
dx
(3x − 2)
P (2, 11), find
6
(i) the equation of the normal to the curve at P,
[3]
(ii) the equation of the curve.
[4]
Relative to an origin O, the position vectors of the points A, B and C are given by
−−→
OA = i − 2j + 4k,
−−→
OB = 3i + 2j + 8k,
−−→
OC = −i − 2j + 10k.
(i) Use a scalar product to find angle ABC .
[6]
(ii) Find the perimeter of triangle ABC , giving your answer correct to 2 decimal places.
[2]
© UCLES 2010
9709/13/M/J/10
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3
7
C
B
D
10 cm
12 cm
O
A
E
F
The diagram shows a metal plate ABCDEF which has been made by removing the two shaded regions
from a circle of radius 10 cm and centre O. The parallel edges AB and ED are both of length 12 cm.
(i) Show that angle DOE is 1.287 radians, correct to 4 significant figures.
[2]
(ii) Find the perimeter of the metal plate.
[3]
(iii) Find the area of the metal plate.
[3]
8
y
B
C (5, 4)
A
(–1, 2)
x
O
D
The diagram shows a rhombus ABCD in which the point A is (−1, 2), the point C is (5, 4) and the
point B lies on the y-axis. Find
(i) the equation of the perpendicular bisector of AC,
[3]
(ii) the coordinates of B and D,
[3]
(iii) the area of the rhombus.
[3]
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9709/13/M/J/10
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4
9
y
y = x + 4x
A
B
y=5
M
x
O
4
The diagram shows part of the curve y = x + which has a minimum point at M . The line y = 5
x
intersects the curve at the points A and B.
(i) Find the coordinates of A, B and M .
[5]
(ii) Find the volume obtained when the shaded region is rotated through 360◦ about the x-axis. [6]
10
The function f : x → 2x2 − 8x + 14 is defined for x ∈ >.
(i) Find the values of the constant k for which the line y + kx = 12 is a tangent to the curve y = f (x).
[4]
(ii) Express f (x) in the form a(x + b)2 + c, where a, b and c are constants.
[3]
(iii) Find the range of f.
[1]
The function g : x → 2x2 − 8x + 14 is defined for x ≥ A.
(iv) Find the smallest value of A for which g has an inverse.
[1]
(v) For this value of A, find an expression for g−1 (x) in terms of x.
[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
9709/13/M/J/10
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UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
General Certificate of Education Advanced Subsidiary Level
9709/21
MATHEMATICS
Paper 2 Pure Mathematics 2 (P2)
May/June 2010
1 hour 15 minutes
*6129767079*
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
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2
1
Solve the inequality | 2x − 3 | > 5.
6
2
Show that ä
0
3
[3]
1
dx = 2 ln 2.
x+2
[4]
(i) Show that the equation tan(x + 45◦ ) = 6 tan x can be written in the form
6 tan2 x − 5 tan x + 1 = 0.
(ii) Hence solve the equation tan(x + 45◦ ) = 6 tan x, for 0◦ < x < 180◦ .
4
5
[3]
[3]
The polynomial x3 + 3x2 + 4x + 2 is denoted by f (x).
(i) Find the quotient and remainder when f (x) is divided by x2 + x − 1.
[4]
(ii) Use the factor theorem to show that (x + 1) is a factor of f (x).
[2]
(i) Given that y = 2x , show that the equation
2x + 3(2−x ) = 4
can be written in the form
y2 − 4y + 3 = 0.
[3]
(ii) Hence solve the equation
2x + 3(2−x ) = 4,
giving the values of x correct to 3 significant figures where appropriate.
6
[3]
The equation of a curve is
x2 y + y2 = 6x.
(i) Show that
dy 6 − 2xy
=
.
dx x2 + 2y
[4]
(ii) Find the equation of the tangent to the curve at the point with coordinates (1, 2), giving your
answer in the form ax + by + c = 0.
[3]
© UCLES 2010
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3
7
(i) By sketching a suitable pair of graphs, show that the equation
e2x = 2 − x
has only one root.
[2]
(ii) Verify by calculation that this root lies between x = 0 and x = 0.5.
[2]
(iii) Show that, if a sequence of values given by the iterative formula
xn+1 = 12 ln(2 − xn )
converges, then it converges to the root of the equation in part (i).
[1]
(iv) Use this iterative formula, with initial value x1 = 0.25, to determine the root correct to 2 decimal
places. Give the result of each iteration to 4 decimal places.
[3]
8
(i) By differentiating
dy
cos x
, show that if y = cot x then
= − cosec2 x.
sin x
dx
[3]
(ii) By expressing cot2 x in terms of cosec2 x and using the result of part (i), show that
1π
2
cot2 x dx = 1 − 14 π .
[4]
(iii) Express cos 2x in terms of sin2 x and hence show that
1
can be expressed as 12 cosec2 x.
1 − cos 2x
ã
1
π
4
Hence, using the result of part (i), find
ä
© UCLES 2010
1
dx.
1 − cos 2x
9709/21/M/J/10
[3]
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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.
9709/21/M/J/10
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UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
General Certificate of Education Advanced Subsidiary Level
9709/22
MATHEMATICS
Paper 2 Pure Mathematics 2 (P2)
May/June 2010
1 hour 15 minutes
*5795319037*
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
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2
1
Given that 13x = (2.8)y , use logarithms to show that y = kx and find the value of k correct to 3 significant
figures.
[3]
2
y
R
O
1
2
3
x
The diagram shows part of the curve y = xe−x . The shaded region R is bounded by the curve and by
the lines x = 2, x = 3 and y = 0.
(i) Use the trapezium rule with two intervals to estimate the area of R, giving your answer correct
to 2 decimal places.
[3]
(ii) State, with a reason, whether the trapezium rule gives an under-estimate or an over-estimate of
the true value of the area of R.
[1]
3
4
Solve the inequality | 2x − 1| < | x + 4 |.
(a) Show that ã
1
π
4
0
[4]
cos 2x dx = 12 .
[2]
(b) By using an appropriate trigonometrical identity, find the exact value of
ã
1
π
3
3 tan2 x dx.
[4]
1
π
6
5
The equation of a curve is y = x3 e−x .
(i) Show that the curve has a stationary point where x = 3.
[3]
(ii) Find the equation of the tangent to the curve at the point where x = 1.
[4]
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3
6
(i) By sketching a suitable pair of graphs, show that the equation
ln x = 2 − x2
has only one root.
[2]
(ii) Verify by calculation that this root lies between x = 1.3 and x = 1.4.
[2]
(iii) Show that, if a sequence of values given by the iterative formula
√
xn+1 = (2 − ln xn )
converges, then it converges to the root of the equation in part (i).
[1]
√
(iv) Use the iterative formula xn+1 = (2 − ln xn ) to determine the root correct to 2 decimal places.
Give the result of each iteration to 4 decimal places.
[3]
7
The polynomial 2x3 + ax2 + bx + 6, where a and b are constants, is denoted by p(x). It is given that
when p(x) is divided by (x − 3) the remainder is 30, and that when p(x) is divided by (x + 1) the
remainder is 18.
(i) Find the values of a and b.
[5]
(ii) When a and b have these values, verify that (x − 2) is a factor of p(x) and hence factorise p(x)
completely.
[4]
8
(i) Prove the identity
√
sin(x − 30◦ ) + cos(x − 60◦ ) ≡ ( 3) sin x.
[3]
(ii) Hence solve the equation
sin(x − 30◦ ) + cos(x − 60◦ ) = 12 sec x,
for 0◦ < x < 360◦ .
© UCLES 2010
[6]
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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.
9709/22/M/J/10
www.maxpapers.com
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
General Certificate of Education Advanced Subsidiary Level
9709/23
MATHEMATICS
Paper 2 Pure Mathematics 2 (P2)
May/June 2010
1 hour 15 minutes
*7552068932*
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.
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2
1
Given that 13x = (2.8)y , use logarithms to show that y = kx and find the value of k correct to 3 significant
figures.
[3]
2
y
R
O
1
2
3
x
The diagram shows part of the curve y = xe−x . The shaded region R is bounded by the curve and by
the lines x = 2, x = 3 and y = 0.
(i) Use the trapezium rule with two intervals to estimate the area of R, giving your answer correct
to 2 decimal places.
[3]
(ii) State, with a reason, whether the trapezium rule gives an under-estimate or an over-estimate of
the true value of the area of R.
[1]
3
4
Solve the inequality | 2x − 1| < | x + 4 |.
(a) Show that ã
1
π
4
0
[4]
cos 2x dx = 12 .
[2]
(b) By using an appropriate trigonometrical identity, find the exact value of
ã
1
π
3
3 tan2 x dx.
[4]
1
π
6
5
The equation of a curve is y = x3 e−x .
(i) Show that the curve has a stationary point where x = 3.
[3]
(ii) Find the equation of the tangent to the curve at the point where x = 1.
[4]
© UCLES 2010
9709/23/M/J/10
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3
6
(i) By sketching a suitable pair of graphs, show that the equation
ln x = 2 − x2
has only one root.
[2]
(ii) Verify by calculation that this root lies between x = 1.3 and x = 1.4.
[2]
(iii) Show that, if a sequence of values given by the iterative formula
√
xn+1 = (2 − ln xn )
converges, then it converges to the root of the equation in part (i).
[1]
√
(iv) Use the iterative formula xn+1 = (2 − ln xn ) to determine the root correct to 2 decimal places.
Give the result of each iteration to 4 decimal places.
[3]
7
The polynomial 2x3 + ax2 + bx + 6, where a and b are constants, is denoted by p(x). It is given that
when p(x) is divided by (x − 3) the remainder is 30, and that when p(x) is divided by (x + 1) the
remainder is 18.
(i) Find the values of a and b.
[5]
(ii) When a and b have these values, verify that (x − 2) is a factor of p(x) and hence factorise p(x)
completely.
[4]
8
(i) Prove the identity
√
sin(x − 30◦ ) + cos(x − 60◦ ) ≡ ( 3) sin x.
[3]
(ii) Hence solve the equation
sin(x − 30◦ ) + cos(x − 60◦ ) = 12 sec x,
for 0◦ < x < 360◦ .
© UCLES 2010
[6]
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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.
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UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
General Certificate of Education Advanced Level
9709/31
MATHEMATICS
Paper 3 Pure Mathematics 3 (P3)
May/June 2010
1 hour 45 minutes
*1590627831*
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
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2
1
Solve the inequality | x + 3a | > 2 | x − 2a |, where a is a positive constant.
2
Solve the equation
[4]
sin θ = 2 cos 2θ + 1,
giving all solutions in the interval 0◦ ≤ θ ≤ 360◦ .
3
4
[6]
The variables x and y satisfy the equation xn y = C, where n and C are constants. When x = 1.10,
y = 5.20, and when x = 3.20, y = 1.05.
(i) Find the values of n and C.
[5]
(ii) Explain why the graph of ln y against ln x is a straight line.
[1]
(i) Using the expansions of cos(3x − x) and cos(3x + x), prove that
1
2 (cos 2x
− cos 4x) ≡ sin 3x sin x.
[3]
(ii) Hence show that
ã
1
π
3
sin 3x sin x dx =
1π
6
5
1
8
√
3.
[3]
Given that y = 0 when x = 1, solve the differential equation
xy
dy
= y2 + 4,
dx
obtaining an expression for y2 in terms of x.
© UCLES 2010
9709/31/M/J/10
[6]
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3
6
C
r
x rad
O
A
B
The diagram shows a semicircle ACB with centre O and radius r. The angle BOC is x radians. The
area of the shaded segment is a quarter of the area of the semicircle.
(i) Show that x satisfies the equation
x = 34 π − sin x.
[3]
(ii) This equation has one root. Verify by calculation that the root lies between 1.3 and 1.5.
[2]
(iii) Use the iterative formula
xn+1 = 34 π − sin xn
to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal
places.
[3]
7
The complex number 2 + 2i is denoted by u.
(i) Find the modulus and argument of u.
[2]
(ii) Sketch an Argand diagram showing the points representing the complex numbers 1, i and u. Shade
the region whose points represent the complex numbers ß which satisfy both the inequalities
|ß − 1| ≤ |ß − i | and |ß − u | ≤ 1.
[4]
(iii) Using your diagram, calculate the value of |ß| for the point in this region for which arg ß is least.
[3]
8
(i) Express
2
in partial fractions.
(x + 1)(x + 3)
[2]
(ii) Using your answer to part (i), show that
2
2
1
1
1
1
≡
−
+
+
.
2
(x + 1)(x + 3)
x + 1 x + 3 (x + 3)2
(x + 1)
1
(iii) Hence show that ä
0
© UCLES 2010
4
dx =
(x + 1) (x + 3)2
2
7
12
− ln 32 .
9709/31/M/J/10
[2]
[5]
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4
9
y
P
O
1
The diagram shows the curve y =
r
1
x
1−x
.
1+x
dy
1−x
, obtain an expression for
in terms of x. Hence show that the
1+x
dx
√
gradient of the normal to the curve at the point (x, y) is (1 + x) (1 − x2 ).
[5]
(i) By first differentiating
(ii) The gradient of the normal to the curve has its maximum value at the point P shown in the
diagram. Find, by differentiation, the x-coordinate of P.
[4]
10
The lines l and m have vector equations
r = i + j + k + s(i − j + 2k)
and
r = 4i + 6j + k + t(2i + 2j + k)
respectively.
(i) Show that l and m intersect.
[4]
(ii) Calculate the acute angle between the lines.
[3]
(iii) Find the equation of the plane containing l and m, giving your answer in the form ax + by + cß = d.
[5]
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
9709/31/M/J/10
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UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
General Certificate of Education Advanced Level
9709/32
MATHEMATICS
Paper 3 Pure Mathematics 3 (P3)
May/June 2010
1 hour 45 minutes
*9385525965*
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.
© UCLES 2010
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2
1
Solve the equation
2x + 1
= 5,
2x − 1
giving your answer correct to 3 significant figures.
[4]
π
2
Show that ã x2 sin x dx = π 2 − 4.
[5]
0
3
It is given that cos a = 35 , where 0◦ < a < 90◦ . Showing your working and without using a calculator
to evaluate a,
(i) find the exact value of sin(a − 30◦ ),
[3]
(ii) find the exact value of tan 2a, and hence find the exact value of tan 3a.
[4]
4
y
1
O
2p
p
x
M
The diagram shows the curve y =
sin x
for 0 < x ≤ 2π , and its minimum point M .
x
(i) Show that the x-coordinate of M satisfies the equation
x = tan x.
[4]
(ii) The iterative formula
xn+1 = tan−1 (xn ) + π
can be used to determine the x-coordinate of M . Use this formula to determine the x-coordinate
of M correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
[3]
5
The polynomial 2x3 + 5x2 + ax + b, where a and b are constants, is denoted by p(x). It is given that
(2x + 1) is a factor of p(x) and that when p(x) is divided by (x + 2) the remainder is 9.
(i) Find the values of a and b.
[5]
(ii) When a and b have these values, factorise p(x) completely.
[3]
© UCLES 2010
9709/32/M/J/10
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3
6
The equation of a curve is
x ln y = 2x + 1.
(i) Show that
y
dy
= − 2.
dx
x
[4]
(ii) Find the equation of the tangent to the curve at the point where y = 1, giving your answer in the
form ax + by + c = 0.
[4]
7
The variables x and t are related by the differential equation
e2t
dx
= cos2 x,
dt
where t ≥ 0. When t = 0, x = 0.
8
(i) Solve the differential equation, obtaining an expression for x in terms of t.
[6]
(ii) State what happens to the value of x when t becomes very large.
[1]
(iii) Explain why x increases as t increases.
[1]
The variable complex number ß is given by
ß = 1 + cos 2θ + i sin 2θ ,
where θ takes all values in the interval − 12 π < θ < 12 π .
(i) Show that the modulus of ß is 2 cos θ and the argument of ß is θ .
(ii) Prove that the real part of
9
1
is constant.
ß
[6]
[3]
The plane p has equation 3x + 2y + 4ß = 13. A second plane q is perpendicular to p and has equation
ax + y + ß = 4, where a is a constant.
(i) Find the value of a.
[3]
(ii) The line with equation r = j − k + λ (i + 2j + 2k) meets the plane p at the point A and the plane q
at the point B. Find the length of AB.
[6]
10
(i) Find the values of the constants A, B, C and D such that
2x3 − 1
B C
D
≡A+ + 2 +
.
x x
2x − 1
x2 (2x − 1)
[5]
(ii) Hence show that
2
ä
1
© UCLES 2010
2x3 − 1
dx = 32 + 12 ln
x2 (2x − 1)
9709/32/M/J/10
16
27 .
[5]
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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.
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UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
General Certificate of Education Advanced Level
9709/33
MATHEMATICS
Paper 3 Pure Mathematics 3 (P3)
May/June 2010
1 hour 45 minutes
*1169365262*
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.
© UCLES 2010
[Turn over
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2
1
Solve the inequality | x − 3 | > 2 | x + 1|.
2
The variables x and y satisfy the equation y3 = Ae2x , where A is a constant. The graph of ln y against
x is a straight line.
[4]
(i) Find the gradient of this line.
[2]
(ii) Given that the line intersects the axis of ln y at the point where ln y = 0.5, find the value of A
correct to 2 decimal places.
[2]
3
Solve the equation
tan(45◦ − x) = 2 tan x,
giving all solutions in the interval 0◦ < x < 180◦ .
4
[5]
Given that x = 1 when t = 0, solve the differential equation
dx 1 x
= − ,
dt x 4
obtaining an expression for x2 in terms of t.
5
[7]
y
M
O
x
p
The diagram shows the curve y = e−x − e−2x and its maximum point M . The x-coordinate of M is
denoted by p.
(i) Find the exact value of p.
[4]
(ii) Show that the area of the shaded region bounded by the curve, the x-axis and the line x = p is
[4]
equal to 81 .
© UCLES 2010
9709/33/M/J/10
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3
6
The curve y =
ln x
has one stationary point.
x+1
(i) Show that the x-coordinate of this point satisfies the equation
x=
x+1
,
ln x
and that this x-coordinate lies between 3 and 4.
[5]
(ii) Use the iterative formula
xn+1 =
xn + 1
ln xn
to determine the x-coordinate correct to 2 decimal places. Give the result of each iteration to
4 decimal places.
[3]
7
(i) Prove the identity cos 3θ ≡ 4 cos3 θ − 3 cos θ .
[4]
(ii) Using this result, find the exact value of
ã
1π
2
cos3 θ dθ .
[4]
1
π
3
8
(a) The equation 2x3 − x2 + 2√x + 12 = 0 has one real root and two complex roots. Showing your
working, verify that 1 + i 3 is one of the complex roots. State the other complex root.
[4]
√
(b) On a sketch of an Argand diagram, show the point representing the complex number 1 + i 3.
On the same diagram, shade the region
the complex numbers ß which
√ whose points represent
1
satisfy both the inequalities |ß − 1 − i 3 | ≤ 1 and arg ß ≤ 3 π .
[5]
9
(i) Express
4 + 5x − x2
in partial fractions.
(1 − 2x)(2 + x)2
(ii) Hence obtain the expansion of
the term in x2 .
10
[5]
4 + 5x − x2
in ascending powers of x, up to and including
(1 − 2x)(2 + x)2
[5]
The straight line l has equation r = 2i − j − 4k + λ (i + 2j + 2k). The plane p has equation 3x − y + 2ß = 9.
The line l intersects the plane p at the point A.
(i) Find the position vector of A.
[3]
(ii) Find the acute angle between l and p.
[4]
(iii) Find an equation for the plane which contains l and is perpendicular to p, giving your answer in
the form ax + by + cß = d .
[5]
© UCLES 2010
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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.
9709/33/M/J/10
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UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
General Certificate of Education
Advanced Subsidiary Level and Advanced Level
9709/41
MATHEMATICS
Paper 4 Mechanics 1 (M1)
May/June 2010
1 hour 15 minutes
*7490804446*
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.
Where a numerical value for the acceleration due to gravity is needed, use 10 m s−2 .
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 4 printed pages.
© UCLES 2010
[Turn over
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2
1
A car of mass 1150 kg travels up a straight hill inclined at 1.2◦ to the horizontal. The resistance to
motion of the car is 975 N. Find the acceleration of the car at an instant when it is moving with speed
16 m s−1 and the engine is working at a power of 35 kW.
[4]
2
v (m s 1)
0.18
O
2
6
8
11
t (s)
The diagram shows the velocity-time graph for the motion of a machine’s cutting tool. The graph
consists of five straight line segments. The tool moves forward for 8 s while cutting and then takes
3 s to return to its starting position. Find
(i) the acceleration of the tool during the first 2 s of the motion,
[1]
(ii) the distance the tool moves forward while cutting,
[2]
(iii) the greatest speed of the tool during the return to its starting position.
[2]
3
7N
45°
A small ring of mass 0.8 kg is threaded on a rough rod which is fixed horizontally. The ring is
in equilibrium, acted on by a force of magnitude 7 N pulling upwards at 45◦ to the horizontal (see
diagram).
(i) Show that the normal component of the contact force acting on the ring has magnitude 3.05 N,
correct to 3 significant figures.
[2]
(ii) The ring is in limiting equilibrium. Find the coefficient of friction between the ring and the rod.
[3]
© UCLES 2010
9709/41/M/J/10
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3
4
y
370 N
160 N
O
a
x
250 N
Coplanar forces of magnitudes 250 N, 160 N and 370 N act at a point O in the directions shown in the
diagram, where the angle α is such that sin α = 0.28 and cos α = 0.96. Calculate the magnitude of the
resultant of the three forces. Calculate also the angle that the resultant makes with the x-direction.
[7]
5
P and Q are fixed points on a line of greatest slope of an inclined plane. The point Q is at a height of
0.45 m above the level of P. A particle of mass 0.3 kg moves upwards along the line PQ.
(i) Given that the plane is smooth and that the particle just reaches Q, find the speed with which it
passes through P.
[3]
(ii) It is given instead that the plane is rough. The particle passes through P with the same speed
as that found in part (i), and just reaches a point R which is between P and Q. The work done
against the frictional force in moving from P to R is 0.39 J. Find the potential energy gained by
the particle in moving from P to R and hence find the height of R above the level of P.
[4]
[Questions 6 and 7 are printed on the next page.]
© UCLES 2010
9709/41/M/J/10
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4
6
2.1 m
A
B
2m
Particles A and B, of masses 0.2 kg and 0.45 kg respectively, are connected by a light inextensible
string of length 2.8 m. The string passes over a small smooth pulley at the edge of a rough horizontal
surface, which is 2 m above the floor. Particle A is held in contact with the surface at a distance of
2.1 m from the pulley and particle B hangs freely (see diagram). The coefficient of friction between
A and the surface is 0.3. Particle A is released and the system begins to move.
(i) Find the acceleration of the particles and show that the speed of B immediately before it hits the
floor is 3.95 m s−1 , correct to 3 significant figures.
[7]
(ii) Given that B remains on the floor, find the speed with which A reaches the pulley.
7
[4]
A vehicle is moving in a straight line. The velocity v m s−1 at time t s after the vehicle starts is given
by
v = A(t − 0.05t2 ) for 0 ≤ t ≤ 15,
B
v = 2 for t ≥ 15,
t
where A and B are constants. The distance travelled by the vehicle between t = 0 and t = 15 is 225 m.
(i) Find the value of A and show that B = 3375.
[5]
(ii) Find an expression in terms of t for the total distance travelled by the vehicle when t ≥ 15.
[3]
(iii) Find the speed of the vehicle when it has travelled a total distance of 315 m.
[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
9709/41/M/J/10
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UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
General Certificate of Education
Advanced Subsidiary Level and Advanced Level
9709/42
MATHEMATICS
Paper 4 Mechanics 1 (M1)
May/June 2010
1 hour 15 minutes
*8896238783*
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.
Where a numerical value for the acceleration due to gravity is needed, use 10 m s−2 .
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 4 printed pages.
© UCLES 2010
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2
1
A car of mass 1150 kg travels up a straight hill inclined at 1.2◦ to the horizontal. The resistance to
motion of the car is 975 N. Find the acceleration of the car at an instant when it is moving with speed
16 m s−1 and the engine is working at a power of 35 kW.
[4]
2
v (m s 1)
0.18
O
2
6
8
11
t (s)
The diagram shows the velocity-time graph for the motion of a machine’s cutting tool. The graph
consists of five straight line segments. The tool moves forward for 8 s while cutting and then takes
3 s to return to its starting position. Find
(i) the acceleration of the tool during the first 2 s of the motion,
[1]
(ii) the distance the tool moves forward while cutting,
[2]
(iii) the greatest speed of the tool during the return to its starting position.
[2]
3
7N
45°
A small ring of mass 0.8 kg is threaded on a rough rod which is fixed horizontally. The ring is
in equilibrium, acted on by a force of magnitude 7 N pulling upwards at 45◦ to the horizontal (see
diagram).
(i) Show that the normal component of the contact force acting on the ring has magnitude 3.05 N,
correct to 3 significant figures.
[2]
(ii) The ring is in limiting equilibrium. Find the coefficient of friction between the ring and the rod.
[3]
© UCLES 2010
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3
4
y
370 N
160 N
O
a
x
250 N
Coplanar forces of magnitudes 250 N, 160 N and 370 N act at a point O in the directions shown in the
diagram, where the angle α is such that sin α = 0.28 and cos α = 0.96. Calculate the magnitude of the
resultant of the three forces. Calculate also the angle that the resultant makes with the x-direction.
[7]
5
P and Q are fixed points on a line of greatest slope of an inclined plane. The point Q is at a height of
0.45 m above the level of P. A particle of mass 0.3 kg moves upwards along the line PQ.
(i) Given that the plane is smooth and that the particle just reaches Q, find the speed with which it
passes through P.
[3]
(ii) It is given instead that the plane is rough. The particle passes through P with the same speed
as that found in part (i), and just reaches a point R which is between P and Q. The work done
against the frictional force in moving from P to R is 0.39 J. Find the potential energy gained by
the particle in moving from P to R and hence find the height of R above the level of P.
[4]
[Questions 6 and 7 are printed on the next page.]
© UCLES 2010
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4
6
2.1 m
A
B
2m
Particles A and B, of masses 0.2 kg and 0.45 kg respectively, are connected by a light inextensible
string of length 2.8 m. The string passes over a small smooth pulley at the edge of a rough horizontal
surface, which is 2 m above the floor. Particle A is held in contact with the surface at a distance of
2.1 m from the pulley and particle B hangs freely (see diagram). The coefficient of friction between
A and the surface is 0.3. Particle A is released and the system begins to move.
(i) Find the acceleration of the particles and show that the speed of B immediately before it hits the
floor is 3.95 m s−1 , correct to 3 significant figures.
[7]
(ii) Given that B remains on the floor, find the speed with which A reaches the pulley.
7
[4]
A vehicle is moving in a straight line. The velocity v m s−1 at time t s after the vehicle starts is given
by
v = A(t − 0.05t2 ) for 0 ≤ t ≤ 15,
B
v = 2 for t ≥ 15,
t
where A and B are constants. The distance travelled by the vehicle between t = 0 and t = 15 is 225 m.
(i) Find the value of A and show that B = 3375.
[5]
(ii) Find an expression in terms of t for the total distance travelled by the vehicle when t ≥ 15.
[3]
(iii) Find the speed of the vehicle when it has travelled a total distance of 315 m.
[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
9709/42/M/J/10
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UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
General Certificate of Education
Advanced Subsidiary Level and Advanced Level
9709/43
MATHEMATICS
Paper 4 Mechanics 1 (M1)
May/June 2010
1 hour 15 minutes
*6164918542*
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.
Where a numerical value for the acceleration due to gravity is needed, use 10 m s−2 .
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 4 printed pages.
© UCLES 2010
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2
1
5.5 N
6.8 N
a°
7.3 N
Three coplanar forces act at a point. The magnitudes of the forces are 5.5 N, 6.8 N and 7.3 N, and the
directions in which the forces act are as shown in the diagram. Given that the resultant of the three
forces is in the same direction as the force of magnitude 6.8 N, find the value of α and the magnitude
of the resultant.
[4]
2
A particle starts at a point O and moves along a straight line. Its velocity t s after leaving O is
(1.2t − 0.12t2 ) m s−1 . Find the displacement of the particle from O when its acceleration is 0.6 m s−2 .
[5]
3
A load is pulled along a horizontal straight track, from A to B, by a force of magnitude P N which
acts at an angle of 30◦ upwards from the horizontal. The distance AB is 80 m. The speed of the load
is constant and equal to 1.2 m s−1 as it moves from A to the mid-point M of AB.
(i) For the motion from A to M the value of P is 25. Calculate the work done by the force as the
load moves from A to M .
[2]
The speed of the load increases from 1.2 m s−1 as it moves from M towards B. For the motion from
M to B the value of P is 50 and the work done against resistance is the same as that for the motion
from A to M . The mass of the load is 35 kg.
(ii) Find the gain in kinetic energy of the load as it moves from M to B and hence find the speed with
which it reaches B.
[5]
© UCLES 2010
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3
4
A
B
60°
60°
The diagram shows a vertical cross-section of a triangular prism which is fixed so that two of its faces
are inclined at 60◦ to the horizontal. One of these faces is smooth and one is rough. Particles A and
B, of masses 0.36 kg and 0.24 kg respectively, are attached to the ends of a light inextensible string
which passes over a small smooth pulley fixed at the highest point of the cross-section. B is held at
rest at a point of the cross-section on the rough face and A hangs freely in contact with the smooth
face (see diagram). B is released and starts to move up the face with acceleration 0.25 m s−2 .
(i) By considering the motion of A, show that the tension in the string is 3.03 N, correct to 3 significant
figures.
[2]
(ii) Find the coefficient of friction between B and the rough face, correct to 2 significant figures.
[6]
5
A ball moves on the horizontal surface of a billiards table with deceleration of constant magnitude
d m s−2 . The ball starts at A with speed 1.4 m s−1 and reaches the edge of the table at B, 1.2 s later,
with speed 1.1 m s−1 .
(i) Find the distance AB and the value of d .
[3]
AB is at right angles to the edge of the table containing B. The table has a low wall along each of its
edges and the ball rebounds from the wall at B and moves directly towards A. The ball comes to rest
at C where the distance BC is 2 m.
(ii) Find the speed with which the ball starts to move towards A and the time taken for the ball to
travel from B to C.
[3]
(iii) Sketch a velocity-time graph for the motion of the ball, from the time the ball leaves A until it
comes to rest at C, showing on the axes the values of the velocity and the time when the ball is
[2]
at A, at B and at C.
6
Particles P and Q move on a line of greatest slope of a smooth inclined plane. P is released from rest
at a point O on the line and 2 s later passes through the point A with speed 3.5 m s−1 .
(i) Find the acceleration of P and the angle of inclination of the plane.
[4]
At the instant that P passes through A the particle Q is released from rest at O. At time t s after Q is
released from O, the particles P and Q are 4.9 m apart.
(ii) Find the value of t.
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[5]
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4
7
A
B
PN
Two rectangular boxes A and B are of identical size. The boxes are at rest on a rough horizontal
floor with A on top of B. Box A has mass 200 kg and box B has mass 250 kg. A horizontal force of
magnitude P N is applied to B (see diagram). The boxes remain at rest if P ≤ 3150 and start to move
if P > 3150.
(i) Find the coefficient of friction between B and the floor.
[3]
The coefficient of friction between the two boxes is 0.2. Given that P > 3150 and that no sliding takes
place between the boxes,
(ii) show that the acceleration of the boxes is not greater than 2 m s−2 ,
[3]
(iii) find the maximum possible value of P.
[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
9709/43/M/J/10
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UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
General Certificate of Education Advanced Level
9709/51
MATHEMATICS
Paper 5 Mechanics 2 (M2)
May/June 2010
1 hour 15 minutes
*2470080101*
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.
Where a numerical value for the acceleration due to gravity is needed, use 10 m s−2 .
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 4 printed pages.
© UCLES 2010
[Turn over
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2
1
20 cm
40 cm
A frame consists of a uniform semicircular wire of radius 20 cm and mass 2 kg, and a uniform straight
wire of length 40 cm and mass 0.9 kg. The ends of the semicircular wire are attached to the ends of
the straight wire (see diagram). Find the distance of the centre of mass of the frame from the straight
wire.
[4]
2
30
cm
m
rc
35°
A uniform solid cone has height 30 cm and base radius r cm. The cone is placed with its axis vertical
on a rough horizontal plane. The plane is slowly tilted and the cone remains in equilibrium until
the angle of inclination of the plane reaches 35◦ , when the cone topples. The diagram shows a
cross-section of the cone.
(i) Find the value of r.
[3]
(ii) Show that the coefficient of friction between the cone and the plane is greater than 0.7.
[2]
© UCLES 2010
9709/51/M/J/10
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3
3
q
2m
A particle of mass 0.24 kg is attached to one end of a light inextensible string of length 2 m. The other
end of the string is attached to a fixed point. The particle moves with constant speed in a horizontal
circle. The string makes an angle θ with the vertical (see diagram), and the tension in the string
is T N. The acceleration of the particle has magnitude 7.5 m s−2 .
(i) Show that tan θ = 0.75 and find the value of T .
[4]
(ii) Find the speed of the particle.
[2]
4
1.5 m
A
30°
B
1.5 m
C
3.5 m
D
A uniform lamina of weight 15 N is in the form of a trapezium ABCD with dimensions as shown in
the diagram. The lamina is freely hinged at A to a fixed point. One end of a light inextensible string
is attached to the lamina at B. The lamina is in equilibrium with AB horizontal; the string is taut and
in the same vertical plane as the lamina, and makes an angle of 30◦ upwards from the horizontal (see
diagram). Find the tension in the string.
[5]
© UCLES 2010
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4
5
A particle is projected from a point O on horizontal ground. The velocity of projection has magnitude
20 m s−1 and direction upwards at an angle θ to the horizontal. The particle passes through the point
which is 7 m above the ground and 16 m horizontally from O, and hits the ground at the point A.
(i) Using the equation of the particle’s trajectory and the identity sec2 θ = 1 + tan2 θ , show that the
possible values of tan θ are 34 and 17
[4]
4.
(ii) Find the distance OA for each of the two possible values of tan θ .
[3]
(iii) Sketch in the same diagram the two possible trajectories.
[2]
6
4.8 m
A
M
B
0.7 m
P
A particle P of mass 0.35 kg is attached to the mid-point of a light elastic string of natural length 4 m.
The ends of the string are attached to fixed points A and B which are 4.8 m apart at the same horizontal
level. P hangs in equilibrium at a point 0.7 m vertically below the mid-point M of AB (see diagram).
(i) Find the tension in the string and hence show that the modulus of elasticity of the string is 25 N.
[4]
P is now held at rest at a point 1.8 m vertically below M , and is then released.
(ii) Find the speed with which P passes through M .
7
[6]
A particle P of mass 0.25 kg moves in a straight line on a smooth horizontal surface. P starts at the
point O with speed 10 m s−1 and moves towards a fixed point A on the line. At time t s the displacement
of P from O is x m and the velocity of P is v m s−1 . A resistive force of magnitude (5 − x) N acts on P
in the direction towards O.
(i) Form a differential equation in v and x. By solving this differential equation, show that v = 10 − 2x.
[6]
(ii) Find x in terms of t, and hence show that the particle is always less than 5 m from O.
[5]
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
9709/51/M/J/10
www.maxpapers.com
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
General Certificate of Education Advanced Level
9709/52
MATHEMATICS
Paper 5 Mechanics 2 (M2)
May/June 2010
1 hour 15 minutes
*7154044207*
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.
Where a numerical value for the acceleration due to gravity is needed, use 10 m s−2 .
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 4 printed pages.
© UCLES 2010
[Turn over
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2
1
20 cm
40 cm
A frame consists of a uniform semicircular wire of radius 20 cm and mass 2 kg, and a uniform straight
wire of length 40 cm and mass 0.9 kg. The ends of the semicircular wire are attached to the ends of
the straight wire (see diagram). Find the distance of the centre of mass of the frame from the straight
wire.
[4]
2
30
cm
m
rc
35°
A uniform solid cone has height 30 cm and base radius r cm. The cone is placed with its axis vertical
on a rough horizontal plane. The plane is slowly tilted and the cone remains in equilibrium until
the angle of inclination of the plane reaches 35◦ , when the cone topples. The diagram shows a
cross-section of the cone.
(i) Find the value of r.
[3]
(ii) Show that the coefficient of friction between the cone and the plane is greater than 0.7.
[2]
© UCLES 2010
9709/52/M/J/10
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3
3
q
2m
A particle of mass 0.24 kg is attached to one end of a light inextensible string of length 2 m. The other
end of the string is attached to a fixed point. The particle moves with constant speed in a horizontal
circle. The string makes an angle θ with the vertical (see diagram), and the tension in the string
is T N. The acceleration of the particle has magnitude 7.5 m s−2 .
(i) Show that tan θ = 0.75 and find the value of T .
[4]
(ii) Find the speed of the particle.
[2]
4
1.5 m
A
30°
B
1.5 m
C
3.5 m
D
A uniform lamina of weight 15 N is in the form of a trapezium ABCD with dimensions as shown in
the diagram. The lamina is freely hinged at A to a fixed point. One end of a light inextensible string
is attached to the lamina at B. The lamina is in equilibrium with AB horizontal; the string is taut and
in the same vertical plane as the lamina, and makes an angle of 30◦ upwards from the horizontal (see
diagram). Find the tension in the string.
[5]
© UCLES 2010
9709/52/M/J/10
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4
5
A particle is projected from a point O on horizontal ground. The velocity of projection has magnitude
20 m s−1 and direction upwards at an angle θ to the horizontal. The particle passes through the point
which is 7 m above the ground and 16 m horizontally from O, and hits the ground at the point A.
(i) Using the equation of the particle’s trajectory and the identity sec2 θ = 1 + tan2 θ , show that the
possible values of tan θ are 34 and 17
[4]
4.
(ii) Find the distance OA for each of the two possible values of tan θ .
[3]
(iii) Sketch in the same diagram the two possible trajectories.
[2]
6
4.8 m
A
M
B
0.7 m
P
A particle P of mass 0.35 kg is attached to the mid-point of a light elastic string of natural length 4 m.
The ends of the string are attached to fixed points A and B which are 4.8 m apart at the same horizontal
level. P hangs in equilibrium at a point 0.7 m vertically below the mid-point M of AB (see diagram).
(i) Find the tension in the string and hence show that the modulus of elasticity of the string is 25 N.
[4]
P is now held at rest at a point 1.8 m vertically below M , and is then released.
(ii) Find the speed with which P passes through M .
7
[6]
A particle P of mass 0.25 kg moves in a straight line on a smooth horizontal surface. P starts at the
point O with speed 10 m s−1 and moves towards a fixed point A on the line. At time t s the displacement
of P from O is x m and the velocity of P is v m s−1 . A resistive force of magnitude (5 − x) N acts on P
in the direction towards O.
(i) Form a differential equation in v and x. By solving this differential equation, show that v = 10 − 2x.
[6]
(ii) Find x in terms of t, and hence show that the particle is always less than 5 m from O.
[5]
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
9709/52/M/J/10
www.maxpapers.com
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
General Certificate of Education Advanced Level
9709/53
MATHEMATICS
Paper 5 Mechanics 2 (M2)
May/June 2010
1 hour 15 minutes
*6690447261*
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.
Where a numerical value for the acceleration due to gravity is needed, use 10 m s−2 .
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
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2
1
A particle is projected horizontally with speed 12 m s−1 from the top of a high cliff. Find the direction
of motion of the particle after 2 s.
[3]
2
3N
Q
20 cm
4 cm
P
A uniform solid cone has height 20 cm and base radius 4 cm. PQ is a diameter of the base of the
cone. The cone is held in equilibrium, with P in contact with a horizontal surface and PQ vertical,
by a force applied at Q. This force has magnitude 3 N and acts parallel to the axis of the cone (see
diagram). Calculate the mass of the cone.
[4]
3
Two particles P and Q are projected simultaneously with speed 40 m s−1 from a point O on a horizontal
plane. Both particles subsequently pass at different times through the point A which has horizontal
and vertically upward displacements from O of 40 m and 15 m respectively.
(i) By considering the equation of the trajectory of a projectile, show that each angle of projection
[3]
satisfies the equation tan2 θ − 8 tan θ + 4 = 0.
(ii) Calculate the distance between the points at which P and Q strike the plane.
[5]
4
B
30°
A
AB is the diameter of a uniform semicircular lamina which has radius 0.3 m and mass 0.4 kg. The
lamina is hinged to a vertical wall at A with AB inclined at 30◦ to the vertical. One end of a light
inextensible string is attached to the lamina at B and the other end of the string is attached to the wall
vertically above A. The lamina is in equilibrium in a vertical plane perpendicular to the wall with the
string horizontal (see diagram).
(i) Show that the tension in the string is 2.00 N correct to 3 significant figures.
[4]
(ii) Find the magnitude and direction of the force exerted on the lamina by the hinge.
[3]
© UCLES 2010
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3
5
Q
30°
B
30°
P
A small ball B of mass 0.4 kg is attached to fixed points P and Q on a vertical axis by two light
inextensible strings of equal length. Both strings are taut and each is inclined at 30◦ to the vertical.
The ball moves in a horizontal circle (see diagram).
(i) It is given that when the ball moves with speed 6 m s−1 the tension in the string QB is three times
the tension in the string PB. Calculate the radius of the circle.
[4]
The ball now moves along this circular path with the minimum possible speed.
(ii) State the tension in the string PB in this case, and find the speed of the ball.
6
A particle P of mass 0.5 kg moves in a straight line on a smooth horizontal surface. At time t s, the
displacement of P from a fixed point on the line is x m and the velocity of P is v m s−√1 . It is given that
when t = 0, x = 0 and v = 9. The motion of P is opposed by a force of magnitude 3 v N.
2
7
[4]
(i) By solving an appropriate differential equation, show that v = (27 − 9x) 3 .
[5]
(ii) Calculate the value of x when t = 0.5.
[4]
One end of a light elastic string of natural length 3 m and modulus of elasticity 24 N is attached to a
fixed point O. A particle P of mass 0.4 kg is attached to the other end of the string. P is projected
vertically downwards from O with initial speed 2 m s−1 . When the extension of the string is x m the
speed of P is v m s−1 .
(i) Show that v2 = 64 + 20x − 20x2 .
[4]
(ii) Find the greatest speed of the particle.
[3]
(iii) Calculate the greatest tension in the string.
[4]
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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
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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
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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
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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]
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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.
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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
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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
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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]
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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.
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UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
General Certificate of Education
Advanced Subsidiary Level and Advanced Level
9709/63
MATHEMATICS
Paper 6 Probability & Statistics 1 (S1)
May/June 2010
1 hour 15 minutes
*7928773839*
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
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2
1
A bottle of sweets contains 13 red sweets, 13 blue sweets, 13 green sweets and 13 yellow sweets.
7 sweets are selected at random. Find the probability that exactly 3 of them are red.
[3]
2
The heights, x cm, of a group of 82 children are summarised as follows.
Σ(x − 130) = −287,
3
4
standard deviation of x = 6.9.
(i) Find the mean height.
[2]
(ii) Find Σ(x − 130)2 .
[2]
Christa takes her dog for a walk every day. The probability that they go to the park on any day is 0.6.
If they go to the park there is a probability of 0.35 that the dog will bark. If they do not go to the park
there is a probability of 0.75 that the dog will bark.
(i) Find the probability that they go to the park on more than 5 of the next 7 days.
[2]
(ii) Find the probability that the dog barks on any particular day.
[2]
(iii) Find the variance of the number of times they go to the park in 30 days.
[1]
Three identical cans of cola, 2 identical cans of green tea and 2 identical cans of orange juice are
arranged in a row. Calculate the number of arrangements if
(i) the first and last cans in the row are the same type of drink,
[3]
(ii) the 3 cans of cola are all next to each other and the 2 cans of green tea are not next to each other.
[5]
5
Set A consists of the ten digits 0, 0, 0, 0, 0, 0, 2, 2, 2, 4.
Set B consists of the seven digits 0, 0, 0, 0, 2, 2, 2.
One digit is chosen at random from each set. The random variable X is defined as the sum of these
two digits.
(i) Show that P(X = 2) = 37 .
[2]
(ii) Tabulate the probability distribution of X .
[2]
(iii) Find E(X ) and Var(X ).
[3]
(iv) Given that X = 2, find the probability that the digit chosen from set A was 2.
[2]
© UCLES 2010
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3
6
The lengths of some insects of the same type from two countries, X and Y , were measured. The
stem-and-leaf diagram shows the results.
Country X
(10)
(18)
(16)
(16)
(11)
Country Y
9766644432
888776655544333220
9998877655322100
8765553322211100
87655443311
80
81
82
83
84
85
86
1122333556789
0012333q4566788
01224444556677789
001244556677789
12r335566788
01223555899
(13)
(15)
(17)
(15)
(12)
(11)
Key: 5 | 81 | 3 means an insect from country X has length 0.815 cm
and an insect from country Y has length 0.813 cm.
(i) Find the median and interquartile range of the lengths of the insects from country X .
[2]
(ii) The interquartile range of the lengths of the insects from country Y is 0.028 cm. Find the values
of q and r.
[2]
(iii) Represent the data by means of a pair of box-and-whisker plots in a single diagram on graph
paper.
[4]
(iv) Compare the lengths of the insects from the two countries.
7
[2]
The heights that children of a particular age can jump have a normal distribution. On average,
8 children out of 10 can jump a height of more than 127 cm, and 1 child out of 3 can jump a height of
more than 135 cm.
(i) Find the mean and standard deviation of the heights the children can jump.
[5]
(ii) Find the probability that a randomly chosen child will not be able to jump a height of 145 cm.
[3]
(iii) Find the probability that, of 8 randomly chosen children, at least 2 will be able to jump a height
of more than 135 cm.
[3]
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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.
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UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
General Certificate of Education Advanced Level
9709/71
MATHEMATICS
Paper 7 Probability & Statistics 2 (S2)
May/June 2010
1 hour 15 minutes
*8793053992*
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
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2
1
f(t)
k
0
12
time waiting (t)
Fred arrives at random times on a station platform. The times in minutes he has to wait for the next
train are modelled by the continuous random variable for which the probability density function f is
shown above.
2
3
(i) State the value of k.
[1]
(ii) Explain briefly what this graph tells you about the arrival times of trains.
[1]
A random sample of n people were questioned about their internet use. 87 of them had a high-speed
internet connection. A confidence interval for the population proportion having a high-speed internet
connection is 0.1129 < p < 0.1771.
(i) Write down the mid-point of this confidence interval and hence find the value of n.
[3]
(ii) This interval is an α % confidence interval. Find α .
[4]
Metal bolts are produced in large numbers and have lengths which are normally distributed with mean
2.62 cm and standard deviation 0.30 cm.
(i) Find the probability that a random sample of 45 bolts will have a mean length of more than
2.55 cm.
[3]
(ii) The machine making these bolts is given an annual service. This may change the mean length
of bolts produced but does not change the standard deviation. To test whether the mean has
changed, a random sample of 30 bolts is taken and their lengths noted. The sample mean length
is m cm. Find the set of values of m which result in rejection at the 10% significance level of the
hypothesis that no change in the mean length has occurred.
[4]
4
The weekly distance in kilometres driven by Mr Parry has a normal distribution with mean 512 and
standard deviation 62. Independently, the weekly distance in kilometres driven by Mrs Parry has a
normal distribution with mean 89 and standard deviation 7.4.
(i) Find the probability that, in a randomly chosen week, Mr Parry drives more than 5 times as far
as Mrs Parry.
[5]
(ii) Find the mean and standard deviation of the total of the weekly distances in miles driven by
Mr Parry and Mrs Parry. Use the approximation 8 kilometres = 5 miles.
[3]
© UCLES 2010
9709/71/M/J/10
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3
5
The random variable T denotes the time in seconds for which a firework burns before exploding. The
probability density function of T is given by
f (t ) = ke0.2t
0
0 ≤ t ≤ 5,
otherwise,
where k is a constant.
(i) Show that k =
6
1
.
5(e − 1)
[3]
(ii) Sketch the probability density function.
[2]
(iii) 80% of fireworks burn for longer than a certain time before they explode. Find this time.
[3]
In restaurant A an average of 2.2% of tablecloths are stained and, independently, in restaurant B an
average of 5.8% of tablecloths are stained.
(i) Random samples of 55 tablecloths are taken from each restaurant. Use a suitable Poisson
approximation to find the probability that a total of more than 2 tablecloths are stained.
[4]
(ii) Random samples of n tablecloths are taken from each restaurant. The probability that at least
one tablecloth is stained is greater than 0.99. Find the least possible value of n.
[4]
7
A hospital patient’s white blood cell count has a Poisson distribution. Before undergoing treatment
the patient had a mean white blood cell count of 5.2. After the treatment a random measurement of
the patient’s white blood cell count is made, and is used to test at the 10% significance level whether
the mean white blood cell count has decreased.
(i) State what is meant by a Type I error in the context of the question, and find the probability that
the test results in a Type I error.
[4]
(ii) Given that the measured value of the white blood cell count after the treatment is 2, carry out the
test.
[3]
(iii) Find the probability of a Type II error if the mean white blood cell count after the treatment is
actually 4.1.
[3]
© UCLES 2010
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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.
9709/71/M/J/10
www.maxpapers.com
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
General Certificate of Education Advanced Level
9709/72
MATHEMATICS
Paper 7 Probability & Statistics 2 (S2)
May/June 2010
1 hour 15 minutes
*3180410318*
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
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2
1
f(t)
k
0
12
time waiting (t)
Fred arrives at random times on a station platform. The times in minutes he has to wait for the next
train are modelled by the continuous random variable for which the probability density function f is
shown above.
2
3
(i) State the value of k.
[1]
(ii) Explain briefly what this graph tells you about the arrival times of trains.
[1]
A random sample of n people were questioned about their internet use. 87 of them had a high-speed
internet connection. A confidence interval for the population proportion having a high-speed internet
connection is 0.1129 < p < 0.1771.
(i) Write down the mid-point of this confidence interval and hence find the value of n.
[3]
(ii) This interval is an α % confidence interval. Find α .
[4]
Metal bolts are produced in large numbers and have lengths which are normally distributed with mean
2.62 cm and standard deviation 0.30 cm.
(i) Find the probability that a random sample of 45 bolts will have a mean length of more than
2.55 cm.
[3]
(ii) The machine making these bolts is given an annual service. This may change the mean length
of bolts produced but does not change the standard deviation. To test whether the mean has
changed, a random sample of 30 bolts is taken and their lengths noted. The sample mean length
is m cm. Find the set of values of m which result in rejection at the 10% significance level of the
hypothesis that no change in the mean length has occurred.
[4]
4
The weekly distance in kilometres driven by Mr Parry has a normal distribution with mean 512 and
standard deviation 62. Independently, the weekly distance in kilometres driven by Mrs Parry has a
normal distribution with mean 89 and standard deviation 7.4.
(i) Find the probability that, in a randomly chosen week, Mr Parry drives more than 5 times as far
as Mrs Parry.
[5]
(ii) Find the mean and standard deviation of the total of the weekly distances in miles driven by
Mr Parry and Mrs Parry. Use the approximation 8 kilometres = 5 miles.
[3]
© UCLES 2010
9709/72/M/J/10
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3
5
The random variable T denotes the time in seconds for which a firework burns before exploding. The
probability density function of T is given by
f (t ) = ke0.2t
0
0 ≤ t ≤ 5,
otherwise,
where k is a constant.
(i) Show that k =
6
1
.
5(e − 1)
[3]
(ii) Sketch the probability density function.
[2]
(iii) 80% of fireworks burn for longer than a certain time before they explode. Find this time.
[3]
In restaurant A an average of 2.2% of tablecloths are stained and, independently, in restaurant B an
average of 5.8% of tablecloths are stained.
(i) Random samples of 55 tablecloths are taken from each restaurant. Use a suitable Poisson
approximation to find the probability that a total of more than 2 tablecloths are stained.
[4]
(ii) Random samples of n tablecloths are taken from each restaurant. The probability that at least
one tablecloth is stained is greater than 0.99. Find the least possible value of n.
[4]
7
A hospital patient’s white blood cell count has a Poisson distribution. Before undergoing treatment
the patient had a mean white blood cell count of 5.2. After the treatment a random measurement of
the patient’s white blood cell count is made, and is used to test at the 10% significance level whether
the mean white blood cell count has decreased.
(i) State what is meant by a Type I error in the context of the question, and find the probability that
the test results in a Type I error.
[4]
(ii) Given that the measured value of the white blood cell count after the treatment is 2, carry out the
test.
[3]
(iii) Find the probability of a Type II error if the mean white blood cell count after the treatment is
actually 4.1.
[3]
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UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
General Certificate of Education Advanced Level
9709/73
MATHEMATICS
Paper 7 Probability & Statistics 2 (S2)
May/June 2010
1 hour 15 minutes
*3115147863*
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.
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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
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1
At the 2009 election, 13 of the voters in Chington voted for the Citizens Party. One year later, a
researcher questioned 20 randomly selected voters in Chington. Exactly 3 of these 20 voters said that
if there were an election next week they would vote for the Citizens Party. Test at the 2.5% significance
level whether there is evidence of a decrease in support for the Citizens Party in Chington, since the
2009 election.
[5]
2
Dipak carries out a test, at the 10% significance level, using a normal distribution. The null hypothesis
is µ = 35 and the alternative hypothesis is µ ≠ 35.
(i) Is this a one-tail or a two-tail test? State briefly how you can tell.
[1]
Dipak finds that the value of the test statistic is ß = −1.750.
(ii) Explain what conclusion he should draw.
[2]
(iii) This result is significant at the α % level. Find the smallest possible value of α , correct to the
nearest whole number.
[2]
3
4
5
The weight, in grams, of a certain type of apple is modelled by the random variable X with mean 62
and standard deviation 8.2. A random sample of 50 apples is selected, and the mean weight in grams,
X , is found.
(i) Describe fully the distribution of X .
[3]
(ii) Find P(X > 64).
[3]
At a power plant, the number of breakdowns per year has a Poisson distribution. In the past the mean
number of breakdowns per year has been 4.8. Following some repairs, the management carry out a
hypothesis test at the 5% significance level to determine whether this mean has decreased. If there is
at most 1 breakdown in the following year, they will conclude that the mean has decreased.
(i) State what is meant by a Type I error in this context.
[1]
(ii) Find the probability of a Type I error.
[2]
(iii) Find the probability of a Type II error if the mean is now 0.9 breakdowns per year.
[3]
The time, in minutes, taken by volunteers to complete a task is modelled by the random variable X
with probability density function given by

 k
f (x) =  x4


0
x ≥ 1,
otherwise.
(i) Show that k = 3.
[2]
(ii) Find E(X ) and Var(X ).
[6]
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3
6
Yu Ming travels to work and returns home once each day. The times, in minutes, that he takes to
travel to work and to return home are represented by the independent random variables W and H with
distributions N(22.4, 4.82 ) and N(20.3, 5.22 ) respectively.
(i) Find the probability that Yu Ming’s total travelling time during a 5-day period is greater than
180 minutes.
[4]
(ii) Find the probability that, on a particular day, Yu Ming takes longer to return home than he takes
to travel to work.
[5]
7
A clinic deals only with flu vaccinations. The number of patients arriving every 15 minutes is modelled
by the random variable X with distribution Po(4.2).
(i) State two assumptions required for the Poisson model to be valid.
[2]
(ii) Find the probability that
(a) at least 1 patient will arrive in a 15-minute period,
[2]
(b) fewer than 4 patients will arrive in a 10-minute period.
[3]
(iii) The clinic is open for 20 hours each week. At the beginning of one week the clinic has enough
vaccine for 370 patients. Use a suitable approximation to find the probability that this will not
be enough vaccine for that week.
[4]
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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.
9709/73/M/J/10