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*8461197174*
Cambridge International ExaminationsCambridge International Advanced Subsidiary and Advanced Level
MATHEMATICS 9709/11
Paper 1 Pure Mathematics 1 (P1) May/June 2015
1 hour 45 minutes
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 an HB pencil for any diagrams or graphs.
Do not use staples, paper clips, glue or correction fluid.
DO NOT WRITE IN ANY BARCODES.
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
The diagram shows part of the curve y =8��3x + 4� . The curve intersects the y-axis at A �0, 4�. The
normal to the curve at A intersects the line x = 4 at the point B.
(i) Find the coordinates of B. [5]
(ii) Show, with all necessary working, that the areas of the regions marked P and Q are equal. [6]
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable
effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will
be pleased to make amends at the earliest possible opportunity.
To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International
Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after
the live examination series.
Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local
Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.
7 The point C lies on the perpendicular bisector of the line joining the points A �4, 6� and B �10, 2�.C also lies on the line parallel to AB through �3, 11�.
(i) Find the equation of the perpendicular bisector of AB. [4]
(ii) Calculate the coordinates of C. [3]
8 (a) The first, second and last terms in an arithmetic progression are 56, 53 and −22 respectively.
Find the sum of all the terms in the progression. [4]
(b) The first, second and third terms of a geometric progression are 2k + 6, 2k and k + 2 respectively,
where k is a positive constant.
(i) Find the value of k. [3]
(ii) Find the sum to infinity of the progression. [2]
9 Relative to an origin O, the position vectors of points A and B are given by
−−→OA = 2i + 4j + 4k and
−−→OB = 3i + j + 4k.
(i) Use a vector method to find angle AOB. [4]
The point C is such that−−→AB =
−−→BC.
(ii) Find the unit vector in the direction of−−→OC. [4]
(iii) Show that triangle OAC is isosceles. [1]
10 The equation of a curve is y =4
2x − 1.
(i) Find, showing all necessary working, the volume obtained when the region bounded by the
curve, the x-axis and the lines x = 1 and x = 2 is rotated through 360Å about the x-axis. [4]
(ii) Given that the line 2y = x + c is a normal to the curve, find the possible values of the constant c.
[6]
11 The function f is defined by f : x → 2x2− 6x + 5 for x ∈ >.
(i) Find the set of values of p for which the equation f�x� = p has no real roots. [3]
The function g is defined by g : x → 2x2− 6x + 5 for 0 ≤ x ≤ 4.
(ii) Express g�x� in the form a�x + b�2+ c, where a, b and c are constants. [3]
(iii) Find the range of g. [2]
The function h is defined by h : x → 2x2− 6x + 5 for k ≤ x ≤ 4, where k is a constant.
(iv) State the smallest value of k for which h has an inverse. [1]
(v) For this value of k, find an expression for h−1�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.
To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International
Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after
the live examination series.
Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local
Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.
In the diagram, OAB is a sector of a circle with centre O and radius r. The point C on OB is such
that angle ACO is a right angle. Angle AOB is ! radians and is such that AC divides the sector into
two regions of equal area.
(i) Show that sin ! cos! = 12!. [4]
It is given that the solution of the equation in part (i) is ! = 0.9477, correct to 4 decimal places.
(ii) Find the ratio
perimeter of region OAC : perimeter of region ACB,
giving your answer in the form k : 1, where k is given correct to 1 decimal place. [5]
(iii) Find angle AOB in degrees. [1]
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable
effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will
be pleased to make amends at the earliest possible opportunity.
To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International
Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after
the live examination series.
Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local
Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.
(ii) Hence, using logarithms, solve the equation �3 × 2y+ 4 � = �3 × 2y
− 11�, giving the answer correct
to 3 significant figures. [2]
2
x
ln y
O
(2, 1.60)
(5, 2.92)
The variables x and y satisfy the equation
y = Aep�x−1�,
where A and p are constants. The graph of ln y against x is a straight line passing through the points
�2, 1.60� and �5, 2.92�, as shown in the diagram. Find the values of A and p correct to 2 significant
figures. [5]
3 The equation of a curve is
y = 6 sin x − 2 cos 2x.
Find the equation of the tangent to the curve at the point�
160, 2
�. Give the answer in the form
y = mx + c, where the values of m and c are correct to 3 significant figures. [5]
4 The polynomials f�x� and g�x� are defined by
f�x� = x3+ ax2
+ b and g�x� = x3+ bx2
− a,
where a and b are constants. It is given that �x + 2� is a factor of f�x�. It is also given that, when g�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, find the greatest possible value of g�x� − f�x� as x varies. [2]
5 (i) Given that Óa
0
�3e12x+ 1�dx = 10, show that the positive constant a satisfies the equation
a = 2 ln
@16 − a
6
A. �5�
(ii) Use the iterative formula an+1 = 2 ln
@16 − an
6
Awith a1 = 2 to find the value of a correct to
3 decimal places. Give the result of each iteration to 5 decimal places. [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.
To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International
Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after
the live examination series.
Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local
Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable
effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will
be pleased to make amends at the earliest possible opportunity.
To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International
Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after
the live examination series.
Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local
Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable
effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will
be pleased to make amends at the earliest possible opportunity.
To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International
Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after
the live examination series.
Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local
Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable
effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will
be pleased to make amends at the earliest possible opportunity.
To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International
Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after
the live examination series.
Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local
Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable
effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will
be pleased to make amends at the earliest possible opportunity.
To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International
Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after
the live examination series.
Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local
Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable
effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will
be pleased to make amends at the earliest possible opportunity.
To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International
Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after
the live examination series.
Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local
Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable
effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will
be pleased to make amends at the earliest possible opportunity.
To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International
Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after
the live examination series.
Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local
Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.
7 A particle P moves on a straight line. It starts at a point O on the line and returns to O 100 s later.
The velocity of P is v m s−1 at time t s after leaving O, where
v = 0.0001t3− 0.015t2
+ 0.5t.
(i) Show that P is instantaneously at rest when t = 0, t = 50 and t = 100. [2]
(ii) Find the values of v at the times for which the acceleration of P is zero, and sketch the velocity-
time graph for P’s motion for 0 ≤ t ≤ 100. [7]
(iii) Find the greatest distance of P from O for 0 ≤ t ≤ 100. [4]
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable
effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will
be pleased to make amends at the earliest possible opportunity.
To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International
Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after
the live examination series.
Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local
Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.
The diagram shows the cross-section OABCDE through the centre of mass of a uniform prism
on a rough inclined plane. The portion ADEO is a rectangle in which AD = OE = 0.6 m and
DE = AO = 0.8 m; the portion BCD is an isosceles triangle in which angle BCD is a right angle,
and A is the mid-point of BD. The plane is inclined at 45Å to the horizontal, BC lies along a line of
greatest slope of the plane and DE is horizontal.
(i) Calculate the distance of the centre of mass of the prism from BD. [3]
The weight of the prism is 21 N, and it is held in equilibrium by a horizontal force of magnitude P N
acting along ED.
(ii) (a) Find the smallest value of P for which the prism does not topple. [2]
(b) It is given that the prism is about to slip for this smallest value of P. Calculate the coefficient
of friction between the prism and the plane. [3]
The value of P is gradually increased until the prism ceases to be in equilibrium.
(iii) Show that the prism topples before it begins to slide, stating the value of P at which equilibrium
is broken. [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.
To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International
Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after
the live examination series.
Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local
Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.
A small ball B is projected with speed U m s−1 at an angle of 1Å above the horizontal from a point O.
At time 2 s after the instant of projection, B strikes a smooth wall which slopes at 60Å to the horizontal.
The speed of B is 18 m s−1 and its direction of motion is perpendicular to the wall at the instant of
impact (see Fig. 1). B bounces off the wall with speed V m s−1 in a direction perpendicular to the wall.
At time 0.8 s after B bounces off the wall, B strikes the wall again at a lower point A (see Fig. 2).
(i) Find U and 1. [5]
(ii) By considering the motion of B after it bounces off the wall, calculate V. [4]
7 A force of magnitude 0.4t N, applied at an angle of 30Å above the horizontal, acts on a particle P,
where t s is the time since the force starts to act. P is at rest on rough horizontal ground when t = 0.
The mass of P is 0.2 kg and the coefficient of friction between P and the ground is -.
(i) Given that P is about to slip when t = 2, find - and the value of t for the instant when P loses
contact with the ground. [5]
(ii) While P is moving on the ground, it has velocity v m s−1 at time t s. Show that
dv
dt= 2.165t − 4.330,
where the coefficients are correct to 4 significant figures. [3]
(iii) Calculate the speed of P when it loses contact with the ground. [4]
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable
effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will
be pleased to make amends at the earliest possible opportunity.
To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International
Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after
the live examination series.
Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local
Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable
effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will
be pleased to make amends at the earliest possible opportunity.
To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International
Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after
the live examination series.
Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local
Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable
effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will
be pleased to make amends at the earliest possible opportunity.
To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International
Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after
the live examination series.
Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local
Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable
effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will
be pleased to make amends at the earliest possible opportunity.
To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International
Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after
the live examination series.
Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local
Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable
effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will
be pleased to make amends at the earliest possible opportunity.
To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International
Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after
the live examination series.
Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local
Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable
effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will
be pleased to make amends at the earliest possible opportunity.
To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International
Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after
the live examination series.
Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local
Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable
effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will
be pleased to make amends at the earliest possible opportunity.
To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International
Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after
the live examination series.
Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local
Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable
effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will
be pleased to make amends at the earliest possible opportunity.
To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International
Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after
the live examination series.
Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local
Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.