A-Level Past Papers – Mathematics A-Level Examinations ... · PDF fileA-Level Past Papers – Mathematics A-Level Examinations October/November 2010 Paper Pages Pure Mathematics

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A-Level Past Papers – MathematicsA-Level Examinations October/November 2010

Paper Pages

Pure MathematicsP1 – 9709/11 2 – 5P1 – 9709/12 6 – 13P1 – 9709/13 14 – 21P2 – 9709/21 22 – 25P2 – 9709/22 26 – 29P2 – 9709/23 30 – 33P3 – 9709/31 34 – 37P3 – 9709/32 38 – 41P3 – 9709/33 42 – 45

MechanicsP4 – 9709/41 46 – 49P4 – 9709/42 50 – 53P4 – 9709/43 54 – 57P5 – 9709/51 58 – 61P5 – 9709/52 62 – 65P5 – 9709/53 66 - 69

Probability & StatisticsP6 – 9709/61 70 – 73P6 – 9709/62 74 – 77P6 – 9709/63 78 – 81P7 – 9709/71 82 – 85P7 – 9709/72 86 – 89P7 – 9709/73 90 – 93

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS

General Certificate of Education

Advanced Subsidiary Level and Advanced Level

MATHEMATICS 9709/11

Paper 1 Pure Mathematics 1 (P1) October/November 2010

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 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.

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2

1 Find ä (x + 1

x)2

dx. [3]

2 In the expansion of (1 + ax)6, where a is a constant, the coefficient of x is −30. Find the coefficient

of x3. [4]

3 Functions f and g are defined for x ∈ > by

f : x → 2x + 3,

g : x → x2 − 2x.

Express gf(x) in the form a(x + b)2 + c, where a, b and c are constants. [5]

4 (i) Prove the identitysin x tan x

1 − cos x≡ 1 + 1

cos x. [3]

(ii) Hence solve the equationsin x tan x

1 − cos x+ 2 = 0, for 0◦ ≤ x ≤ 360◦. [3]

5

O

C

B

A

8 cm

6 cm

10 cm

i

jk

The diagram shows a pyramid OABC with a horizontal base OAB where OA = 6 cm, OB = 8 cm and

angle AOB = 90◦. The point C is vertically above O and OC = 10 cm. Unit vectors i, j and k are

parallel to OA, OB and OC as shown.

Use a scalar product to find angle ACB. [6]

6 (a) The fifth term of an arithmetic progression is 18 and the sum of the first 5 terms is 75. Find the

first term and the common difference. [4]

(b) The first term of a geometric progression is 16 and the fourth term is 274

. Find the sum to infinity

of the progression. [3]

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3

7 A function f is defined by f : x → 3 − 2 tan(12x) for 0 ≤ x < π.

(i) State the range of f. [1]

(ii) State the exact value of f(23π). [1]

(iii) Sketch the graph of y = f(x). [2]

(iv) Obtain an expression, in terms of x, for f−1(x). [3]

8

x cm

x cmy cm

The diagram shows a metal plate consisting of a rectangle with sides x cm and y cm and a quarter-circle

of radius x cm. The perimeter of the plate is 60 cm.

(i) Express y in terms of x. [2]

(ii) Show that the area of the plate, A cm2, is given by A = 30x − x2. [2]

Given that x can vary,

(iii) find the value of x at which A is stationary, [2]

(iv) find this stationary value of A, and determine whether it is a maximum or a minimum value. [2]

[Questions 9, 10 and 11 are printed on the next page.]

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4

9

2 cm

8 cm

R S

Q

P

T

C1

C2

The diagram shows two circles, C1

and C2, touching at the point T . Circle C

1has centre P and radius

8 cm; circle C2

has centre Q and radius 2 cm. Points R and S lie on C1

and C2

respectively, and RS is

a tangent to both circles.

(i) Show that RS = 8 cm. [2]

(ii) Find angle RPQ in radians correct to 4 significant figures. [2]

(iii) Find the area of the shaded region. [4]

10 The equation of a curve is y = 3 + 4x − x2.

(i) Show that the equation of the normal to the curve at the point (3, 6) is 2y = x + 9. [4]

(ii) Given that the normal meets the coordinate axes at points A and B, find the coordinates of the

mid-point of AB. [2]

(iii) Find the coordinates of the point at which the normal meets the curve again. [4]

11 The equation of a curve is y = 9

2 − x.

(i) Find an expression fordy

dxand determine, with a reason, whether the curve has any stationary

points. [3]

(ii) Find the volume obtained when the region bounded by the curve, the coordinate axes and the

line x = 1 is rotated through 360◦ about the x-axis. [4]

(iii) Find the set of values of k for which the line y = x + k intersects the curve at two distinct points.

[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.

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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MATHEMATICS 9709/12


1 hour 45 minutes


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2

1 (i) Find the first 3 terms in the expansion, in ascending powers of x, of (1 − 2x2)8. [2]

(ii) Find the coefficient of x4 in the expansion of (2 − x2)(1 − 2x2)8. [2]

2 Prove the identity

tan2 x − sin2 x ≡ tan2 x sin2 x. [4]

3 The length, x metres, of a Green Anaconda snake which is t years old is given approximately by the

formula

x = 0.7√(2t − 1),

where 1 ≤ t ≤ 10. Using this formula, find

(i)dx

dt, [2]

(ii) the rate of growth of a Green Anaconda snake which is 5 years old. [2]

4

3 cm

3 cm

3 cm

2.3 rad 2.3 rad

A B

C

O

P

The diagram shows points A, C, B, P on the circumference of a circle with centre O and radius 3 cm.

Angle AOC = angle BOC = 2.3 radians.

(i) Find angle AOB in radians, correct to 4 significant figures. [1]

(ii) Find the area of the shaded region ACBP, correct to 3 significant figures. [4]

5 (a) The first and second terms of an arithmetic progression are 161 and 154 respectively. The sum

of the first m terms is zero. Find the value of m. [3]

(b) A geometric progression, in which all the terms are positive, has common ratio r. The sum of

the first n terms is less than 90% of the sum to infinity. Show that rn > 0.1. [3]

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6 A curve has equation y = kx2 + 1 and a line has equation y = kx, where k is a non-zero constant.

(i) Find the set of values of k for which the curve and the line have no common points. [3]

(ii) State the value of k for which the line is a tangent to the curve and, for this case, find the

coordinates of the point where the line touches the curve. [4]

7 The function f is defined by

f(x) = x2 − 4x + 7 for x > 2.

(i) Express f(x) in the form (x − a)2 + b and hence state the range of f. [3]

(ii) Obtain an expression for f−1(x) and state the domain of f−1. [3]

The function g is defined by

g(x) = x − 2 for x > 2.

The function h is such that f = hg and the domain of h is x > 0.

(iii) Obtain an expression for h(x). [1]

8

Ox

y

A

B

y x= 3 + 4

y =2

1 – x

The diagram shows part of the curve y = 2

1 − xand the line y = 3x + 4. The curve and the line meet at

points A and B.

(i) Find the coordinates of A and B. [4]

(ii) Find the length of the line AB and the coordinates of the mid-point of AB. [3]


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9

O A

BE

G F

D

P

C

i

jk

10 cm

6 cm6 cm

10 cm

a cm

10 cm

The diagram shows a pyramid OABCP in which the horizontal base OABC is a square of side 10 cm

and the vertex P is 10 cm vertically above O. The points D, E, F, G lie on OP, AP, BP, CP

respectively and DEFG is a horizontal square of side 6 cm. The height of DEFG above the base is

a cm. Unit vectors i, j and k are parallel to OA, OC and OD respectively.

(i) Show that a = 4. [2]

(ii) Express the vector−−→BG in terms of i, j and k. [2]

(iii) Use a scalar product to find angle GBA. [4]

10

x

h

x12

x54

x45

The diagram shows an open rectangular tank of height h metres covered with a lid. The base of the

tank has sides of length x metres and 12x metres and the lid is a rectangle with sides of length 5

4x metres

and 45x metres. When full the tank holds 4 m3 of water. The material from which the tank is made is

of negligible thickness. The external surface area of the tank together with the area of the top of the

lid is A m2.

(i) Express h in terms of x and hence show that A = 32x2 + 24

x. [5]

(ii) Given that x can vary, find the value of x for which A is a minimum, showing clearly that A is a

minimum and not a maximum. [5]

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5

11

Ox

y

A

B

x = 5

y =1

(3 + 1)x14

The diagram shows part of the curve y = 1

(3x + 1)14

. The curve cuts the y-axis at A and the line x = 5

at B.

(i) Show that the equation of the line AB is y = − 110

x + 1. [4]

(ii) Find the volume obtained when the shaded region is rotated through 360◦ about the x-axis. [9]

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MATHEMATICS 9709/13


1 hour 45 minutes


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2

1 Find the term independent of x in the expansion of (x −1

x2)9

. [3]

2 Points A, B and C have coordinates (2, 5), (5, −1) and (8, 6) respectively.

(i) Find the coordinates of the mid-point of AB. [1]

(ii) Find the equation of the line through C perpendicular to AB. Give your answer in the form

ax + by + c = 0. [3]

3 Solve the equation 15 sin2 x = 13 + cos x for 0◦ ≤ x ≤ 180◦. [4]

4 (i) Sketch the curve y = 2 sin x for 0 ≤ x ≤ 2π. [1]

(ii) By adding a suitable straight line to your sketch, determine the number of real roots of the

equation

2π sin x = π − x.

State the equation of the straight line. [3]

5 A curve has equation y =1

x − 3+ x.

(i) Finddy

dxand

d2y

dx2. [2]

(ii) Find the coordinates of the maximum point A and the minimum point B on the curve. [5]

6 A curve has equation y = f(x). It is given that f ′(x) = 3x2 + 2x − 5.

(i) Find the set of values of x for which f is an increasing function. [3]

(ii) Given that the curve passes through (1, 3), find f(x). [4]

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7

Ox

y

The diagram shows the function f defined for 0 ≤ x ≤ 6 by

x → 12x2 for 0 ≤ x ≤ 2,

x → 12x + 1 for 2 < x ≤ 6.

(i) State the range of f. [1]

(ii) Copy the diagram and on your copy sketch the graph of y = f−1(x). [2]

(iii) Obtain expressions to define f−1(x), giving the set of values of x for which each expression is

valid. [4]

8A B

CD

P

Q

The diagram shows a rhombus ABCD. Points P and Q lie on the diagonal AC such that BPD is an

arc of a circle with centre C and BQD is an arc of a circle with centre A. Each side of the rhombus

has length 5 cm and angle BAD = 1.2 radians.

(i) Find the area of the shaded region BPDQ. [4]

(ii) Find the length of PQ. [4]


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4

9 (a) A geometric progression has first term 100 and sum to infinity 2000. Find the second term. [3]

(b) An arithmetic progression has third term 90 and fifth term 80.

(i) Find the first term and the common difference. [2]

(ii) Find the value of m given that the sum of the first m terms is equal to the sum of the first

(m + 1) terms. [2]

(iii) Find the value of n given that the sum of the first n terms is zero. [2]

10

A

B

O

C

The diagram shows triangle OAB, in which the position vectors of A and B with respect to O are

given by

−−→OA = 2i + j − 3k and

−−→OB = −3i + 2j − 4k.

C is a point on OA such that−−→OC = p

−−→OA, where p is a constant.

(i) Find angle AOB. [4]

(ii) Find−−→BC in terms of p and vectors i, j and k. [1]

(iii) Find the value of p given that BC is perpendicular to OA. [4]

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5

11

Ox

y

a b

P

Q

y x= 9 –3

y =8

x3

The diagram shows parts of the curves y = 9 − x3 and y =8

x3and their points of intersection P and Q.

The x-coordinates of P and Q are a and b respectively.

(i) Show that x = a and x = b are roots of the equation x6 − 9x3 + 8 = 0. Solve this equation and

hence state the value of a and the value of b. [4]

(ii) Find the area of the shaded region between the two curves. [5]

(iii) The tangents to the two curves at x = c (where a < c < b) are parallel to each other. Find the

value of c. [4]

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General Certificate of Education Advanced Subsidiary Level

MATHEMATICS 9709/21


1 hour 15 minutes


Graph Paper


















This document consists of 3 printed pages and 1 blank page.


*7876081842*

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2

1 Solve the inequality |x + 1| > |x − 4 |. [3]

2 Use logarithms to solve the equation 5x = 22x+1, giving your answer correct to 3 significant figures.

[4]

3 Show that ã1

0

(ex + 1)2 dx = 12e2 + 2e − 3

2. [5]

4 The parametric equations of a curve are

x = 1 + ln(t − 2), y = t + 9

t, for t > 2.

(i) Show thatdy

dx= (t2 − 9)(t − 2)

t2. [3]

(ii) Find the coordinates of the only point on the curve at which the gradient is equal to 0. [3]

5 Solve the equation 8 + cot θ = 2 cosec2θ, giving all solutions in the interval 0◦ ≤ θ ≤ 360◦. [6]

6 The curve with equation y = 6

x2intersects the line y = x + 1 at the point P.

(i) Verify by calculation that the x-coordinate of P lies between 1.4 and 1.6. [2]

(ii) Show that the x-coordinate of P satisfies the equation

x =√( 6

x + 1). [2]

(iii) Use the iterative formula

xn+1

=√( 6

xn+ 1

),

with initial value x1= 1.5, to determine the x-coordinate of P correct to 2 decimal places. Give

the result of each iteration to 4 decimal places. [3]

7 The polynomial 3x3 + 2x2 + ax + b, where a and b are constants, is denoted by p(x). It is given that

(x − 1) is a factor of p(x), and that when p(x) is divided by (x − 2) the remainder is 10.

(i) Find the values of a and b. [5]

(ii) When a and b have these values, solve the equation p(x) = 0. [4]

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8

O

Q

x

y

p

The diagram shows the curve y = x sin x, for 0 ≤ x ≤ π. The point Q (12π, 1

2π) lies on the curve.

(i) Show that the normal to the curve at Q passes through the point (π, 0). [5]

(ii) Findd

dx(sin x − x cos x). [2]

(iii) Hence evaluate ã12π

0

x sin x dx. [3]

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MATHEMATICS 9709/22


1 hour 15 minutes


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*8627712723*

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2

1 Solve the inequality |x + 1| > |x − 4 |. [3]

2 Use logarithms to solve the equation 5x = 22x+1, giving your answer correct to 3 significant figures.

[4]

3 Show that ã1

0

(ex + 1)2 dx = 12e2 + 2e − 3

2. [5]


x = 1 + ln(t − 2), y = t + 9

t, for t > 2.

(i) Show thatdy

dx= (t2 − 9)(t − 2)

t2. [3]

(ii) Find the coordinates of the only point on the curve at which the gradient is equal to 0. [3]

5 Solve the equation 8 + cot θ = 2 cosec2θ, giving all solutions in the interval 0◦ ≤ θ ≤ 360◦. [6]

6 The curve with equation y = 6

x2intersects the line y = x + 1 at the point P.

(i) Verify by calculation that the x-coordinate of P lies between 1.4 and 1.6. [2]

(ii) Show that the x-coordinate of P satisfies the equation

x =√( 6

x + 1). [2]


xn+1

=√( 6

xn+ 1

),

with initial value x1= 1.5, to determine the x-coordinate of P correct to 2 decimal places. Give

the result of each iteration to 4 decimal places. [3]

7 The polynomial 3x3 + 2x2 + ax + b, where a and b are constants, is denoted by p(x). It is given that

(x − 1) is a factor of p(x), and that when p(x) is divided by (x − 2) the remainder is 10.

(i) Find the values of a and b. [5]

(ii) When a and b have these values, solve the equation p(x) = 0. [4]

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8

O

Q

x

y

p

The diagram shows the curve y = x sin x, for 0 ≤ x ≤ π. The point Q (12π, 1

2π) lies on the curve.

(i) Show that the normal to the curve at Q passes through the point (π, 0). [5]

(ii) Findd

dx(sin x − x cos x). [2]

(iii) Hence evaluate ã12π

0

x sin x dx. [3]

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MATHEMATICS 9709/23


1 hour 15 minutes


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2

1 Solve the inequality |3x + 1| > 8. [3]

2 The sequence of values given by the iterative formula

xn+1

= 7xn

8+ 5

2x4n

,

with initial value x1= 1.7, converges to α.

(i) Use this iterative formula to determine α correct to 2 decimal places, giving the result of each

iteration to 4 decimal places. [3]

(ii) State an equation that is satisfied by α and hence show that α = 5√

20. [2]

3 The polynomial x3 + 4x2 + ax + 2, where a is a constant, is denoted by p(x). It is given that the

remainder when p(x) is divided by (x + 1) is equal to the remainder when p(x) is divided by (x − 2).(i) Find the value of a. [3]

(ii) When a has this value, show that (x − 1) is a factor of p(x) and find the quotient when p(x) is

divided by (x − 1). [3]

4 (a) Find ã e1−2x dx. [2]

(b) Express sin2 3x in terms of cos 6x and hence find ã sin2 3x dx. [4]

5

Ox

ln y

(1.4, 0.8)

(2.2, 1.2)

The variables x and y satisfy the equation y = A(bx), where A and b are constants. The graph of

ln y against x is a straight line passing through the points (1.4, 0.8) and (2.2, 1.2), as shown in the

diagram. Find the values of A and b, correct to 2 decimal places. [6]

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6 (i) Express 2 sin θ − cos θ in the form R sin(θ − α), where R > 0 and 0◦ < α < 90◦, giving the exact

value of R and the value of α correct to 2 decimal places. [3]

(ii) Hence solve the equation

2 sin θ − cos θ = −0.4,

giving all solutions in the interval 0◦ ≤ θ ≤ 360◦. [4]

7

Ox

y

1

M

The diagram shows the curve y = ln x

x2and its maximum point M.

(i) Find the exact coordinates of M. [5]

(ii) Use the trapezium rule with three intervals to estimate the value of

ä 4

1

ln x

x2dx,

giving your answer correct to 2 decimal places. [3]

8 The equation of a curve is

x2 + 2xy − y2 + 8 = 0.

(i) Show that the tangent to the curve at the point (−2, 2) is parallel to the x-axis. [4]

(ii) Find the equation of the tangent to the curve at the other point on the curve for which x = −2,

giving your answer in the form y = mx + c. [5]

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General Certificate of Education Advanced Level

MATHEMATICS 9709/31


1 hour 45 minutes


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*3454847368*

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2

1 Solve the inequality 2|x − 3 | > |3x + 1|. [4]

2 Solve the equation

ln(1 + x2) = 1 + 2 ln x,

giving your answer correct to 3 significant figures. [4]


cos(θ + 60◦) = 2 sin θ,


4 (i) By sketching suitable graphs, show that the equation

4x2 − 1 = cot x

has only one root in the interval 0 < x < 12π. [2]

(ii) Verify by calculation that this root lies between 0.6 and 1. [2]


xn+1

= 12

√(1 + cot xn)

to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal

places. [3]

5 Let I = ä 1

0

x2√(4 − x2) dx.

(i) Using the substitution x = 2 sin θ, show that

I = ã 16π

0

4 sin2θ dθ. [3]

(ii) Hence find the exact value of I. [4]

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3

6 The complex number ß is given by

ß = (√3) + i.

(i) Find the modulus and argument of ß. [2]

(ii) The complex conjugate of ß is denoted by ß*. Showing your working, express in the form x + iy,

where x and y are real,

(a) 2ß + ß*,

(b)iß*

ß .

[4]

(iii) On a sketch of an Argand diagram with origin O, show the points A and B representing the

complex numbers ß and iß* respectively. Prove that angle AOB = 16π. [3]

7 With respect to the origin O, the points A and B have position vectors given by−−→OA = i + 2j + 2k and−−→

OB = 3i + 4j. The point P lies on the line AB and OP is perpendicular to AB.

(i) Find a vector equation for the line AB. [1]

(ii) Find the position vector of P. [4]

(iii) Find the equation of the plane which contains AB and which is perpendicular to the plane OAB,

giving your answer in the form ax + by + cß = d. [4]

8 Let f(x) = 3x

(1 + x)(1 + 2x2) .

(i) Express f(x) in partial fractions. [5]

(ii) Hence obtain the expansion of f(x) in ascending powers of x, up to and including the term in x3.

[5]

[Questions 9 and 10 are printed on the next page.]


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4

9

Ox

y

M2

The diagram shows the curve y = x3 ln x and its minimum point M.


(ii) Find the exact area of the shaded region bounded by the curve, the x-axis and the line x = 2. [5]

10 A certain substance is formed in a chemical reaction. The mass of substance formed t seconds after

the start of the reaction is x grams. At any time the rate of formation of the substance is proportional

to (20 − x). When t = 0, x = 0 anddx

dt= 1.

(i) Show that x and t satisfy the differential equation

dx

dt= 0.05(20 − x). [2]

(ii) Find, in any form, the solution of this differential equation. [5]

(iii) Find x when t = 10, giving your answer correct to 1 decimal place. [2]

(iv) State what happens to the value of x as t becomes very large. [1]






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MATHEMATICS 9709/32


1 hour 45 minutes


Graph Paper




















*4772556275*

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2

1 Solve the inequality 2|x − 3 | > |3x + 1|. [4]


ln(1 + x2) = 1 + 2 ln x,

giving your answer correct to 3 significant figures. [4]


cos(θ + 60◦) = 2 sin θ,


4 (i) By sketching suitable graphs, show that the equation

4x2 − 1 = cot x

has only one root in the interval 0 < x < 12π. [2]

(ii) Verify by calculation that this root lies between 0.6 and 1. [2]


xn+1

= 12

√(1 + cot xn)

to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal

places. [3]

5 Let I = ä 1

0

x2√(4 − x2) dx.

(i) Using the substitution x = 2 sin θ, show that

I = ã 16π

0

4 sin2θ dθ. [3]

(ii) Hence find the exact value of I. [4]

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3

6 The complex number ß is given by

ß = (√3) + i.

(i) Find the modulus and argument of ß. [2]

(ii) The complex conjugate of ß is denoted by ß*. Showing your working, express in the form x + iy,

where x and y are real,

(a) 2ß + ß*,

(b)iß*

ß .

[4]

(iii) On a sketch of an Argand diagram with origin O, show the points A and B representing the

complex numbers ß and iß* respectively. Prove that angle AOB = 16π. [3]

7 With respect to the origin O, the points A and B have position vectors given by−−→OA = i + 2j + 2k and−−→

OB = 3i + 4j. The point P lies on the line AB and OP is perpendicular to AB.

(i) Find a vector equation for the line AB. [1]

(ii) Find the position vector of P. [4]

(iii) Find the equation of the plane which contains AB and which is perpendicular to the plane OAB,

giving your answer in the form ax + by + cß = d. [4]

8 Let f(x) = 3x

(1 + x)(1 + 2x2) .

(i) Express f(x) in partial fractions. [5]

(ii) Hence obtain the expansion of f(x) in ascending powers of x, up to and including the term in x3.

[5]



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4

9

Ox

y

M2

The diagram shows the curve y = x3 ln x and its minimum point M.


(ii) Find the exact area of the shaded region bounded by the curve, the x-axis and the line x = 2. [5]

10 A certain substance is formed in a chemical reaction. The mass of substance formed t seconds after

the start of the reaction is x grams. At any time the rate of formation of the substance is proportional

to (20 − x). When t = 0, x = 0 anddx

dt= 1.

(i) Show that x and t satisfy the differential equation

dx

dt= 0.05(20 − x). [2]

(ii) Find, in any form, the solution of this differential equation. [5]

(iii) Find x when t = 10, giving your answer correct to 1 decimal place. [2]

(iv) State what happens to the value of x as t becomes very large. [1]






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MATHEMATICS 9709/33


1 hour 45 minutes


Graph Paper




















*3712617301*

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2

1 Expand (1 + 2x)−3 in ascending powers of x, up to and including the term in x2, simplifying the

coefficients. [3]


x = t

2t + 3, y = e−2t.

Find the gradient of the curve at the point for which t = 0. [5]

3 The complex number w is defined by w = 2 + i.

(i) Showing your working, express w2 in the form x + iy, where x and y are real. Find the modulus

of w2. [3]

(ii) Shade on an Argand diagram the region whose points represent the complex numbers ß which

satisfy

|ß − w2| ≤ |w2|. [3]

4 It is given that f(x) = 4 cos2 3x.

(i) Find the exact value of f ′(19π). [3]

(ii) Find ã f(x) dx. [3]

5 Show that ä7

0

2x + 7(2x + 1)(x + 2) dx = ln 50. [7]

6 The straight line l passes through the points with coordinates (−5, 3, 6) and (5, 8, 1). The plane p

has equation 2x − y + 4ß = 9.

(i) Find the coordinates of the point of intersection of l and p. [4]

(ii) Find the acute angle between l and p. [4]

7 (i) Given that äa

1

ln x

x2dx = 2

5, show that a = 5

3(1 + ln a). [5]

(ii) Use an iteration formula based on the equation a = 53(1 + ln a) to find the value of a correct to

2 decimal places. Use an initial value of 4 and give the result of each iteration to 4 decimal

places. [3]

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3

8 (i) Express (√ 6) cos θ + (√10) sin θ in the form R cos(θ − α), where R > 0 and 0◦ < α < 90◦. Give

the value of α correct to 2 decimal places. [3]

(ii) Hence, in each of the following cases, find the smallest positive angle θ which satisfies the

equation

(a) (√6) cos θ + (√10) sin θ = −4, [2]

(b) (√6) cos 12θ + (√10) sin 1

2θ = 3. [4]

9 A biologist is investigating the spread of a weed in a particular region. At time t weeks after the

start of the investigation, the area covered by the weed is A m2. The biologist claims that the rate of

increase of A is proportional to√(2A − 5).

(i) Write down a differential equation representing the biologist’s claim. [1]

(ii) At the start of the investigation, the area covered by the weed was 7 m2 and, 10 weeks later, the

area covered was 27 m2 . Assuming that the biologist’s claim is correct, find the area covered

20 weeks after the start of the investigation. [9]

10 The polynomial p(ß) is defined by

p(ß) = ß3 + mß2 + 24ß + 32,

where m is a constant. It is given that (ß + 2) is a factor of p(ß).(i) Find the value of m. [2]

(ii) Hence, showing all your working, find

(a) the three roots of the equation p(ß) = 0, [5]

(b) the six roots of the equation p(ß2) = 0. [6]

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MATHEMATICS 9709/41

Paper 4 Mechanics 1 (M1) October/November 2010

1 hour 15 minutes


Graph Paper











Where a numerical value for the acceleration due to gravity is needed, use 10 m s−2.










*7969954022*

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2

1

O

V

t (s)

v (m s )–1

2 4

P

Q

Two particles P and Q move vertically under gravity. The graphs show the upward velocity v m s−1

of the particles at time t s, for 0 ≤ t ≤ 4. P starts with velocity V m s−1 and Q starts from rest.

(i) Find the value of V . [2]

Given that Q reaches the horizontal ground when t = 4, find

(ii) the speed with which Q reaches the ground, [1]

(iii) the height of Q above the ground when t = 0. [2]

2 A car of mass 600 kg travels along a horizontal straight road, with its engine working at a rate of

40 kW. The resistance to motion of the car is constant and equal to 800 N. The car passes through the

point A on the road with speed 25 m s−1. The car’s acceleration at the point B on the road is half its

acceleration at A. Find the speed of the car at B. [5]

3P1 P2

X

A

B

C

The diagram shows three particles A, B and C hanging freely in equilibrium, each being attached to

the end of a string. The other ends of the three strings are tied together and are at the point X. The

strings carrying A and C pass over smooth fixed horizontal pegs P1

and P2

respectively. The weights

of A, B and C are 5.5 N, 7.3 N and W N respectively, and the angle P1XP

2is a right angle. Find the

angle AP1X and the value of W . [5]

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3

4 A particle P starts from a fixed point O at time t = 0, where t is in seconds, and moves with constant

acceleration in a straight line. The initial velocity of P is 1.5 m s−1 and its velocity when t = 10 is

3.5 m s−1.

(i) Find the displacement of P from O when t = 10. [2]

Another particle Q also starts from O when t = 0 and moves along the same straight line as P. The

acceleration of Q at time t is 0.03t m s−2.

(ii) Given that Q has the same velocity as P when t = 10, show that it also has the same displacement

from O as P when t = 10. [5]

5 A particle of mass 0.8 kg slides down a rough inclined plane along a line of greatest slope AB. The

distance AB is 8 m. The particle starts at A with speed 3 m s−1 and moves with constant acceleration

2.5 m s−2.

(i) Find the speed of the particle at the instant it reaches B. [2]

(ii) Given that the work done against the frictional force as the particle moves from A to B is 7 J, find

the angle of inclination of the plane. [4]

When the particle is at the point X its speed is the same as the average speed for the motion from A

to B.

(iii) Find the work done by the frictional force for the particle’s motion from A to X. [3]

6A

X

B

3 mH m

h m

A smooth slide AB is fixed so that its highest point A is 3 m above horizontal ground. B is h m above

the ground. A particle P of mass 0.2 kg is released from rest at a point on the slide. The particle

moves down the slide and, after passing B, continues moving until it hits the ground (see diagram).

The speed of P at B is vB

and the speed at which P hits the ground is vG

.

(i) In the case that P is released at A, it is given that the kinetic energy of P at B is 1.6 J. Find

(a) the value of h, [3]

(b) the kinetic energy of the particle immediately before it reaches the ground, [1]

(c) the ratio vG

: vB. [2]

(ii) In the case that P is released at the point X of the slide, which is H m above the ground (see

diagram), it is given that vG

: vB= 2.55. Find the value of H correct to 2 significant figures. [3]


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4

7

30°

3.2 N

Q

P

Particles P and Q, of masses 0.2 kg and 0.5 kg respectively, are connected by a light inextensible

string. The string passes over a smooth pulley at the edge of a rough horizontal table. P hangs freely

and Q is in contact with the table. A force of magnitude 3.2 N acts on Q, upwards and away from the

pulley, at an angle of 30◦ to the horizontal (see diagram).

(i) The system is in limiting equilibrium with P about to move upwards. Find the coefficient of

friction between Q and the table. [6]

The force of magnitude 3.2 N is now removed and P starts to move downwards.

(ii) Find the acceleration of the particles and the tension in the string. [4]






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MATHEMATICS 9709/42


1 hour 15 minutes


Graph Paper





















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2

1 A block of mass 400 kg rests in limiting equilibrium on horizontal ground. A force of magnitude

2000 N acts on the block at an angle of 15◦ to the upwards vertical. Find the coefficient of friction

between the block and the ground, correct to 2 significant figures. [5]

2 A cyclist, working at a constant rate of 400 W, travels along a straight road which is inclined at 2◦ to

the horizontal. The total mass of the cyclist and his cycle is 80 kg. Ignoring any resistance to motion,

find, correct to 1 decimal place, the acceleration of the cyclist when he is travelling

(i) uphill at 4 m s−1,

(ii) downhill at 4 m s−1.

[5]

3

PF N

F N

5 N

6 N

a°

A particle P is in equilibrium on a smooth horizontal table under the action of four horizontal forces

of magnitudes 6 N, 5 N, F N and F N acting in the directions shown. Find the values of α and F. [6]

4 A block of mass 20 kg is pulled from the bottom to the top of a slope. The slope has length 10 m and

is inclined at 4.5◦ to the horizontal. The speed of the block is 2.5 m s−1 at the bottom of the slope and

1.5 m s−1 at the top of the slope.

(i) Find the loss of kinetic energy and the gain in potential energy of the block. [3]

(ii) Given that the work done against the resistance to motion is 50 J, find the work done by the

pulling force acting on the block. [2]

(iii) Given also that the pulling force is constant and acts at an angle of 15◦ upwards from the slope,

find its magnitude. [2]

5 Particles P and Q are projected vertically upwards, from different points on horizontal ground, with

velocities of 20 m s−1 and 25 m s−1 respectively. Q is projected 0.4 s later than P. Find

(i) the time for which P’s height above the ground is greater than 15 m, [3]

(ii) the velocities of P and Q at the instant when the particles are at the same height. [5]

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3

6

t (s)

v (m s )–1

V

O 2.5 4.5 14.5

The diagram shows the velocity-time graph for a particle P which travels on a straight line AB, where

v m s−1 is the velocity of P at time t s. The graph consists of five straight line segments. The particle

starts from rest when t = 0 at a point X on the line between A and B and moves towards A. The

particle comes to rest at A when t = 2.5.

(i) Given that the distance XA is 4 m, find the greatest speed reached by P during this stage of the

motion. [2]

In the second stage, P starts from rest at A when t = 2.5 and moves towards B. The distance AB is

48 m. The particle takes 12 s to travel from A to B and comes to rest at B. For the first 2 s of this stage

P accelerates at 3 m s−2, reaching a velocity of V m s−1. Find

(ii) the value of V , [2]

(iii) the value of t at which P starts to decelerate during this stage, [3]

(iv) the deceleration of P immediately before it reaches B. [2]

7 A particle P travels in a straight line. It passes through the point O of the line with velocity 5 m s−1 at

time t = 0, where t is in seconds. P’s velocity after leaving O is given by

(0.002t3 − 0.12t2 + 1.8t + 5)m s−1.

The velocity of P is increasing when 0 < t < T1

and when t > T2, and the velocity of P is decreasing

when T1< t < T

2.

(i) Find the values of T1

and T2

and the distance OP when t = T2. [7]

(ii) Find the velocity of P when t = T2

and sketch the velocity-time graph for the motion of P. [3]

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MATHEMATICS 9709/43


1 hour 15 minutes


Graph Paper





















*6604966683*

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2

1 A particle P is released from rest at a point on a smooth plane inclined at 30◦ to the horizontal. Find

the speed of P

(i) when it has travelled 0.9 m,

(ii) 0.8 s after it is released.

[4]

2

1.8 m

A

B C

The diagram shows the vertical cross-section ABC of a fixed surface. AB is a curve and BC is a

horizontal straight line. The part of the surface containing AB is smooth and the part containing BC

is rough. A is at a height of 1.8 m above BC. A particle of mass 0.5 kg is released from rest at A and

travels along the surface to C.

(i) Find the speed of the particle at B. [2]

(ii) Given that the particle reaches C with a speed of 5 m s−1, find the work done against the resistance

to motion as the particle moves from B to C. [2]

3A

B

P Q

C30°

3 3 NÖ

A small smooth pulley is fixed at the highest point A of a cross-section ABC of a triangular prism.

Angle ABC = 90◦ and angle BCA = 30◦. The prism is fixed with the face containing BC in contact

with a horizontal surface. Particles P and Q are attached to opposite ends of a light inextensible

string, which passes over the pulley. The particles are in equilibrium with P hanging vertically below

the pulley and Q in contact with AC. The resultant force exerted on the pulley by the string is 3√

3 N

(see diagram).

(i) Show that the tension in the string is 3 N. [2]

The coefficient of friction between Q and the prism is 0.75.

(ii) Given that Q is in limiting equilibrium and on the point of moving upwards, find its mass. [5]

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3

4 A particle starts from rest at a point X and moves in a straight line until, 60 seconds later, it reaches a

point Y . At time t s after leaving X, the acceleration of the particle is

0.75 m s−2 for 0 < t < 4,

0 m s−2 for 4 < t < 54,

−0.5 m s−2 for 54 < t < 60.

(i) Find the velocity of the particle when t = 4 and when t = 60, and sketch the velocity-time graph.

[5]

(ii) Find the distance XY . [2]

5 A force of magnitude F N acts in a horizontal plane and has components 27.5 N and −24 N in the

x-direction and the y-direction respectively. The force acts at an angle of α◦ below the x-axis.

(i) Find the values of F and α. [4]

A second force, of magnitude 87.6 N, acts in the same plane at 90◦ anticlockwise from the force of

magnitude F N. The resultant of the two forces has magnitude R N and makes an angle of θ◦ with the

positive x-axis.

(ii) Find the values of R and θ. [3]

6 A particle travels along a straight line. It starts from rest at a point A on the line and comes to rest

again, 10 seconds later, at another point B on the line. The velocity t seconds after leaving A is

0.72t2 − 0.096t3 for 0 ≤ t ≤ 5,

2.4t − 0.24t2 for 5 ≤ t ≤ 10.

(i) Show that there is no instantaneous change in the acceleration of the particle when t = 5. [4]

(ii) Find the distance AB. [4]

[Question 7 is printed on the next page.]


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4

7 A car of mass 1250 kg travels along a horizontal straight road. The power of the car’s engine is

constant and equal to 24 kW and the resistance to the car’s motion is constant and equal to R N. The

car passes through the point A on the road with speed 20 m s−1 and acceleration 0.32 m s−2.

(i) Find the value of R. [3]

The car continues with increasing speed, passing through the point B on the road with speed 29.9 m s−1.

The car subsequently passes through the point C.

(ii) Find the acceleration of the car at B, giving the answer in m s−2 correct to 3 decimal places. [2]

(iii) Show that, while the car’s speed is increasing, it cannot reach 30 m s−1. [2]

(iv) Explain why the speed of the car is approximately constant between B and C. [1]

(v) State a value of the approximately constant speed, and the maximum possible error in this value

at any point between B and C. [1]

The work done by the car’s engine during the motion from B to C is 1200 kJ.

(vi) By assuming the speed of the car is constant from B to C, find, in either order,

(a) the approximate time taken for the car to travel from B to C,

(b) an approximation for the distance BC.

[4]






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MATHEMATICS 9709/51


1 hour 15 minutes


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*6534434770*

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2

1 A horizontal circular disc rotates with constant angular speed 9 rad s−1 about its centre O. A particle

of mass 0.05 kg is placed on the disc at a distance 0.4 m from O. The particle moves with the disc

and no sliding takes place. Calculate the magnitude of the resultant force exerted on the particle by

the disc. [3]

2

O

B

A

0.8 m

radp2

3

A bow consists of a uniform curved portion AB of mass 1.4 kg, and a uniform taut string of mass m kg

which joins A and B. The curved portion AB is an arc of a circle centre O and radius 0.8 m. Angle

AOB is 23π radians (see diagram). The centre of mass of the bow (including the string) is 0.65 m

from O. Calculate m. [6]

3

0.2 m

A

P30°

One end of a light inextensible string of length 0.2 m is attached to a fixed point A which is above a

smooth horizontal surface. A particle P of mass 0.6 kg is attached to the other end of the string. P

moves in a circle on the surface with constant speed v m s−1, with the string taut and making an angle

of 30◦ to the horizontal (see diagram).

(i) Given that v = 1.5, calculate the magnitude of the force that the surface exerts on P. [4]

(ii) Given instead that P moves with its greatest possible speed while remaining in contact with the

surface, find v. [3]

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3

4

A

B

1.7 m

0.8 m 70 N

220 N

A uniform beam AB has length 2 m and weight 70 N. The beam is hinged at A to a fixed point on a

vertical wall, and is held in equilibrium by a light inextensible rope. One end of the rope is attached

to the wall at a point 1.7 m vertically above the hinge. The other end of the rope is attached to the

beam at a point 0.8 m from A. The rope is at right angles to AB. The beam carries a load of weight

220 N at B (see diagram).

(i) Find the tension in the rope. [3]

(ii) Find the direction of the force exerted on the beam at A. [4]

5 A particle P of mass 0.28 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 at the same horizontal level and

4.8 m apart. P is released from rest at the mid-point of AB. In the subsequent motion, the acceleration

of P is zero when P is at a distance 0.7 m below AB.

(i) Show that the modulus of elasticity of the string is 20 N. [4]

(ii) Calculate the maximum speed of P. [3]

6 A cyclist and his bicycle have a total mass of 81 kg. The cyclist starts from rest and rides in a

straight line. The cyclist exerts a constant force of 135 N and the motion is opposed by a resistance

of magnitude 9v N, where v m s−1 is the cyclist’s speed at time t s after starting.

(i) Show that9

15 − v

dv

dt= 1. [2]

(ii) Solve this differential equation to show that v = 15(1 − e−

19t). [4]

(iii) Find the distance travelled by the cyclist in the first 9 s of the motion. [4]


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4

7

O

A10 m s–1

45° 30°

A particle P is projected from a point O with initial speed 10 m s−1 at an angle of 45◦ above the

horizontal. P subsequently passes through the point A which is at an angle of elevation of 30◦ from

O (see diagram). At time t s after projection the horizontal and vertically upward displacements of P

from O are x m and y m respectively.

(i) Write down expressions for x and y in terms of t, and hence obtain the equation of the trajectory

of P. [3]

(ii) Calculate the value of x when P is at A. [3]

(iii) Find the angle the trajectory makes with the horizontal when P is at A. [4]






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MATHEMATICS 9709/52


1 hour 15 minutes


Graph Paper





















*7222159052*

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2

1 A horizontal circular disc rotates with constant angular speed 9 rad s−1 about its centre O. A particle

of mass 0.05 kg is placed on the disc at a distance 0.4 m from O. The particle moves with the disc

and no sliding takes place. Calculate the magnitude of the resultant force exerted on the particle by

the disc. [3]

2

O

B

A

0.8 m

radp2

3

A bow consists of a uniform curved portion AB of mass 1.4 kg, and a uniform taut string of mass m kg

which joins A and B. The curved portion AB is an arc of a circle centre O and radius 0.8 m. Angle

AOB is 23π radians (see diagram). The centre of mass of the bow (including the string) is 0.65 m

from O. Calculate m. [6]

3

0.2 m

A

P30°

One end of a light inextensible string of length 0.2 m is attached to a fixed point A which is above a

smooth horizontal surface. A particle P of mass 0.6 kg is attached to the other end of the string. P

moves in a circle on the surface with constant speed v m s−1, with the string taut and making an angle

of 30◦ to the horizontal (see diagram).

(i) Given that v = 1.5, calculate the magnitude of the force that the surface exerts on P. [4]

(ii) Given instead that P moves with its greatest possible speed while remaining in contact with the

surface, find v. [3]

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3

4

A

B

1.7 m

0.8 m 70 N

220 N

A uniform beam AB has length 2 m and weight 70 N. The beam is hinged at A to a fixed point on a

vertical wall, and is held in equilibrium by a light inextensible rope. One end of the rope is attached

to the wall at a point 1.7 m vertically above the hinge. The other end of the rope is attached to the

beam at a point 0.8 m from A. The rope is at right angles to AB. The beam carries a load of weight

220 N at B (see diagram).

(i) Find the tension in the rope. [3]

(ii) Find the direction of the force exerted on the beam at A. [4]

5 A particle P of mass 0.28 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 at the same horizontal level and

4.8 m apart. P is released from rest at the mid-point of AB. In the subsequent motion, the acceleration

of P is zero when P is at a distance 0.7 m below AB.

(i) Show that the modulus of elasticity of the string is 20 N. [4]

(ii) Calculate the maximum speed of P. [3]

6 A cyclist and his bicycle have a total mass of 81 kg. The cyclist starts from rest and rides in a

straight line. The cyclist exerts a constant force of 135 N and the motion is opposed by a resistance

of magnitude 9v N, where v m s−1 is the cyclist’s speed at time t s after starting.

(i) Show that9

15 − v

dv

dt= 1. [2]

(ii) Solve this differential equation to show that v = 15(1 − e−

19t). [4]

(iii) Find the distance travelled by the cyclist in the first 9 s of the motion. [4]


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4

7

O

A10 m s–1

45° 30°

A particle P is projected from a point O with initial speed 10 m s−1 at an angle of 45◦ above the

horizontal. P subsequently passes through the point A which is at an angle of elevation of 30◦ from

O (see diagram). At time t s after projection the horizontal and vertically upward displacements of P

from O are x m and y m respectively.

(i) Write down expressions for x and y in terms of t, and hence obtain the equation of the trajectory

of P. [3]

(ii) Calculate the value of x when P is at A. [3]

(iii) Find the angle the trajectory makes with the horizontal when P is at A. [4]






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MATHEMATICS 9709/53


1 hour 15 minutes


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2

1

A B

CD0.9 m

0.9 m

1.8 m

ABCD is a uniform lamina with AB = 1.8 m, AD = DC = 0.9 m, and AD perpendicular to AB and DC

(see diagram).

(i) Find the distance of the centre of mass of the lamina from AB and the distance from AD. [4]

The lamina is freely suspended at A and hangs in equilibrium.

(ii) Calculate the angle between AB and the vertical. [2]

2 A particle P is projected with speed 26 m s−1 at an angle of 30◦ above the horizontal from a point O

on a horizontal plane.

(i) For the instant when the vertical component of the velocity of P is 5 m s−1 downwards, find the

direction of motion of P and the height of P above the plane. [4]

(ii) P strikes the plane at the point A. Calculate the time taken by P to travel from O to A and the

distance OA. [3]

3A

P

QB0.3 m

5 rad s–1

a°

Particles P and Q have masses 0.8 kg and 0.4 kg respectively. P is attached to a fixed point A by

a light inextensible string which is inclined at an angle α◦ to the vertical. Q is attached to a fixed

point B, which is vertically below A, by a light inextensible string of length 0.3 m. The string BQ

is horizontal. P and Q are joined to each other by a light inextensible string which is vertical. The

particles rotate in horizontal circles of radius 0.3 m about the axis through A and B with constant

angular speed 5 rad s−1 (see diagram).

(i) By considering the motion of Q, find the tensions in the strings PQ and BQ. [3]

(ii) Find the tension in the string AP and the value of α. [5]

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3

4

A

B

1.2 m

30°

A uniform rod AB has weight 15 N and length 1.2 m. The end A of the rod is in contact with a rough

plane inclined at 30◦ to the horizontal, and the rod is perpendicular to the plane. The rod is held in

equilibrium in this position by means of a horizontal force applied at B, acting in the vertical plane

containing the rod (see diagram).

(i) Show that the magnitude of the force applied at B is 4.33 N, correct to 3 significant figures. [3]

(ii) Find the magnitude of the frictional force exerted by the plane on the rod. [2]

(iii) Given that the rod is in limiting equilibrium, calculate the coefficient of friction between the rod

and the plane. [3]

5

0.5 m

2.4 m

A B

P

A light elastic string has natural length 2 m and modulus of elasticity λ N. The ends of the string are

attached to fixed points A and B which are at the same horizontal level and 2.4 m apart. A particle

P of mass 0.6 kg is attached to the mid-point of the string and hangs in equilibrium at a point 0.5 m

below AB (see diagram).

(i) Show that λ = 26. [4]

P is projected vertically downwards from the equilibrium position, and comes to instantaneous rest

at a point 0.9 m below AB.

(ii) Calculate the speed of projection of P. [5]

[Question 6 is printed on the next page.]


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6

30°

P

2 m s–1

A particle P of mass 0.2 kg is projected with velocity 2 m s−1 upwards along a line of greatest slope on

a plane inclined at 30◦ to the horizontal (see diagram). Air resistance of magnitude 0.5v N opposes the

motion of P, where v m s−1 is the velocity of P at time t s after projection. The coefficient of friction

between P and the plane is1

2√

3. The particle P reaches a position of instantaneous rest when t = T .

(i) Show that, while P is moving up the plane,dv

dt= −2.5(3 + v). [3]

(ii) Calculate T . [4]

(iii) Calculate the speed of P when t = 2T . [5]






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MATHEMATICS 9709/61

Paper 6 Probability & Statistics 1 (S1) October/November 2010

1 hour 15 minutes


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*3792937618*

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2

1 Anita made observations of the maximum temperature, t◦C, on 50 days. Her results are summarised

by Σ t = 910 and Σ(t − t)2 = 876, where t denotes the mean of the 50 observations. Calculate t and

the standard deviation of the observations. [3]

2 On average, 2 apples out of 15 are classified as being underweight. Find the probability that in a

random sample of 200 apples, the number of apples which are underweight is more than 21 and less

than 35. [5]

3 The times taken by students to get up in the morning can be modelled by a normal distribution with

mean 26.4 minutes and standard deviation 3.7 minutes.

(i) For a random sample of 350 students, find the number who would be expected to take longer

than 20 minutes to get up in the morning. [3]

(ii) ‘Very slow’ students are students whose time to get up is more than 1.645 standard deviations

above the mean. Find the probability that fewer than 3 students from a random sample of 8

students are ‘very slow’. [4]

4 The weights in grams of a number of stones, measured correct to the nearest gram, are represented in

the following table.

Weight (grams) 1 − 10 11 − 20 21 − 25 26 − 30 31 − 50 51 − 70

Frequency 2x 4x 3x 5x 4x x

A histogram is drawn with a scale of 1 cm to 1 unit on the vertical axis, which represents frequency

density. The 1 − 10 rectangle has height 3 cm.

(i) Calculate the value of x and the height of the 51 − 70 rectangle. [4]

(ii) Calculate an estimate of the mean weight of the stones. [3]

5 Three friends, Rick, Brenda and Ali, go to a football match but forget to say which entrance to the

ground they will meet at. There are four entrances, A, B, C and D. Each friend chooses an entrance

independently.

• The probability that Rick chooses entrance A is 13. The probabilities that he chooses entrances

B, C or D are all equal.

• Brenda is equally likely to choose any of the four entrances.

• The probability that Ali chooses entrance C is 27

and the probability that he chooses entrance D

is 35. The probabilities that he chooses the other two entrances are equal.

(i) Find the probability that at least 2 friends will choose entrance B. [4]

(ii) Find the probability that the three friends will all choose the same entrance. [4]

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6

Pegs are to be placed in the four holes shown, one in each hole. The pegs come in different colours

and pegs of the same colour are identical. Calculate how many different arrangements of coloured

pegs in the four holes can be made using

(i) 6 pegs, all of different colours, [1]

(ii) 4 pegs consisting of 2 blue pegs, 1 orange peg and 1 yellow peg. [1]

Beryl has 12 pegs consisting of 2 red, 2 blue, 2 green, 2 orange, 2 yellow and 2 black pegs. Calculate

how many different arrangements of coloured pegs in the 4 holes Beryl can make using

(iii) 4 different colours, [1]

(iv) 3 different colours, [3]

(v) any of her 12 pegs. [3]

7 Sanket plays a game using a biased die which is twice as likely to land on an even number as on an

odd number. The probabilities for the three even numbers are all equal and the probabilities for the

three odd numbers are all equal.

(i) Find the probability of throwing an odd number with this die. [2]

Sanket throws the die once and calculates his score by the following method.

• If the number thrown is 3 or less he multiplies the number thrown by 3 and adds 1.

• If the number thrown is more than 3 he multiplies the number thrown by 2 and subtracts 4.

The random variable X is Sanket’s score.

(ii) Show that P(X = 8) = 29. [2]

The table shows the probability distribution of X.

x 4 6 7 8 10

P(X = x) 39

19

29

29

19

(iii) Given that E(X) = 589

, find Var(X). [2]

Sanket throws the die twice.

(iv) Find the probability that the total of the scores on the two throws is 16. [2]

(v) Given that the total of the scores on the two throws is 16, find the probability that the score on

the first throw was 6. [3]

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MATHEMATICS 9709/62


1 hour 15 minutes


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2

1 The discrete random variable X takes the values 1, 4, 5, 7 and 9 only. The probability distribution of

X is shown in the table.

x 1 4 5 7 9

P(X = x) 4p 5p2 1.5p 2.5p 1.5p

Find p. [3]

2 Esme noted the test marks, x, of 16 people in a class. She found that Σ x = 824 and that the standard

deviation of x was 6.5.

(i) Calculate Σ(x − 50) and Σ(x − 50)2. [3]

(ii) One person did the test later and her mark was 72. Calculate the new mean and standard deviation

of the marks of all 17 people. [3]

3 A fair five-sided spinner has sides numbered 1, 2, 3, 4, 5. Raj spins the spinner and throws two fair

dice. He calculates his score as follows.

• If the spinner lands on an even-numbered side, Raj multiplies the two numbers showing on

the dice to get his score.

• If the spinner lands on an odd-numbered side, Raj adds the numbers showing on the dice to

get his score.

Given that Raj’s score is 12, find the probability that the spinner landed on an even-numbered side.

[6]

4 The weights in kilograms of 11 bags of sugar and 7 bags of flour are as follows.

Sugar: 1.961 1.983 2.008 2.014 1.968 1.994 2.011 2.017 1.977 1.984 1.989

Flour: 1.945 1.962 1.949 1.977 1.964 1.941 1.953

(i) Represent this information on a back-to-back stem-and-leaf diagram with sugar on the left-hand

side. [4]

(ii) Find the median and interquartile range of the weights of the bags of sugar. [3]

5 The distance the Zotoc car can travel on 20 litres of fuel is normally distributed with mean 320 km and

standard deviation 21.6 km. The distance the Ganmor car can travel on 20 litres of fuel is normally

distributed with mean 350 km and standard deviation 7.5 km. Both cars are filled with 20 litres of fuel

and are driven towards a place 367 km away.

(i) For each car, find the probability that it runs out of fuel before it has travelled 367 km. [3]

(ii) The probability that a Zotoc car can travel at least (320 + d) km on 20 litres of fuel is 0.409. Find

the value of d. [4]

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3

6 (i) State three conditions that must be satisfied for a situation to be modelled by a binomial

distribution. [2]

On any day, there is a probability of 0.3 that Julie’s train is late.

(ii) Nine days are chosen at random. Find the probability that Julie’s train is late on more than 7 days

or fewer than 2 days. [3]

(iii) 90 days are chosen at random. Find the probability that Julie’s train is late on more than 35 days

or fewer than 27 days. [5]

7 A committee of 6 people, which must contain at least 4 men and at least 1 woman, is to be chosen

from 10 men and 9 women.

(i) Find the number of possible committees that can be chosen. [3]

(ii) Find the probability that one particular man, Albert, and one particular woman, Tracey, are both

on the committee. [2]

(iii) Find the number of possible committees that include either Albert or Tracey but not both. [3]

(iv) The committee that is chosen consists of 4 men and 2 women. They queue up randomly in a line

for refreshments. Find the probability that the women are not next to each other in the queue.

[3]

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MATHEMATICS 9709/63


1 hour 15 minutes


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*8831179177*

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2

1 Name the distribution and suggest suitable numerical parameters that you could use to model the

weights in kilograms of female 18-year-old students. [2]

2 In a probability distribution the random variable X takes the value x with probability kx, where x takes

values 1, 2, 3, 4, 5 only.

(i) Draw up a probability distribution table for X, in terms of k, and find the value of k. [3]

(ii) Find E(X). [2]

3 It was found that 68% of the passengers on a train used a cell phone during their train journey. Of

those using a cell phone, 70% were under 30 years old, 25% were between 30 and 65 years old and

the rest were over 65 years old. Of those not using a cell phone, 26% were under 30 years old and

64% were over 65 years old.

(i) Draw a tree diagram to represent this information, giving all probabilities as decimals. [2]

(ii) Given that one of the passengers is 45 years old, find the probability of this passenger using a

cell phone during the journey. [3]

4 Delip measured the speeds, x km per hour, of 70 cars on a road where the speed limit is 60 km per hour.

His results are summarised by Σ(x − 60) = 245.

(i) Calculate the mean speed of these 70 cars. [2]

His friend Sachim used values of (x − 50) to calculate the mean.

(ii) Find Σ(x − 50). [2]

(iii) The standard deviation of the speeds is 10.6 km per hour. Calculate Σ(x − 50)2. [2]

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3

5 The following histogram illustrates the distribution of times, in minutes, that some students spent

taking a shower.

0 2 4 6 8 10 12 14 16 18 20

5

0

10

15

20

25

30

35

40

Frequencydensity

Time inminutes

(i) Copy and complete the following frequency table for the data. [3]

Time (t minutes) 2 < t ≤ 4 4 < t ≤ 6 6 < t ≤ 7 7 < t ≤ 8 8 < t ≤ 10 10 < t ≤ 16

Frequency

(ii) Calculate an estimate of the mean time to take a shower. [2]

(iii) Two of these students are chosen at random. Find the probability that exactly one takes between

7 and 10 minutes to take a shower. [3]



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4

6Windows

Windows

AisleFrontBack

A small aeroplane has 14 seats for passengers. The seats are arranged in 4 rows of 3 seats and a back

row of 2 seats (see diagram). 12 passengers board the aeroplane.

(i) How many possible seating arrangements are there for the 12 passengers? Give your answer

correct to 3 significant figures. [2]

These 12 passengers consist of 2 married couples (Mr and Mrs Lin and Mr and Mrs Brown), 5 students

and 3 business people.

(ii) The 3 business people sit in the front row. The 5 students each sit at a window seat. Mr and Mrs

Lin sit in the same row on the same side of the aisle. Mr and Mrs Brown sit in another row on

the same side of the aisle. How many possible seating arrangements are there? [4]

(iii) If, instead, the 12 passengers are seated randomly, find the probability that Mrs Lin sits directly

behind a student and Mrs Brown sits in the front row. [4]

7 The times spent by people visiting a certain dentist are independent and normally distributed with a

mean of 8.2 minutes. 79% of people who visit this dentist have visits lasting less than 10 minutes.

(i) Find the standard deviation of the times spent by people visiting this dentist. [3]

(ii) Find the probability that the time spent visiting this dentist by a randomly chosen person deviates

from the mean by more than 1 minute. [3]

(iii) Find the probability that, of 6 randomly chosen people, more than 2 have visits lasting longer

than 10 minutes. [3]

(iv) Find the probability that, of 35 randomly chosen people, fewer than 16 have visits lasting less

than 8.2 minutes. [5]






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MATHEMATICS 9709/71


1 hour 15 minutes


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2

1 In a survey of 1000 randomly chosen adults, 605 said that they used email. Calculate a 90% confidence

interval for the proportion of adults in the whole population who use email. [3]

2 People arrive randomly and independently at a supermarket checkout at an average rate of 2 people

every 3 minutes.

(i) Find the probability that exactly 4 people arrive in a 5-minute period. [2]

At another checkout in the same supermarket, people arrive randomly and independently at an average

rate of 1 person each minute.

(ii) Find the probability that a total of fewer than 3 people arrive at the two checkouts in a 3-minute

period. [3]

3 A book contains 40 000 words. For each word, the probability that it is printed wrongly is 0.0001 and

these errors occur independently. The number of words printed wrongly in the book is represented

by the random variable X.

(i) State the exact distribution of X, including the values of any parameters. [1]

(ii) State an approximate distribution for X, including the values of any parameters, and explain why

this approximate distribution is appropriate. [3]

(iii) Use this approximate distribution to find the probability that there are more than 3 words printed

wrongly in the book. [3]

4

x

f( )x

0

0.5

1

0 1 2

The diagram shows the graph of the probability density function, f, of a random variable X which

takes values between 0 and 2 only.

(i) Find P(1 < X < 1.5). [2]

(ii) Find the median of X. [3]

(iii) Find E(X). [2]

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3

5 The marks of candidates in Mathematics and English in 2009 were represented by the independent

random variables X and Y with distributions N(28, 5.62) and N(52, 12.42) respectively. Each

candidate’s marks were combined to give a final mark F, where F = X + 12Y .

(i) Find E(F) and Var(F). [3]

(ii) The final marks of a random sample of 10 candidates from Grinford in 2009 had a mean of

49. Test at the 5% significance level whether this result suggests that the mean final mark of all

candidates from Grinford in 2009 was lower than elsewhere. [5]

6 It is claimed that a certain 6-sided die is biased so that it is more likely to show a six than if it was fair.

In order to test this claim at the 10% significance level, the die is thrown 10 times and the number of

sixes is noted.

(i) Given that the die shows a six on 3 of the 10 throws, carry out the test. [5]

On another occasion the same test is carried out again.

(ii) Find the probability of a Type I error. [3]

(iii) Explain what is meant by a Type II error in this context. [1]

7 (a) Give a reason why sampling would be required in order to reach a conclusion about

(i) the mean height of adult males in England, [1]

(ii) the mean weight that can be supported by a single cable of a certain type without the cable

breaking. [1]

(b) The weights, in kg, of sacks of potatoes are represented by the random variable X with mean µ

and standard deviation σ. The weights of a random sample of 500 sacks of potatoes are found

and the results are summarised below.

n = 500, Σ x = 9850, Σ x2 = 194 125.

(i) Calculate unbiased estimates of µ and σ2. [3]

(ii) A further random sample of 60 sacks of potatoes is taken. Using your values from part (b) (i),

find the probability that the mean weight of this sample exceeds 19.73 kg. [4]

(iii) Explain whether it was necessary to use the Central Limit Theorem in your calculation in

part (b) (ii). [2]

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MATHEMATICS 9709/72


1 hour 15 minutes


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2

1 In a survey of 1000 randomly chosen adults, 605 said that they used email. Calculate a 90% confidence

interval for the proportion of adults in the whole population who use email. [3]

2 People arrive randomly and independently at a supermarket checkout at an average rate of 2 people

every 3 minutes.

(i) Find the probability that exactly 4 people arrive in a 5-minute period. [2]

At another checkout in the same supermarket, people arrive randomly and independently at an average

rate of 1 person each minute.

(ii) Find the probability that a total of fewer than 3 people arrive at the two checkouts in a 3-minute

period. [3]

3 A book contains 40 000 words. For each word, the probability that it is printed wrongly is 0.0001 and

these errors occur independently. The number of words printed wrongly in the book is represented

by the random variable X.

(i) State the exact distribution of X, including the values of any parameters. [1]

(ii) State an approximate distribution for X, including the values of any parameters, and explain why

this approximate distribution is appropriate. [3]

(iii) Use this approximate distribution to find the probability that there are more than 3 words printed

wrongly in the book. [3]

4

x

f( )x

0

0.5

1

0 1 2

The diagram shows the graph of the probability density function, f, of a random variable X which

takes values between 0 and 2 only.

(i) Find P(1 < X < 1.5). [2]


(iii) Find E(X). [2]

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3

5 The marks of candidates in Mathematics and English in 2009 were represented by the independent

random variables X and Y with distributions N(28, 5.62) and N(52, 12.42) respectively. Each

candidate’s marks were combined to give a final mark F, where F = X + 12Y .

(i) Find E(F) and Var(F). [3]

(ii) The final marks of a random sample of 10 candidates from Grinford in 2009 had a mean of

49. Test at the 5% significance level whether this result suggests that the mean final mark of all

candidates from Grinford in 2009 was lower than elsewhere. [5]

6 It is claimed that a certain 6-sided die is biased so that it is more likely to show a six than if it was fair.

In order to test this claim at the 10% significance level, the die is thrown 10 times and the number of

sixes is noted.

(i) Given that the die shows a six on 3 of the 10 throws, carry out the test. [5]

On another occasion the same test is carried out again.

(ii) Find the probability of a Type I error. [3]

(iii) Explain what is meant by a Type II error in this context. [1]

7 (a) Give a reason why sampling would be required in order to reach a conclusion about

(i) the mean height of adult males in England, [1]

(ii) the mean weight that can be supported by a single cable of a certain type without the cable

breaking. [1]

(b) The weights, in kg, of sacks of potatoes are represented by the random variable X with mean µ

and standard deviation σ. The weights of a random sample of 500 sacks of potatoes are found

and the results are summarised below.

n = 500, Σ x = 9850, Σ x2 = 194 125.

(i) Calculate unbiased estimates of µ and σ2. [3]

(ii) A further random sample of 60 sacks of potatoes is taken. Using your values from part (b) (i),

find the probability that the mean weight of this sample exceeds 19.73 kg. [4]

(iii) Explain whether it was necessary to use the Central Limit Theorem in your calculation in

part (b) (ii). [2]

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MATHEMATICS 9709/73


1 hour 15 minutes


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2

1 A random variable has the distribution Po(31). Name an appropriate approximating distribution and

state the mean and standard deviation of this approximating distribution. [3]

2 The editor of a magazine wishes to obtain the views of a random sample of readers about the future

of the magazine.

(i) A sub-editor proposes that they include in one issue of the magazine a questionnaire for readers

to complete and return. Give two reasons why the readers who return the questionnaire would

not form a random sample. [2]

The editor decides to use a table of random numbers to select a random sample of 50 readers from

the 7302 regular readers. These regular readers are numbered from 1 to 7302. The first few random

numbers which the editor obtains from the table are as follows.

49757 80239 52038 60882

(ii) Use these random numbers to select the first three members in the sample. [2]

3 The masses of sweets produced by a machine are normally distributed with mean µ grams and standard

deviation 1.0 grams. A random sample of 65 sweets produced by the machine has a mean mass of

29.6 grams.

(i) Find a 99% confidence interval for µ. [3]

The manufacturer claims that the machine produces sweets with a mean mass of 30 grams.

(ii) Use the confidence interval found in part (i) to draw a conclusion about this claim. [2]

(iii) Another random sample of 65 sweets produced by the machine is taken. This sample gives a 99%

confidence interval that leads to a different conclusion from that found in part (ii). Assuming

that the value of µ has not changed, explain how this can be possible. [1]

4 The masses, in milligrams, of three minerals found in 1 tonne of a certain kind of rock are modelled

by three independent random variables P, Q and R, where P ∼ N(46, 192), Q ∼ N(53, 232) and

R ∼ N(25, 102). The total value of the minerals found in 1 tonne of rock is modelled by the random

variable V , where V = P + Q + 2R. Use the model to find the probability of finding minerals with a

value of at least 93 in a randomly chosen tonne of rock. [7]

5 A continuous random variable X has probability density function given by

f(x) = { 16x 2 ≤ x ≤ 4,

0 otherwise.

(i) Find E(X). [3]


(iii) Two independent values of X are chosen at random. Find the probability that both these values

are greater than 3. [3]

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3

6 A clinic monitors the amount, X milligrams per litre, of a certain chemical in the blood stream of

patients. For patients who are taking drug A, it has been found that the mean value of X is 0.336. A

random sample of 100 patients taking a new drug, B, was selected and the values of X were found.

The results are summarised below.

n = 100, Σ x = 43.5, Σ x2 = 31.56.

(i) Test at the 1% significance level whether the mean amount of the chemical in the blood stream

of patients taking drug B is different from that of patients taking drug A. [8]

(ii) For the test to be valid, is it necessary to assume a normal distribution for the amount of chemical

in the blood stream of patients taking drug B? Justify your answer. [2]

7 In the past, the number of house sales completed per week by a building company has been modelled

by a random variable which has the distribution Po(0.8). Following a publicity campaign, the builders

hope that the mean number of sales per week will increase. In order to test at the 5% significance

level whether this is the case, the total number of sales during the first 3 weeks after the campaign is

noted. It is assumed that a Poisson model is still appropriate.

(i) Given that the total number of sales during the 3 weeks is 5, carry out the test. [6]

(ii) During the following 3 weeks the same test is carried out again, using the same significance

level. Find the probability of a Type I error. [3]

(iii) Explain what is meant by a Type I error in this context. [1]

(iv) State what further information would be required in order to find the probability of a Type II

error. [1]

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A-Level Past Papers – Mathematics A-Level Examinations ... · PDF fileA-Level Past Papers – Mathematics A-Level Examinations October/November 2010 Paper Pages Pure Mathematics