Cambridge International
AS & A Level Mathematics:
Pure Mathematics 1
Practice Book
,Contents
How to use this book
1 Quadratics
2 Functions
3 Coordinate geometry
4 Circular measure
5 Trigonometry
6 Series
7 Differentiation
8 Further differentiation
9 Integration
Answers
, How to use this book
Throughout this book you will notice particular features that are designed to help your learning.
This section provides a brief overview of these features.
■ Find the equation of a straight line given sufficient information.
■ Interpret and use any of the forms in
solving problems.
■ Understand that the equation represents the circle with centre (a,
b) and radius r.
Learning objectives indicate the important concepts within each chapter and help you to navigate
through the practice book.
TIP
Substitute the value of r that minimises the quantity back into the original expression to find
the minimum value of that quantity.
Tip boxes contain helpful guidance about calculating or checking your answers.
WORKED EXAMPLE 1.3
Solve the equation Write your answers as exact values (in surd form).
Answer
Using a = 2, b = −2 and c = −1 in the quadratic
formula.
Simplify.
Simplify the surd.
Worked examples provide step-by-step approaches to answering questions. The left side shows a fully
worked solution, while the right side contains a commentary explaining each step in the working.
END-OF-CHAPTER REVIEW EXERCISE 7
1 Find of gradient of the graph of at the point where the y-coordinate is 3.
2 f(x) = ax3 + bx−2 where a and b are constants. f′(1) = 18 and f″(1) = 18. Find a and b.
, 3 Find the values of x for which the gradient of f(x) is 9.
The End-of-chapter review exercise contains exam-style questions covering all topics in the chapter.
You can use this to check your understanding of the topics you have covered.
Throughout each chapter there are multiple exercises containing practice questions. The questions are
coded:
PS These questions focus on problemsolving.
P These questions focus on proofs.
M These questions focus on modelling.
You should not use a calculator for these questions.
,Chapter 1
Quadratics
■ Carry out the process of completing the square for a quadratic polynomial ax2 + bx + c and
use a completed square form.
■ Find the discriminant of a quadratic polynomial ax2 + bx + c and use the discriminant.
■ Solve quadratic equations, and quadratic inequalities, in one unknown.
■ Solve by substitution a pair of simultaneous equations of which one is linear and one is
quadratic.
■ Recognise and solve equations in x that are quadratic in some function of x.
■ Understand the relationship between a graph of a quadratic function and its associated
algebraic equation, and use the relationship between points of intersection of graphs and
solutions of equations.
, 1.1 Solving quadratic equations by factorisation
WORKED EXAMPLE 1.1
PS Lim walked 12 km from A to B at a steady speed of x km/h.
His average speed for the return was 2 km/h slower.
a Write down, in terms of x, the total time taken for the complete journey.
b If the total time he took was 3.5 hours, write an equation in x and solve it to find his
speed from A to B.
Answer
a
From using
b
Multiply both sides by
12(x − 2) + 12x = 3.5x(x − 2) Expand brackets and rearrange.
12x − 24 + 12x = 3.5x2 − 7x Multiply both sides by 2 and rearrange.
7x2 − 62x + 48 = 0
Factorise.
(7x − 6) (x − 8) = 0 Solve.
The value of is a solution to the
equation, but it gives a negative speed
for the return journey.
Lim’s speed from A to B is 8 km/h.
EXERCISE 1A
1 By factorising, solve the following equations:
a i 3x2 + 2x = x2 + 3x + 6
ii 2x2 + 3 = 17x − 7 − x2
b i 9x2 = 24x − 16
ii 18x2 = 2x2 − 40x − 25
c i (x − 3) (x + 2) = 14
ii (2x + 3) (x − 1) = 12
d i
ii
2 Solve the following equations. (In most cases, multiplication by an appropriate
expression will turn the equation into a form you should recognise.)
a
b
c
, d
e
f
g
h
3 Solve algebraically:
(2x − 3) (x − 5) = (x − 3)2
4 Solve the equation x2 + 8k2 = 6kx, giving your answer in terms of k.
5 Find the exact solutions of the equation
6 Solve the equation
PS 7 The product of two positive, consecutive even integers is 168. Use this information to
form a quadratic equation and solve it to find the two integers.
PS 8 Two men A and B working together can complete a task in 4 days. If B completes the
task on his own, he takes 6 more days than if A did the task on his own.
Use the information to form an equation.
Solve the equation to find the time that A takes to complete the task on his own.
9 Solve by factorisation.
a
b
c
d
PS 10 If 5 is a root of the equation , find the value of c and the second root of
the equation.
,1.2 Completing the square
WORKED EXAMPLE 1.2
a Write in completed square form.
b Complete the square .
c Express in the form .
d i Express in the form where a and b are constants.
ii Hence, state the maximum value of .
Answer
a
Halve the coefficient of x and complete
the square.
Remove the brackets .
Simplify.
b Take out the factor of 2 from the terms
that involve x.
Complete the square.
Simplify.
Simplify.
c Rearrange.
Take out the factor of –2 from the terms
that involve x.
Complete the square.
Simplify.
Simplify.
Rearrange.
d i Complete the square.
Simplify.
, ii The maximum value of this fraction is
when the denominator has the
minimum value. This occurs when
.
Evaluate the expression.
EXERCISE 1B
1 Express the following in completed square form.
a
b
c
d
e
f
g
h
i
2 Use the completed square form to factorise the following expressions.
a
b
c
d
e
f
3 Solve the following quadratic equations. Leave surds in your answer.
a
b
c
d
e
f
PS 4 A recycling firm collects aluminium cans from a number of sites. It crushes them and
then sells the aluminium back to a manufacturer.
The profit from processing t tonnes of cans each week is , where
.
By completing the square, find the greatest profit the firm can make each week, and
, how many tonnes of cans it has to collect and crush each week to achieve this profit.
P 5 By writing the left-hand side in the form , show that the equation
has no real roots.
6 The quadratic function passes through the points and
. Its maximum y value is 48. Find the values of a, b and c.
7 a Write in the form .
b Hence, or otherwise, find the maximum value of .
PS 8 Two cars are travelling along two straight roads that are perpendicular to each other
and meet at the point O, as shown in the diagram. The first car starts 50 km west of O
and travels east at the constant speed of 20 km/h. At the same time, the second car
starts 30 km south of O and travels north at the constant speed of 15 km/h.
a Show that at time t (hours) the distance d(km) between the two cars satisfies
.
b Hence find the closest distance between the two cars.
AS & A Level Mathematics:
Pure Mathematics 1
Practice Book
,Contents
How to use this book
1 Quadratics
2 Functions
3 Coordinate geometry
4 Circular measure
5 Trigonometry
6 Series
7 Differentiation
8 Further differentiation
9 Integration
Answers
, How to use this book
Throughout this book you will notice particular features that are designed to help your learning.
This section provides a brief overview of these features.
■ Find the equation of a straight line given sufficient information.
■ Interpret and use any of the forms in
solving problems.
■ Understand that the equation represents the circle with centre (a,
b) and radius r.
Learning objectives indicate the important concepts within each chapter and help you to navigate
through the practice book.
TIP
Substitute the value of r that minimises the quantity back into the original expression to find
the minimum value of that quantity.
Tip boxes contain helpful guidance about calculating or checking your answers.
WORKED EXAMPLE 1.3
Solve the equation Write your answers as exact values (in surd form).
Answer
Using a = 2, b = −2 and c = −1 in the quadratic
formula.
Simplify.
Simplify the surd.
Worked examples provide step-by-step approaches to answering questions. The left side shows a fully
worked solution, while the right side contains a commentary explaining each step in the working.
END-OF-CHAPTER REVIEW EXERCISE 7
1 Find of gradient of the graph of at the point where the y-coordinate is 3.
2 f(x) = ax3 + bx−2 where a and b are constants. f′(1) = 18 and f″(1) = 18. Find a and b.
, 3 Find the values of x for which the gradient of f(x) is 9.
The End-of-chapter review exercise contains exam-style questions covering all topics in the chapter.
You can use this to check your understanding of the topics you have covered.
Throughout each chapter there are multiple exercises containing practice questions. The questions are
coded:
PS These questions focus on problemsolving.
P These questions focus on proofs.
M These questions focus on modelling.
You should not use a calculator for these questions.
,Chapter 1
Quadratics
■ Carry out the process of completing the square for a quadratic polynomial ax2 + bx + c and
use a completed square form.
■ Find the discriminant of a quadratic polynomial ax2 + bx + c and use the discriminant.
■ Solve quadratic equations, and quadratic inequalities, in one unknown.
■ Solve by substitution a pair of simultaneous equations of which one is linear and one is
quadratic.
■ Recognise and solve equations in x that are quadratic in some function of x.
■ Understand the relationship between a graph of a quadratic function and its associated
algebraic equation, and use the relationship between points of intersection of graphs and
solutions of equations.
, 1.1 Solving quadratic equations by factorisation
WORKED EXAMPLE 1.1
PS Lim walked 12 km from A to B at a steady speed of x km/h.
His average speed for the return was 2 km/h slower.
a Write down, in terms of x, the total time taken for the complete journey.
b If the total time he took was 3.5 hours, write an equation in x and solve it to find his
speed from A to B.
Answer
a
From using
b
Multiply both sides by
12(x − 2) + 12x = 3.5x(x − 2) Expand brackets and rearrange.
12x − 24 + 12x = 3.5x2 − 7x Multiply both sides by 2 and rearrange.
7x2 − 62x + 48 = 0
Factorise.
(7x − 6) (x − 8) = 0 Solve.
The value of is a solution to the
equation, but it gives a negative speed
for the return journey.
Lim’s speed from A to B is 8 km/h.
EXERCISE 1A
1 By factorising, solve the following equations:
a i 3x2 + 2x = x2 + 3x + 6
ii 2x2 + 3 = 17x − 7 − x2
b i 9x2 = 24x − 16
ii 18x2 = 2x2 − 40x − 25
c i (x − 3) (x + 2) = 14
ii (2x + 3) (x − 1) = 12
d i
ii
2 Solve the following equations. (In most cases, multiplication by an appropriate
expression will turn the equation into a form you should recognise.)
a
b
c
, d
e
f
g
h
3 Solve algebraically:
(2x − 3) (x − 5) = (x − 3)2
4 Solve the equation x2 + 8k2 = 6kx, giving your answer in terms of k.
5 Find the exact solutions of the equation
6 Solve the equation
PS 7 The product of two positive, consecutive even integers is 168. Use this information to
form a quadratic equation and solve it to find the two integers.
PS 8 Two men A and B working together can complete a task in 4 days. If B completes the
task on his own, he takes 6 more days than if A did the task on his own.
Use the information to form an equation.
Solve the equation to find the time that A takes to complete the task on his own.
9 Solve by factorisation.
a
b
c
d
PS 10 If 5 is a root of the equation , find the value of c and the second root of
the equation.
,1.2 Completing the square
WORKED EXAMPLE 1.2
a Write in completed square form.
b Complete the square .
c Express in the form .
d i Express in the form where a and b are constants.
ii Hence, state the maximum value of .
Answer
a
Halve the coefficient of x and complete
the square.
Remove the brackets .
Simplify.
b Take out the factor of 2 from the terms
that involve x.
Complete the square.
Simplify.
Simplify.
c Rearrange.
Take out the factor of –2 from the terms
that involve x.
Complete the square.
Simplify.
Simplify.
Rearrange.
d i Complete the square.
Simplify.
, ii The maximum value of this fraction is
when the denominator has the
minimum value. This occurs when
.
Evaluate the expression.
EXERCISE 1B
1 Express the following in completed square form.
a
b
c
d
e
f
g
h
i
2 Use the completed square form to factorise the following expressions.
a
b
c
d
e
f
3 Solve the following quadratic equations. Leave surds in your answer.
a
b
c
d
e
f
PS 4 A recycling firm collects aluminium cans from a number of sites. It crushes them and
then sells the aluminium back to a manufacturer.
The profit from processing t tonnes of cans each week is , where
.
By completing the square, find the greatest profit the firm can make each week, and
, how many tonnes of cans it has to collect and crush each week to achieve this profit.
P 5 By writing the left-hand side in the form , show that the equation
has no real roots.
6 The quadratic function passes through the points and
. Its maximum y value is 48. Find the values of a, b and c.
7 a Write in the form .
b Hence, or otherwise, find the maximum value of .
PS 8 Two cars are travelling along two straight roads that are perpendicular to each other
and meet at the point O, as shown in the diagram. The first car starts 50 km west of O
and travels east at the constant speed of 20 km/h. At the same time, the second car
starts 30 km south of O and travels north at the constant speed of 15 km/h.
a Show that at time t (hours) the distance d(km) between the two cars satisfies
.
b Hence find the closest distance between the two cars.
