Throughout this book you will notice particular features that are designed to help your learning.
This section provides a brief overview of these features.
■ Differentiate products and quotients.
■ Use the derivatives of e
x
, ln x, sin x, cos x, tan x, together with constant multiples, su...
Throughout this book you will notice particular features that are designed to help your learning.
This section provides a brief overview of these features.
■ Differentiate products and quotients.
■ Use the derivatives of ex, ln x, sin x, cos x, tan x, together with constant multiples, sums,
differences and composites.
■ Find and use the first derivative of a function which is defined parametrically or implicitly.
Learning objectives indicate the important concepts within each chapter and help you to navigate
through the practice book.
TIP
log10 x could also be written as log x or lg x.
Tip boxes contain helpful guidance about calculating or checking your answers.
END-OF-CHAPTER REVIEW EXERCISE 3
1 Find the exact solution of the following equations for −π ⩽ θ ⩽ π.
a i cosec θ = −2
ii cosec θ = −1
b i cot θ =
ii cot θ = 1
c i sec θ = 1
ii sec θ = −
d i cot θ = 0
ii cot θ = −1
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.
WORKED EXAMPLE 2.2
On average, flaws occur in a roll of cloth at the rate of 3.6 per metre.
Assuming a Poisson distribution is appropriate, find the probability of:
a exactly nine flaws in three metres of cloth
b less than three flaws in half a metre of cloth.
Answer
a
Use the interval to determine the mean,
λ. For three metres, λ = 3 × 3.6 = 10.8.
, b
For half a metre, λ = × 3.6 = 1.8.
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.
Throughout each chapter there are exercises containing practice questions. The questions are coded:
PS These questions focus on problem-solving.
P These questions focus on proofs.
M These questions focus on modelling.
You should not use a calculator for these questions.
You can use a calculator for these questions.
This book covers both Pure Mathematics 2 and Pure Mathematics 3. One topic (5.5 The trapezium rule) is
only covered in Pure Mathematics 2 and this section is marked with the icon P2. Chapters 7–11 are only
covered in Pure Mathematics 3 and these are marked with the icon P3 . The icons appear in the Contents
list and in the relevant sections of the book.
,Chapter 1
Algebra
■ Understand the meaning of , sketch the graph of and use relations such as
and in the course of solving
equations and inequalities.
■ Divide a polynomial, of degree not exceeding , by a linear or quadratic polynomial, and
identify the quotient and remainder (which may be zero).
■ Use the factor theorem and the remainder theorem.
,1.1 The modulus function
WORKED EXAMPLE 1.1
Solve:
a
b
Answer
a
Split the equation into two parts.
Using equation (1), expand brackets.
Solve.
Solve using equation (2).
The solution is:
b
Subtract from both sides.
Divide both sides by -5.
Split the equation into two parts.
Using equation (1), rearrange.
Factorise.
Using equation (2), rearrange.
Factorise.
, Check.
EXERCISE 1A
1 Solve:
a
b
c
d
e
f
g
h
TIP
Remember:
2 Solve these equations.
a
b
c
3 Solve these equations.
a
b
c
4 Solve these equations.
a
b
5 Solve:
a
b
c
d
e
f
6 Solve the simultaneous equations.
, a
b
7 Solve the equation
8 a Solve the equation
b Sketch the graph of .
c Write down the equation of the line of symmetry of the curve.
9 Solve the equation
PS 10 Solve the equation
,1.2 Graphs of where is linear
WORKED EXAMPLE 1.2
a Sketch the graph of , showing the points where the graph meets the
axes. Use your graph to express in an alternative form.
b Use your answer to part a to solve graphically .
Answer
a
First sketch the graph of
Reflect in the -axis the part of the line
that is below the -axis.
The line has gradient and a -intercept of .
The graph shows that can be written as:
b On the same axes, draw .
Find the points of intersection of the
lines and .
There are two points of intersection of
the lines and so
there are two roots.
The solutions to are
and .
EXERCISE 1B
1 Sketch the graphs of each of the following functions showing the coordinates of the
points where the graph meets the axes.
a
b
c
2 Sketch the following graphs.
a
b
c
d
e
, f
g
h
i
j
k
l
3 Describe fully the transformation (or combination of transformations) that maps the
graph of onto each of these functions.
a
b
c
d
e
f
4 Sketch each of the following sets of graphs.
a and
b and
c and
d and and
e and
5 for . Find the range of function f.
6 a Sketch the graph of for showing the coordinates of
the vertex and the -intercept.
b On the same diagram, sketch the graph of .
c Use your graph to solve the equation
7 a Sketch the graph of for showing the coordinates of the
vertex and the -intercept.
b On the same diagram, sketch the graph of .
c Use your graph to solve the equation
8 a Sketch the graph of .
b Use your graph to solve the equation
9 Write the equation of each graph in the form .
a
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