Exam (elaborations)

Cambridge International AS & A Level Pure Mathematics 2 & 3 Practice Book

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Course
AS & A Level pure Mathematics

Institution
123 University

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...

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cambridge international as a level mathematics
pure mathematics 2 3 practice book

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Cambridge International
AS & A Level Mathematics:

Pure Mathematics 2 &
3
Practice Book

, Contents
How to use this book

1 Algebra

2 Logarithmic and exponential functions

3 Trigonometry

4 Differentiation

5 Integration

6 Numerical solutions of equations

P3 7 Further algebra

P3 8 Further calculus

P3 9 Vectors

P3 10 Differential equations

P3 11 Complex numbers

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.

■ 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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