A Level Further Mathematics for OCR A Additional Pure Student Book (AS/A Level) [Cambridge Elevate ed.] 9781108431866, 9781108431910

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A Level Mathematics for OCR A Additional Pure Student Book (AS/A Level) John Sykes

Contents Introduction How to use this resource 1 Sequences and series Section 1: Recurrence relations, properties of sequences, Fibonacci and related numbers Section 2: Solving recurrence relations Mixed practice 1 2 Number theory Section 1: Number bases Section 2: Divisibility tests Section 3: Modular (or finite) arithmetic Section 4: Prime numbers Mixed practice 2 3 Groups Section 1: Binary operations Section 2: Definition of a group Section 3: Subgroups, cyclic groups, isomorphism and structure of groups Mixed practice 3 Cross-topic review exercise 1 4 Further vectors Section 1: Vector product Mixed practice 4 5 Surfaces and partial differentiation Section 1: Three-dimensional (3-D) surfaces Section 2: Partial differentiation Section 3: Stationary points Section 4: Tangent planes Mixed practice 5 6 Further calculus Section 1: Integration by reduction Section 2: Arc lengths and surface areas Mixed practice 6 Cross-topic review exercise 2 Practice paper Formulae Answers Acknowledgements Copyright

Introduction You have probably been told that mathematics is very useful, yet it can often seem like a lot of techniques that just have to be learnt to answer examination questions. You are now getting to the point where you will start to see how some of these techniques can be applied in solving real problems. However, as well as seeing how maths can be useful we hope that anyone working through this resource will realise that it can also be incredibly frustrating, surprising and ultimately beautiful. The resource is woven around three key themes from the new curriculum. Proof Maths is valued because it trains you to think logically and communicate precisely. At a high level, maths is far less concerned about answers and more about the clear communication of ideas. It is not about being neat – although that might help! It is about creating a coherent argument that other people can easily follow but find difficult to refute. Have you ever tried looking at your own work? If you cannot follow it yourself it is unlikely anybody else will be able to understand it. In maths we communicate using a variety of means – feel free to use combinations of diagrams, words and algebra to aid your argument. And once you have attempted a proof, try presenting it to your peers. Look critically (but positively) at some other people’s attempts. It is only through having your own attempts evaluated and trying to find flaws in other proofs that you will develop sophisticated mathematical thinking. Problem solving Maths is valued because it trains you to look at situations in unusual, creative ways, to persevere and to evaluate solutions along the way. We have been heavily influenced by a great mathematician and maths educator George Polya, who believed that students were not just born with problem-solving skills – they were developed by seeing problems being solved and reflecting on their solutions before trying similar problems. You might not realise it but good mathematicians spend most of their time being stuck. You need to spend some time on problems you can’t do, trying out different possibilities. If after a while you have not cracked it then look at the solution and try a similar problem. Don’t be disheartened if you cannot get it immediately – in fact, the longer you spend puzzling over a problem the more you will learn from the solution. You may never need to integrate a rational function in future, but we firmly believe that the problem-solving skills you will develop by trying it can be applied to many other situations. Modelling Maths is valued because it helps us solve real-world problems. However, maths describes ideal situations and the real world is messy! Modelling is about deciding on the important features needed to describe the essence of a situation and turning that into a mathematical form, then using it to make predictions, compare to reality and possibly improve the model. In many situations the technical maths is actually the easy part – especially with modern technology. Deciding which features of reality to include or ignore and anticipating the consequences of these decisions is the hard part. Yet it is amazing how some fairly drastic assumptions – such as pretending a car is a single point or that people’s votes are independent – can result in models that are surprisingly accurate. More than anything else, this resource is about making links. Links between the different chapters, the topics covered and the themes above, links to other subjects and links to the real world. We hope that you will grow to see maths as one great complex but beautiful web of interlinking ideas. Maths is about so much more than examinations, but we hope that if you take on board these ideas (and do plenty of practice!) you will find maths examinations a much more approachable and possibly even enjoyable experience. However, always remember that the result of what you write down in a few hours by yourself in silence under exam conditions is not the only measure you should consider when judging your mathematical ability – it is only one variable in a much more complicated mathematical model!



How to use this resource Throughout this resource you will notice particular features that are designed to aid your learning. This section provides a brief overview of these features.

In this chapter you will learn how to: work with general sequences use induction to prove results relating to sequences and series describe behaviour of sequences use the limits of sequences work with Fibonacci and Lucas numbers solve first-order recurrence relations solve second-order recurrence relations.

Learning objectives A short summary of the content that you will learn in each chapter.

Before you start… A Level Mathematics Student Book 2, Chapter 4

You should be able to use the following formulae for geometric progressions: term of a G.P.

1 Given that a sequence has terms find: a the

term

b the sum of the first five terms. 2 Given that a sequence has terms find: a the

term

b the sum of the first

terms

c the sum to infinity. Pure Core Student Book 1, Chapter 6

You should be able to use the method of induction: check true for initial conditions, e.g. show true for (assuming true for

)

hence explain true for all positive integers.

3 Prove by induction that the sum of the first integers is

.

Before you start Points you should know from your previous learning and questions to check that you're ready to start the chapter.

WORKED EXAMPLE

The left-hand side shows you how to set out your working. The right-hand side explains the more difficult steps and helps you understand why a particular method was chosen.

PROOF

Step-by-step walkthroughs of standard proofs and methods of proof.

Key point A summary of the most important methods, facts and formulae.

Explore Ideas for activities and investigations to extend your understanding of the topic.

Tip Useful guidance, including on ways of calculating or checking and use of technology. Each chapter ends with a Checklist of learning and understanding and a Mixed practice exercise, which includes past paper questions marked with the icon

.

In between chapters, you will find extra sections that bring together topics in a more synoptic way. CROSS-TOPIC REVIEW EXERCISE Questions covering topics from across the preceding chapters, testing your ability to apply what you have learnt.

You will find practice paper questions towards the end of the resource, as well as a glossary of key terms (picked out in colour within the chapters), and answers. Maths is all about making links, which is why throughout this resource you will find signposts emphasising connections between different topics, applications and suggestions for further research.

Rewind Reminders of where to find useful information from earlier in your study.

Fast forward

Links to topics that you might cover in greater detail later in your study.

Did you know? Interesting or historical information and links with other subjects to improve your awareness about how mathematics contributes to society.

Colour-coding of exercises The questions in the exercises are designed to provide careful progression, ranging from basic fluency to practice questions. They are uniquely colour-coded, as shown here.

1

Find the recurrence relation for each of these sequences. a b c d

5

The Fibonacci sequence is defined by

.

a Find the first eight terms of the sequence. b Find the first seven terms of the sequence obtained by taking the differences between consecutive terms of the Fibonacci sequence. c Write down the first seven terms of the sequence obtained by taking quotients of consecutive terms of the Fibonacci sequence (as decimals to decimal places). d A rectangle has area square unit. Its length is and its width is

.

i Form a quadratic equation in and solve it to find the value of . ii Write down the name of this number. 7

In 2017 the population of the United Kingdom is taken to be million people. The annual growth rate is . In addition, it is expected that the net immigration will be people per year. Set up a recurrence system to model the population and solve it to estimate the number of people in years’ time. Show that the population of the United Kingdom in 2022 will be million people.

13

A sequence is defined by Prove by induction that

, with

.

for all positive integers .

Verify that your proof is valid for the first five terms of the sequence. 15

A sequence is defined by Prove by induction that

, with

.

for all positive integers .

Verify that your proof is valid for the first five terms of the sequence. 17

In base , a number, when converted to decimal, is divisible by if it ends in zero. Show that this is correct by:

a dividing

by

b dividing

by

c dividing

by

d dividing

by

Black – drill questions; for essential skills practice. Green – practice questions at a basic level. Yellow – designed to encourage reflection and discussion. Blue – practice questions at an intermediate level. Red – practice questions at an advanced level. Purple – challenging questions that apply the concepts of the current chapter across other areas of maths.

– indicates content that is for A Level students only – indicates content that is for AS Level students only



1 Sequences and series In this chapter you will learn how to: work with general sequences use induction to prove results relating to sequences and series describe behaviour of sequences use the limits of sequences work with Fibonacci and Lucas numbers solve first-order recurrence relations solve second-order recurrence relations.

Before you start… A Level Mathematics Student Book 2, Chapter 4

You should be able to use the following formulae for geometric progressions: term of a G.P.

1 Given that a sequence has terms find: a the

term

b the sum of the first five terms. 2 Given that a sequence has terms find: a the

term

b the sum of the first

terms

c the sum to infinity. Pure Core Student Book 1, Chapter 6

You should be able to use the method of induction: check true for initial conditions, e.g. show true for (assuming true for

)

hence explain true for all positive integers.

3 Prove by induction that the sum of the first integers is

.

Section 1: Recurrence relations, properties of sequences, Fibonacci and related numbers Key point 1.1 A sequence is a list of terms. A series is the sum of the terms. The terms of a sequence may be defined as: recurrence relation , e.g. position-to-term formula, e.g.

. .

Induction may be used to prove results relating to both sequences and series. Behaviour of sequences: Periodic: Terms of the sequence are repeated, e.g.

Oscillating: Convergent:

Divergent: Monotonic:

Particular sequences Fibonacci numbers:

The number of repeated terms is called the period in this example). Periodic with two terms, e.g. (For a sequence) the terms of the sequence get closer to some limiting value; (for a series) the sum to infinity of the terms of the sequence has a finite value. Not convergent and the sum of the sequence is not finite. A sequence is monotonic increasing if each term is larger than or the same value as the previous one, i.e. , and monotonic decreasing if each term is less than or the same value as the previous one, i.e. .

. The ratio of each term to the previous one converges to . is the golden ratio, where

Lucas numbers:

.

.

Tip The Fibonacci sequence is a recurrence system (see Key point 1.2) and would normally have a range of the values of , in a similar way that functions have a domain. With functions, when the domain is ‘obviously’ the set of real numbers, the domain is often omitted from the definition. Similarly for recurrence systems, when the range is ‘obviously’ the set of natural numbers then this may be omitted.

WORKED EXAMPLE 1.1 a Write down the first five terms of the sequence defined by b Find the position-to-term rule for the sequence c Describe the relationship between

. which has

term

and .

d Describe the behaviour of these sequences. Each term is more than the previous one.

a Sequence is:

b

, … has first difference

The difference between consecutive terms is always . This first difference gives the coefficient of .

c

By inspection.

d Both sequences are monotonic increasing.

WORKED EXAMPLE 1.2

A sequence is defined by

with

Prove by induction that

and

.

for all positive integers . This is the standard method for induction. Check true for initial conditions.

Assume true for for , i.e. . For

, i.e.

and true

Show true for

:

Since it was true for

Assume true for

and if true for

it

and

.

.

Hence true for all positive integers.

is true for then, by the principal of induction, it is true for all positive integers.

Rewind The method of induction was covered in Pure Core Student Book 1, Chapter 6. Worked example 1.2 shows an extension of the method of induction to a 2nd order recurrence relation.

WORKED EXAMPLE 1.3

Consider the sequences

and

.

a Write down the first five terms of each sequence. b State the value to which each sequence converges. c Find the sum of the first five terms and the sum to infinity of each sequence. d Describe each sequence.

a

Substituting

and

b They both converge to . c

As

is a geometric sequence:

and is not geometric: Add all the terms. and the sum to infinity d

is a monotonic decreasing sequence and a convergent series. is a monotonic decreasing sequence and a divergent series and it is called a harmonic series.

Explore The second series in Worked example 1.3 is called the harmonic series. Determine why the sum to infinity of the harmonic series is

.

Tip Do not confuse

with .

is the value of the term, is the position of the term.

EXERCISE 1A 1

Find the recurrence relation for each of these sequences. a b c d

2

For each of these sequences, state the position-to-term rule. a b c d

3

For each of the following recurrence relations, write down the first five terms of the sequence

and find a position-to-term formula. a b c d 4

For each of the following position-to-term formulae, write down the first five terms of the sequence and find a recurrence relation. a b c d

5

The Fibonacci sequence is defined by

.

a Find the first eight terms of the sequence. b Find the first seven terms of the sequence obtained by taking the differences between consecutive terms of the Fibonacci sequence. c Write down the first seven terms of the sequence obtained by taking quotients of consecutive terms of the Fibonacci sequence (as decimals to decimal places). d A rectangle has area square unit. Its length is and its width is

.

i Form a quadratic equation in and solve it to find the value of . ii Write down the name of this number. 6

Evaluate the first six terms, i.e.

7

The Lucas sequence is defined by

to

, of the sequence with position-to-term rule

.

a Find the first eight terms of the sequence. The

Lucas number is denoted by

and the

Fibonacci number is denoted by

.

b Verify that: i

,

ii iii 8

, ,

The terms of a sequence are given by

.

a Using the tables or spreadsheet facility on your calculator, calculate the values of the terms to . b Describe the behaviour of the series. Repeat the question for: c

d 9

Describe the sequence given by

10

A sequence of terms is given by a If

, find

. .

and describe the sequence.

b Find another value of

that leads to a fixed point sequence.

c Determine the behaviour of the sequence if: i ii 11

a The sequence

is

Find the position-to-term rule for this sequence. b The sequence

is

Find the recurrence relation for this sequence. c Hence find

.

d Describe the sequence given by 12

The sequence a Evaluate

is defined by and

and and verify that

.

.

is defined by

The sequence c Find

and

and verify that

The sequence b Evaluate

for all positive integers .

and

.

.

is defined by

.

and , expressing your answers as fractions, and verify that

.

d By writing your answers to question part c as decimals, suggest the value to which this sequence of terms is converging. (The sequence Pell–Lucas numbers.) 13

A sequence is defined by Prove by induction that

gives the Pell numbers. The sequence

, with

.

for all positive integers .

Verify that your proof is valid for the first five terms of the sequence. 14

A sequence is defined by Prove by induction that

, with

.

for all positive integers .

Verify that your proof is valid for the first five terms of the sequence. 15

A sequence is defined by Prove by induction that

, with

.

for all positive integers .

Verify that your proof is valid for the first five terms of the sequence.

gives the

16

a Write down the first five terms of

, with

b Write down the first five terms of

.

.

c Prove by induction that the sequence defined by for all positive integers, . 17

The sequence

is defined by

Prove by induction that

and

, with

has

.

.

© OCR, AS GCE Mathematics: Further Pure Mathematics 1, Paper 4725, January 2011 18

The sequence a Show that

is defined by

and

.

.

b Hence suggest an expression for

.

c Use induction to prove that your answer to part b is correct. © OCR, AS GCE Mathematics: Further Pure Mathematics 1, Paper 4725, January 2008 19

The sequence a Find

and

is defined by and verify that

b Hence suggest an expression for

and

.

. .

c Use induction to prove that your answer to part b is correct. © OCR, AS GCE Mathematics: Further Pure Mathematics 1, Paper 4725, June 2009 20

The sequence a Find

and

is defined by and show that

b Hence suggest an expression for

and

for

.

. .

c Use induction to prove that your answer to part b is correct. © OCR, AS GCE Mathematics: Further Pure Mathematics 1, Paper 4725, January 2013

Explore This is an image of the Mandelbrot set, which is a geometric fractal. That is, a set of points in the complex plane that have different colours according to an algorithm. Investigate how fractals can be used to describe and simulate naturally occurring objects and phenomena.





Section 2: Solving recurrence relations First-order recurrence relations Key point 1.2 A linear recurrence relation is one in which each term of a sequence is a linear function of a previous term or terms, e.g. . Homogeneous: a first-order recurrence relation is homogeneous if it is of the form . Non-homogeneous: a first-order recurrence relation is non-homogeneous if it is of the form . ( will be polynomial or of the form .) Closed form solution: a solution in position-to-term form, i.e. , is written in terms of and not referring to previous terms. Recurrence system: a system consists of the recurrence relation (e.g. initial condition (e.g. ) and a range of the variable .

), an

Method The solution is made up of two parts: the complementary function . Consider the first-order recurrence relation

and the particular solution

.

Start with the reduced equation and replace by (where is a non-zero constant). This gives the auxiliary equation (or characteristic). Solving this gives a value for . The complementary function is then Next find the particular solution

.

by replacing

by a general function of the same form:

i.e. let

be

number, e.g. linear, e.g. quadratic, e.g. power, e.g. where the constants

and are found by substitution into the full equation.

Finally, the general (closed form) solution is

.

The final solution is found by applying the initial condition to determine . WORKED EXAMPLE 1.4

Solve the recurrence relation

, with

.

Reduced equation:

No

term.

Auxiliary equation:

Replace

by

, etc. and simplify.

There is no

Complementary function: The general solution is

term.

Using

.

Final solution:

Apply initial conditions.

substituting

WORKED EXAMPLE 1.5

Solve the recurrence relation

, with

.

Complementary function:

Same as Worked example 1.4.

Particular solution: let

Substitute Since

in the full equation. .

General solution:

Using

.

Final solution: substituting

Apply initial conditions.

Check:

It is wise to check the answers. with

WORKED EXAMPLE 1.6

Solve the recurrence relation

, with

Complementary function: Particular solution: let

.

Same as Worked example 1.4. ,

Substitute equation. Note: if

coefficients of

, etc. in the full then

Compare coefficients on each side.

numbers: General solution:

Using

.

Final solution: substituting

Apply initial conditions.

Check:

It is wise to check the answers. with

Tip When finding

you substitute

When finding

you replace, for example,

Do not confuse these and put

with

and

and

with when finding

.

WORKED EXAMPLE 1.7

Solve the recurrence relation

, with

.

Complementary function:

Same as Worked example 1.4.

Particular solution: let

Substitute Note: if

in the full equation. then

Solve.

General solution:

Using

.

Final solution: substituting

Apply initial conditions.

Check:

It is wise to check the answers. with

Failure case Note: the failure case is when multiply by .

is of the same form as the complementary function. In this situation,

WORKED EXAMPLE 1.8

Solve the recurrence relation

, with

.

Complementary function:

Same as Worked example 1.4.

Particular solution:

Putting gives the same form as (i.e. failure case) so substitute in the full equation. Note: if , then .

let

Solve.

General solution:

Using

.

Final solution: substituting

Apply initial conditions.

Check:

It is wise to check the answers. with

Tip It is common not to notice a failure case. If you do this, cannot be found (you obtain the equation ). This tells you to reflect on your solution and recognise whether or not it is a failure case.

Second-order recurrence relations Key point 1.3 A second-order recurrence relation is one in which each term of a sequence is a linear function of two previous terms, e.g. . Homogeneous : a second-order recurrence relation is homogeneous if it is of the form where and are constants Non-homogeneous: a second-order recurrence relation is non-homogeneous if it is of the form . ( will be polynomial or of the form .)

Method As with first-order recurrence relations, the solution is given by the sum of the complementary function and the particular solution

, i.e.

.

Consider the second-order recurrence relation Start with the reduced equation and replace characteristic) equation. Solving this gives values and .

. by

. This gives the auxiliary (or

To find the complementary function: real and distinct roots and repeated real roots (i.e.

)

complex roots and Next find the particular solution

by replacing

Again, for the failure case, multiply

as for solving first-order relations.

by .

Finally, the general (closed form) solution is

.

The final solution is found by applying the initial conditions to determine and . When a recurrence relation converges to a fixed value (which is usually zero) it has reached a steady state. This happens if and only if and but may also depend on the value of . [The method is comparable with the one used for solving second-order differential equations.]

Rewind You will have solved second-order differential equations in Pure Core Student Book 2, Chapter 11.

WORKED EXAMPLE 1.9

Solve the second-order recurrence relation

, with

and

.

Reduced equation:

Reduced equation is found by putting

Auxiliary equation:

Replace

Complementary function:

Using

The general solution is

Using (since the recurrence relation is homogeneous there is no term).

Final solution: substituting

Apply initial conditions.

by

and simplify.

(i) (ii) Solving simultaneously and Hence Check:

Check the answers. , with

and

.

.

WORKED EXAMPLE 1.10

Solve the second-order recurrence relation

with

and

Reduced equation:

Reduced equation is found by putting

So the complementary function is .

Same as Worked example 1.9.

Particular solution: let

Substitute if , then

. .

in the full equation. Note: .

Compare coefficients on each side. coefficients of numbers: Hence: Using

The general solution is

.

. Final solution: substituting

Apply initial conditions.

(i) (ii) Solving simultaneously

and

Check:

Check the answers. , with

and

WORKED EXAMPLE 1.11

Solve the second-order recurrence relation

with

Reduced equation:

Put

.

Auxiliary equation:

Replace

by

and

and simplify.

(equal roots) The complementary function is:

Using

.

.

Substitute

Particular solution: let

in the full equation.

Note: if

, then, .

Solve.

The general solution is

Using

.

Final solution: substituting

.

Apply initial conditions. (i) (ii)

Solving simultaneously: and

Check:

Check the answers. , with

and

WORKED EXAMPLE 1.12

Solve the second-order recurrence relation

with

Reduced equation:

Put

.

Auxiliary equation becomes:

Replace

by

The complementary function is:

Using

Particular solution: let

Substitute

and

.

and simplify.

. Solve.

      Hence: The general solution is

Using

.

Final solution: substituting

(i) (ii)

(i)

Apply initial conditions. The initial conditions do not have to be complex numbers for questions with complex roots.

(ii) and

Hence Check:

.

Check the answers.

Substituting Substituting

gives gives

. . .

WORKED EXAMPLE 1.13

Solve the second-order recurrence relation

, with

Reduced equation:

Put

.

Auxiliary equation:

Replace

by

and

.

and simplify.

or Complementary function:

Using

Particular solution: failure case so let

.

Putting gives the same form as one of the terms in (i.e. failure case) so put in the full equation. Note: if

, then . Solve.

The general solution is

Using

.

Final solution: substituting

Apply initial conditions. (i) (ii)

Solving simultaneously

and

Hence or Check:

Check the answers. with

and

EXERCISE 1B 1

Solve the following first-order recurrence equations. Check your solutions by writing down the first five terms of the original relation and of its solution. a b c d

2

Solve the following first-order recurrence equations. Check your solutions by writing down the first five terms of the original relation and of its solution.

a b c d 3

Solve the following first-order recurrence equations. Check your solutions by writing down the first five terms of the original relation and of its solution. a b c d

4

Solve the following first-order recurrence equations. Check your solutions by writing down the first five terms of the original relation and of its solution. a b c d

5

Solve the following first-order recurrence equations. Check your solutions by writing down the first five terms of the original relation and of its solution. a b

, with , with

c d 6

Solve the following first-order recurrence equations. Check your solutions by writing down the first five terms of the original relation and of its solution. a b

, with , with

c d 7

In 2017 the population of the United Kingdom is taken to be million people. The annual growth rate is . In addition, it is expected that the net immigration will be people per year. Set up a recurrence system to model the population and solve it to estimate the number of people in years’ time. Show that the population of the United Kingdom in 2022 will be million people.

8

The population of red squirrels in a large wood is found to be decreasing by a constant rate of each year. To overcome this, a conservation group introduces squirrels a year. At the moment, there are believed to be squirrels. a Using the first model, set up a recurrence relation and solve it to find the population in four

years’ time. What would this model predict for the population growth over time? b In a second model, the recurrence relation is , where is the integer value of (i.e. the whole number part). Use this model to determine the population in four years’ time. What would this second model predict for the population growth over time? 9

The Tower of Hanoi is a problem involving three rods and a number of discs of different sizes. Only one disc is moved at a time and may be placed on the top of any disc of a larger size on any other rod. The purpose is to reconstruct the pile of discs on another rod, in the mimimum number of moves. Obtain a recurrence system for the number of moves required to solve the puzzle for discs. Show that for three discs the puzzle can be solved in seven moves.

10

Solve the following second-order recurrence equations. Check your solutions by writing down the first five terms of the original relation and of its solution. a b c d

11

Solve the following second-order recurrence equations. Check your solutions by calculating the first five terms of the solution. a b c d

12

Solve the following second-order recurrence equations. Check your solutions by writing down the first five terms of the original relation and of its solution. a b c d

13

Solve the following second-order recurrence equations. Check your solutions by writing down the first five terms of the original relation and of its solution. a b c d

14

The diagram shows a

grid.

It is to be filled with tiles of size

or

.

a If is the number of ways that a grid of size and

and find

and

can be filled with

or

tiles show that

.

b Find the second order recurrence relation and show that the solution is

15

Let

be the number of arrangements in a string of binary numbers, excluding the cases where

there are at least two consecutive zeros. a By writing out the numbers show that

and

b Explain why the recurrence relation is c Obtain the equation for

and

.

.

in terms of and show that it can be written as

is the th Fibonacci number and 16

and find

where

is the th Lucas number.

In my local pond, there are a number of caddis flies. A webcam films the process of them hatching out. In the first hour of the film, there are caddis flies. One hour later, the number has doubled and there are more. In the next hour, this number is increased by doubling the number of the previous hour plus more. a If

is the number of caddis flies at any particular hour explain why .

b It is given that

. Find

and show that .

c With reasons, explain whether this is a sensible model over a long period of time. 17

A farmer has observed the number of foxes and geese on his land. In any year, the number of foxes is and the number of geese is . The populations in the following year are given by: and

.

a Show that a recurrence relation for the number of foxes is given by . b Hence find a general equation for the population of foxes in the th year. c In 2018, there were

foxes and

geese. Find an equation for the number of foxes.

d Explain what happens to the population of foxes with time. 18

Show that the solution of the Fibonacci recurrence system Binet’s formula:

with

and

Obtain a similar solution for the Lucas recurrence system

with

and

is

and verify that your solution gives 19

.

Show that the recurrence system

with

and

has solution:

Prove by induction that the result is correct. 20

The number is a palindrome as it is the same when read forwards or backwards. Similarly, we can find all the integer sums for a particular number and determine the number of palindromes. For example, can be written as or , that is, in two palindromic ways. Similarly can be written as

or

, that is, in four palindromic ways.

Find the number of palindromic ways in which each of the numbers to can be written and write down a recurrence relation satisfying the results. Show that the solution is:

Checklist of learning and understanding Sequences: Recurrence relations are term-to-term rules of the form Position-to-term rules take the form

.

.

Behaviour of sequences: Periodic is where terms of the sequence are repeated regularly. The number of repeated terms is called the period. Oscillating is periodic with two terms. Convergent is where the terms of the sequence get closer to some limiting value. Divergent is when the sequence is not convergent and the sum of the sequence is not finite. A series is monotonically increasing (or decreasing) when each term is larger or the same value (or smaller or the same value) as the previous one. Fibonacci and Lucas numbers: The Fibonacci recurrence relation is

.

The Fibonacci sequence is The ratio of each term to its previous one converges to the golden ratio . The golden ratio

.

The Lucas numbers are generated from the recurrence relation . The Lucas sequence is First-order recurrence relations: A first-order linear relation is one in which each term of the sequence is a linear function of a previous term such as

.

A first-order recurrence relation is homogeneous if it is of the form Solutions are made up of complementary function To find

replace

To find

replace

by

and a particular solution

in the reduced equation

by a function of the same form.

The general solution is

.

.

then put

. .

Apply the boundary conditions to find any constants. Second-order recurrence relations: A second-order linear relation is one in which each term of the sequence is a linear function of two previous terms, e.g.

.

A second-order recurrence relation is homogeneous if it is of the form Solutions are made up of complementary function To find replace by characteristic) equation.

and a particular solution

in the reduced equation

If the auxiliary equation has real roots then If the auxiliary equation has equal roots then If the auxiliary equation has complex roots and then To find

replace

by a function of the same form.

The general solution is

.

Apply the boundary conditions to find any constants.



. .

to give the auxiliary (or . .

Mixed practice 1 1

A sequence

is defined by

a Given that b Using

with

, show that

.

, show that

c Prove by induction that

.

. for all positive integers .

d Verify that your proof is valid for the first three terms of each sequence. 2

The sequence

is defined by

a If

, verify that

with and show that

b Prove by induction that c Show that 3

. .

.

is divisible by .

A sequence has recurrence relation a By putting

.

, show that the particular solution is

b Solve the recurrence relation to show that c Explain why the terms of the sequence

. .

are odd when is odd and even when is

even. 4

a Solve the recurrence relation

, with

.

b Prove your answer is correct by induction. 5

The population of fish in a pond is decreasing by each year through fishing. Each year the pond is re-stocked by having an extra fish added. The initial population of fish is a Write down an equation to model the population b Solve this equation to find the population

of fish in years’ time in terms of

in terms of and .

c Find the value of to maintain a constant number of fish at the start of each year. d If 6

find the number of fish there will be at the end of the tenth year.

A second-order recurrence equation is

a By writing

as

show that the complementary function is

b Show that the particular solution is

.

c Hence find the solution of the recurrence relation. 7

Solve the second-order recurrence equation

8

Solve the second-order recurrence equation

9

A recurrence system has equation:

.

. .

with

.

Solve the recurrence equation for:

10

a

, showing your answer is in terms of the golden ratio

b

, showing your answer is in terms of the silver ratio

In a tactical game the number of points a player scores in a session is times the previous score minus times the score before that plus times the number of the session. Show that, after sessions, the score

After the first session the player has

is given by:

points. After the second he has

points.

Solve the recurrence relation and find the number of sessions played for the score to exceed . 11

Solve

given that

Show that your solution and the recurrence relation both give 12

In chaos theory, the quadratic recurrence relation

and

.

. is used

.

Depending on the long-term behaviour of and the starting value of the position of the coordinates of on an Argand diagram coloured by a specific colour. This leads to such diagrams as those named after Julia or Mandelbrot. An attractor is a number to which the iteration converges. a Put i ii

and investigate what happens when , .

b Repeat a with



.

c Repeat a with

.

d Repeat a with

.

2 Number theory In this chapter you will learn how to: understand and be able to work with number bases use divisibility tests use the division algorithm understand and use finite (modular) arithmetic solve linear congruences solve simultaneous linear congruences calculate quadratic residues and solve equations using them understand prime numbers use Euclid’s lemma use Fermat’s little theorem use the order of a modulo use the binomial theorem.

Before you start… GCSE

GCSE

You should be able to divide by single and double digit numbers. Division to include long division:

You should be able to carry out long multiplication.

1 Without a calculator: a Divide

by .

b Divide

by

.

2 Without a calculator multiply .

by

For example:

GCSE

You should be able to find prime factors of a number. A prime factor is a prime number that divides into the given number without a remainder.

3 Express factors.

as a product of its prime



A Level Mathematics

You should know the binomial expansion.

Student Book 1, Chapter 9

The binomial expansion for positive integer is:

4 Find the binomial expansion of .

Section 1: Number bases Key point 2.1 Decimal numbers are numbers to the base The decimal number

. .

For a base , We shall consider numbers with bases from to In numbers up to base , we use

. . .

Decimal numbers have digits to . Binary numbers have digits and . Hexadecimal numbers have digits to . Numbers to base have digits to . Base is called binary and base

is hexadecimal (both used in computing).

WORKED EXAMPLE 2.1

Convert the hexadecimal number

to a decimal number. is in the units column, is in the is in the column.

column and

WORKED EXAMPLE 2.2

Convert the decimal

to binary (base ).

Method 1: consider powers of , i.e. .

Work out how many of each power of is needed to make , i.e. we require the answer to be of the form:

i.e. Method 2: repeatedly divide by

Each time you divide the quotient by , the remainder forms part of the answer. Repeat the divisions until the quotient is zero. By doing this the remainders are the values of in:

Reading the sequence from the bottom to the top gives . Check: e.g. which is correct.

Tip

Check that your answer is correct.

It is a common error to write the number in the reverse order.

WORKED EXAMPLE 2.3

Convert

to base .

First convert

Now convert

Use the powers of .

to decimal

to base

Use Method 2 as in Worked example 2.2.

WORKED EXAMPLE 2.4

Find

, giving your answer in base . i.e. down, to carry since i.e. down, to carry

WORKED EXAMPLE 2.5

Find

, giving your answer in base .

Check your answer by converting the numbers to decimals.

Start by , noting that Put down then Put

down then

Check:

Do the multiplication then convert back to base .

Tip Always convert the numbers to the base being used. In Worked example 2.5 it is a common

error to write

rather than

.

WORKED EXAMPLE 2.6

Calculate

.

Check your answer by converting the numbers to decimals. Working in base .

Or Convert to base

:

and

Do the division then convert back to base .

EXERCISE 2A 1

Convert the following numbers to decimal numbers: a b c d

2

Write the following numbers as binary numbers (base ): a b c d

3

Change the following numbers to base : a b c d

4

Convert the following numbers to hexadecimal numbers (base

):

a b c d 5

Find the following numbers as numbers in base : a b c d

6

Convert the numbers to the base given: a b c d

7

to base to base to base to base

Do the following calculations in the base given. You might wish to check your answers by converting to decimal numbers and back again. a b c

8

Carry out the following calculations in the base given. You might wish to check your answers by converting to decimal numbers and back again. a b c

9

Evaluate the following products, giving your answers to the same base as that in the question. You might wish to check your answers by converting to decimal numbers and back again. a b c

10

Do the following divisions, writing your answers in the given base. You might wish to check your answers by converting to decimal numbers and back again. a b c

11

Carry out the following calculations:

a

giving your answer in base

b

giving your answer in base

c 12

giving your answer in base

Carry out the following calculations: a

giving your answer in base

b

giving your answer in base

c 13

Carry out the following calculations: a

giving your answer in base

b

giving your answer in base

c 14

15

giving your answer in base

giving your answer in base

Carry out the following calculations: a

giving your answer in base

b

giving your answer in base

c

giving your answer in base

a Write

as a quadratic in .

b Write

as a quadratic in , simplifying your answer.

c Given that . d Express 16

show that

as a decimal number for the found in part c.

Base is also called octal. Apply the same idea developed in question 15 to these problems: a Convert

to octal.

b Convert c Convert d Convert 17

and hence find the value of given that

to octal. to binary. to binary.

In base , a number, when converted to decimal, is divisible by if it ends in zero. Show that this is correct by: a dividing

by

b dividing

by

c dividing

by

d dividing

by

Prove the result that a number in base converted to a decimal is divisible by if it ends in zero. 18

In base , if a number is divisible by a factor of , the number ends in a digit that is also divisible by that factor. Show that this is correct by:

a dividing

by

b dividing

by

c dividing

by

d dividing

by

Prove this result. 19

In base , a number is divisible by if the last digit is divisible by only if is divisible by . a Show this result is true when b Show that

is divided by .

is not divisible by .

Prove this result. 20

In base , if the sum of the digits is divisible by a factor of divisible by that factor. Show that this is correct by: a dividing

by

b dividing

by

c dividing

by

d dividing



by

, then the number itself is also

Section 2: Divisibility tests Key point 2.2 Notation: means that divides exactly into , i.e. is a factor of or is a multiple of . Standard tests for divisibility: Divides by

Rule

Example

last digit divisible by sum of digits divisible by and the number formed by the last digits divisible by last digit or

last digit is

the number formed by the last digits divisible by sum of digits divisible by and beginning at the left most digit, add and subtract digits in an alternating pattern; the value found (this could be or ) must be divisible by Algorithm for divisibility of a number by any prime number (you will only be required to test for divisibility by primes less than ). The purpose here is to produce a smaller number that is easier to test for divisibility by the prime. Express in the form (or as appropriate), where and are integers. For example, Write down a second expression for in terms of and , such that the two expressions in and can be combined to give a multiple of . A method for finding is: find any multiple of ending in set . It follows that if then . For example, to find whether

: put

and which means that

and use and .

Now set up an iterative equation: If is written as and , let and put where . Carry out the iteration until is small enough to determine whether it is a multiple of and if then , i.e. . Divisibility of composite numbers: a number is divisible by a composite number if it is divisible by all the factors of the composite number. The division algorithm (the fundamental theorem of Euclid): If is divided by where then where is the quotient and is the residue. If then .

WORKED EXAMPLE 2.7

WORKED EXAMPLE 2.7

Without performing the division, show that

is divisible by

.

Check if divisible by .

is a composite number: i.e.

Check if divisible by

.

Hence

WORKED EXAMPLE 2.8

Find the quotient and residue when

is divided by .

By the division algorithm:

WORKED EXAMPLE 2.9

Use a suitable algorithm to determine whether Let

is divisible by

.

Since

and

Set up the iteration

with

substitute into substitute into substitute into (multiple of

)

Hence

WORKED EXAMPLE 2.10

By putting , where is any odd integer, prove that the square of takes the form where is an integer.

,

By the division algorithm, let the integer be given by . Since is odd then

or

.

As is odd then must be odd and [Check with, say,

. ]

Hence, the square of any odd integer is of the form .

EXERCISE 2B 1

Without performing the division, determine whether: a

is divisible by

b

is divisible by

c

is divisible by

d

is divisible by

e

is divisible by

.

2

Show that

3

Without performing the division, determine whether:

.

a

and

are both divisible by and/or

b

and

are both divisible by and/or

c

and

are both divisible by and/or

d

and

are both divisible by and/or

.

4

Show that

5

Use the algorithm described in Key Point 2.2 to determine whether: a b

is divisible by is divisible by

c d 6

.

(using (using

is divisible by is divisible by

and and

)

, stating the expressions for and and explaining why they work , stating the expressions for and and explaining why they work.

Without doing the division, show that Find the residue when

)

is not a factor of

is divided by

.

.

7

Given that

8

Show that

9

Use the division algorithm to find the quotient and the remainder when: a b

divides into

is divided by is divided by

, find and . .

c

is divided by

d

is divided by

10

Find the residue when

11

Find the remainders when

. is divided by

.

and are divided by .

Hence find the residue when

is divided by .

12

Find the residue when

13

Find the smallest number that has a residue of when divided by and a residue of when divided by or .

14

By putting or

15

Given that is an integer and by putting

is divided by

.

, where is an integer, prove that every square number is of the form , where is an integer. prove that

is an integer when it is

divided by . 16

Given that is an integer and by putting

show that

Hence prove that every cube number is of the form 17

By putting

show that

Hence prove

is of the form

18

Show that

19

Show that

20

a Explain why

has the form

, where is an integer.

. or

or

, where is an integer. , where is an integer.

. , where and are integers, is divisible by

b Use a suitable algorithm to determine whether



or

.

is divisible by

.

.

Section 3: Modular (or finite) arithmetic Key point 2.3 From the division algorithm,

, from which:

i.e. is the residue when a number is divided by a number . e.g. [This is sometimes referred to as clock arithmetic, e.g. hours is the same as p.m.] Rules If

and

, then: for any non-negative integer .

If If

is a polynomial in with integer coefficients, then and , where and have no common factors greater than , then:

Solving linear congruences A linear congruence is an equation of the form

.

It has a solution if, and only if, where is the highest common factor of and . In this case, there are exactly different (incongruent) solutions given by: where is a solution found by inspections and . Note: if is prime, then will have a solution (since ). Numbers such as, for example, are all congruent modulo since they are all . Numbers such as , ,

are all incongruent modulo since they are all different (i.e. , .

WORKED EXAMPLE 2.11 a Show that

has no solution.

b Solve the linear congruence a is not a multiple of , so no solution b

. For a solution requires Simplify by changing i.e.

where to base

,

is the number of incongruent solutions. Solutions are:

By inspection, set:

Note: If divide by to give with solutions

Using

, giving the solution

Note: since

.

, all solutions lie between and

it is possible to , where is an integer

.

Solving simultaneous linear congruences WORKED EXAMPLE 2.12

Solve the simultaneous linear congruences: (i) (ii) (i) is

Where is an integer.

Write

Use the modulo of the other equation. Since (ii) is etc.

Simplify and solve. Because

Rewriting this gives an integer)

.

.

( is

Substituting in (i)

Hence the solution to the simultaneous equations is:

To check the answer note that: and

In the above example we could have started with

and then written this in

modulo and proceeded in the same way.

Tip Most errors occur when solving the linear congruences, e.g. . Always check this answer before completing the problem.

WORKED EXAMPLE 2.13

Solve the simultaneous linear congruences: (i) (ii)

should give

(iii) (i) is

Where is an integer.

Write

Equating (i) and (ii) and using the modulo of (ii). Simplify and solve.

Hence ( is an integer) and substituting in (*)

Hence the solution to (i) and (ii) is: . …(iv) Now solve the simultaneous equations (iii) and (iv): (iii) and Since Hence

( is an integer)

and substituting in (iii):

To check the answer note that: and

Hence the solution to all three equations is:

Tip Always check that your answer to the solution of the first two congruences satisfies both of them before moving on to complete the problem.

Condition for a solution of two linear congruences Each congruence must be solvable. If

and , then the condition for an integer solution to exist is , since and . Subtracting these gives Dividing by gives the result. WORKED EXAMPLE 2.14

Explain why

have no integer solutions. Find where Find

is not a multiple of . No solution.

.

.

Check whether

is a multiple of .

, where .

Tip Remember that you are checking that

and not

and

.

Condition for a solution of three linear congruences Each congruence must be solvable. If all three moduli have no common factors (other than ) then there will be a unique solution. Check whether each pair of congruences has a solution (as above). WORKED EXAMPLE 2.15

Explain whether the following have solutions: a

and

b

and are coprime

a Yes b

is odd is even Hence no solution. Alternatively: Consider

and

Find where

Find Check whether

is not a multiple of . No solution.

is a multiple of .

Another alternative is:

Hence

, since is even

But

is odd,

hence no solution.

Quadratic residues If the congruence

has a solution, then is called the quadratic residue

If the congruence has no solution, then is a quadratic non-residue. WORKED EXAMPLE 2.16

.

WORKED EXAMPLE 2.16

Find all the quadratic residues for integers modulo Write a table with all possible values for modulo .

to

. are all values modulo

.

Note the symmetry of the table. (Reason is that, for example, so , etc.). Quadratic residues modulo

in modulo

This means that has solutions only for these quadratic residues.

are:

Tip It is easy to make an error in converting to the given modulo. Always check the symmetry of the values in the table. If you use the symmetry of the table always check your values are correct – it is a common error to take symmetry about the wrong ‘middle’ number.

WORKED EXAMPLE 2.17

Solve the congruence

.

Write a table with all possible values for modulo .

From the table:

or

to are all values modulo .

Possible values are

or

WORKED EXAMPLE 2.18

Prove that the congruence Write a table with all possible values for modulo .

From the table there are no solutions satisfying

has no solutions. to are all values modulo .

Only

, or

have solutions.

Or would be an appropriate way to express the answer.

Tip If you exclude the trivial value from the tables in the examples above, you will notice that the pattern of numbers is symmetrical.

EXERCISE 2C 1

Write the numbers in the given modulo. a

modulo

b

modulo

2

Write

in modulo and

in modulo . Hence find

modulo .

3

Write

in modulo and

in modulo . Hence find

modulo .

4

Write

5

Write

in modulo and

in modulo . Hence find

modulo .

6

Write

in modulo and

in modulo . Hence find

modulo .

7

If

8

Find the value of

9

Find the highest common factor of and

in modulo and

in modulo . Hence find

find the value of

.

.

10

Find the highest common factor of

11

Solve the following linear congruences:

and

. Hence solve the linear congruence:

. Hence solve the linear congruence:

a b 12

Solve the following simultaneous linear congruences: a b

13

Explain why the following have no integer solutions: a b

14

Solve the following simultaneous linear congruences: a b

modulo .

15

Find the quadratic residues for: a modulo b modulo c modulo d modulo

.

16

Show that is a quadratic residue of

17

Solve the following congruences:

and is a non-residue of

.

a b If and 18

, where

, has one solution

, state another solution in terms of

.

Prove that the following have no solutions: a b

19

List all the quadratic residues and all the quadratic non-residues for: a modulo b modulo . State the number of quadratic residues and non-residues for modulo where is prime.

20

If is an odd prime number and and have no common factors, then Euler proved that is a quadratic residue of if and only if

.

Demonstrate that this result is correct for all the quadratic residues modulo You will be able to prove this result when you have studied the next section.



.

Section 4: Prime numbers Key point 2.4 Prime: an integer is prime if it has no divisors except and itself. Composite: a composite number has at least one divisor other than and itself. Highest common factor: the (also called the greatest common divisor, gcd) is the highest factor of two or more numbers. Coprime: two or more integers are coprime (or relatively prime) if is their only common factor. The fundamental theorem of arithmetic (also known as the unique prime factorisation theorem) states that every integer greater than is either prime or the product of primes in exactly one way (apart from arrangements). Useful results for integers and : If and are coprime and and , then . If and , then . If and , then . If and , then , where and are integers. The highest common factor of and can be found by making the smallest possible integer that can be written as . Note, if the is then and are coprime.

WORKED EXAMPLE 2.19

Given that

and

show that:

a b Use the result of part a to show that is the highest common factor of a If

then

.

If

then

.

Hence

. b Let and

, so

Putting and gives , and since this is the smallest possible value then is the .

WORKED EXAMPLE 2.20 a Show that and b Show that

are coprime. is irreducible for all integers .

c Find the values of for which

is reducible.

and

.

a Let

.

Coprime = no common factors.

Then Putting

and

gives

.

i.e. the highest common factor of and is . Hence and

If , then and other than .

are coprime.

b Let

have no common factors

Irreducible = cannot cancel down.

Then

.

Putting

and

gives

.

i.e. the highest common factor of and Hence

is . Numerator and denominator have no common factors (other than ) so the fraction cannot cancel down.

is irreducible.

c Let

Reducible = can cancel down.

Then Putting

. and

gives

.

i.e. the highest common factor of and is . Hence

Numerator and denominator have common factor so the fraction can cancel down.

is reducible.

We now require

and

to

be divisible by .

and These conditions are satisfied for Check:

( an integer).

which can then be reduced.

Euclid’s lemma If a prime number divides into the composite number one of to . A result that can be concluded from Euclid’s lemma is that if , then

.

WORKED EXAMPLE 2.21

and are both factors of

.

then must divide into at least

, where and are coprime, i.e.

Explain why

is also a factor of

. Euclid’s lemma

since

then

and are coprime, and hence

Fermat’s little theorem This can be written in two forms: If is prime and

, then

If is prime, then

.

.

Note that if is prime, then this result is true. It does not follow that if this result is true then is prime. If is a composite number and

then is called a pseudo-prime to base

WORKED EXAMPLE 2.22

Show that Putting

and hence find the value of and

Using Fermat’s little theorem, Note that and are coprime.

(prime)

gives

.

.

Tip It is a common error to use

.

Remember Fermat’s little theorem can be stated in two forms, that

or

.

The order of modulo The order of modulo is the smallest positive integer such that Comparing with Fermat’s little theorem, which states that necessarily the least value of , for example, Note also that

, , note that even though

.

WORKED EXAMPLE 2.23

Use Fermat’s little theorem to find a value of for which Find any other values of

satisfying

. .

. is not .

Fermat’s little theorem: satisfies

Test other values of :

The order of modulo

Smallest value of is .

is .

The binomial theorem , where is prime Proof: This is obtained by:

The th term has coefficient: Since is prime, this expression (which is an integer) is a multiple of , since all factors in the denominator are

and coprime to it (also, exclude first and last terms). So

Hence: WORKED EXAMPLE 2.24

Solve Using But

Fermat’s little theorem

Make a table

Hence



or

EXERCISE 2D 1

Use a suitable algorithm, showing each step, to show that Hence write the composite number

2

It is given that

is a factor of

.

in terms of its prime factors.

and

.

Show that, for integer values of greater than , and are both composite numbers. 3

Factorise

4

Find a counter-example to show that the statement ‘

5

Use a step-by-step algorithm to show that

and hence explain why

State another factor of

.

is a composite number for all integers is prime’ is false.

is a factor of

.

Explain why

is also a factor of

6

If

7

a Find and given that

.

, prove that

. .

Hence state the highest common factor of b By writing down multiples of . 8

Use the fact that if

, then and

b

is the highest common factor of

and

is the highest common factor of

and

is the highest common factor of and

.

Given that is a positive integer: a prove that

is irreducible

b find the values of for which c determine whether 10

is the highest common factor of

, where and are integers, to show that:

is the highest common factor of

d

.

, show that

a

c

9

and

and

and

is reducible

has any integer values.

Evaluate the following using Fermat’s little theorem: a b c d

11

Evaluate the following using Fermat’s little theorem: a b c d

Tip Put

12

Evaluate Show that

13

.

and state whether this means that and

are factors of

.

Solve the following using Fermat’s little theorem: a b c

is a prime number.

and

d 14

Find the order of the following numbers, each with order . a b c d For each question show that

15

.

Prove that

, where is prime, and hence write down the congruence for:

Find, for modulo , the values of

and and hence make a conjecture for the value of

.

Prove your conjecture by induction and hence obtain a proof of Fermat’s little theorem. 16

If today is Monday what day will it be in

17

Solve

18

It is given that

days’ time?

.

a Prove that

and if

.

Tip Define

and

b Explain why, if any value of

and set up the iteration is divisible by

, is divisible by

with

.

.

c Demonstrate, step by step, how an algorithm based on this iteration can be used to show that . 19

Solve

20

By considering

and

, or otherwise, show that

and

are coprime, where and

are positive integers, and is odd.

Checklist of learning and understanding Number bases: Columns are powers of the base, e.g. Divisibility tests: To divide by , the last digit is divisible by . To divide by , the sum of the digits is divisible by . To divide by , the number formed by the last digits is divisible by . To divide by , the last digit is or . To divide by , the number formed by the last digits is divisible by .

.

To divide by , the sum of the digits is divisible by . To divide by

, beginning at the left most digit, add and subtract digits in an alternating

pattern; the value found (this could be or ) must be divisible by

.

Division algorithm: If is divided by , where residue. If

, then

, then

, where is the quotient and is the

.

Finite (modular arithmetic): If

then

.

Linear congruences: A linear congruence is an equation of the form

.

Linear congruences have a solution if, and only if, factor of and . The solutions are and

, , , …,

, where

, where is the highest common is a solution found by inspection

.

If is prime, then

will have a solution (since

).

Quadratic residues: If the congruence

has a solution, then is a quadratic residue

.

Prime numbers: An integer

is prime if it has no divisors except and itself.

A composite number has at least one divisor other than and itself. Two or more integers are coprime (or relatively prime) if is their only common factor. The fundamental theorem of arithmetic states that every integer greater than is either prime or the product of primes in exactly one way (apart from arrangements). Euclid’s lemma: If a prime number divides into the composite number divide into one of

to

, then must

.

Fermat’s little theorem: This can be written in two forms: If is prime and If is prime, then

, then

.

.

The order of modulo : This is the smallest positive integer such that Binomial theorem:



,

.

Mixed practice 2 1

a Write

as a quadratic in .

b Given that . c Express d Show that

show that

and hence find the value of where

as a decimal number. is a composite number.

2

Use the standard divisibility tests to show that

3

By putting , where is an integer, prove that the square of takes the form where is an integer.

4

Solve the linear congruence

5

Solve the system of linear congruences:

6

Solve the simultaneous linear congruences:

7

If

and

is divisible by

. or

.

find suitable values for and such that a linear

combination of and is a multiple of

.

Hence use a step-by-step algorithm to show that works. 8

and

, explaining why your method

a Explain why the simultaneous linear congruences: and have no integer solutions. b Solve the system of linear congruences: and

9

a Use the binomial expansion to prove that

modulo .

b Hence use Fermat’s little theorem to simplify

modulo .

c Use quadratic residues to find the values of that make a perfect square. 10

Let

11

a i Use the binomial theorem to show that

and

. Prove that

Hence show that if of .

if, and only if,

modulo

.

modulo . Show that

modulo .

, where is an integer, then is odd and is a multiple

Use these results to find . ii It is given that . By considering the values of and finding quadratic residues show that cannot be an integer.

modulo

and

b i Use Fermat’s little theorem to show that and . ii Find the values in modulo and of

modulo . Find the values, modulo , of

and

.

iii Hence find the integer value satisfying 12

a Show that is not a factor of

if is even.

b Use Fermat’s little theorem to find

modulo , where is an odd prime.

c Find the set of values of ( is a positive integer) for which d Prove your answer to c by induction.

.

.

3 Groups In this chapter you will learn how to: work with binary operations construct group tables define a group find the order of a group and element find subgroups use Lagrange’s theorem for subgroups find cyclic groups determine whether groups are isomorphic work with abstract groups.

Before you start… Chapter 2

You should be able to carry out modular addition and multiplication.

Do the following calculations, giving your answer in the modulo stated:

For example:

1 2 3 4 5

Pure Core Student Book 1, Chapter 4

You should be able to use complex number operations and roots of unity. For example: addition and multiplication of complex numbers i ii

rationalisation by multiplying by the conjugate

Given that 6 7 8 9 Rationalise .

and

find:

iii

Pure Core Student Book 1, Chapter 1

You should be able to add and multiply

If

and

find:

matrices. 10 For example:

11

addition of matrices

12 13 14

multiplication of matrices

You should be able to find the inverse of a matrix

(provided

)

Pure Core

You should be able to use matrices to

Find the matrix for the following

Student Book 1, Chapter 3

describe transformations by finding the image of the unit square. The matrix is the column vectors of the points that

transformations:

and

are mapped onto.

e.g. for reflection in

, 15 A reflection in the -axis. 16 A reflection in the -axis. 17 A reflection in the line

the transformation matrix is

.

The transformation matrix for a rotation of (or radians) anticlockwise around the origin is

18 A rotation of the origin.

clockwise about

19 A rotation of

clockwise.

.

20 A transformation has matrix .

Find the least positive integer if , where is the identity matrix.



Section 1: Binary operations Key point 3.1 A binary operation is a process involving two members of a set. For example, the operation ○ might be defined as ○ Definitions Consider members and of the set

.

Closed: a set is closed under the operation if, for all

.

Commutative: the operation is commutative if, for all Associative: the operation is associative if, for all Identity element: if there exists where identity element for the operation .

such that

. . , then is the

Inverse: for all non-identity elements if there exists , where and where is the identity element, then is the inverse of for the operation . Self-inverse: is self-inverse if

, and so

, where is the identity element.

Cayley table, group table or operation table: a two-way table that illustrates the operation. Latin square: a square two-way Cayley table in which each element appears exactly once in each row and exactly once in each column.

WORKED EXAMPLE 3.1

It is given that

, where

(the set of non-negative integers

, …).

Find: a

and

b

and show that the operation is commutative

and

and show that the operation is associative.

Determine whether the operation is closed, has an identity and whether each element has an inverse. a

,

Does

,

One example does not give the general case, so: and , so the operation * is commutative. b

Does

, ,

.

In general: and The operation * is associative. Closure: for all operation is closed.

so the

Identity: identity is

if

Valid since

. Hence the

.

.

Inverse: if

then , which is not possible for

, so

none of the elements has an inverse.

Tip Giving specific numerical examples such as for commutativity or for associativity does not prove these conditions. The proofs need to be general, i.e. or However, a counterexample with numbers is sufficient evidence to show that the operation is not commutative or associative.

WORKED EXAMPLE 3.2

The operation ○ is defined by ○

, where

.

Determine whether the operation is a closed, b commutative, c associative, d has an identity element and e determine whether each element has an inverse. a

b

The sum of and is real.

and , so the operation ○ is closed.

Does ○

○

○

, so the operation ○

○

is commutative. c

○ ○

Does ○ ○

○

○ ○

,

○ ○

so the operation ○ is associative. d

○

if

, so the identity

Valid since

.

element is . e

○ if so every element has an inverse.

Valid since

.

WORKED EXAMPLE 3.3

Make a Cayley table to show multiplication modulo for the set

.

a Show that operation of multiplication modulo on the elements of set is closed. b Use the table to explain why multiplication on the elements of is commutative. c Determine the identity element of . d Write down the inverse of each element of .

For example,

a The operation is closed.

b

Cayley table

All the elements in the table are in .

for any pair of elements. The operation of multiplication for

The elements in the table are symmetrical about the leading diagonal.

the elements of is commutative. c Multiplying by leaves the heading unchanged, i.e. the identity element is .

Find the row that is identical to the header.

d Thus:

For example that is, is the inverse of (easily read off from the table).

Tip is not but is found using

(where is the identity element).

WORKED EXAMPLE 3.4

A set of elements under multiplication contain an identity and elements and . a Prove that b Given that

. , prove that

, where Using

a

.

Pre-multiply by

.

Pre-multiply by

.

b Pre-multiply by . Post-multiply by .

Tip The order of the multiplication is important.

Tip

.

; it is not

, unless commutative.

EXERCISE 3A 1

The operation * is defined by a Find

and

b Find

, where

(the set of real numbers).

. Determine whether or not the operation is commutative.

and

. Determine whether or not the operation is associative.

c Determine whether or not the operation is closed. 2

Given that

, where

.

a Show that the operation is closed. b Find

and

. Prove that the operation is commutative.

c Determine whether or not the operation is associative. 3

Given that a Find

, where and

b Find

.

. Prove that the operation is commutative. and

. Prove that the operation is associative.

c Find the identity . d Find the inverse of

.

e Determine whether the operation is closed. 4

If a Find

, where and

b Find

.

. Prove that the operation is commutative. and

. Prove that the operation is associative.

c Find the identity . d Find the inverse of . e Determine whether the operation is closed. 5

The operation is defined by

, where and are real.

a Prove that the operation is commutative and associative. b Find the identity element and show that the inverse of is

.

c Show that the operation is closed. 6

It is given that

, where and are complex numbers with integer components.

a Show that the identity element is . b Find the inverse of . c Is the operation * closed? 7

It is given that a Find

and

modulo . . Prove that the operation is commutative.

b Show that the operation is associative.

c Find the identity element of the operation. d Find the inverses of 8

Given that

and . for

, show that:

a the operation is commutative b the operation is associative c the operation has an identity d each element has an inverse. 9

Given that

for

, show that:

a the operation is closed b the operation is commutative c the operation is not associative d the operation does not have an identity for all 10

.

The binary operation * is defined on by

Give reasons why: a the operation * is commutative b the operation * is associative c there is an identity element with respect to *. 11

The set S consists of all numbers of the form

, where and are integers.

Give reasons why: a

is closed under addition and multiplication

b there is an identity in for addition and also an identity in for multiplication c not every element of has an inverse with respect to multiplication. 12

Binary operations ○ and are defined on , the set of complex numbers, as follows: for ○ and , where and are non-zero constants. a Write down the identity in with respect to *. b Prove that every element of , with one exception, has an inverse in with respect to . c Prove that the operation ○ is associative for just one value of , which is to be found.

13

A set consists of matrices of the form

, where

and are real numbers.

a Determine whether the operation of addition on the members of is commutative and associative. b State, with a reason, the identity element for addition. c Explain why every element has an inverse.

Rewind

You learnt how to work with matrices in Pure Core Student Book 1, Chapter 3.

14

A set consists of matrices of the form

, where

and are real numbers.

a Prove that, in general, the operation of multiplication on the members of is not commutative. b Write down the identity element for multiplication. c State the condition needed for each member of to have an inverse under multiplication. 15

A set consists of matrices of the form

, where is a non-zero real number.

a Prove that is closed under multiplication. b Find the inverse of each element for multiplication, defining the condition needed for the inverse to exist. 16

A set consists of matrices of the form

, where is a non-zero integer.

a Prove that is closed under multiplication. b Prove that multiplication of members of is associative. c Find the identity element and determine whether or not each element has an inverse. 17

Make a Cayley table to show addition modulo for the set

.

a Show that addition on the elements of is closed. b Using the table, show that addition on the elements of is commutative. c Determine the identity element of . d Write down the inverse of each member of . 18

Make a Cayley table to show multiplication modulo for the set a Show that multiplication on the elements of is closed. b Using the table, explain why multiplication on the elements of is commutative. c Determine the identity element of . d Write down the inverse of each member of .

19

It is given that

and that

a Make a Cayley table for

.

for the set .

b Show that the operation * is closed and commutative. c State the identity element and write down the inverse of each element of . d Explain why the operation * is not closed for the subset

.

e Give three subsets of for which the operation * is closed. 20

a For the infinite group of non-zero complex numbers under multiplication, state the identity element and the inverse of

, giving your answers in the form

b The group is now redefined such that combined under the operation of addition modulo .

and that these elements are

Find the identity element of the operation and the inverse of



.

Section 2: Definition of a group Key point 3.2 A set under a binary operation forms a group if: is closed: i.e. for all the operation is associative: i.e. for all has an identity element such that

for all

each non-identity element of has an inverse in : i.e. for all that and .

, there exists

such

If the operation is also commutative , then the set forms an Abelian group. The order of a group is the number of elements it contains. The order of an element is the power to which the element has to be raised to give the identity element, i.e. the order of the element is if , where is the identity element.

WORKED EXAMPLE 3.5

Determine whether

forms a group under addition modulo five.

If so, state the order of the group and the order of each element. The Cayley table for is The top row is the same as the header so

.

The identity appears in every row so each element has an inverse.

Closed: all the elements in the table are in , so the operation is closed. Associative: addition is associative.

Since

.

Identity: Inverse:

i.e. each element has an inverse. Hence, forms a group under addition .

All four conditions are satisfied.

Note that the operation is commutative so forms an

Since

.

Abelian group. The order of the group is .

The group has five elements.

The order of each element is:

The number of times each element is added to itself to give .

WORKED EXAMPLE 3.6

A set

and

.

a Draw a group table for multiplication and explain why it forms a Latin square. b Show that the set is closed under multiplication. c Find the identity and state the inverse of each element. d Assuming multiplication is associative, state whether S forms a group under multiplication. e State the order of the group and the order of each element. a

The table is square and no element appears exactly once in each row and exactly once in each column. b

is closed.

c Identity

All elements in the table appear in the set. .

Inverses:

Row for , and column for 1 also, is the same as the header, since multiplication is commutative and the row for and column for is the same as the header. Read off from the table.

i.e. each element has an inverse. d The operation is closed, associative, has an identity and each element has an inverse, so forms a group under multiplication. e The group has order Order: Note that the order of each element is a factor of the order of the group.

Tip When finding the order of an element, remember to count the element itself. In Worked example 3.6, means that is of order . It is a common error to say that has been multiplied by itself once more and therefore it is of order this is not true!

EXERCISE 3B 1

The operation * is defined by a Given that

modulo . , copy and complete this operation table.

b Determine whether the operation on is closed. c Show that the operation on is associative. d State the identity and write down the inverse of each element. e State the order of the group and the order of each element. f

Explain why forms a group under the operation.

g Explain why forms an Abelian group under the operation. 2

Make an operation table for

under multiplication modulo .

a Explain, with reasons, whether the table forms a Latin square. b Is the operation on closed? c Show that the operation on is associative. d State the identity element and write down, where possible, the inverse of each element. e Solve f

modulo .

State the order of and, where possible, the order of each element.

g Determine whether forms a group under multiplication modulo . 3

Make an operation table for

under addition modulo .

a Is the operation on closed? b Show that the operation on is associative. c State the identity and write down the inverse of each element. d Solve

modulo .

e State the order of the group and the order of each element. f

Determine whether forms a group under this operation.

4

An operation is defined by

modulo

on the set

.

a Draw an operation table such that is closed under . b State the identity element. c Find the inverse of . d State the order of each element. e Determine whether the operation is commutative and associative. f 5

Explain why forms an Abelian group under .

It is given that

modulo on the set , where

.

a Make an operation table for . b Determine whether the operation on is commutative. c Solve

.

d Determine whether the operation on is closed. e By using numerical examples, determine whether the operation is associative. f

Determine whether forms a group under the operation .

6

Make a Cayley table for multiplication modulo for the set , where . State, with reasons, whether the table forms a Latin square. Show that is a group.

7

Show that the set

8

Show that the infinite set of complex numbers under addition.

9

A rectangle has four symmetries (including rotations and reflections). Define the symmetries and show that they form a group.

forms a group under addition modulo . , where and are real, forms a group

10

An equilateral triangle has six symmetries (including rotations and reflections). Define the symmetries and show that they form a group.

11

Find the cube roots of unity and show that they form a group under multiplication.

12

It is given that

,

. Show that these

four functions form a group under the operation of composite functions, where . 13

Show that

and

group under the operation of composite functions, where 14

A set consists of matrices of the form

form a .

, where is a non-zero real number. Show that

forms a group under multiplication. 15

is the set of matrices of the form

, where is a real number. It is assumed that

matrix addition and multiplication are associative. a Show that is not a group under matrix addition.

b Show that is a group under matrix multiplication. 16

a Write down the matrices and , where represents a reflection in the -axis, represents a reflection in the -axis and represents a rotation of about the origin. b Show that the set , where I is the identity matrix, forms an Abelian group under the operation ○, where ○ means the transformation followed by the transformation for all .

17

It is well known that mattresses on beds should be flipped or spun to ensure more even wear. Let the flips be and , where is a rotation of the mattress about the axis parallel to the longest edge and is a rotation about the axis parallel to the next longest edge. Let the spin be , which is a rotation about the axis parallel to the shortest edge. Show that the set

18

, where is not moving the mattress, forms a group.

a Write down the matrix for a rotation of . b Calculate

and

.

c Form a multiplicative group which contains and (where is the identity matrix). d State the order of the group. e Find the inverse and order of each element of the group. 19

a The operation is defined by real constant.

where and are real numbers and is a

i Prove that the set of real numbers, together with the operation , forms a group. ii State, with a reason, whether the group is commutative. iii Prove that there are no elements of order . b The operation ○ is defined by ○ , where and are positive real numbers. By giving a numerical example in each case, show that two of the basic properties are not necessarily satisfied. © OCR, AS GCE Mathematics: Further Pure Mathematics 3, Paper 4727, January 2009 20

The operation ○ on real numbers is defined by ○

.

a Show that ○ is not commutative. b Prove that ○ is associative. c Determine whether the set of real numbers, under the operation ○, forms a group. © OCR, AS GCE Mathematics: Further Pure Mathematics 3, Paper 4727, June 2008



Section 3: Subgroups, cyclic groups, isomorphism and structure of groups Key point 3.3 If is a subset of the group and is also a group then is a subgroup of . The trivial subgroup of is , where is the identity element. The whole of group, itself, is a non-proper subgroup. Proper subgroups are the other subgroups of . If the operation on is associative, then it can be assumed that the operation on any subgroup is also associative. The remaining conditions required on are: must have an identity element , each element of must have an inverse, the inverse of each element in must also be in , and must be closed. Lagrange’s theorem states that the order of a subgroup is a factor of the order of the group i ii iii

It follows that:

the order of each element of is a factor of the order of if has order , where is prime, then has no proper subgroups if has order , where is prime, then each non-identity element of will have order .

WORKED EXAMPLE 3.7

The table below shows matrix multiplication modulo on the set , where is

.

a State the order of the group and possible orders of any subgroups. b Write down the order of each element. c List the proper subgroups. a

has order . The orders of the subgroups are and .

elements in the group. By Lagrange’s theorem, the orders of the subgroups are factors of the order of the group.

The trivial subgroup has order . The non-proper subgroup, itself, has order . Proper subgroups will have orders and . b c Subgroup of order is

The order of an element is the number of times that element has to be multiplied by itself to give the identity .

i.e.

Since has order , it may be a member of a group with elements.

Subgroup of order is

Since and have order , they may be members of a group with elements.

i.e.

Key point 3.4 A group is cyclic if each element of is of the form

, where

and

.

is called a generator of the group. There may be more than one possible generator in a group. Properties of cyclic groups: Commutative. If the group is of order , then there is an element for which identity. At least one member of the group must be of order .

, where is the

If the order of is , where is prime, then is cyclic.

WORKED EXAMPLE 3.8

Show that the elements of

under multiplication modulo form a cyclic group. All modulo .

, ( and the sequence repeats…) Thus, the operation table is:

The operation table is a Latin square.

a Closed

All numbers in the table are in the group.

b

Row is the same as the header.

c Each element has the identity is . d Multiplication is associative

, so

Each row contains the identity .

True for multiplication.

e Commutative

True for multiplication.

So, forms an Abelian group with a generator , and hence is a cyclic group.

So is also a generator.

Key point 3.5 Two groups and are isomorphic, i.e. have the same form, if they have the same structure. Elements of can be matched to elements of with a one-to-one correspondence.

WORKED EXAMPLE 3.9

The group is under multiplication modulo and the group is , where , , and are operations on a square; is the identity, is a rotation of , is a rotation of and is a rotation of

.

Show that and are isomorphic. The operation tables are:

See Worked example 3.7 for the first table. For the second table, it is helpful to make a sketch. and I are the identity elements.

Orders of the elements:

There are two possible one-to-one correspondences of the elements:

or Either of these shows that and are isomorphic, i.e. .

Key point 3.6 Groups with orders 1 to 7 have the following structures: Order : the only group is . Orders and : Since these orders are prime numbers, all groups with these orders are cyclic (so they are Abelian and have a generator). Order :

If at least one element has order , the group will be cyclic. If none of the elements has order , it is a Klein group. Order : If at least one element has order , the group will be cyclic. If none of the elements has order , then it forms a symmetry group where and are order and and are order .

of the following form,

(Note: the operation, *, is not commutative so this is not an Abelian group.) Higher orders: groups of any order may be considered but knowledge of their properties is not required.

EXERCISE 3C 1

Let

where is the identity element and each element

and is self-inverse.

a Copy and complete this operation table.

b Show that is a group under the operation . c State the order of each element. 2

The set

has the operation table shown below.

a Show that forms an Abelian group of order . b State the order of each element. c Determine whether the group is cyclic. d Write down all the subgroups of order 3

Make an operation table for a Determine whether forms a group.

under multiplication modulo

.

b State the identity and write down the inverse and order of each element. c State whether is cyclic. Give a reason. d Write down all the proper subgroups. 4

A group of order contains , and . is the identity and and are both self-inverse. a Write out the operation table. b Write down the inverse and order of each element. c Determine whether this forms a group and if so what type of group. d Write down all the proper subgroups.

5

Write out the operation table for the cyclic group under complex multiplication with generator i. a Show that the conditions for being a group are satisfied. b State a subgroup, showing that it satisfies all the conditions of a group.

6

Show that modulo .

7

a Given that

under addition modulo is isomorphic to

,

,

under addition

,

i Copy and complete the operation table for

ii State the order of each element. iii With justification, state whether is a group. b Write out the operation table for

under multiplication modulo .

i State the order of each element. ii Explain why is a group. c Are and isomorphic? Justify your answer. 8

is the multiplicative group

, where

.

a Write out the operation table for . b State the order of each element. c Prove that is a cyclic group. is the multiplicative group

under multiplication modulo .

d Write out the operation table for . e State the order of each element. f

Prove that is a cyclic group.

g State whether or not and are isomorphic.

9

It is given that

.

a Show that is a group under addition modulo . It is also given that

where:

b Show that forms a group under matrix multiplication. c Show that and are isomorphic. 10

Find the elements of your answers in a group table.

under multiplication modulo , displaying

a Show that the order of each element of is a factor of the order of . b Find all the proper subgroups of and show that the orders of these subgroups are factors of the order of . 11

a

,

and

i Find and and show that

. .

ii Copy and complete the operation table for

.

iii State the order of each element. iv With justification, state whether is a group. b Write out the operation table for

under multiplication modulo

.

i State the order of each element. ii Explain why is a group. c Determine whether or not and are isomorphic and if so give the correspondence between the elements. 12

A set has elements

.

a Show that the set forms a group under multiplication modulo associativity. b Explain how to obtain the order of .

. You may assume

using modular arithmetic such that

c Find the order of each element. d Write down all the subgroups of order . e Write down three subgroups of order . f 13

Explain why there are no subgroups of order .

a The diagram shows part of a group table for the set under multiplication modulo .

Copy and complete the table. b Write down the order and inverse of each element. c Explain whether the operation forms a group on the set . The set consists of elements of the form

where takes integer values from to .

d Explain why is a cyclic group. e Explain whether and are isomorphic, justifying your comment. 14

A group has an element

.

a Find all the elements of . b Prove that is a cyclic group of order . c Find a proper subgroup of . is the set of integers

.

d Find the order of each element of under multiplication modulo . e Determine, with reasons, whether and are isomorphic. 15

A multiplicative group of order has distinct elements and , both of which have order . The group is commutative, the identity element is , and it is given that . a Write down the elements of a proper subgroup of : i which does not contain ii which does not contain . b Find the order of each of the elements

and

, justifying your answers.

c State the possible order(s) of proper subgroups of . d Find two proper subgroups of which are distinct from those in part (a), simplifying the elements. © OCR, AS GCE Mathematics: Further Pure Mathematics 1, Paper 4725, January 2007 16

Elements of the set

a Verify that

are combined according to the operation table below.

.

b Assuming that the associative property holds for all elements, prove that the set with the operation table shown, forms a group .

c A multiplicative group is isomorphic to the group . The identity element of is and another element is . Write down the elements of in terms of and . © OCR, AS GCE Mathematics: Further Pure Mathematics 1, Paper 4725, January 2008

17

A multiplicative group of order has elements

, where is the

identity. The elements have the properties that a Prove that

and that

and

.

b Find the order of each of the elements c Prove that

.

.

is a subgroup of .

d Determine whether is a commutative group. © OCR, AS GCE Mathematics: Further Pure Mathematics 1, Paper 4725, January 2009 18

A set of matrices M is defined by: , where and

,

,

,

,

,

are the complex cube roots of . It is given that is a group under matrix

multiplication. a Write down the elements of a subgroup of order . b Explain why there is no element of the group, other than , which satisfies the equation c By finding

. and

, verify the closure property for the pair of elements and

d Find the inverses of and . e Determine whether the group is isomorphic to the group , which is defined as the set of numbers under multiplication modulo . Justify your answer clearly. © OCR, AS GCE Mathematics: Further Pure Mathematics 1, Paper 4725, January 2010 19

is a multiplicative group of order

.

a Two elements of are and . It is given that has order and that orders of the elements

. Find the

and .

The table below shows the number of elements of with each possible order. Order of element Number of elements and are the non-cyclic groups of order and , respectively. b Construct two tables, similar to the one above, to show the number of elements with each possible order for the groups and . Hence explain why there are no non-cyclic proper subgroups of . © OCR, AS GCE Mathematics: Further Pure Mathematics 1, Paper 4725, January 2011 20

A multiplicative group has the elements identity, elements and have orders and , respectively, and

where is the .

a Show that

and also that

b Hence show that each of

.

and

has order .

c Find two non-cyclic subgroups of of order , and show that they are non-cyclic. © OCR, AS GCE Mathematics: Further Pure Mathematics 1, Paper 4725, January 2013

Checklist of learning and understanding Conditions for a group: A set forms a group if it is closed, associative, there is an identity element and each other member of has an inverse. A set is closed under the operation if, for all

.

In an operations table for a closed set, all the elements in the table appear in the header. The operation is associative if, for all

, then

The identity element for a set is the element for which

, for all in

. An element has an inverse

if ○

○

.

An Abelian group is a group for which the operation is commutative, for all An operation is commutative if

.

.

A Cayley table, group table or operation table is a two-way table that gives the results of the operation between each pair of elements. The order of a group is the number of elements it contains. The order of an element is the power to which the element has to be raised to give the identity element. Subgroups: If is a subset of the group and is also a group, then is a subgroup of . The trivial subgroup of is , where is the identity element. The whole of group, itself, is a non-proper subgroup. Other subgroups are proper subgroups. If the operation on is associative, then it can be assumed that the operation on any subgroup is also associative. The order of a subgroup is a factor of the order of group . Cyclic groups: A group of order is cyclic if there is an element in , for which

.

is called the generator. Cyclic groups are commutative. If is prime, then the group is cyclic. Isomorphic groups: If two groups and are isomorphic they have the same form and there is a one-toone correspondence between the elements. Structure of groups: If the order of the group is , then it is

.

If the order of the group is , , or , then the group is cyclic. If the order of the group is , then either the group will be cyclic when at least one element has order or it will be a Klein group when no element has order . If the group has order , then it will be cyclic if one element has order . If no elements have order , it forms a symmetrical group.



Mixed practice 3 1

The binary operation is defined on the set of all real numbers by: a Show that is associative. b Find the identity element of the operation. c Prove that every element has an inverse. d Explain why the set forms a group under addition.

2

The operation is defined on the set of all real numbers by:

a Prove that the operation is commutative. b Prove that the operation is associative. c Find the identity element. d State whether the set forms a group under justifying your answer. 3

A set has elements of . a Draw a group table for

.

represents multiplication modulo of the elements

.

b Show that forms a group under multiplication modulo . c Write down the order of the elements. A group is cyclic if it is commutative and has a generator where the group and the identity element.

being the order of

d State whether the group is cyclic, justifying your answer. 4

is a set with elements

.

a Show that forms a group under multiplication modulo . b State the order of each element. c Write down all the proper subgroups of . 5

a Copy and complete this group table, where is the identity,

and

b Write down a proper subgroup. 6

Elements of the set

under a binary operation have the following Cayley table:

a Assuming that the operation is associative prove that the set

forms an Abelian

group under the binary operation . b Find the order of each element. c Write down any proper subgroups of the set

7

.

The set consists of matrices with a generator

.

a Show that forms a group under matrix multiplication. You may assume that matrix multiplication is associative. b State the order of the group . c State the order of each element of the group . d Write down all the proper subgroups of . e State whether forms a cyclic group, justifying your answer. 8

The set consists of the six matrices

, where

. It is given that forms

a group under matrix multiplication, with numerical addition and multiplication both being carried out modulo . a Determine whether

is a commutative group, justifying your answer.

b Write down the identity element of the group and find the inverse of c State the order of

and give a reason why

.

has no subgroup of order .

d The multiplicative group has order . All the elements of , apart from the identity, have order or . Determine whether is isomorphic to , justifying your answer. © OCR, AS GCE Mathematics: Further Pure Mathematics 3, Paper 4727, January 2012 9

is a set with elements for a Draw an operation table for

and

represents multiplication modulo .

.

b State the order of the group and write down the order of each element of . c List all the proper subgroups of . d State the number of proper subgroups of sets with order where and are distinct prime numbers. 10

and with order

,

Let be any multiplicative group. is a subset of . consists of all elements such that for every element in . a Prove that is a subgroup of . Now consider the case where is given by the following table:

b Show that consists of just the identity element. © OCR, AS GCE Mathematics: Further Pure Mathematics 3, Paper 4727, June 2015 11

When using cryptography, permutations are employed. For example, when encoding three objects a matrix may be used such as:

to show that the number is replaced by , is replaced by and is replaced by . Write down all six permutation matrices, labelling them The inverse of the matrix above is

, i.e. the inverse is found by swapping the rows.

Rewriting this matrix with the first row in order gives Make a Cayley table for permutation 12

on

.

means first do the permutation

then do the

. Show that the operation is not commutative.

The set consists of all where

where

.

non-singular real matrices A satisfying the condition

.

a Prove that the form of each matrix is b Find the conditions on a such that

. is in .

c Assuming the condition in and that matrix multiplication is associative prove that is a group under the operation of matrix multiplication.

CROSS-TOPIC REVIEW EXERCISE 1 1

Solve the second-order recurrence relation:

given that 2

and

.

a Solve the recurrence relation b Write down the value of

3

The sequence a Find

and

,with

.

.

is defined by

with

and show that

.

.

b Hence suggest an expression for

in terms of .

c Use induction to prove that your answer to part b is correct. d Use your answer to part b to prove that 4

Express

and

and

.

a Use a step-by-step algorithm to show that b Given that

7

.

as a decimal number and then use the standard divisibility tests to show

that it is divisible by 6

.

Solve the second-order recurrence relation:

given that 5

as

, find

is a factor of

.

.

A set consists of matrices of the form

, where and are real numbers and

. a Show that is an Abelian group under multiplication. You may assume that matrix multiplication is associative. b 8

is a subset of of order . Find the elements of .

a Solve

.

b i Explain why

and

ii Solve 9

A set has elements

and

has no integer solution. .

.

a Draw a table for multiplication modulo of the elements of . b Write down the inverse of each element of . c Write down the order of each element of . d State, with a reason, whether is cyclic. is a group with an element , where

and is the identity.

e State whether and are isomorphic. 10

a Show that the set

under multiplication modulo

is the smallest set that contains the element

b Show that is a group. c Determine, with reasons, whether and are isomorphic.

.

forms a group.

4 Further vectors In this chapter you will learn how to: understand and use the vector product be aware of the properties of the vector product understand the significance of use the vector product to calculate areas of triangles and parallelograms use the vector product to calculate volumes of tetrahedra and parallelepipeds use the scalar triple product.

Before you start… Pure Core Student Book 1, Chapter 2

You should be able to use the scalar product of two vectors: where and

, etc.

1 Given that

and

find the value of . Hence find the acute angle between and .

Pure Core Student Book 1, Chapter 2

You should be able to write the vector equation of a straight line in dimensions. A straight line has vector equation:

2 Find the equation of the straight line passing through the points and .

where is any point on the line and is the direction of the line.

Pure Core Student Book 2, Chapter 4

You should be able to write the equation of a plane in dimensions. A plane has vector equation:

where is a vector perpendicular to the plane and is a constant found by calculating the value of where is a point on the plane.

Pure Core Student

You should be able to find the

3 A plane contains the points and . Find a vector perpendicular to and to . Hence find the vector equation of the plane.

4 Find the determinant of

Book 1, Chapter 1

determinant of a

matrix. .

If

then the

determinant of is det where

and

(the

determinant may be calculated using other rows or columns).



Section 1: Vector product Key point 4.1 Consider the vectors and , where and . Scalar product: , where is the angle between and and and

. , where is a unit vector (i.e. a vector of length one unit)

Vector product:

perpendicular to and . The direction of is found using the right hand rule. Placing the first and second fingers and the thumb mutually perpendicular, if a is in the direction of the first finger and b is in the direction of the second finger then the thumb indicates the direction of . is the angle between and .

Tip Note that most calculators can be used to calculate the vector product.

Rewind You learnt how to find the scalar product and the vector product in Pure Core Student Book 1, Chapter 2. Notes on vector product: 1 The magnitude or length of 2 If , provided or collinear. 3

or

is , then

. , i.e.

, which means that and are parallel

, i.e. the vector product is not commutative.

4

, i.e. the vector product is distributive over addition.

5

Straight line: the general vector equation of a straight line is: , where is the position vector of any point on the line and is the direction vector of the line. Then

, and hence

This is a standard vector equation for a straight line.

(since

).

Rewind You learnt how to write the vector equation of a line in Pure Core Student Book 1, Chapter 2. Areas of triangles: from the diagram, the area of the triangle is . Let

and

.

Areas of parallelograms: from the diagram, the area of the parallelogram is . Let

and

.

Volumes of tetrahedra: if the base has sides and and the third adjacent side is then: the area of the base is

and the height is

.

Volumes of parallelepipeds: if the base has sides and and the third adjacent side is and the height of the parallelepiped is

Scalar triple product

.

:

The vectors can move in a cyclic order such that

.

gives the volume of a parallelepiped. If this volume is zero (i.e. ) then , and must be coplanar. If and are parallel or collinear, then

Also,

, since

.

.

Explore Investigate how the vector product is used in Physics, for example, in calculations involving torque or the magnetic force on a moving charge. What is the right-hand rule used for? WORKED EXAMPLE 4.1

WORKED EXAMPLE 4.1

A triangle has vertices

and

. Find the angle

.

A diagram helps.

Using

Using

then

and

.

WORKED EXAMPLE 4.2

It is given that a Find the vector product

and

.

.

b Hence, find the angle between the vectors and .

a

Using

b Use

Using

Tip Be careful when finding the component of the vector product. It is A common error is to use .

or

WORKED EXAMPLE 4.3

A straight line passes through the points and the form and use this to verify that The straight line has direction vector

. Find the equation of the line in is a point on the line.

.

If and are the position vectors of two points then a direction vector joining them is .

,

Using so the line has equation

Substituting

.

.

gives Since satisfies the equation of the line it must be on the line.

and hence .

is on the line through

and

WORKED EXAMPLE 4.4

Find the area of the triangle with vertices

and

.

A diagram helps.

Let

and

. Then:

WORKED EXAMPLE 4.5

Find the area of the parallelogram with vertices

and A diagram helps.

Let

and

. Then:

WORKED EXAMPLE 4.6

Calculate the volume of a tetrahedron with vertices

and

.

A diagram helps.

Let

,

and

. Then:

Volume of a tetrahedron is

WORKED EXAMPLE 4.7

A parallelepiped with coordinates of vertices and

is shown below.

Find the volume of the parallelepiped. Let

and

. Then: Volume of a tetrahedron is .

Tip It is a common error to have two of the vectors is not the case.

EXERCISE 4A

and to be parallel. Always check that this

1

For each pair of vectors and , calculate the value of the scalar product

.

a

b

c d

,

e

,

f

2

,

Use the scalar product to find the acute angle between the vectors and .

Tip The scalar product gives the angle between the positive directions of a and b. This may not be acute. In general, if a question asks for the angle between two vectors, the acute angle is usually found.

a

b

c d

,

e

,

f

3

,

A triangle has vertices

and . Find the angle

a

is

, is

and is

b

is

, is

and is

c

is

, is

d

is

, is

and is and is

, where:

4

e

is

f

is

, is , is

and is and is

Calculate the following vector products. a

b

c d

,

e

,

f

5

,

Find a unit vector perpendicular to: a

b

c d e f 6

For each pair of vectors and , calculate the value of the vector product angle between the vectors and . a

b

c d

,

, and hence find the

e

,

f

7

,

You are given that a Find i

and

and ii

b Find i

.

. Is the vector product commutative? and ii

. Is the vector product distributive over vector addition?

c Prove that, in general,

.

d Prove that, in general,

.

8

Find a vector perpendicular to plane:

9

Find, in vector form, the equations of the planes which pass through the following points. a

and

b

and

c

11

and

Find the equations of the straight lines, in the form following sets of points. a

and

b

and

c

and

d

and

Find the point of intersection of the following pairs of lines. and

a

and

b

12

Find the area of the triangles with vertices: a

and

b

and

c d 13

, and hence find a vector equation of the

and

d 10

and

and and

Find the area of the parallelograms

with vertices:

, which pass through the

a b c d 14

Calculate the volume of tetrahedron a

and

b

and

c d 15

with vertices:

and and

Calculate the volume of parallelepiped PQ RSTU VW with vertices:

a

and

b

and

c

and

d

and

16

Given that

and

17

Determine whether

18

A pavilion is made such that its base forms a cuboid and the roof is a right pyramid.

, show that:

and

are coplanar.

The length the roof is

of the base is

, its width

is

and its height

is

and the apex of

above the ground.

a Taking A as the origin and find vectors for

,

,

in the -direction, and

in the -direction and

in the -direction

.

b Hence find the volume of the pavilion. 19

a Given that is perpendicular to and find

.

b Given that is a unit vector find 20

.

A triangle has vertices

and

a Find the area of the triangle

.

.

b Find a vector perpendicular to the triangle. c Find the vector perpendicular to the triangle with length equal to the area of the triangle. d Find the vector perpendicular to the triangle with vertices length equal to the area of the triangle.

and

with

Checklist of learning and understanding The scalar product is

, where is the angle between and and and

.

The vector product is and is the angle between and .

, where is a unit vector perpendicular to and

The vector product is calculated using

.

The straight line

may be written in the form

The area of a triangle

, where

The area of a parallelogram The volume of a tetrahedron

and

, where , where

.

, is and

. , is

.

and

The volume of a parallelepiped with three non-parallel sides is

, is

. and

. The scalar triple product is The scalar triple product can be found using

. .



Mixed practice 4 1 2

Find the exact area of a triangle with vertices A triangle has vertices

and

Calculate the size of the angle 3

and

.

. The area of the triangle is

.

are the vertices of a parallelogram where

and

.

a Find the coordinates of . b Show that the area of the parallelogram is 4

A parallelogram has vertices

. and

.

a Find the equation of the line through and of the form

5

b Given that the area of the parallelogram is

, find angle

A tetrahedron has vertices

and

. . .

Find its volume. 6

A tetrahedron

has base

The vertex is

where is

, is

and is

.

above the centre of the base.

a Show that has coordinates

.

b Find the volume of the tetrahedron. 7

A parallelepiped has one face

. The face opposite this is labelled TU VW as shown.

and

.

a Find the coordinates of the vertices and . b Calculate the volume of the parallelepiped. 8

Given that

and

find:

a b Show that the results are the same. 9

It is given that a Find

.

and .

b Interpret your answer geometrically.

.

10

The vectors of a parallelepiped

and

represent three edges

and

.

a Find the volume of the parallelepiped. b State what the answer to part a implies about 11

The vectors and are perpendicular. a Explain why b Evaluate

. .

c Hence simplify 12

and .

giving a reason for each step of your solution.

a A triangle has vertices

and . The lengths of the opposite sides are

and , respectively.

Use the vector product form for the area of the triangle to obtain the sine rule:

b A pyramid

has vectors

The vectors and the area of each face. Show that

and

.

are perpendicular to each of the faces and of magnitude equal to

.

5 Surfaces and partial differentiation In this chapter you will learn how to: work with functions of two variables sketch sections and contours find first and second partial derivatives find the coordinates and types of stationary points in 3-D find the equation of the tangent plane to a 3-D curve.

Before you start… A Level Mathematics Student Book 2, Chapter 10

You should be able to differentiate a range of functions. For example: i polynomials

Differentiate the following with respect to : 1 2 3

ii exponential functions

4 5 6

iii logarithmic functions

7 8

iv trigonometric functions

9 10

v products

vi quotients

viiimplicit functions e.g.

Pure Core Student Book 2,

You should know that a plane has Cartesian equation:

11 Write the vector equation of the

Chapter 4

plane: where is the vector normal to the plane. in Cartesian form.



Section 1: Three-dimensional (3-D) surfaces Key point 5.1 In three dimensions (3-D) a surface is described by the form: or , where is a constant, i.e. the height above the plane is found directly or indirectly from the given equation. Conventionally, the plane is horizontal and the -axis is vertical (using the right-hand system). Plotting all possible points gives the surface. Sections: these are cross-sections of the surface for defined values of or : i.e. graphs of or for specific values of or . Contours: these are effectively a plan view of the surface looking from above the

plane

for defined values of (as on an Ordnance Survey map): i.e. graphs of for specific values of . Multi-variable functions: it is possible to have more variables such as

,

where is a variable.

WORKED EXAMPLE 5.1

Consider the surface

.

a On separate diagrams sketch the sections i b Draw the contours for a i

for taking the values

. and

.

These are equivalent to sketching .

Section

ii Section

b Contour diagrams for

and ii

.

and

To visualise the 3-D shape of the surface , using the above:

parabola in parabola in

plane plane

ellipses for each value of Combining all three of these gives the paraboloid shown.

EXERCISE 5A In this exercise you are encouraged to use graph plotting software or a graphical calculator. 1

For the surface

:

a on separate diagrams, sketch the sections i b draw the contours for 2

For the surface

for taking values

and

.

:

a on separate diagrams, sketch the sections i for b draw the contours for

and ii

and

for taking values

ii for and

.

and

3

A surface has equation

.

a Sketch the section with section with the plane

and state the coordinates of the points of intersection of this .

b Sketch the section with section with the plane

and state the coordinates of the points of intersection of this .

c Draw the contours for 4

for taking values

A surface has equation

.

.

a Sketch the sections with , and the coordinates of the points of intersection. b Draw the contours for 5

and

. Where the sections intersect the

for taking values

A surface has equation

and

plane find

.

.

a Sketch the sections with , and . Where the sections intersect the plane find the coordinates of the points of intersection.

6

b Draw the contours for

for taking values

A surface has equation

.

a Determine whether the point ( b Find whether the point (

) is on, above or below the surface. for taking values

A surface has equation

and ii for

b Find a point on which lies in the

plane.

c Draw the contours for

for taking values

For the surface

b draw the contours for For the surface

for taking values

and

For the surface

and .

for taking values

and ii for and

.

:

a on separate diagrams, sketch the sections i for

and ii for

b verify that the two sections in a intersect where the point on the surface of this intersection

and

c draw the contours for For the surface

.

:

b draw the contours for

11

.

and ii

a on separate diagrams, sketch the sections i for

10

.

:

a on separate diagrams, sketch the sections i

9

and

.

a On separate diagrams, sketch the sections i for

8

.

) lies on the surface.

c Draw the contours for 7

and

for taking values

and state the coordinates of

and

:

a on separate diagrams, sketch the sections i for b draw the contours for

for taking values

and ii for and

.

.

12

A surface has equation

.

a Draw sections of for

,

,

.

b State which of the sections in a intersect the plane 13

For the surface

:

a on separate diagrams, sketch the sections i for b draw the contours for 14

For the surface

for taking values

and ii for and .

:

a on separate diagrams, sketch the sections i for b draw the contours for 15

.

for taking values

A surface has equation

and ii for and .

.

a Sketch the sections of when

and

.

b State which of the sections in a intersect the plane

.

c Use a graph plotting package to sketch the contour of when 16

.

The surface of a three-dimensional object has equation a State the equation of the section of for which

.

and sketch this section.

b Find the coordinates of the points where the section intersects the plane c Find the coordinates of the turning points on this section of . 17

For the surface

:

a on separate diagrams, sketch the sections when i b draw the contours for

and ii

for

c use a graph plotter to draw the contours for 18

For the surface

and

:

a on separate diagrams, sketch the sections i b draw the contours for

19

For the surface

for taking values

b draw the contours for

For the surface,

and .

:

b draw the contours for

For the surface

and ii for

for taking values

a on separate diagrams, sketch the sections i for

21

and

:

a on separate diagrams, sketch the sections i for

20

and ii

and ii for

for taking values

:

and .

.

.

a on separate diagrams, sketch the sections with i

b draw the contours for

,

for taking values

and

and ii

,

and

.

Explore Investigate some of the software that is available on the internet for producing 3-D graphs and modelling.



Section 2: Partial differentiation Key point 5.2 If a 3-D surface has equation

, then:

differentiate with respect to , assuming is constant which gives the rate of change on the surface of as changes differentiate with respect to , assuming is constant differentiate with respect to , assuming is constant differentiate with respect to , assuming is constant differentiate with respect to , now assuming is constant differentiate with respect to , now assuming is constant. Note: the mixed derivative theorem for most well-behaved, continuous functions, states that:

WORKED EXAMPLE 5.2

Given that

, find:

a b c d Also show that

.

a

Differentiate with respect to , assuming constant.

b

Differentiate with respect to , assuming constant. Differentiate with respect to , assuming constant.

c

Differentiate with respect to , assuming constant.

d

Differentiate with respect to , assuming constant.

Also,

and

so

,

.

WORKED EXAMPLE 5.3

Differentiate with respect to , assuming constant.

Verifies the mixed derivative theorem.

Given that

, find:

a b c d a

Differentiate with respect to , assuming and constant.

b

Differentiate with respect to , assuming and constant.

c

Differentiate with respect to , assuming and constant. Differentiate with respect to , assuming and constant.

d

Differentiate constant.

with respect to , assuming and

Tip It is a common error to confuse with though, for this course, they will always yield the same result and may be used interchangeably. Remember that

and

.

EXERCISE 5B For questions 1 to 15, find a , b , c 1 2 3 4 5 6 7 8 9

and d

. Also show that

.

10 11 12 13 14 15 16

Given that

17

If

, find

and

, show that, at the point

and show that :

a b c d e 18

.

Show, for the point

on the curve

, that:

a b

c 19

If

, show that, at the point

:

a b c 20

Given that a b c d



, find:

.

Section 3: Stationary points Rewind You learnt about stationary points in A Level Mathematics Student Book 1, Chapter 14.

Key point 5.3 If is a stationary point on the surface then and at . There are three types of stationary point: local maximum, local minimum and saddle point. At a local maximum, the surface is at its greatest height (like at the top of a hill) in the immediate neighbourhood, and at a local minimum it is at its least height (like the bottom of a valley). A saddle point looks similar to a horse’s saddle, in that in some directions moving away from it the height decreases, whilst in other directions it increases. The nature of a turning point can be determined by means of the sign of the determinant of the matrix of second partial derivatives (the Hessian matrix). The Hessian matrix is defined as i

Local maximum

ii

If and Local minimum

. It follows that

there is a local maximum.

.

If

and

there is a local minimum.

iii Saddle point

If Note: If

there is a saddle point. then the nature of the stationary point cannot be determined by this test.

WORKED EXAMPLE 5.4

For the surface given by

find:

a the coordinates of the stationary points b

the nature of the stationary points.

a Let

For a stationary point

.

Using partial differentiation rules.

and

Applying conditions for stationary points.

or and When

,

Finding the coordinates of the stationary points.

giving stationary points

and

When giving stationary points

. ,

and

. Using values of turning points.



b At



to determine the nature of the

, and and stationary point At

maximum



and and stationary point At

minimum



saddle point At



saddle point

EXERCISE 5C 1

A surface S has equation

.

Find the coordinates and nature of the stationary point and show that it is a minimum. 2

Find the coordinates and nature of the stationary point on the surface

3

A surface S is defined by

4

Find the coordinates and nature of the stationary point on

5

Find the coordinates of the four stationary points on and determine whether each stationary point is a maximum, a minimum or a saddle point.

6

Show that

.

. Show that the only stationary point is a saddle point. .

has three stationary points.

Find the coordinates of each stationary point and determine whether each is a maximum, a minimum or a saddle point. 7

The diagram shows a stand used in a museum display.

The surface of the stand is given by the equation four saddle points and one minimum stationary point.

. Show that the stand has

Find the coordinates of all these points. 8

A modern building has a roof surface with equation:

a Find the position of the highest point and the lowest point on the roof. b The roof has two supports placed at the saddle points. Find the coordinates of the saddle points. 9

A design department of the manufacturer of corn snacks produces a template of the following shape:

This surface has the equation:

and it has two stationary points.

Show that the coordinates of these are and each point is a maximum, a minimum or a saddle point. 10

Show that

and determine whether

has nine stationary points.

Verify that four of these are saddle points. For the other five points, determine whether each is a maximum or a minimum stationary point.

11

Find the coordinates of the stationary point on

12

Given that

13

A roof light has a surface given by the equation where all lengths are in metres.

and prove that it is a saddle point.

, show that there is a minimum stationary point at , for

a Show that the roof light has a maximum height of b Sketch the section of for which

. and

,

and that this occurs at its centre.

.

c Draw a contour map for for values

and

.

14

Standing waves in a water tank can be modelled by the equation . Show that the surface of these waves bounded by and has two stationary points. Find the coordinates of these points and determine whether each is a maximum, a minimum or a saddle point.

15

Find the coordinates of the stationary points on each is a maximum, a minimum or a saddle point.

16

To reduce drag the nose cone of a Formula racing car looks like the one in the diagram.

and determine whether

Given that the equation of this surface is stationary point at ( ). 17

Show that minimum stationary point.

, show that there is a minimum

has two saddle points, one maximum stationary point and one

Find the coordinates of all these points. 18

A surface has equation a Show that the

.

coordinates

,

,

,

give stationary points on .

b Determine whether each is a maximum, a minimum or a saddle point. 19

Show that

is a stationary point on

.

Determine the type of stationary point depending on the values of . 20

A surface has equation

for

and

.

Show that has maximum turning points at points at and



and .

and minimum turning

Section 4: Tangent planes Key point 5.4 For a 3-D surface with equation the point is:

, the equation of the tangent plane to the surface at

The tangent plane is a 3-D version of a tangent to a curve.

WORKED EXAMPLE 5.5

Find the equation of the tangent plane to Let Given that

at the point where

and

.

. and , so

.

To find when

and

.

Now find and

, so The equation of the tangent plane is then:

Apply the general equation for the tangent plane. This diagram illustrates the plane.

or

EXERCISE 5D 1

Show that the equation of the tangent plane to

2

A surface S has equation and .

3

Find the equation of the tangent plane to

4

Show that the point

at the point

is

.

. Find the equation of the tangent plane at the point where

at the point

lies on the surface

. .

Find the equation of the tangent plane at . 5

Show that the equation of the tangent plane to

at the point

is

. 6

Find the equation of the tangent plane to

7

Find the equation of the tangent plane to .

8

Find the equation of the plane that is tangential to

9

Find the equation of the tangent plane to

at the point where

and

at the point where

at the point

at the point where

10

A surface has equation the point .

11

Show that the equation of the tangent plane to .

12

Find the equation of the tangent plane to

13

Show that the equation of the tangent plane to .

and

.

and

.

. Find the equation of the plane that is tangential to at

at the point where

at the point where

and

at the point

and

is

. is

.

14

Find the equation of the plane that is tangential to the surface

at the point

. 15

Given that a surface has equation to at the point where

16

, find the equation of the plane that is tangential

and

.

Show that the equation of the plane that is tangential to the surface with equation at the point

17

Show that the point

18

Determine whether the plane

is

.

lies on the plane that is tangential to

at

.

is parallel to the plane that is tangential to the surface

at the point where

and

19

Find the equation of the tangent plane to

20

A surface has equation

. at the point where

and

.

.

a Show that the plane that is tangential to at the point b Sketch a section of for which

has equation

.

c Sketch a contour diagram for values of

.

Checklist of learning and understanding A surface is a 3-D shape defined by

or

.

The section of a surface is a cross-section found using, for example,

or

for specific values of and . The contours of a surface are plan views found by putting The partial derivative

, for specific values of .

is found by differentiating the function with respect to

assuming all other variables are constant. The partial derivative

is found by differentiating the function with respect to

assuming all other variables are constant. The partial derivative

is found by differentiating the function with

respect to assuming all other variables are constant. The mixed derivative theorem states that If (

.

) is a stationary point on the surface

then

A maximum stationary point has

and

A minimum stationary point has

and

A saddle point has

and . .

.

The Hessian matrix has determinant

.

A tangent plane is a plane tangential to a surface at a given point. A tangent plane to

at the point

has equation:

at (

).



Mixed practice 5 1

A curve has equation

.

a Sketch sections: i ii

.

b Sketch contours for 2

Show that

a

, where

.

for:

noting that

b

noting that

3

Find the equation of the tangent plane to

4

Show that the curve

5

A surface has equation

and

has a minimum turning point at where

Show that has a saddle point at

. .

. .

6

Find the equation of the tangent plane to

7

It is given that a Show that

at the point where

at the point where

and

.

. .

b Find the stationary point on the surface minimum or saddle point.

and explain why it is a maximum,

c The surface has sections , where is a constant greater than zero. Find, in terms of , the coordinates of the turning point of this section. Sketch this section. 8

A surface has equation

, where

.

a Find the stationary points on the curve and determine whether each is a minimum, maximum or saddle point. b Sketch a section for which 9

An open-topped box has volume

. . Its base has dimensions

and its height is

a Find an expression for the surface area in terms of and . b Use partial differentiation to prove that the surface area is a minimum when and

,

.

10

Find the equation of the tangent plane to at the point where writing your answer in the form , where is a constant.

11

Given that and are functions of and and that

, find

and

,

and show that

.

6 Further calculus This chapter is for A Level students only. In this chapter you will learn how to: use reduction formulae to find integrals and then evaluate them find arc lengths find surfaces of revolution.

Before you start… A Level Mathematics Student Book 2, Chapter 11

You should be able to carry out integration by parts:

A Level Mathematics

You should be able to differentiate trigonometric functions:

Student Book 2, Chapter 9

1

Differentiate with respect to : 2 3 4

A Level Mathematics Student Book 2, Chapter 9

You should be able to integrate trigonometric functions:

Integrate:

5

6

Pure Core Student Book 2, Chapter 8

You should be able to differentiate hyperbolic functions:

Differentiate with respect to : 7 8 9

Pure Core

You should be able to integrate

Student Book 2, Chapter 8

hyperbolic functions:

Integrate:

10

11

Pure Core

You should be able to recall the

Student Book 2, Chapter 8

following hyperbolic identities:

12 Solve , giving your answers in logarithmic form. 13 If terms of .



, express

in

Section 1: Integration by reduction Key point 6.1 If part of a function to be integrated is raised to a high power, then use integration by parts: i.e. to reduce the power. This leads to a recursive method to simplify the integral.

Rewind Integration by parts was covered in A Level Mathematics Student Book 2, Chapter 11.

WORKED EXAMPLE 6.1

Given that

show that

.

Hence find: a b

, giving your answers in terms of . Let

Hence

a

and

and substitute into

Note that

Using

.

with

.

This recursive method is repeated to reduce the power from to . but Find by integration.

Putting these together gives:

b

WORKED EXAMPLE 6.2

Finding in terms of leads to the solution.

WORKED EXAMPLE 6.2

Given that

obtain the reduction formula:

and use it to evaluate

.

This is an important technique to create integration by parts.

Now use

to give

Split the integral into two parts and use and

.

and hence

from which

or

For

First simplify the reduction formula by applying the limits.

then giving and hence

Writing

but:

and hence:

as gives:

This formula is used to reduce the power from to .

Find by integration.

Tip Remember that

. It is a common error to forget the

.

Tip For

, put

and recall that

Remember that

. (You may verify this by differentiating the

right-hand side.)

EXERCISE 6A

1

It is given that

for

.

a Use the formula

with

and

to prove that

.

b Use your answer to find . 2

a By using

with

b Hence show that if

, where

c Show that 3

By writing

, find

.

, the reduction relation is

.

.

as

and

and using integration by parts show that if

then

.

Hence find the exact value of .

4

Find a reduction formula for

5

Given that

and use it to find the value of .

, put

the identity

Given that

a

and use integration by parts to apply

and obtain the reduction formula

Use this reduction formula to evaluate

6

as

.

, obtain a reduction formula and use it to evaluate:



.

b

7

.

Given that

, with

, show that

and use it to evaluate

.

8

Given that

, show that

.

Use this result to find the exact value of .

9

Given that

, show, by integrating by parts twice, that the reduction formula

is

.

Use your answer to find

10

.

Given that

, prove that, for

Use your answer to find

11

,

.

Given that

, show that, for

,

evaluate:

12

a



b

.

Obtain a reduction formula for

a

b

13

a If

, find

.

.

and use it to find:

and use it to

b Given that

c Find i

14

, show that

and ii

Given that

for

.

.

, show that

for

.

Find: a



b

.

15

If

, show that, for

16

a Find

.

,

and use this to find the exact value of .

b By integrating the expression you found in part a, show that:

where

.

c Use your result to find . 17

a Find

.

b By integrating the expression you found in part a, show that:

where

.

c Use your result to find . 18

a If

, show that

b Given that

.

and using

for

, show that:

.

c Find .

19

Obtain a reduction formula for

and use it with

to show that:

20

a Show that b If c Find .



. , show that, for

,

.

Section 2: Arc lengths and surface areas Key point 6.2



Arc length If

use

If

use

Cartesian equations Area of surface of revolution when rotated about the -axis If use

If

use

Area of surface of revolution when rotated about the -axis If use

If

use

If

Arc length

use

Parametric equations Area of surface of revolution when rotated about the -axis

Area of surface of revolution when rotated about the -axis

Rewind You met solids of revolution in Pure Core Student Book 2, Chapter 9.

Tip It is essential that you take particular care reading the question as it is a common error to rotate about the wrong axis.

WORKED EXAMPLE 6.3

The arc of the curve with equation using the substitution .

Given that,

from

to

is labelled . Find the length of

, so the arc

length is given by:

Using:

Let

To integrate this, a hyperbolic substitution is required.

, making

Therefore:

Using

.

Using



and

Tip In questions such as Worked example 6.3, it is a common error to:

1

forget to change the limits

2

evaluate

using a calculator.

If you have

or

, remember to rewrite in terms of

and/or

before applying

the limits or writing in terms of .

WORKED EXAMPLE 6.4

The part of the curve with equation from to is labelled . Find the area of the surface generated when is rotated one full revolution about the -axis.

Given that

,

, so the

area of the surface is given by:

Using:

WORKED EXAMPLE 6.5

A fruit bowl is made by rotating part of the curve

from

to

about the -

axis. Find the area of the curved surface generated. You are given that . The surface area is:

, so

As the equation is in the form

, use:

WORKED EXAMPLE 6.6

A curve is defined parametrically by the equations Show that the length of the arc of the curve from

Given that

,

and

and to

. is

.

Differentiate parametrically then use the parametric equation for the arc length:

,

WORKED EXAMPLE 6.7

A curve is defined parametrically by the equations

and

Find the area of the surface generated when the arc of the curve from full revolution about the -axis. Given that

and and

and so

EXERCISE 6B

,

. to

is rotated one

Differentiate parametrically and then use the parametric equation for the surface area:

EXERCISE 6B In this exercise, your solutions should show full details of methods and working but you might wish to check your numerical answers using a graphical calculator or computer package.

1

Using a b c

find the lengths of the arcs of the following curves. from from

to

from

to from

d

2

to

to

Using

find the lengths of the arcs of the following parametrically

defined curves. a

from

b

from

c

from

d 3

to to

to

from

to

Find the areas of the surfaces generated when the following arcs are rotated one revolution about the given axis using the results that if

the -axis, if

then

b

4

for rotation about

for rotation about the -axis and if

then a

then

for rotation about the -axis. from

to

from

about the -axis to

about the -axis

c

from

to

about the -axis

d

from

to

about the -axis

Using

for rotation about the -axis or

for rotation about the -axis find the areas of the surfaces generated when the following parametrically defined arcs are rotated one revolution about the given axis. a b c

from from from

to

about the -axis to

to

about the -axis about the -axis

d 5

from

to

about the -axis

A curve is defined parametrically by

.

a Show that

and hence that the length of the arc of the curve from

the origin to

is

.

b Use the substitution 6

a Write

and

to show that

.

in exponential form, hence find

and

b The curve has parametric equations given by surface generated when the part of , from to

.

. Show that the area of the is rotated about the -axis is given by

. You are not asked to evaluate this integral. 7

A curve has equation the curve from to

. Show that the area of the surface generated when the arc of is rotated about the -axis is given by:

Using the identity 8

show that

Show that the surface area generated when the arc on the curve rotated about the -axis is

9

. from

A curve has parametric equations

.

.

b Find the length of the arc of the curve from the point , where c Show that the area of the surface generated when the arc . A curve is defined parametrically by

, to the point , where

c Show that the area of the surface generated when the arc

,

.

.

b Hence prove that the length of from

to

is

.

is rotated one full revolution

.

A curve has equation

a Show that

is rotated about the -axis is

.

b Find the length of the curve from the origin , where

11

, to the point , where

.

a Show that

about the -axis is

is

.

a Show that

10

to

.

.

12

A curve is defined parametrically by .

from the origin to the point , where

a Show that the length of the curve is

.

b Show that the area of the surface generated when the arc about the -axis is given by

is rotated one full revolution

.

c Use integration by parts to find the exact value of this integral. 13

A curve has equation

.

a Show that

.

b Hence show that the area of the surface generated when the arc from one full revolution about the -axis is

14

Use to

15

to

.

to show that the area generated when the arc of can be written as

Use

is rotated

from

.

to show that the area generated when the arc of the curve

from

to

can be written as

.

Use integration by parts, or otherwise, to find the exact value of this integral. 16

Show that the arc of the curve with equation

Use the substitution

17

from

to show that:

A parabolic reflector is part of a surface with equation

, as shown.

to

is given by:

A supporting metal strip is attached along the length of the contour of in the to 18

plane, from

. Show that the length of this strip is

A designer creates a wing for a model plane using the equation

The designer needs to know the length of the part of the wing defined by the contour for which . a Sketch the contour for which b By using the substitution to is

. , or otherwise, show that the length of this contour from

. 19

The prow of a boat is part of a surface given by the equation

.

The contour of in the plane

20

is designed to be the Plimsoll line.

Show that the length of this line from

to

A bowl has a surface defined by

for

The base of the bowl is where

is

. .

and the lip of the bowl is where

a Show that the length of the perimeter of the lip of the bowl is

. .

b Find the length of the shortest distance along the bowl from its lip to its base. c Show that the area of the outside surface of the bowl is .

Explore This is an image of Gabriel’s horn, which has a finite volume but an infinite surface area. Gabriel’s horn is a shape formed by rotating the graph of

around the -axis (for values of

). Would it be possible to coat this surface with a finite amount of paint? Investigate this paradox and how it led to a dispute about the nature of infinity involving many philosophers of the seventeenth century.

Checklist of learning and understanding Reduction formulae:

To obtain a reduction formula use integration by parts. To obtain a reduction formula for

, write the integral as

To obtain a reduction formula for

, put

. .

Arc lengths with Cartesian equations:

If

, use

.

If

, use

.

Arc lengths with parametric equations:

Use

.

Area of a surface of revolution when rotated about the -axis with Cartesian equations:

If

, use

.

If

, use

.

Area of a surface of revolution when rotated about the -axis with Cartesian equations:

If

, use

.

If

, use

.

Area of a surface of revolution with parametric equations:



Use

if rotated about the -axis.

Use

if rotated about the -axis.

Mixed practice 6 1

It is given that

for

.

a Use the formula

with

and

to prove that

. b By writing

as , show that

c Find the exact value of

2

.

It is given that a By using

for

.

in

prove that

b By writing

3

.

, show that

It is given that

for

a By substituting

and

.

.

. into

, prove that

. b Show that

4

.

It is given that

for

.

a Use integration by parts to prove that b Show that

5

a Find

.

.

.

An arc is part of the curve with

from

b Show that the length of the arc is 6

.

.

A curve is defined by parametric equations Show that the length of the arc from

7

to

to

. is

.

A catenary is a curve made when a flexible wire or chain has its ends attached to fixed points and then it is allowed to hang freely under gravity. A catenary has equation

and it is rotated about the -axis from

to

Find the area of the surface generated. 8

A bowl is made by rotating the curve

about the -axis from

a Show that the area of the curved surface formed is:

to

.

.

b Hence show that the outside curved surface area plus the base have a total area:

9

A design student is making a vase using a computer aided design machine. He decides that the arc of the curve with parametric equations

from

to

makes an aesthetically pleasing shape.

a Show that the length of is . The vase is formed by rotating about the -axis. b Show that the area of the vase is

.

c Show that the exact value of this area is:

10

The tip of a bullet is made by rotating the arc of the curve with parametric equations from

to

.

a Show that the surface area of the tip of the bullet is given by:

b Hence find the exact value of the area of .

11

The Gamma function is defined by a Show that b Prove that



.

, hence find a reduction formula for . !

c Prove that 12

.

Gabriel’s horn is a shape found by rotating the graph of

for values of

.

Calculate the volume of paint needed to fill the horn but show that you will never have enough paint to cover its surface.

CROSS-TOPIC REVIEW EXERCISE 2 1

Find the volume of the tetrahedron .

2

A parallelepiped has one face

where

and

. The face opposite this is labelled

and

as shown.

.

a Find the coordinates of the vertices and . b Calculate the volume of the parallelepiped. 3

Given that

4

A surface has equation

and

, show that:

, where

.

a On separate diagrams sketch the sections i b Draw the contours for

and ii

for taking values

c Prove that has a minimum turning point at 5

Given that

6

A surface has equation

.

, show that

and

.

.

. , where

.

a Find and . b Show that has a stationary point at minimum or saddle point.

and determine whether it is a maximum,

c The section of , where is a constant, has exactly one stationary point. Find the value of and hence write down the equation of this section. 7

It is given that



.

a Prove that the reduction formula is b Find the exact value of c Show that if [You may assume that as

.

. , then

. .]

8

It is given that



a Find

.

.

b Obtain a reduction formula for

.

c Find the exact value of

9

A curve C has equation

.

, for

to

.

a Show that the length of the curve is given by: . b Use the substitution

to find the exact value of this integral, expressing your

answer in logarithmic form. c Show that the area of the surface generated when is rotated about the -axis is . 10

A curve has parametric equations

.

a Show that the length of , from

to

, can be written as:

and find

the exact value of the length of the curve. b Show that the area of the curved surface formed when , from about the -axis is

.

to

, is rotated

PRACTICE PAPER Time allowed: 1 hour 30 minutes. The total number of marks is

.

The marks for each question are shown in brackets [ ]. You are reminded of the need for clear presentation in your answers. 1

The sequence

is defined by

a Show that

, with

. [4]

.

b Prove by induction that

[4]

.

c To what value does the sequence converge? 2

[2]

Solve the second-order recurrence equations: [8]

3

a Given that

, find the value of and write the number in base

.

b Solve the simultaneous linear congruences: 4

A cyclic group has generator where

[4] .

[6]

, being the identity element.

a i Draw a group table for .

[2]

ii State the order of each element of . is a group with elements

[2]

.

b i Draw a group table for multiplication of the elements of .

[2]

ii State the order of each element of .

[1]

State whether and are isomorphic, justifying your answer. 5

The points

,

and

[2]

are three vertices on the parallelogram

.

a Find the coordinates of the point . b Using the vector product of

[2]

and

c Explain why a straight line

, find the area of the parallelogram.

[3]

can be written in the form: [2]

d Express the equation of the line that passes through and in the form this to show that lies on the line. 6

Find the equation of the tangent to the plane

7

It is given that

and

. [6]

.

a Prove that, for b Find

at the point where

and use [3]

. .

[4] [3]

c By making a suitable substitution, show that 8

A cycloid has parametric equations a Show that the length of one cycle, that is, from

[4]

. . to

, is:

Hence find the length of one cycle. b Show that the surface area generated when this cycle is rotated about the -axis is

[5] .

[6] [75]



FORMULAE

AS and A Level formulae The following formulae for Additional Pure Mathematics will be given on the AS and A Level assessment papers. Vector product , where a, b, in that order form a right-handed triple.

A Level-only formulae The following formulae for Additional Pure Mathematics will be given on the A Level assessment paper only. Surfaces For 3-D surfaces given in the form

, the Hessian Matrix is given by

.

At a stationary point of the surface: 1 if

and

, there is a (local) minimum;

2 if

and

, there is a (local) maximum;

3 if

there is a saddle point;

4 if

then the nature of the stationary point cannot be determined by this test.

The equation of a tangent plane to the curve at a given point . Calculus

Arc length

Surface area of revolution

     

  



is

Answers Chapter 1 Before you start… 1 a b 2 a

b c

3 true Assume true for

Add next term

true for Hence true for all integer values . Exercise 1A 1 a b c d 2 a b c d

3 a b c d 4 a b c d 5 a b c d e Golden ratio 6

i.e. Fibonacci numbers

7 a b i Proof ii Proof iii Proof 8 a b Oscillating (or periodic) c Divergent d Convergent 9

Periodic

10 a

, fixed point

b c i

Diverges

ii 11 a b c d Fixed point of

Diverges

12 a b c d 13 Proof. 14 Proof. 15 Proof. 16 a b c Proof 17 Proof 18 a Proof b c Proof 19 a

, proof

b c Proof 20 a

, proof

b c Proof Exercise 1B 1 a b c d 2 a b c

d

3 a

b

c

d

4 a

b

c

d

5 a

b

c

d

6 a

b

c

d

7

million

Proof 8 a Population tends to 60. b Population becomes stable at 9

with (Mersenne numbers) Proof

10 a

b

c

d

11 a

b

c

d

12 a

b

c

d

13 a

b

.

c

d

14 a Proof, b Proof 15 a b Each value is the sum if the previous two terms. c 16 a At each hour, number previous number doubled plus more, i.e. another . b c No, number of caddis flies increases indefinitely. 17 a Proof b c d The population of foxes decreases.

18 19 Proof 20 with Proof Mixed practice 1 1 a Proof b Proof c Proof d Proof 2 a Proof b Proof c Proof 3 a Proof b Proof

and

plus the number from the hour before that

c

is always even If is odd: If is even:

is odd, so is even so

4 a b Proof 5 a b c d 6 a Proof b Proof c 7 8 9 a b 10 Proof

games 11 Proof 12 a i diverges ii converges to b i diverges ii oscillates between c i diverges ii oscillates between d i diverges ii chaotic

and

is odd. is even.

Answers Chapter 2 Before you start… 1 a b 2 3 4 Exercise 2A 1 a b c d 2 a b c d 3 a b c d 4 a b c d 5 a b c d 6 a b c

d 7 a b c 8 a b c 9 a b c 10 a b c 11 a b c 12 a b c 13 a b c 14 a b c 15 a b c Proof, d 16 a b c d

17 a Proof b Proof c Proof d Proof Proof 18 a Proof b Proof c Proof d Proof Proof 19 a Proof b Proof Proof 20 a Proof b Proof c Proof d Proof Exercise 2B 1 a Yes b No c Yes d No e Yes 2 Proof 3 a Neither b Neither c

and

d Neither 4 Proof 5 a Yes b Yes c No,

and

d Yes,

and

6 Proof;

then then

. .

7

or

or

8 Proof 9 a b c d 10 Note that 11 has same residue as , i.e. 12 First five powers of have residues

which then repeat.

has same residue as , i.e. 13

answer has form Answer

14 Proof 15 Proof 16 Proof 17 Proof 18 Proof 19 Proof 20 a Hence b Set up the iteration

(Multiple of

)

with

using part a

Hence Exercise 2C 1 a b 2 3 4 5 6 7 8 9

solutions

10

solutions

11 a b 12 a b 13 a

is not a multiple of

b

and is not a multiple of

14 a b 15 a b c d 16 Proof 17 a b

18 Proof

and and

19 a res b res res

; non-res ; non-res ; non-res

20 Proof Exercise 2D 1

2 Proof , hence it is a composite number with factors

and

3 4

composite

5 Proof, , Euclid’s lemma 6 Proof 7 a



b Proof 8 a Proof b Proof c Proof d Proof 9 a Proof b

which gives

, for integer

c Since 10 a b c d 11 a

has no integer solutions,

has no integer values.

.

b c d 12 , no, pseudo-prime; proof 13 a b c d 14 a



b



. Proof

c



. Proof

d

. Proof



. Proof

15 Proof.

. Proof

16 , i.e. Friday 17 18 a If

then

Repeat the process for starting assumption b If

then, by part a,

c

Hence 19 20 Proof Mixed practice 2 1 a b Proof. c d Proof

, etc. until

, i.e.

.

2 Proof 3 Proof 4 5

or

6 7

such that

8 a

, proof.

is not a multiple of

b 9 a Proof b

modulo 7

c

, where

10 Proof 11 a i Proof but

, so

ii Proof b i Proof.

and

ii

and and

iii

by Fermat’s little theorem, so and is even. and

is a multiple of .

Clearly, of

and

A check on a calculator shows that

is an even multiple

.

12 a Proof b

by Fermat’s little theorem, hence

c If is an odd multiple of , then for example if d Proof

since

,

Answers Chapter 3 Before you start… 1 2 3 4 5 6 7 8 9 10

11

12

13

14

15

16

17

18

19

20 is a rotation of

about the origin, so

Exercise 3A 1 a b

Not commutative Not associative

c Closed 2 a Proof b

. Proof

c Not associative 3 a

. Proof

b

. Proof

c d e Closed 4 a

. Proof

b

. Proof

c d e Closed 5 a Proof . Proof

b c Proof 6 a Proof b

c Not closed 7 a

. Proof

b Proof c d 8 a Proof b Proof c Proof d Proof 9 a Proof

.

b Proof c Proof d Proof , commutative

10 a b

Associative c

if identity element is

11 a closed under addition since

and

closed under multiplication since b

are integers.

and

are integers.

if identity under addition is if

and

identity under multiplication is c inverse of

is

which is only an integer when

or

Hence, not every element of S has an inverse. 12 a b Proof. Not c Proof. 13 a Commutative, not associative b

provided Identity under addition is

c Inverse is 14 a Proof b

are integers.

c 15 a Proof b

provided

16 a Proof b Proof c

No inverses

17

a Proof b Symmetry c d

element inverse

18

a Proof b Symmetry c d

element inverse

19 a

b Symmetry element c inverse d e

mod

is not in subset

20 a b Exercise 3B 1 a

b Closed c Proof element

d

inverse e Group order element order f

Satisfies closed, associative, identity, inverses

g Commutative 2

a Not a Latin square, as some elements appear more than once in a row or a column. b No c Proof d

element inverse

e

or

f

Group order element order

g 3

does not form a group

a Yes b Proof c

element inverse

d e Group order element order f

forms a group

4 a

b c d

element order

e Commutative and associative f

Closed, associative, identity, inverses

5 a

b Commutative c d Closed e Not associative f 6

Not a group

The table is a Latin square as no element appears more than once in each row or column. Proof 7 Proof 8 Proof 9

reflection in major axis reflection in minor axis rotation

about centre

identity Proof 10 For triangle rotation

about centre

rotation

about centre

reflection in line of symmetry through reflection in line of symmetry through reflection in line of symmetry through identity Proof 11 Proof 12 Proof 13 Proof 14 Proof 15 a Proof b Proof 16 a

b Proof

17 Proof

18 a

b

c d Order e

element inverse order

19 a i Proof ii For any real numbers iii Proof b Proof 20 a Proof b Proof c

, no unique identity, not a group.

Exercise 3C 1 a

b Proof c

element order

2 a Proof b

element order

c No, no element has order . d 3

: commutative

a Yes b Identity element inverse order c Not cyclic, no element has order . d 4 a

b

element inverse order

c Abelian group d 5

a Proof b Closed, associative, identity , each element has inverse, commutative: Abelian group 6 Proof 7 a i

ii

element order

iii Closed, associative, commutative, identity , each element has inverse: forms an Abelian Klein group. b

i

element order

ii Closed, associative, commutative, identity , each element has inverse: forms an Abelian Klein group. c

e.g.

8 a

b

element order

c Proof d

e

element order

f

Proof

g with 9 a Proof b Proof c Proof 10

a Proof b

, Proof , Proof

11 a i

. Proof

ii

iii

element order

iv Closed, associative, identity , each element has inverse: forms a group. b

i

element order

ii Closed, associative, identity , each element has inverse: forms a group. c Yes, where

12 a Proof b so c

element order

is order

d e f

has order . is not a factor of

13 a

b

element order inverse

c Closed, associative, identity , each element has inverse: is a group. d

is a group of rotations of radians. has a generator

.

is a cyclic group. e

is a cyclic group with generator or . Hence and are isomorphic.

14 a b Proof c d

, , ,

e

and are not isomorphic since the orders of the elements are different.

15 a i ii b

order ,

c Order d 16 a Proof b Proof c 17 a Proof

order

b

element order

c Proof d e.g. Not commutative. 18 a b

is order , no element has order .

c

, so closed.

d e

is not commutative, is. Not isomorphic.

19 a

element order

b

(order ) order of elements number of elements (order ) order of elements number of elements Only non-cyclic subgroups of have order and (order must be non-prime), i.e. and . But and have elements with order . has only one element with order . So no non-cyclic subgroups of .

20 a Proof b Proof c Proof Mixed practice 3 1 a Proof b identity c Proof d closed, associative, has identity, each element has inverse is a group; 2 a Proof b Proof c d

has no inverse so not a group.

3 a

b Proof c

element order

d cyclic, elements order generator

or

4 a Proof b

element order

c 5 a

b 6 a Proof b

element order

c 7 a Proof b c

element order

d e Cyclic, with generator 8 a commutative

b c order d

is not a factor of or

are order

Not isomorphic 9 a

b group has order element order c d

and

10 a Proof b Proof 11

Proof 12 a Proof b

since is non-singular

c Proof

Answers Cross-topic review exercise 1 1 2 a b proof

3 a b

c Proof d Proof 4 5

, proof

6 a Proof b 7 a Proof b 8 a b i

is not a multiple of

ii 9 a

b c

element inverse element order

d

is cyclic, generator or

e

and both cyclic order .

Hence isomorphic. 10 a Proof b Proof c

is not cyclic. is cyclic, generator . and are not isomorphic.

Answers Chapter 4 Before you start… 1

2

3

∴ perpendicular vector is

(or use the vector product

Vector equation is:

4

Exercise 4A 1 a b c d e f 2 a b c d e f

which is parallel to

).

3 a b c d e f 4 a b c d e f 5 a b c d e f 6 a b c d e f 7 a i ii No b i ii Yes c Proof d Proof 8

9 a b c d

10 a

b

c

d

11 a b 12 a b c d 13 a b c d 14 a b c d 15 a b c d 16 Proof 17

, not coplanar

18 a

,

,

b 19 a b 20 a b c d Mixed practice 4 1 2 3 a b Proof

4 a b 5 6 a Proof b 7 a b 8 a b

9 a b

is parallel to

10 a b

and are coplanar.

,

11 a

b

12 a Let the sides have vectors Area

b Proof

and .

Answers Chapter 5 Before you start… 1 2 3 4 5 6 7 8 9

10 11 Exercise 5A 1 a i

ii

b

2 a i

ii

b

3 a

b

c

4 a

b

5 a

b

6 a Yes b

so above the surface.

c

7 a i

ii

b e.g. c

8 a i

ii

b

9 a i

ii

b

10 a i

ii

b

i.e. intersect at c

11 a i

ii

b

12 a

b 13 a i

ii

b

14 a i

ii

b

15 a

b c

16 a

b c 17 a i

ii

b

c

18 a i

ii

b

This is an ellipsoid:

19 a i

ii

b

This is an elliptic cone:

20 a i

ii

b

hyperboloid of one sheet 21 a i

ii

b

hyperboloid of two sheets Exercise 5B 1 a b c d Proof 2 a b c d Proof 3 a b c d Proof 4 a b

c d , proof 5 a b c d Proof 6 a b c d Proof 7 a b c d Proof 8 a b c d Proof 9 a b c d Proof 10 a b

c d Proof 11 a b c d Proof 12 a b c d Proof 13 a b c d Proof 14 a b c d Proof 15 a b c d Proof 16 17 Proof 18 Proof 19 Proof

. Proof

20 a b c d Exercise 5C 1

proof

2

max

3 Proof 4

min,

5

saddle

max,

min,

saddle

6 Proof min,

min,

saddle 7 Proof saddle,

min

8 a

max,

min

b

and

saddle

9 Proof and

saddle

10 Proof max, min, saddle 11

, proof

12 Proof 13 a Proof b



c

14 Proof max, 15

saddle saddle saddle

16 Proof 17 Proof max,

min,

saddle

18 a Proof b 19 Proof. 20 Proof Exercise 5D 1 Proof 2 3 4 Proof

5 Proof 6 7 8 9 10 11 Proof 12

saddle, min if

min, , saddle if

max,

saddle

13 Proof 14 15 16 Proof 17 Proof 18 Yes, parallel to 19 20 a Proof b

c

Mixed practice 5

1 a i

ii

b

2 a Proof b Proof 3 4 Proof 5 Proof 6 7 a Proof b

, maximum since

c

8 a

min,

min

b

9 a b Proof 10 11

, proof

Answers Chapter 6 Before you start…

1

2 3 4 5 6 7 8 9 10 11 12

or 13

Exercise 6A 1 a Proof

b

2 a b Proof c Proof 3 Proof a 4 5 6 a b 7 Proof a 8 Proof a 9 Proof a 10 Proof a 11 Proof a b

12 a b

13 a b Proof c i

ii 14 Proof a b 15 Proof

16 a b Proof c 17 a b Proof c 18 a Proof b Proof c 19 Proof 20 a Proof b Proof c Exercise 6B 1 a b c d 2 a b c d 3 a

b c d 4 a b c d 5 a Proof b Proof 6 a

b Proof 7 Proof 8 Proof 9 a Proof b c Proof 10 a Proof b c Proof 11 a Proof b Proof 12 a Proof b Proof c 13 a Proof b Proof 14 Proof 15 Proof

16 Proof 17 Proof 18 a

b Proof 19 Proof 20 a

(circle)

b c Proof Mixed practice 6 1 a Proof b Proof c 2 a Proof b Proof 3 a Proof b Proof 4 a Proof b Proof

5 a b Proof 6 Proof 7 8 a Proof

b Proof 9 a Proof b Proof c Proof 10 a Proof b 11 a Proof

b Proof c Proof

12 Proof

Answers Cross-topic review exercise 2 1 2 a b 3 Proof 4 a i

ii

b

c Proof 5 Proof 6 a b Proof, saddle point c

7 a Proof b c Proof 8 a b c 9 a Proof b c Proof 10 a Proof, b Proof

Answers Practice paper 1 a Proof b Proof c 2 3 a b 4 a i

ii

element order

b i







ii





element order

and are isomorphic: orders of elements correspond, e.g. 5 a b c d Proof 6 7 a Proof b

c Proof 8 a Proof, b Proof

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