JC Success - Maths 1 - Website Sample

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Steps to Success SAMPLE

Introduction 03

Maths Exam Language 09

Topic 1 | Working with Numbers 11

Topic 2 | Working with Letters 29

Topic 3 | Working with Geometry & Constructions 45

Topic 4 | Working with Triangles 87

Topic 5 | Working with Formulae 113

Topic 6 | Working with Money & Arithmetic 141

Meet the authors

Brendan O’Sullivan

Brendan O’Sullivan teaches at Davis College, Mallow, Co. Cork. He is an experienced examiner, creator of mock papers and designer of professional development. A regular contributor at educational conferences, he has a special interest in mathematics textbooks, having conducted one of the largest analyses of textbook tasks nationally and internationally as part of his PhD at DCU.

Eoghan O’Leary

Throughout the book, QR codes are used. Once scanned, these will allow you access to our bank of online resources to further assist you in your exam preparation.

You will also encounter the Key to Success information boxes. These boxes contain expert analysis, explanations, and hints and tips from the authors that will help you maximise your grade in the exam.

Ciarán Duffy

Since 2018, Ciaran has been Deputy Principal of St Kevin's Community College, Clondalkin. He has lectured Junior Cycle mathematics methodologies to trainee teachers in DCU, has corrected all levels of mathematics papers with the SEC since 2002 and has written numerous mock papers and marking schemes. Since 2012, Ciarán has served on the committee of the Dublin Branch of the Irish Mathematics Teachers' Association and has been national chairperson of the IMTA since 2019.

Eoghan is an Assistant Principal and teacher of Mathematics and Economics. Eoghan was shortlisted for the Award for Irish Teachers of Mathematics in both 2017/18 and 2018/19. He is a former Chairperson of the Cork Maths Teachers Association. Eoghan is also the Founder of and Director of Education at The Tuition Centre, one of the most popular and best-regarded sources of online tuition for secondary school students in Ireland.

New Junior Cycle Grading System

The following are the general instructions found on the exam paper: “There are [x] questions on this examination paper. Answer all questions. Questions do not necessarily carry equal marks. To help you manage your time during this examination, a maximum time for each question is suggested. If you remain within these times you should have about 10 minutes left to review your work.”

Time Allocation: We recommend that you follow the timings as given on the exam paper. It’s a good idea to jot down, on the exam paper, your planned end time for each question as you go along. We strongly advise you to practise past papers and sample papers of this type to get used to this kind of time management. Try doing a full paper under exam conditions and with a time limit of 2 hours, and see how that goes for you. It is essential that you answer all questions, so make sure you do not run out of time.

Answer all questions.

Always show your work. Some questions will not require this. These questions usually say ‘write down…’; otherwise, supporting work is always needed.

One Paper (2 hours) – 270 marks

Put everything on the exam booklet. While you are entitled to as much extra paper as you like, we recommend that you use the exam booklet itself to do all of your work. There is no such thing as ‘rough work’ – all work is part of your solution.

is the front cover of your exam. You fill in your exam number and the day and month of your birth ONLY. Never write your name or school on ANYTHING.

Understanding How the Marking Process Works

Your examiner must read everything you do. If you try a question two different ways, your examiner will have to correct both and give you the higher mark of the two attempts. So, don’t scribble things out, don’t use Tippex, don’t use pencil (so you can’t rub things out).

Consider the example below.

(a) Jane buys a laptop online for $699, plus a shipping cost of $30 The exchange rate is $1 = €0.90 Work out in euro the total cost to Jane of buying the laptop online.

7 10 4 10 10 10

699 + 30 = 729 - 0.9 = €810

699 x 0.9 = €629.10

699 + 30 = 729 x 0.9 = €656.10

If all three attempts are present in your answer box, then you will get the marks for the best solution. This does not apply, however, when asked a question with only two choices as shown below.

The ratio of students to teachers in Liam's school was 15:1. The school hired one extra teacher, while the number of students stayed the same. The new ratio of students to teachers was a:1 where a ∈ Q.

(c) Put a tick ( ) in the correct box to show shich statement is true. Tick one box only. Give an example to support your answer.

a > 15 a = 15 a < 15

Example:

If you tick more than one of the boxes then you will get 0 marks. If you make a mistake, just make it clear to your examiner which one is your answer. Maybe put a circle around it and say ‘this is my answer’ beside it and/or cross out the wrong answer. This is the only time you should ever scribble out work. Otherwise, put a light line through your work and do it again on the side. There should be plenty of room.

Mark Allocation: Part a) is usually easier than part b), which in turn is usually easier than part c), etc. However, there is no fixed allocation of marks across parts a), b), c), etc. In fact,

Other top tips

Diagrams: Drawing a diagram can be really helpful to visualise something that is difficult. You don’t need to be told to draw one, but doing so can make your life easier. If there is a diagram given in the question, it can be useful to draw in any known measurements or angles, particularly 90° angles. Be careful not to assume something is a right angle.

Right angles: Students often mistake angles as being right angled or 90° when they are not. You must be told it is a right angle or given some additional information to deduce that it is. This kind of information includes:

> Perpendicular lines

> Angle of elevation

> Angle in a triangle in a semi-circle (must go through the centre point to be a semi-circle).

> Real-life scenarios in trigonometry such as walls, trees, etc., which we are allowed to assume are perpendicular to the ground.

> If a triangle’s measurements satisfy a2+b2=c2, then the triangle must be right-angled. It is a major error to assume a triangle is right-

many questions have more than three parts. Don’t assume that the later parts of the question will carry the highest marks. Marks may possibly be allocated more heavily towards the earlier parts of some questions. However, this is not decided until the final marking schemes are approved by the chief examiner, long after the exams are over. You simply can’t make any assumptions about where the marks will be allocated! So the recommended approach is to take care to pick up all marks on the easy stuff, and if something seems difficult, at least have a go at it. “Attempt” marks can contribute a lot to your grade.

angled, so be very careful.

Blank boxes: No matter what the question is, write something down. If you have no idea how to do it, look at the absolute basic principles of the question and see what can be done.

Examples include:

> If you see brackets, multiply at least two things together.

> Label a point as (x,y) or (x1,y1) and (x2,y2) if given two points.

> Draw a right-angled symbol on a diagram.

> Write down a formula and change one letter to a given value.

> Write down an answer from a previous part which might be useful in the new box.

> Is a percentage given? Multiply AND divide by any other number in the question.

> Is there a decimal or fraction or percentage mentioned? Change it to a different form using your calculator e.g. 3 4 = 75% = 0.75

> Can you identify a radius or a diagram or half a diameter to get a radius?

> In probability if a fraction is given, take it away from 1 on your calculator.

> Can you do something with the opposite of the situation given?

> Indicate two equal sides and/or angles in a triangle if the word Isosceles is mentioned.

> If you see a horrible-looking expression and a random number elsewhere, just sub the number in. Somewhere, anywhere!

Read the questions carefully: Take in as much of the information given and crosscheck with any diagrams to see if it backs up your understanding of the problem posed. If not, allow the diagram to correct your misunderstandings and proceed from there. Every word on the paper is there for a reason – use them all wisely. At the same time, if the examiner needed to tell you something that you think is missing, then maybe your thought was incorrect. Do not make assumptions (*except in real-life right-angled situations mentioned earlier).

Does your answer make sense? Are you working on a probability question and get an answer bigger than 1? This is not possible, so you must have done something wrong. Or, did you work out a rate of tax to be 75% or an hourly rate of pay to be €1.36 or €235,057? Do the three angles in your triangle add to more than 180° or does the height of a person work out to be 5 metres tall? Check that everything makes sense and have an idea what the answer should/could be or not be. Knowledge and skills that you have acquired outside the classroom are valid and to be used where possible.

It is worth looking at marking schemes from previous years in order to gain an understanding of what marks tend to be awarded for. The process won’t change that much in the future.

That’s what the calculator said: Calculators are operated by humans! If your monthly tax bill is €20 it is likely you’ve forgotten a zero or two somewhere along the way. Be familiar with your own calculator and how it works. Practise all of your classwork and homework with the same calculator where possible. Do you know how to use your calculator to do the following?

> Find prime factors

> Use table mode for drawing graphs

> Change fractions to percentages to decimals

> Work with time (hours and minutes)

> Use Sin, Cos and Tan for ratios and their inverses for angles

> Square root and squaring

> Cubed root and cubing

> Mixed fractions

> Change angle mode from Degrees to Radians (this is only important if you share your calculator with a student in 5th or 6th year). Radians mode is not a concept on the JC course but it is on the LC HL course and could mess up all of your trigonometry answers if you don’t know how to change it.

> Know how to get π on your calculator.

Show work: Correct answers without work usually earn high partial credit (approx. 70% of the marks). Any calculation or step shown before the answer will count as showing work. If a question asks you to label a point or use your graph to get an answer, then you must do so for full marks.

Formula & Tables book: Be familiar with all of the relevant pages in the

tables book. Reading the contents page can be very helpful. It is not a good idea to use a newly discovered formula on the day of an exam that you have never used in class before. There is probably a reason you have never seen it before!

Order of questions: Do questions in any order you like but as you have to do them all there’s not a huge advantage to moving back and forth. Some people like to start with questions they like to settle themselves into an exam; that choice is up to you. There is a danger that you skip some pages and miss questions by moving back and forth. The choice is yours!

I’ve never seen this before: You will come across questions that look unusual and nothing like questions you’ve practised in class. This is the same for everyone doing the exam, so take it on the chin and do your best. Apply all of your mathematical knowledge and skills to this unforeseen question and do your best. Most mathematicians don’t know if the method they have chosen to start a question is the correct one. This only becomes apparent after you’ve started. So, start the question and see what happens. If you get inspired half way through to do it differently, then do so. If you run into a brick wall, change your starting position or your tactics and see if something new emerges.

Units: Ensure you are working in the same units when solving some problems involving time, distance, speed, interest, money, etc. If two different units are given in a question, it is a good idea to check what form the answer should be in. Then change the other unit to this. E.g., if we are told the length of a rectangle is 25 cm and the width is 43 mm and we want to know the area in cm2, then we change the mm into cm. Of course, it can be done the other way, but this involves two conversions, is more time consuming and way more prone to error. It is also important to include the correct units as part of your final answer. You can lose 1

mark every time you leave them out. This doesn’t sound like a lot, but they add up and why do it to yourself after doing all the hard work so well?

But I got the right answer: Maths isn’t about answers, weirdly enough. It is all about method and process. It is possible to get the correct answer doing something completely illegal, mathematically speaking!

Example 1: Simplify 26 65

The correct answer is 2 5 after you divide top and bottom by 13 However, you get the same answer if you cross out the 6 on top and bottom, which of course is extremely hurtful to all of the laws of mathematics and would result in 0 marks in an exam.

Example 2: Evaluate (25)(92)

If the student says (25)(92) = 2592 instead of saying (25)(92)=32×81=2592, they would be correct but they’ve totally ignored the powers. Coincidentally it is the correct answer.

Example 3: Simplify x2-y2 x-y

If a student does this by "cancelling" the x's and the y's on top and bottom, to get: x-y -

and then concludes that "two negatives make a positive", so the final answer has to be x+y, this gives a correct answer but uses really offensive mathematics!

Stop focusing on the answer. Your method and work are the most important thing in an exam. The best way to approach this is to practise showing work as often as you can in all maths problems you work on.

But I don't know how to study maths! The best way to study maths is to do maths. Find questions, even ones you've done before, and practise, practise, practise!

MATHS EXAM LANGUAGE

Give an account of the similarities and (or) differences between two (or more) items or situations, referring to both (all) of them throughout

Using only the prescribed tools of compass, set square, ruler, pencil and/or protractor to accurately draw properties of shapes and geometric results

Change from one form to another

State or calculate a rough value for a particular quantity

Give a reasoned account, showing how causes lead to outcomes

Judge the relative quality or validity of something, which may include analysing, comparing and contrasting, criticising, defending or judging. When working with numbers it can also simply mean 'find the value of'.

Use the answer from a previous part to solve this part. If you don’t, you can still get high marks, just not full marks.

We suggest you use the answer from the last part to solve this part, but if you can find a different way then that’s ok for full marks.

Use knowledge and understanding to explain the meaning of something in context.

Give valid reasons or evidence to support an answer or conclusion.

Give a deductive argument to demonstrate that a particular statement is true, including reasons for each step in the argument.

Give the number in the required form (for example, a multiple of 10, or a number with two significant figures) that is closest in absolute terms to a particular number.

Any correct method can be used to show that two things are the same (except measuring something with a ruler unless specified).

Draw a rough diagram or graph without using geometrical tools.

Provide a concise statement with little or no supporting argument.

To get full marks you must show the examiner that you have used the graph given or that you have drawn. This is best done by drawing feint lines on the graph and marking clearly any points of intersection of your line and the graph.

Little or no supporting work is required.

Demonstrate that a statement is true.

WORKING WITH LETTERS SAMPLE

What you need to remember

x + y = ?

Types of Questions

Simplifying – often, this is when you remove brackets and add like terms to each other.

Expand and simplify –(2x + 3) (x + 4) = 2x2 + 8x + 3x + 12 = 2x2 + 11x + 12

Factorising – this is when you break the expression into terms that can be multiplied together to give the original expression. It involves taking out common factors. 2x + 6 = 2(x + 3) 9a2 − 16b2 = (3a − 4)(3a + 4) pq pr + 3q −3r = (p + 3)(q r) x2 + 5x + 6 = (x + 2)(x + 3)

Squaring – means multiplied by itself. (x + 3)2 = (x + 3) (x + 3) = x2 + 3x + 3x + 9 = x2 + 6x + 9

Evaluating – this is when you substitute a given value for a letter. x 2 x 3 when x = 1 (1) 2 (1) 3 = 1 6

More complicated questions involving letters

Algebraic Fractions

Adding two fractions together y 4 + y 5 = 9y 20

Solving an equation involving fractions x + 1 4 + 2x 3 11 = 3

Dividing above and below x2 16 x2 + 5x + 4 = (x 4) (x + 4) (x + 1) (x + 4) = (x − 4) (x + 4) (x + 1) (x + 4) = (x − 4) (x + 1)

Term on the bottom divides into the term on top once so we cross them out.

Long Division Divide x3 + 6x2 + 11x + 6 by x + 2 x + 2 x2 + 4x + 3 x3 + 6x2 + 11x + 6 x3 + 2x2 4x2 + 11x + 6 4x2 + 8x 3x + 6 3x + 6 0

Letters can occur in different contexts like perimeter/area questions. Write down an expression in x for the perimeter and area of this rectangle. It follows the same rules as if you were working with exact values but often people find it confusing once terms involving letters are introduced.

2x2 + 7x 15 x + 5

Simultaneous Equations

You can only work with one letter at a time when it comes to solving equations. Two sets of equations with two different letters are known as simultaneous equations. It is necessary to eliminate one and then solve for the other before subbing back in to end up with both values. 2x + y = 5 x + 3y = 10

We will look at solutions to each of these types of questions throughout this chapter.

Worked Example 1

Multiply out and simplify (2x 3) (4 − 5x + x2)

We can also use the array method as follows for the same answer.

When tidying up, like goes with like. This means that we total the x3 , x2 , x and constant terms. 2x3 and −12 are not like any other term so remain unchanged.

Terms can be presented in any order — it is the signs and powers that you need to be careful with.

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Worked Example 2

(a) Multiply out and simplify (x + 5)2

(b) Hence, or otherwise, show that the following expression is always divisible by 4 (x + 5)2 − (x − 5)2

Solution

(a) Write out the expression as (x + 5) (x + 5) first.

Break up the first bracket into two parts x and 5 then multiply them by the second bracket as shown below.

= x(x + 5) + 5(x + 5) = x2 + 5x + 5x + 25 = x2 + 10x + 25

(b) We focus on the second part first as we have the first part done in part (a). (x − 5)2 = (x − 5) (x − 5) = x(x − 5) − 5 (x − 5) = x2 − 5x − 5x + 25 = x2 − 10x + 25 (x + 5)2 (x 5)2 = x2 + 10x + 25 − (x2 − 10x + 25) = x2 + 10x + 25 − x2 + 10x − 25 = 20x = 4(5x) 20x is divisible by 4 so (x + 5)2 − (x − 5)2 is always divisible by 4.

Scan for Video − 5x − 5x causes huge issues in exams. Work on this.

A minus in front of a bracket changes all the signs. It’s like multiplying by an unseen −1

Be careful to include 1. This causes errors.

We take out 4x2 because it is the highest common factor of both terms. We have to get identical brackets in the second line if we are to have a common factor.

Factorise p2q − 2rs − pr + 2pqs

As it is written at the moment, we cannot factorise the first two terms. Change the order so that the first two terms have a common factor. Remember that we are seeking identical brackets.

p2q − 2rs − pr + 2pqs

= p2q − pr + 2pqs − 2rs = p( pq − r) + 2s( pq − r) = ( p + 2s) ( pq r)

Factorising a quadratic

Factorise 11x2 + 75x − 14

First Method: The Guide Number method

Multiply the first coefficient by the last (11 times −14 = −154) List the factors of −154 until you find a pair whose sum is equal to the middle coefficient 75 154 × −1, 77 × −2, 22 × −7, 11 × −14 We choose the factors 77 and −2 as 77 − 2 = 75. We now rewrite the quadratic as: 11x2 + 77x − 2x − 14

= 11x(x + 7) − 2(x + 7) = (11x − 2) (x + 7)

Second Method: Trial and Improvement

11x2 + 75x − 14

Begin by factorising the first term 11x2 . This gives (11x)(x) Now consider the last term 14 and go through the factors (1, 14) or (2, 7) It’s pretty clear that 1 and 14 are too far apart to give the sum of 75x So, concentrate on combinations of 2 and 7 We can test the effectiveness by multiplying the two farthest away terms and the two nearest. Test the following (i) (11x + 7) (x −2) (ii) (11x + 2) (x − 7) (iii) (11x − 2) (x + 7)

(i) 11x(−2) + 7(x) = −22x + 7x = −15x This doesn’t work (ii) 11x(−7) + 2(x) = −77x + 2x = −75x This doesn’t work (iii) 11x(7) − 2(x) = 77x − 2x = 75x This works

We use (11x − 2) (x + 7)

A quadratic is any expression where x2 is the highest power.

The more you practise these the less you have to guess, but guessing is fine!

(a)

Worked Example 3

Factorise 5x 15 and 6 2x

If A and B are variable quantities, we say that A is proportional to B if A B is a constant.

(b) Using your answers to part (a) above, show that 5x 15 is proportional to 6 2x

Solution

(a) For this part, we’re taking out the common factors.

5x − 15 = 5(x − 3)

6 − 2x = −2x + 6 = −2(x − 3)

It’s useful to write the terms in a similar order (letters first, constants second) before factorising.

(b) We focus on the second part first as we have the first part done in part (a).

5x − 15 6 − 2x = 5(x − 3) −2(x − 3)

5(x − 3) −2(x − 3) = −5 2

This is a constant, so it is proportional.

A variable is a symbol (usually a letter) in the place of an unknown numerical value.

Worked Example 4

Factorise fully 5fh 2h2 6h + 15f.

Solution

This type of question is known as a ‘group of four’ or ‘factors by grouping’. We attempt to take out what is common from the first pair, then the second. If the first pair have nothing in common, we arrange them to get something in common.

5fh − 2h2 − 6h + 15f h(5f 2h) + 3(− 2h + 5f ) (h + 3) (5f − 2h)

The order does not make a difference once the signs are correct for each term.

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Partially Worked Example 5

This is a question that tests a lot of skills. It is important to practise it. Can you complete the following?

Factorise fully 9a2 − 6ab + 12ac − 8bc Scan

Solution

Worked Example 6

Solution

Factorise

This is of the form a2 − b2 = (a − b)(a + b) and is known as "the difference of two squares".

What must be squared to get 9x2 ? → 3x

What must be squared to get 16y2 ? → 4y

9x2 − 16y2 = (3x − 4y) (3x + 4y)

The key here is that both terms involve known squares; we would not use this method with 3x2 − 16x, which would be x(3x − 16).

Students often fail to recognise this question in the exams. Don’t fall into the same trap! Keep revising so it’s in your head!

Worked Example 7

Factorise 8x2 + 45x − 18

Solution

We answer this using the guide number method. Multiply the first and last numbers together.

List the factors of −144; we must bear in mind that we are looking for a factor pair that will sum to the middle value of +45 For this, the bigger factor must be positive.

We choose −3 and 48 as they sum to 45. We replace 45x with −3x and 48x. Now it is a group of four.

(8)(−18) = −144

Factors of −144

−1 × 144, −2 × 72, −3 × 48, −4 × 36 −6 × 24, −8 × 18, −9 × 16, −12 × 12

8x2 − 3x + 48x − 18 x(8x − 3) + 6(8x − 3) (x + 6)(8x − 3)

Note the difference between solving a quadratic equation and factorising a quadratic expression. You have the option to use the ‘-b formula’ when it comes to solving the quadratic equation, but you really need to know how to factorise in order to answer the kind of question asked here.

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Worked Example 8

Use factorisation to simplify 4e2 − 9 2e2 + 3e − 9

Solution

Start off with the top line 4e2 − 9 (2e − 3) (2e + 3)

Now the bottom line

2e2 + 3e − 9 (2)(−9) = −18 −1 × 18, −2 × 9, −3 × 6 2e2 − 3e + 6e − 9 e(2e − 3) + 3(2e − 3) (e + 3)(2e − 3)

Now we put one answer over the other. (2e − 3)(2e + 3) (e + 3)(2e − 3) = 2e + 3 e + 3

Difference of two squares

Factorising a quadratic

Once all terms are in brackets (i.e. factorised) we divide above and below by the term 2e − 3. Once this is complete, no further simplification is possible or necessary.

Solution

There are two methods used here.

Method 1

The first term of the cubic is divided by the first term of the factor. This result is then multiplied by the factor and the difference between terms is found. It is repeated until the remainder is zero. x2 + 2x − 35

Method 2

method has a different presentation but similar approach.

Worked Example 10

A rectangle has sides of length x 3 units and ax2 + bx + c units, where a, b, c ∈ ℤ. The area of the rectangle is 2x3 13x2 + 25x − 12 square units. Find the value of a, b and c. We will divide the area by the length of the side that is known; this will enable us to find the length of the missing side.

Area 2x3 − 13x2 + 25x − 12

x − 3 ax2 + bx + c

Solution

We know what to do straight away when working with a rectangle, for example, of area 35 m2. If the width of the rectangle is 5 m, we can find the length through division 35 5 = 7 m.

See how we also use division in algebra. Use the same approach when letters are involved.

There are two methods used here. The first term of the cubic is divided by the first term of the factor. This result is then multiplied by the factor and the difference between terms is found. It is repeated until the remainder is zero.

2x2 − 7x + 4 x 3 ∙⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ 2x3 13x2 + 25x − 12 2x3 − 6x2 −7x2 + 25x − 12 −7x2 + 21x 4x − 12 4x − 12 0

a = 2, b = −7, c = 4

Array method: 2x2 −7x 4 x 2x3 −7x2 4x −3 −6x2 21x −12

a = 2, b = −7, c = 4

Resist the temptation to go straight to the calculator; show the examiner what you are subbing into the expression before you use the calculator.

SAMPLE

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We combine the two denominators 2x + 1 and 4 to get the common denominator.

Each numerator is adjusted for the new denominator. The first numerator is multiplied by 4 and the second by (2x + 1). Like the previous example, it has taken on the sign in the middle. Be careful when multiplying out here, mistakes are often made due to the sign.

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Each term is multiplied by the common denominator (2)(3) = 6

The fractions are gone and leave us with a simpler equation to solve. Remember, letters to one side, numbers to the other. Give your answer in the form m n . Don't be put off by this. It only means write your answer as a fraction. It is only relevant at the end and should never prevent you from attempting a question.

Worked Example 15

Solve the simultaneous equations:

2x − 3y = 18

5x + 9y = −10

Solution

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5x + 9y = −10

6x − 9y = 54

2x − 3y = 18 (×3) x = 44 11

5x + 9y = −10 11x = 44 x = 4

Now sub the x value into one of the original lines to find the y variable.

2x − 3y = 18 2(4) − 3y = 18 8 − 3y = 18 −3y = 18 − 8 −3y = 10 y = −10 3

We eliminate one variable first.

To use elimination, we multiply the first line by 3, to eliminate the y's. This allows for the x terms to be our sole focus. The y's are eliminated and leave us with an equation in x to solve.

If you were asked to verify your answer, sub the values for x and y into the line you didn't use. If the sides are equal, it’s verified.

5x + 9y = −10 5(4) + 9 ∙ −10 3 ∙ = −10 20 − 30 = −10 −10 = −10

Over to You

Q1 Write down the four factors of 12k + 8, apart from 1 and 12k + 8 Two of the factors should be in terms of k.

Q2 Factorise a2 16n2

Q3 Show that 2y + 3 is a factor of 2y3 9y2 −28y − 15.

Q4 Work out the value of 3p − 4t2, when p = 6 and t = 5.

Q5 Factorise fully 10de − df −5ef + 2d2

Q6 Write the following as a single fraction in its simplest form. 2 n 3 − 5 2n + 5

Q7 Use factors to simplify 2n2 + n − 15 n2 9 .

Q8 For all values of a, b, and x ∈ ℝ: (x + a) (x + b) = x2 + (a + b) x + ab. Using this fact, or otherwise, answer the following question: Solve the following equation in x, where a and b are constants. Give your answer in terms of a or b. x2 + (a + b) x + ab = 0

Q9 Simplify [x2 + (a + b) x + ab] ÷ (x + a).

Q10 Factorise n2 − 11n + 18.

Q11 Factorise fully wy − y − 1 + w

Q12 Find the value of 5 3x − 2 − 7 6x − 12 ,when x = 4.

Q13 Solve the following equation. 2x + 4 3 − 5x − 7 2 = 5

Q14 Factorise the following.

(i) 25x2 −49n2

(ii) 2x2 −9x −18

Reflect on Your Learning

Identify three things you can do very well in this topic.

Identify three things you need more practice on in this topic.

1. 2. 3. Write down some useful websites you have found to help with your revision.

1. 2. 3.