head start lecture mathematics extension 1 · • mathematics is a skill based subject – your...

Head Start Lecture

Mathematics Extension 1

Presented by:Leo Su

• 2018 Graduate from Baulkham Hills High School

• HSC All-Rounder, achieving a Band 6 across 12 units of

subjects:

– Biology

– Business Studies

– English (Advanced)

– Mathematics Extension 1

– Mathematics Extension 2

– Modern History

• 99.45 ATAR

• Internally ranked 1/187 for Maths Ext 1

• 100/100 HSC Assessment Mark and 99/100 Overall HSC

Mark for Maths Ext 1

Who am I?

2

• Section 1:

– Syllabus Overview

– Study tips and advice

– Some special tricks for Permutations and Combinations

• Section 2

– Mathematical Induction

• What is induction?

• Equality and Divisibility Type Questions

• False Proof by Mathematical Induction

• Section 3

– Trigonometry

• Auxiliary Angle Method

• T-formulae

• Trigonometric Identities

• Derivatives of Inverse Trigonometric Functions

• Primitive of sin2 𝑥 and cos2 𝑥

Today’s Schedule

SYLLABUS

OVERVIEW

I’m sorry but the old syllabus can’t come to the phone right

now.

Syllabus Overview

I’m sorry but the old syllabus can’t come to the phone right

now. Why? Oh, ‘cause it’s dead!

Syllabus Overview

• The syllabus is your best friend!

• You are first cohort to do the new syllabus therefore it is

crucial that you have a clear understanding of what is on

it

– You don’t want to be surprised in the exam!

• It is important to know what topics are new and what

topics have been moved in order to utilise past

resources effectively

Syllabus Overview

Syllabus Overview

Syllabus Overview

• Let’s compare it with the old syllabus

• Also keep in mind that the 3U course is done

concurrently with the 2U course so you’ll need to be

familiar with 2U content in order to do some 3U topics

• E.g. Integral calculus (Integration)

Syllabus

Yr 12 topics:

• Proof (Mathematical Induction)

• Vectors

• Trigonometric functions

• Calculus (which includes differential equations)

• Statistical Analysis

*RED = new topic

Syllabus Overview

Examples of what has changed?

• The 4U Graphs topic has been moved into the 3U Yr 11 course

• Volumes of revolution have been moved from the 2U/4U course into

the 3U Yr 12 course

• Vectors is a new topic but there is slight overlap with 4U complex

numbers

• Pigeonhole Principle added to existing 3U Yr 11 perms and combs

topic

• Slope fields added to existing 3U differential calculus topic

• Simple Harmonic Motion moved to 4U

• Circle geometry, parametric/locus focused questions, binomial

identity proofs among other topics are now GONE

Syllabus Overview

STUDY TIPS

• You’ve already had Year 11 to experiment with different

ways of studying

• Different people learn more effectively in different ways

e.g. visual learner, auditory learner, kinaesthetic learner

• I’ve compiled some tips which I found useful but the

most important thing is that you do what works for you!

Study Tips

How I studied for maths:

• Consistent effort for maths throughout the entire year =

less stress come exam time

• I didn’t use the reference sheet but I made sure I knew

what was on it just in case!

• Before exams, my routine for studying maths purely

consisted of doing past papers!

• Maths was the subject I dedicated the least time towards

in the lead up to Trials and the HSC

Study Tips

Effective studying:

• Mathematics is a skill based subject

– Your brain is a muscle that must be trained

– Reading over notes will not cut it to get a Band 6 in Mathematics

– Effective study involves:

Practice

Practice

Practice

Balance in studying:

ApplyConsolidate

• Topic focused

revision

• Past papers

• Reteach yourself

content

(exercises)

• Teach others!

• Derive formulas

Tips on past papers:

1. Grab a Paper

2. Read in 15 minutes

Circle questions that you don’t know

how to do

3. Use circled questions as exercise

Study should never be easy!

Using the formula sheet: Pros/Cons

• The formula sheet:

• Good: Less to remember, less silly mistakes, less stress

• Bad: Over-reliance, lose time, may lead to harder exams

• Don’t rely on your formula sheet! Use it as a backup!

• Deriving your formulae is another great way to

study...

• Some other ways to study for Mathematics:

1. Record Video Lessons for Your Own Revision Later (or

watch our free ATARNotes Videos!)

2. Race Through Quick, Short Exercises

3. Derive your Formulas

• Past papers/practice will always be the best way!

Other helpful methods of study:

• Make a list in a notebook/exercise book of questions you

continually get wrong, challenging questions, fast

methods of solving questions etc.

• Keep track of the silly mistakes you make to avoid

making them again!

• Continually add, revise and read this list until you are

confident with these topics

Note: this works well for ALL your subjects, not just MX1

’Dumb question’ bank:

Some Special

Tricks for

Perms &

Combs

• So you’ve covered permutations and combinations as

part of your combinatorics topics in the Year 11 course

• Before we dive into new Year 12 content, let’s discuss

some special things/tricks you should be aware of to

consolidate your knowledge!

Permutations and Combinations

Very quick recap:

• Recall: nPr = 𝑛!

𝑛−𝑟 !

nCr = 𝑛!

𝑛−𝑟 !𝑟!

• You can arrange ‘n’ different objects in a line in n! ways

• You can arrange ‘n’ different objects in a circle in (n-1)!

ways

• Divide by r! for ‘r’ number of repeated objects

Permutations and Combinations

A common area where people make mistakes in perms and

combs is when you are asked to divide things into equal

groups.

Perms and Combs – Equal Groups

How many ways can 5 people be chosen from a group of

10?10C5= 252 ways

How many ways can a group of 10 people be split into two

teams for a game of soccer?

10C5 x 5C5 = 252 ways?

Common silly mistake!


WHY?

Let’s simplify the example to four people: A B C D

There are 4C2 = 6 ways of choosing 2 people out of the 4

1) A & B

2) A & C

3) A & D

4) B & C

5) B & D

6) C & D


But look what happens when we are asked to divide

them into equal teams of two…

1) A & B VS C & D

2) A & C VS B & D

3) A & D VS B & C These are the same!

4) B & C VS A & D

5) B & D VS A & C

6) C & D VS A & B

There are only actually 3 different ways of dividing 4 people into 2 teams.

The other half are just repeating what we already have!


REMEMBER: When splitting into equal groups, divide by

the (number of groups)!

How many ways can a group of 4 people be split into equal

teams of two?

Answer: 4C2 x 2C2

2!= 3 ways


• Alternatively, you can also think this way:

• “From the perspective of one particular person, choose a

number of people from the remaining group to form a

team”

Let’s consider A B C D again from the perspective of A.

To be divided into equal teams, A needs one teammate. A

can choose from B, C or D. A has to choose 1 person from

3 i.e. 3C1 = 3 ways.

This gives us the same result as 4C2 x 2C2

2!= 3 using the

previous method


So to answer that question… How many ways can a group

of 10 people be split into two teams for a game of soccer?

10C5 x 5C5

2!= 126 ways

Alternative method: 9C4 = 126 ways

Same answer!


REMEMBER: This only applies when you are dividing into

equal groups!!!!

So for unequal groups…

E.g. How many ways can you divide 10 people into a group

of 8 and a group of 2?

10C8 x 2C2 = 45 ways


• ‘Dividers’ or ‘slotting’ techniques are useful for some

types of perms and combs questions

Perms and Combs – ‘Dividers’ and ‘Slotting’

• ‘Dividers’ or ‘slotting’ techniques are useful for some

types of perms and combs questions

Ariana Grande asks, Q: How many ways can 7 different

rings be placed on the 5 fingers of a hand if:

i) There is no restriction on how many rings a finger can

have

ii) Each finger must have at least one ring

Give it a shot!


Q: How many ways can 7 different rings be placed on the 5 fingers of a hand if:

i) There is no restriction on how many rings a finger can have

• For part i) we need to use ‘dividers’

• Permutate A B C D E F G | | | |

E.g. C|DAG|FE|B| = ||EAC|BGD|F =


(‘|’ represents the space between a finger)

Q: How many ways can 7 different rings be placed on the 5 fingers of a hand if:

ii) Each finger must have at least one ring

• For part ii) we need to use ‘slotting’/’insertion’

• Permutate A B C D E F G

• Choose the spots to ‘slot’ the dividers in

A_B_C_D_E_F_G

• E.g. D_E_A_G_B_C_F =


37

BREAK TIME!

Come ask questions

Check out our range of course notes outside –

All updated for the new syllabus!

MATHEMATICAL

INDUCTION

When you hear the word ‘induction’, several images often

spring into mind.

Mathematical Induction

Induction cooking?

Workplace induction?

But what about Mathematical Induction?


But what about Mathematical Induction?

• Mathematical Induction is a technique used in

mathematics to prove statements that are asserted by

natural numbers

– For example that 1+3+5+…+(2n-1)=n2 for all integers n≥1

• It works by proving that a statement is true for an integer

if it is true for the integer before it, thus creating a chain

effect

• Think about dominos!


• There are three main types of induction questions

– Equality

– Divisibility

– Inequality

• All 3 types were previously tested in the HSC for Maths

Ext 1

• However, in the new syllabus, inequality type induction

questions have been moved to the Extension 2 course

• You will only need to worry about equality and

divisibility induction questions in Extension 1


• There are four main steps to any induction question:

1. Prove true for the first integer of the domain (usually n=1)

2. Assume true for the general term n=k

3. Prove true for the general term n=k+1 USING the assumption

from part 2

4. Write a conclusion to sum up what is going on

• Step 1 forms the base element while Step 2 and 3 forms

the inductive element

• It is beneficial to have a strong, clear structure in your

working so that the marker can see that you understand

the rationale behind how mathematical induction works


• Let’s see it in action by proving the geometric series

summation formula!


𝑎 + 𝑎𝑟 + 𝑎𝑟2 +⋯+ 𝑎𝑟𝑛−1 =𝑎(𝑟𝑛 − 1)

𝑟 − 1

• ‘Short’ VS ‘Long’ Conclusion

• Short: By the principle of mathematical induction, the

statement is true.

• Long: The statement is true for n=k+1 if it is true for n=k.

Since it is true for n=1, it must be true for n=2 and n=3

and so on for all integers n≥1.


Short Long

PROS • Takes less time to write

• Strong potential to still attain full marks

• Shows the marker you understand

what is going on

• Strengthens your proof

• Helps you to remember the actual

induction steps!

CONS • Slim risk of losing marks, especially if your

earlier sections were a little dodgy

• Takes more time to write

• “Unnecessary” and a “waste of

effort”?

• Now, let’s try a divisibility type question.

Prove using mathematical induction that 23𝑛 − 1 is

divisible by 7 for all n≥1


Let’s do another question!

a) Show that 𝑘 + 3 3 = 𝑘3 + 9𝑘2 + 27𝑘 + 27

b) Prove that the sum of the cubes of three consecutive

positive integers is divisible by 9


• Something to keep in mind is that n=1 isn’t necessarily

the default

• Always read the question carefully!

• E.g. take this question straight from a HSC paper


• Remember: Induction only works if you’ve used all the

steps and satisfied both the base and the inductive

elements

• In the inductive element, you need to use the assumption

somewhere in Step 3

• You also can’t just do the inductive element by itself either

False proof by mathematical induction

Let’s say you were trying to prove:

1 + 2 + 3 +⋯+ 𝑛 =1

2(𝑛 − 1)(𝑛 + 2)


50

BREAK TIME!

Come ask questions

Check out our range of course notes outside –

All updated for the new syllabus!

TRIGONOMETRY

First, to get us in the mood for some trig, let’s start with a

fun mnemonic!

Trigonometry

Hey kids! Mnemonics are fun!

We all know the trigonometric identity: sin2 𝑥 + cos2 𝑥 = 1

By dividing both sides of this identity with cos2 𝑥 or sin2 𝑥 we

can also get these other two identities respectively:

• tan2 𝑥 + 1 = 𝑠𝑒𝑐2 x

• 1 + cot2 𝑥 = 𝑐𝑜𝑠𝑒𝑐2𝑥

All three of them are very useful in a variety of maths

problems and it is beneficial to be able to recall them in a

cinch.

If you have difficulty memorising these latter two identities,

there’s an easy mnemonic to help you remember them!

But first…a fun mnemonic!

53

Just remember:

“The one with the tan is sexy”

1 + tan2 𝑥 = sec2 𝑥

AND

“The one in the cot is cosy”

1 + cot2 𝑥 = 𝑐𝑜𝑠𝑒𝑐2𝑥

But first…a fun mnemonic!

• Recall from Year 11 course:

Sum and difference expansions

Double angle formulae

Trigonometry

• The auxiliary angle method of solving trigonometric

equations involves changing an equation with a sine and

cosine component into another form to make it easier to

solve

• asin 𝑥 ± 𝑏𝑐𝑜𝑠 𝑥 = 𝑐 is changed into 𝑅𝑠𝑖𝑛 𝑥 ± α = 𝑐

• a𝑐𝑜𝑠 𝑥 ± 𝑏𝑠𝑖𝑛 𝑥 = 𝑐 is changed into 𝑅𝑐𝑜𝑠 𝑥 ∓ α = 𝑐

• This requires the use of sum or difference expansions

• a, b, and R are positive numbers

• α is called the auxiliary angle

Trigonometry – Auxiliary Angle Method

Solve:

3 sin 𝑥 − cos 𝑥 = 1 , for 0 ≤ 𝑥 ≤ 2𝜋


The auxiliary angle method is also useful for graphing!

Let’s say we are asked to graph the function

y= 3 sin 𝑥 − cos 𝑥 , for 0 ≤ 𝑥 ≤ 2𝜋


• Recall from Year 11 course:

Trigonometry – t formulae

• Remember that it doesn’t just have to be for θ and θ/2!

E.g. you can write sin(4θ) as 2𝑡

1+𝑡2where t=tan(2θ)

• An easy way to remember the t formulae is through the

tan double angle formula

• tan 2𝑥 =2tan x

1−tan2 𝑥


• The t formulae can also be used to solve trigonometric

equations!

• But a very important thing you must remember is that

tan(𝜃

2) is undefined for

𝜃

2=±

𝜋

2, ±

3𝜋

2, ±

5𝜋

2etc. so you’ll

need to test 𝜃=±𝜋,±3𝜋,±5𝜋 etc. independently to see if

it is a solution


• Let’s do the previous question again but with the t

formulae this time:

3 sin 𝑥 − cos 𝑥 = 1 , for 0 ≤ 𝑥 ≤ 2𝜋


TIP: As you saw in that previous question, a common hint

that there are actually more solutions is if the 𝑡2 cancels

out.


• You’ll need all those formulas you’ve learnt to be able to

prove trigonometric identities and solve equations!

Trigonometric Identities

• You’ll need all those formulas you’ve learnt to be able to

prove trigonometric identities and solve equations!

Let’s do this question:

a) Show that cos 3𝑥 = 4 cos3 𝑥 − 3 cos 𝑥

b) Hence solve cos 3𝑥 + cos 2𝑥 + cos 𝑥 = 0(for 0 ≤ 𝑥 ≤ 2𝜋)

Trigonometric Identities

• In the Year 11 course, you learnt about differential

calculus and how to find the derivative of an equation

A syllabus dot-point in the Yr 12 course is that you:

Differentiating Inverse Trigonometric Functions

How do we differentiate inverse trigonometric functions?

Let me show you…



By following the steps I did through the document

camera, we arrive at these results:

Now that we know that, we can go the other way and

integrate those values to get some of our standard

integrals:


• In the Yr 12 3U course you’ll need to know how to

integrate sin2 𝑥 and cos2 𝑥

• But how do we do it?

Primitive of sin2 𝑥 and cos2 𝑥

• In the Yr 12 3U course you’ll need to know how to

integrate sin2 𝑥 and cos2 𝑥

• But how do we do it?

The answer lies in the cosine double angle formula.

Let me show you how…


Cosine Double Angle Formula

• cos 2𝑥 = 2 cos2 𝑥 − 1

= 1 − 2 sin2 𝑥

Can be rearranged to be:

• cos2 𝑥 =1

2(1 + cos 2𝑥 )

• sin2 𝑥 =1

2(1 − cos 2𝑥 )

• Know these formulas by memory! (Or at least how to

derive them)

• These used to not be on the reference sheet but they are

now for the new syllabus


Solve:

3cos2(3𝑥)𝑑𝑥


• Good luck as you begin your Year 12 course!

• Your school assessments will count now

• This is your time to shine!

• Put in the effort and don’t give yourself any chance for

regret!

Good Luck!

Thanks for

coming!!

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