AS Double Mathematics - ingridflynn.weebly.com · AS Double Mathematics Blue Block P1, ... (x, y) plane Topic 3 ... Probability: Tree Diagrams – Given (Conditional Probability)

Name:________________________

AS Double Mathematics

Blue Block

P1, P2, Stats+Mech Teacher:IngridFlynn

2

Checklist for Completed Assignments:

o The assignment cover sheet has boxes ‘Done’ and ‘Ready’ ticked for every question, and none of these ticks are a lie

o Each question is started on a new side of A4 o Question numbers are written as a large title at the top of every page and underlined

twice: e.g. “Question 1 ”. o All questions are in order o Equals signs are all in a straight vertical line down the page (no snaking!) o All questions are written neatly and all working is shown o Mistakes are boxed off neatly and scored out o Answers are underlined twice and checked (show it has been checked by ticking it) o All pages are stapled together in the top left corner

Example:

Assignment Test

You will have an 30min assignment test on the day you hand in your assignment. There will be 5 questions which are identical to the questions in the assignment. Therefore, everyone should pass this test.

3

Your Exams The Pearson Edexcel Level 3 Advanced GCE in Mathematics consists of three externally-examined papers:

Paper 1: Pure Mathematics 1 (*Paper code: 9MA0/01) Paper 2: Pure Mathematics 2 (*Paper code: 9MA0/02) Paper 3: Statistics and Mechanics (*Paper code: 9MA0/03) Each paper is: 2-hour written exam with calculator; 33.33% of the qualification; 100 marks. To get an E you need an average of about 45% in the exams. To get an A grade you need an average of roughly 85%.

PURE (Paper 1 and Paper 2) ● Topic 1 – Algebra and functions ● Topic 2 – Coordinate geometry in the (x, y) plane ● Topic 3 – Sequences and series ● Topic 4 – Trigonometry ● Topic 5 – Proof ● Topic 6 – Exponentials and logarithms ● Topic 7 – Differentiation ● Topic 8 – Integration ● Topic 9 – Numerical methods ● Topic 10 – Vectors STATISTICS (Paper 3, Section A) ● Topic 1 – Statistical sampling ● Topic 2 – Data presentation and interpretation ● Topic 3 – Probability ● Topic 4 – Statistical distributions ● Topic 5 – Statistical hypothesis testing MECHANICS (Paper 3, Section B) ● Topic 6 – Quantities and units in mechanics ● Topic 7 – Kinematics ● Topic 8 – Forces and Newton’s laws ● Topic 9 – Moments

Next Year’s Exams: Further Maths A Level (4 in total, each 1.5hr and 25%):

Compulsory: Core Pure 1, Core Pure 2 Options x2: Choose one from { FP1,FS1,FM1,D1} and {FP2,FS1,FM1,D1,FS2,FM2, FS1,D2}

4

Your Teacher: Ingrid www.ingridflynn.weebly.com

Warning: I am very strict with homework so don’t even bother trying to get away with not doing all of the work I set (which is a lot)! I am also very helpful… I am (almost) always in room 3 or room 11 to give help if you need it.

Your Lessons

BRING THIS PACK TO EVERY LESSON PLEASE

In the first 3 half terms (about 19 weeks) we will study the course content, then for the next two half terms (about 12 weeks) we will practice the techniques, consolidate your learning and prepare for the exams.

Before each lesson you will have watched a video introducing a new topic. In total, this is usually about an hour’s work per week. In the lesson you will, for most of the time, be working rather than listening to me talk. You will be practicing basic mathematical skills and strengthening your understanding of the new topic by working through exercises, together with developing your problem solving skills by attempting to solve complicated problems using the maths you have learned.

Calculators

You need a calculator for this course. The recommended calculator is the

Casio Fx-991ex Classwiz (~£22).

Some doubles students choose to buy the Casio fx9860GII, an expensive but very good graphical calculator (around £70 from www.calculatorsdirect.co.uk) which is a huge advantage in the exams as it will solve all the equations for you, so you can check your answers. Slightly better (same functions but colour screen and nicer graph sketcher) is

the Casio fx-CG20 (around £100). Please talk to me if you are worried about buying a calculator.

Expectations. You will…

1. Attend all lessons and contact me as soon as possible if a lesson needs to be missed. You will check the Absence Box when you return to catch up on any missed work

2. Come to each lesson on time 3. Work hard in lessons 4. Hand in a complete, well presented assignment on last lesson of each week 5. Prepare fully for the weekly assignment test by practising 6. Ask for help if you need it, not wait for me to come to you and offer help 7. If an assignment test is not passed, you will need to re-do the incorrect questions twice each

and also find two similar questions to do (for each incorrect question). These will be handed in with the next assignment.

Assignments You will be set 1 assignment per week. It will always have the same format. You will have 9 hours of maths lessons per week, and are expected to do 9 hours of study out of lessons also in order to keep up with the pace of the course. Some of you will complete the assignment in 2 hours. Some of you will take 6 or 7 hours to complete it. It is your responsibility to make sure you start early enough to ensure you meet the deadline. Videos

Eachweek,youwillbesetvideostowatchtointroduceyoutoanewtechnique.Asyouwatchthevideo,

completetheassociatedpackpagesthentickitoff.Feelfreetogetahead!Videoscanbeaccessedusingthe

QRcodesoronmywebsite.

5

Wk Wk begins Videos Half Term 1 Mins Watch

by Page ü

0 11/9

Pure: Simultaneous Equations 20 9-10 Stats: Sampling Pure: Equation of a Line 14 11 Pure: Quadratics: Discriminant 6 12 Pure: Quadratics: Completing the square a≠1 8 13 Mech: Constant Acceleration: VT graphs 7 130

1 18/9

Pure: Trigonometry: Radians measure and application 17 14-15 Stats: Measures of Location and Spread: Median and IQR of a list 9 99-100 Stats: Measures of Location and Spread: Mean and S.D of grouped data (by hand)

10 101-102

Stats: Measures of Location and Spread: Mean and S.D of grouped data (using a calculator)

3 103

Stats: Measures of Location and Spread: Mean and IQR of grouped data (Interpolation)

104

Stats: Measures of Location and Spread: Percentiles 7 105 Pure: Graphs and transformations: Sketching Cubics 24 16-18 Pure: Graphs and transformations: Translating and stretching graphs 6 19 Pure: Quadratics and Inequalities 20 20-21 Mech: Constant Acceleration: SUVAT 5 151

2 25/9

Pure: Trigonometry: Mini Trig Equations 13 22-23 Pure: Trigonometry: Reciprocal Trig Functions 3 24 Pure: Trigonometry: Reciprocal Trig Graphs 13 25 Pure: Trigonometry: Pythagorean Trig Identities - Proof 6 26 Pure: Trigonometry: Using Trig Identities to Solve Equations 2 27 Stats: Linear Coding 6 106 Stats: Combined mean 2 107 Pure: Differentiation: Gradient Function 17 28-29 Pure: Differentiation: Equations of Tangents and Normals 10 30 Pure: Differentiation: 1st Principles 6 31 Mech: Projectiles 17 152

3 2/10

Pure: Arithmetic Series and Proof 25 32-34 Pure: Geometric Series and Proof 17 35-37 Stats: Histograms: Intro 3 108 Stats: Histograms: Dimensions of Bars 6 109 Pure: Differentiation: 2nd Derivative and Classifiation of Turning Points

11 38

Pure: Differentiation: Increasing and Decreasing Functions 6 39 Pure: Differentiation: Sketching Gradient Functions 5 40

4 9/10

Pure: Binomial Expansion: Finite 41-43 Pure: Recurrence Relation 6 44 Pure: Sigma Notation 22 45 Stats: Probability: Venn Diagrams: Union 5 110 Stats: Probability: Venn Diagrams: Intersection 5 111 Stats: Venn Diagrams: Addition Rule 8 112 Pure: Circles: Perpendicular Bisector 8 46 Pure: Circles: Equation of a Circle 11 47 Pure: Circles: Solving Problems 10 48 Mech: Forces: Free Body (Forces) Diagram 6 153-

154

Mech: Forces: Newton’s Laws 6 155 5 16/10 Pure: Factor Theorem

6

Pure: Algebraic Division 4 49 Stats: Probability: Venn Diagrams: Given (Conditional Probability) 10 113 Stats: Probability: Tree Diagrams 6 114 Stats: Probability: Tree Diagrams – Given (Conditional Probability) 6 115 Pure: Differentiation: Optimisation 12 50


by Page ü

6 30/10

Pure: Algebraic Fractions 4 51 Pure: Partial Fractions 16 52-53 Pure: Integration – Definite and Indefinite 17 54-55 Pure: Integration – Area Under a Curve (Easy) 15 56-57 Pure: Integration – Area Under a Curve (Harder) 11 58 Pure: Integration – Area Under a Curve (Even harder) 13 59-60 Mech: Forces and Motion – Resolving Forces on a Slope 7 156

7 6/11

Pure: Proof by Contradiction and Deduction 13 61-62 Pure: Proof by Exhaustion 9 63 Stats: Probability – Mutually Exclusive and Independent Event 5 116 Pure: Vectors – Distance Between Points 15 64-65 Pure: Vectors – Position Vectors 15 66-67 Mech: Forces and Motion – Connected Particles - Pulleys 12 157 Mech: Forces and Motion – Connected Particles – On a Slope 16 158

9 20/11

Pure: Trigonometry – Compound Angle Formulae 9 68 Pure: Trigonometry – Double Angle Formulae 3 69 Stats: Statistical Distributions – DRVs 7 117 Stats: Statistical Distributions – Discrete uniform distribution 3 118 Pure: Functions: The Modulus – Modulus Function and its Graph 7 70 Pure: Functions: The Modulus – Solving Modulus Equations 12 71-72

10 17/11

Stats: Statistical Distributions – Binomial Distribution 10 119 Pure: Functions – Functions and Mapping 5 73 Pure: Functions – Domain and Range 10 74 Pure: Functions – Composite Functions 6 75 Pure: Functions – Inverse Functions 12 76-77 Mech: Forces and Motion – Friction 8 159 Mech: Forces and Motion – Single Particle 7 160

11 4/12

Pure: Exponentials and Logs – The Basics 14 78-79 Pure: Exponentials and Logs – Sketching e^x and lnx 6 80-81 Pure: Exponentials and Logs – Solving Equations 8 82-84 Pure: Exponentials and Logs – Modelling 5 85 Pure: Differentiation – of sinx from 1st principles 5 86 Pure: Differentiation – of cosx from 1st principles 5 87 Pure: Differentiation – Chain Rule 13 88-89 Pure: Differentiation – Product Rule 8 90

12 11/12

Stats: Hypothesis Testing – Binomial Distribution 10 120 Stats: Hypothesis Testing – Lower Tails Test 9 121 Stats: Hypothesis Testing – Upper Tails Test 7 122 Pure: Differentiation – Quotient Rule 5 91

7

Pure

● Topic 1 – Algebra and functions ● Topic 2 – Coordinate geometry in the (x, y) plane ● Topic 3 – Sequences and series ● Topic 4 – Trigonometry ● Topic 5 – Proof ● Topic 6 – Exponentials and logarithms ● Topic 7 – Differentiation ● Topic 8 – Integration ● Topic 9 – Numerical methods ● Topic 10 – Vectors


by Page ü

13 1/1

Pure: Trigonometry – Pythagorean Identities to Solve Equations 2 92 Pure: Trigonometry – Inverse Trig inc. Graphs 5 93 Stats: Hypothesis Testing – Critical Values - Lower Tail Test 8 123 Stats: Hypothesis Testing – Critical Values - Upper Tail Test 8 124 Stats: Hypothesis Testing – Critical Regions - Two Tail Test 16 125 Pure: Differentiation – !! 9 94 Pure: Differentiation – !"# 7 95 Pure: Differentiation – Practice 7 96

8

Pure - Simultaneous Equations Linear and quadratic simultaneous equations Equations and inequalities 20 min

Write down the easier equation Rearrange into y = or x = Sub that the harder equation Solve to find y (or x) Use the easy equation to find x (or y)

Level 1 2041032

=+

=+

yxyx

Little sketch of what you’re finding:

Level 2 2041032

2 =+

=+

yx

yx Little sketch of what you’re finding:

9

10

Write down the easier equation Rearrange into y = or x = Sub that the harder equation Solve to find y (or x) Use the easy equation to find x (or y)

Level 3 2041032

22 =+

=+

yx

yx Little sketch of what you’re finding:

11

Pure - Equation of a line Straight line graphs 14 min

(start watching the video from 11 mins in)

What is the equation of the line passing through (1, -3) with gradient 1/2?

What is the equation of the line passing through (1, -3) and (-4, -1)?

What is the equation of the line passing through (4, -1) parallel to 3x+y-1=0 ?

What is the equation of the line through (4, -1) perpendicular to x+2y-3=0 ?

12

Pure – Quadratics Discriminant 6 min

Quadratic Function Value of the Discriminant Corresponding graph of Quadratic function

Number and Type of Roots

13

Pure - Quadratics Completing the square where a is not equal to 1 8 min

Example 1:

Complete the square in order to solve 2!! − 8! + 7 = 0

Example 2:

Complete the square in order to solve 5!! + 8! − 2 = 0

14

Pure - Trigonometry Radian measure and its applications (TOOLS) 17 min

T Triangle Area

O Sector Area O Arc Length

L Cosine Rule

S Sine Rule

15

16

Pure - Graphs and transformations Sketching factorised cubic functions 24 min

FactorisedCubicsActivity1

Foreachoftheequationsbelow,

(1) Putx=0tofindoutwherethecubiccrossestheyaxis.(2) Puty=0tofindoutwherethegraphcrossesthexaxis.

( )( )132 +−= xxy Crossesyaxisat: Crossesxaxisat:

( )( )( )211 −+−= xxxy Crossesyaxisat: Crossesxaxisat:

( ) ( )123 2 ++= xxy Crossesyaxisat: Crossesxaxisat:

17

FactorisedCubicsActivity2

Cubicgraphswhichalsohavequadraticand/orlineartermsinthemhaveaslightly

morecomplicatedshapebecausethequadratic/lineartermsmakethegraph‘twist’inthemiddle.

Highlightthe‘roots’ofthesecubicsbyputtingdotswherethegraphscrossthexaxis:

3roots……………………………………………………………………………..→

• Usedotstoindicateeachofthe3roots

2roots-onesingleroot(wherethecurveslicesthroughthexaxis)andonedoubleroot(wherethecurvetouchesthexaxisbutdoesn’tcutthroughit)………………→

• UseadottoindicatetheSINGLEroot• UseacrosstoindicatetheDOUBLEroot

1root…………………………………………………………………………….…..→

• Useadottoindicatetheroot

18

FactorisedCubicsActivity3Tickifthecubichasthefeaturedescribed

Cubic Double Root? Root at x = 1? Crosses y axis at - 6? ( )( )32 −−−= xxxy

( )31−= xy

( )( )16 −+= xxxy

( )( )216 +−= xxy

( )( )( )312 ++−= xxxy

( ) ( )13 2 −+= xxy

( ) ( )61 2 +−= xxy

( )( )16 −+= xxxy

FactorisedCubicsActivity4Sketcheachofthecubicsfromthepreviousactivityby‘joiningthedots’

( )( )32 −−−= xxxy ( )( )612 +−= xxy

( )31−= xy ( )( )16 −+= xxxy

( )( )216 +−= xxy ( )( )( )312 ++−= xxxy

( ) ( )13 2 −+= xxy ( ) ( )61 2 +−= xxy

19

Pure - Graphs and transformations Translating and stretching graphs 6 min

Describe the transformations and give an example of each type

‘Outside’ Transformations are changes to y ‘Inside’ Transformations are changes to x

axf +)(

)( axf +

axf −)(

)( axf −

)(xaf

)(axf

)(xf−

)( xf −

20

Pure - Quadratics and Inequalities 20 min

Equation to solve Highlight the solutions on the graph Solution

0542 =−− xx

3542 =−− xx

9542 −=−− xx

0542 <−− xx

0542 ≤−− xx

0542 >−− xx

21

0542 ≥−− xx

Solving two inequalities

Solve these inequalities 1) 024112 <+− xx 2) 01272 ≥++ xx

3) 0384 2 ≤+− xx 4) 010116 2 >−+ xx

5) 1753and01072 <+<+− xxx

6) 2173and0122 ≥+>−− xxx

22

Pure - Trigonometry Graphs of standard trig functions 13 min and solving mini trig equations

23

sin =x 3600 ≤≤ x

23

sin =x π20 ≤≤ x

23

23

cos −=x 3600 ≤≤ x

2 tan x = - 2 3600 ≤≤ x

24

Pure - Trigonometry Reciprocal trig functions 3 min

Write down the three reciprocal trig functions

Secant ! (Sec !) =

Cosecant ! (Cosec !) =

Cotangent (Cot !) =

Work out the value of this function:

Sec 60° =

Top Tip for remembering which is which:

Circle the first letter: sin x cos x tan x

Circle the third letter: cosec x sec x cot x

25

Pure - Trigonometry Reciprocal trig graphs 13 min

y = cosec !

y = sec !

y = cot !

26

Pure - Trigonometry Pythagorean trig identities – proof 6 min

Prove that:1+cot2!=cosec2 !

Divide through by sin2 !

Prove that:tan2!+1=sec2!

Divide through by cos2 !

27

Pure - Trigonometry Using Pythagorean Trig Identities to Solve Equations 2 min

This is an example of using a pythagorean trig identity to solve an equation.

Solve 4!"#$!!! − 9 = !"#$ !"# 0 ≤ ! ≤ 360

28

Pure - Differentiation Gradient function 17 min

Examples

29

Differentiate:

1 xx

y232 −

= 2 243 xxy −=

3 x

xy14 2 −= 4 23 += xy

5 xxy 23 3 −= 6 2

32xx

y−

=

30

Pure - Differentiation Equations of tangents and normals 10 min

122 23 −+−= xxxy

What is the equation of the tangent at x = -1? What is the equation of the normal at x = -2?

523 2 +−= xxy What is the equation of the tangent at x = 2? What is the equation of the normal at x = -2?

31

Pure - Differentiation From 1st principles 6 min

Complete the diagram below and use it to find from 1st principles the gradient function for y=x2

32

Pure - Arithmetic Series and Proof 25 min

What do all these terms add up to?

Proof of the sum of an Arithmetic Series (to learn)

33

Translate the information into maths using the two equations!

34

Translate the following information

1 The 5th term is 11 2 The 8th term is 7 3 The sum of the first 8 terms is 12 4 The sum of the first 18 terms is -2 5 The 12th term is -17 6 The sum of the first 9 terms is 15 7 The 17th term is 91 8 The sum of the first 52 terms is 500 9 The 11th term is 9 10 The sum of the first 6 terms is -20

35

Pure - Geometric Series 17 min

2,4,8,16,…,256,…,3276 What do all these terms add up to?

Proof of the sum of a Geometric Series (to learn)

36

Translate the information into maths using the two equations!

8

37

Translate the following information

1 The 5th term is 11 2 The 8th term is 7 3 The sum of the first 8 terms is 12 4 The sum of infinite terms is -2 5 The 12th term is -17 6 The sum of the first 9 terms is 15 7 The 17th term is 91 8 The sum of infinite terms is 500 9 The 11th term is 9 10 The sum of the first 6 terms is -20

38

Pure - Differentiation 2nd derivative and classification of turning points 11 min

105272 23 +−+= xxxy

Find the coordinates of the stationary points

105272 23 +−+= xxxy

Find and classify the stationary points It is a maximum if: It is a minimum if:

39

Pure - Differentiation Increasing/decreasing functions 6 min

Using the graph below indicate (with different colours) where the function is INCREASING and where it is DECREASING. Write the inequalities which give the range of values of x where each is true.

40

Pure - Differentiation Sketching gradient functions 5 min

Sketch gradient functions for the following with notes explaining the shape:-

41

Pure: Binomial Expansion

( )

( )

( )

( )

( )

( )7

4

3

2

1

0

ba

ba

ba

ba

ba

ba

+

+

+

+

+

+

Finding the coefficients

Method 1:

Method 2:

Method 3:

42

( )432 x+

( )71 x−

4

22 ⎟

⎠

⎞⎜⎝

⎛ −x

43

( )42 px+

( )nx21+

Why is 0! = 1? Where does the rnC formula come from?! Why is it called n ‘choose’ r? Watch these

videos to find out!

44

Pure: Recurrence Relations This is an example of how to interpret recurrence 6 min relation notation

4,5 11 =+=+ uuu nn

45

Pure: Sigma Notation 22 min

I haven’t left you much room here sorry – you need to make notes on what the notation is telling you to do and how you would answer the problem but don’t attempt to copy out everything from the screen! Just the key points to enable you to answer one of these questions.

( )∑=

−6

212

kk

( )∑=

+142

1027

rr

∑=

4

2

2r

r

∑=

10

4

3k

ku

4,5 11 =+=+ uuu nn

46

Pure - Circles Equation of the perpendicular bisector 8 min of a line segment

MidpointofAB=

GradientofAB=

∴Perp.Gradient=

∴Equationofperpendicularbisectoris:

47

Pure - Circles Equation of a circle 11 min

Circles

Mostimportanttoptip:

48

Pure - Circles Using circle properties to solve problems 10 min on coordinate grids

i)Centre

ii)Gradient,m=

iii)CP=

49

Pure: Algebraic Division 4 min

Copytheexample !!!!!!!!!!!!"!!!!!!!

50

Pure - Differentiation Optimisation problems 12 min

51

Pure: Algebraic Fractions 4 min

Copy the two examples:

1) Simplify!!!!!!!!!!!

!!

2) Simplify!!!!!!!!

!" ÷!!!!!"

52

Pure: Partial Fractions https://youtu.be/OeUCqui7bu0 16 min

Write!!!!

!!! !!! inpartialfractions.Showthefullmethod

Write!!!!!!"! !!!!! ! !!!! inpartialfractions.Showthefullmethod

….continuedonnextpage

53

Partial Fractions Page 2

Write!!!!!!!!!!! !!! inpartialfractions.Showthefullmethod.

54

Pure - Integration Indefinite and definite integration 17 min

324 2 +−= xxdxdy

What was y?

dxxx 324 2 +−∫

dxx

x∫

−

312

xx

dxdy +

−=12 find y.

55

( )( )2121 −−= xxdxdy

when x = – 1, y = 3. Find y.

Isn’t calculus lovely? If you’re interested, watch these videos which explain a bit more about where calculus came from and what integration actually is:

56

x

y

C

O

P

A

R

Pure - Integration Area under a curve (easy) 15 min

Find the area bound by the graph y = x2 + 2, the lines x = 1, x = 5 and the x axis

Find the shaded area

2 33 12 4

y x x= −

Find the shaded area

57

y

O x2

C

1 5

( )( )1 5y x x x= − −

58

y

x

C

L

R

O

A

R

O

B

x

y

Pure - Integration Area under a curve (harder) 11 min

Find the area shaded between the curves9y x= − and 2 2 3y x x= − +

R is the region bounded by 26y x x= − and 2y x= Find the area of R

59

x

y

O

A

B

C

Pure - Integration Area under a curve (even harder) 13 min

The curve322 6 10, 0y x x x= − + ≥ passes through the point A(1, 6)

and has a minimum turning point at B. Find the shaded area.

60

y

x

AB

NO

y x x x= – 8 + 203 2

R

Find the shaded area (there is an error in this video: 20 is written when it should be 20x)

61

Pure - Algebraic Methods - Proof Contradiction (Counter-example) 8 min

Definition of proof by contradiction:

Write down the proof that 2 is irrational

If !! is even then ! is even. Why is this true?

62

Pure - Algebraic Methods - Proof Deduction 5 min

Definition of the proof by deduction:

Write down the proof that the sum of any two consecutive odd numbers is a multiple of 4:

Write down the useful definitions of:

Even numbers

Odd numbers

63

Pure - Algebraic Methods - Proof Exhaustion 9 min

Definition of the proof by exhaustion:

Write the proof of the conjecture that 97 is a prime number:

As there are no factors < 97 ………

(Make sure you conclude your proof)

64

Pure – Vectors – Distance Between Points Defining and representing vectors in 3D 15 mins

i is the unit vector in the x direction

j is the unit vector in the y direction

k is the unit vector in the z direction

A is the point (1, 4, 7)

The position vector of A is:

Extension of Pythagoras to 3D - Length of a vector

The distance of O to A (2, 4, -3), or |a|, can be found by Pythagoras in 3D.

|a| =

In general, the length (modulus/magnitude) of a vector xi + yj + zk is:

Find the distance from the origin to the point P(4, -7, -1)

What mistake do students often make? Not you, you wouldn’t do this, I mean other students.

The distance between 2 points A and B is equal to the length of the vector AB

Read this ⇑ . He doesn’t explain it like this but I think it makes more sense?

The distance between ( )111 ,, zyxA and ( )222 ,, zyxB is

65

Example

Find the distance between the points A(1, 3, 4) and B(8, 6, -5) giving the answer correct to 1 dp.

Example

The coordinates of A and B are (5, 0, 3) and (4, 2, k) respectively. Given that the distance from A to B is 3 units, find the possible values of k.

66

Pure – Vectors 3D – Position Vectors Position vectors in 3 dimensions 15 mins

A, B, C, D are:

a, b, c, d are:

a = !" b = !" c = !" d = !"

Find vectors !"

!"

!"

!"

!"

!"

A position vector for the point A is:

This diagram is important. Annotate it as he does in the video:

!" means

!" means

67

In the diagram the points A and B have position vectors a and b respectively (referred to the origin O).

The point P divides AB in the ratio 1:2.

Find the position vector of P.

68

Pure: Trigonometry - Compound Angle Formulae https://youtu.be/DyqQG7MzOPU 9 min

sin(A±B)=sinAcosB±cosAsinB

cos(A±B)=cosAcosB∓sinAsinB

tan(A±B)=!"#!±!"#! ! ∓!"#$ !"#$

69

Pure: Trigonometry - The Double Angle Formulae https://youtu.be/upkil94kk_g 3 min

sin2A=2sinAcosA

tan2A=! !"#!!!!"#! !

cos2A=cos! ! − sin! !

Therearetwootherformulaeforcos2A.

cos2A=2 cos! ! − 1

cos2A=1 − 2 sin! !

70

Pure: Functions: The Modulus Function and its Graph Objectives: • Know what the modulus function does and what its graph looks like 7 min

What does mean?

x

23 −= xy 23 −= xy 1032 −−= xxy

1032 −−= xxy

xy sin=

xy sin= xy = 1+= xy

Whydoestheleft

handsidelooklike

this?

Whydoestheright

handsidelooklike

this?

71

Pure: Functions: The Modulus – Solving Modulus Equations Objectives: • Be able to solve equations involving the modulus function 12 min

For each modulus equation, draw an appropriate sketch and use it to find the solution(s) to the equation. Explain the method to your future self who will have forgotten how to do it.

a) |x + 1| = x + 4

b) |x| + 1 = x + 4 **he doesn’t do this one – see if you can do it**

c) |2x + 3| = 3x – 2

d) |2x + 3| = 6 – x

Exaggerate the steepness of the steeper function to make sure you get all the intercepts

72

e) |x2 – 4x| = 3x – 6

f) |2x – 4| = |x| + 2 **he doesn’t do this one – see if you can do it**

73

Pure - Functions Functions and mappings 5 min

74

Pure - Functions Domain and Range 10 min

The DOMAIN of is:

The RANGE of is:

Example:

Domain:

Note: For Edexcel you only need to write

or

Range:

Note: For Edexcel you only need to write

or

Example:

Domain:

Range:

Domain and Range song

)(xfy =

)(xfy =

{ }3| ≥xx

3≥x [ )∞,3

{ }0| ≥yy

0≥y [ )∞,0

3)( += xxf

=− )1(f

=)2(f

⇒

Seewhattheyvalueis?!

BING!!!!!

75

Pure - Functions Composite functions 6 min

What do we mean when we write ?

Examples

You try!

3)(,)( 2 −== xxgxxf

)(xgf

)2(gf )2(2f

)(xfg )(xgf

1)(,5)( 2 −=+= xxgxxf

)2(fg )(xgf )1(2g

76

Pure - Functions Inverse functions 12 min

Write out the steps for finding this inverse function

1)( += xxf

1)(1 −=− xxf

77

Write out the steps for finding this inverse function

What’s the standard trick in this example?

Write out the steps for finding this inverse function.

What’s the standard trick in this example?

xx

xf2)( +

=

12)(1−

=−

xxf

12)( 2 −+= xxxf

11)(1 ++−=− xxf

78

Pure - Exponentials and logs The basics 14 min

79

1. 2log3 =x 2. 416log =x 3. 2log4=x

4.21

log9 =x 5.31

2log −=x 6. x3log3 =

7. 2log31

x= 8.21

log16 −=x 9. x=25log5

10. x4log21=−

80

Pure - Exponentials and logs Sketching !! and !"# 6 min

y = ex Check on your calculator!

y = e2x y = ex+1 y = e-x

y = ex+1 y = 2ex y =- ex

Remember to show the x-intercept and/or y-intercept

Always label the equation of the asymptote!

81

y = lnx Check on your calculator!

y = ln(x+1) y = ln(-x) y = ln(2x)

y = lnx+1 y=-lnx

Remember to show the x-intercept and/or y-intercept

Always label the equation of the asymptote!

82

Pure - Exponentials and logs Solving equations using logs and powers 8 min

Useful Facts: Formula 1: Formula 2: Formula 3: Formula 4: 1. 4log2 =x 2. 185 =x 3. 024log8log4 =−− xx 4. 56772 =+ xx

83

5. 3loglog 39 =+ xx 6. 03339 1 =+−− + xxx

84

7. ( ) 24loglog2 33 =+− xx

8. ⎟⎠

⎞⎜⎝

⎛−+41

log6log5log 101010

9. ( ) ( ) 02196log32log2 2

33 =+−−− xxx Answers

85

Pure - Exponentials and Logs Modelling using logarithmic and power relationships 5 min

Sarah Swift got a speeding ticket on her way home from work. If she pays the fine now, there will be no added penalty. If she delays her payment, then a penalty will be assessed for the number of months, that she delays paying her fine. Her total fine, f in Euros is indicated in the table below. These numbers represent an exponential function.

Number of months t payment is delayed

Amount F of the fine

1 300

2 450

3 675

4 1012.50

What is the common ratio of consecutive values of F?

Write the formula for this function F =

What is the fine in Euros for Sarah’s speeding ticket if she pays it on time?

86

Pure - Differentiation Proof of the differentiation of !"# ! 5 min from 1st principles

In the 2nd year you will learn about the Small Angle Approximation for angles measured in RADIANS. This states that:

!"#$! ≈ ! (where x is measured in Radians and ≈ means approximately)

You will also learn about the Compound Angle Formulae one of which states that:

sin(A+B)=sinAcosB + cosAsinB

87

Pure - Differentiation Proof of the differentiation of !"# ! 5 min from 1st principles

In the 2nd year you will learn about the Small Angle Approximation for angles measured in RADIANS. This states that:

!"#$! ≈ ! (where x is measured in RADIANS and ≈ means approximately)

You will also learn about the Compound Angle Formulae one of which states that:

cos(A+B)=cosAcosB - sinAsinB

88

Pure - Differentiation Chain rule 13 min In words, how do we differentiate without expanding it out?

What does the chain rule say you can do?

Examples Chain Rule Songs

I haven’t done these in the video but you should try them to check you’re ok with this:

IMPORTANT You’re in an exam (not really, just pretend that you are) and you’ve just differentiated something. How can you check your answer? Write your idea here and we will discuss in class:

How can you write the chain rule down as a rule without mentioning a baby?

( )14sin += xy

12 +xedxd

⎟⎠

⎞⎜⎝

⎛− 31xdx

d

( )2sin xdxd

xdxd 2sin

( )1013 −= xy ( )xy 3sin2=

xy 2tan= 123 −= xy

89

Write out the example of differentiating ‘properly’ using t = 4x + 1.

Write out the example of differentiating ‘properly’ using

Useful (and fun) application of the chain rule:

what is in terms of x? Explain each step!

( )14sin += xy

12+= xey 12 += xt

yx tan=dxdy

90

Pure - Differentiation Product rule 8 min

What is the product rule?

Any self-respecting mathematician will want to see a proof of this formula. Here it is

Examples

Product rule song

How can you write the product rule down as a rule?

IMPORTANT You’re in an exam (not really, just pretend that you are) and you’ve just differentiated something. How can you check your answer? You should know this!

⇒

( )432 += xxy

( ) xxxy sin4 +=

( ) xxxy cossin12 +=

91

Pure - Differentiation Quotient rule 5 min

What is the quotient rule?

Any self-respecting mathematician will want to see a proof of this formula. Here it is

Examples

Quotient Rule

Songs

IMPORTANT You’re in an exam (not really, just pretend that you are) and you’ve just differentiated something. How can you check your answer? You should know this!

⇒

⇓

21

2 −

+=xx

yxx

y21

sin+

=

xx

ysincos4 +

=

92

Pure - Trigonometry Pythagorean trig identities – use in solving equations 2 min

This is an example of using a pythagorean trig identity to solve an equation.

Solve 4!"#$!!! − 9 = !"#$ !"# 0 ≤ ! ≤ 360

93

Pure: Trigonometry – Inverse Trig inc. Graphs https://youtu.be/hklOnHJx1t4

Whatisthedifferencebetweeny=sin-1xandy=(sinx)-1?…………………………………………………………………………………………………………………………Sketchthegraphsofy=arcsinx(y=sin-1x) y=arccosx(y=cos-1x)y=arctanx(y=tan-1x)

94

Pure - Differentiation Differentiating !! 9 min

! = !!! = !!!! = 5 + !!! = 3 − 5!!! = 3!!4

!"!" =

!"!" =

!"!" =

!"!" =

!"!" =

Findtheequationofthetangenttothecurve! = 3 − 2!!5 atthepointwhere! = 0.

Giveyouranswerintheform!" + !" + ! = 0where!, ! and !areintegers.

95

Pure - Differentiation Differentiating !" ! 7 min

! = !"#! = 2 + !"#! = 5 − 2!"#! = 2!"#3

!"!" =

!"!" =

!"!" =

!"!" =

Findthecoordinatesofthestationarypointonthecurve! = 34 ! −

3!"#4

Anyoneinterestedindoingmathsatuniversityshouldbeinterestedintheproofwhichishere!

96

Pure - Differentiation Differentiating !"!" 7 min

! = 3 tan 2! − 1 Find !"!"

97

Statistics (Paper 3, Section A)

● Topic 1 – Statistical sampling ● Topic 2 – Data presentation and interpretation ● Topic 3 – Probability ● Topic 4 – Statistical distributions ● Topic 5 – Statistical hypothesis testing

98

99

Statistics - Measures of location and spread Median and Interquartile Range (IQR) 9 min

2|5means25

0 6 7 8 1 0 2 3 4 7 7 7 8 9 2 1 3 4 5 5 7 3 1 1 2 6 6 9 4 1 5 5 6 9 5 6 7 9 Findthelowerquartile,median,upperquartile,IQRanddecideifthereareoutliers.

Ifitisawholenumber…………………………………………………………………….

Ifitisnotawholenumber…………………………………………………………………

100

Keystem=10s

9 0 6 8 3 5 7 7 1 6 6 6 0 2 2 4 5 1 1 2 4 3 4 7 8 3 5 7 2 1 6 Findthelowerquartile,median,upperquartile,IQRanddecideifthereareoutliers

101

Statistics - Measures of location and spread Mean and S.D with grouped data 10 min

Ungroupedfrequency

Maths test mark

No of people

1 6 2 5 3 7 4 4

Findthemeanandstandarddeviation

Groupedfrequency

height frequency 0-4 2 5-10 4 11-16 6 17-20 5 21-30 5

Findthemeanandstandarddeviation

102

Ungroupedfrequency

No of pets owned

No of people

1 4 2 6 3 2 4 2 Findthemeanandstandarddeviation

Groupedfrequency

English mark

Frequency

5-14 3 15-19 4 20-29 5 30-34 2

Calculatethemeanandstandarddeviation

103

Statistics - Measures of location and spread Mean and S.D of grouped data (by calculator) 3 min

x f

1 7

2 10

3 13

4 9

5 4

The buttons I need to press to calculate the mean and sd are:

104

Statistics - Measures of location and spread Interpolation Median and Interquartile Range (IQR) of grouped data

105

Statistics - Measures of location and spread Percentiles 7 min

Example 1 Calculate the 50th percentile for Bethany

43,54,56,61,62,66,68,69,69, 70,71,72,77,78,79,

85,87,88,89,93,95,96,98,99,99

Example 2 Calculate the 40th percentile for DeKwanye East

43,54,56,61,62,66,68,69,69, 70,71,72,77,78,79,

85,87,88,89,93,95,96,98,99,99

106

Statistics - Measures of location and spread Linear Coding 6 min

107

Statistics - Measures of location and spread Combined mean 2 min

Nadir asked 15 students about their AP results, their mean was 62.

He later asked 25 students about their AP results, their mean was 71.

Work out their combined mean.

108

Statistics – Histograms Intro This video shows you the key features of Histograms 3 min

109

Statistics – Histograms - Dimensions of Bars This video shows you how to find the dimensions of a Histogram 6 min

IMPORTANT!!!

CHECK THE BAR WIDTH!

110

Statistics: Probability: Venn Diagrams: Union

P(A∪B) 5 min Watch the examples then complete the questions for the Venn diagram shown (showing all working!) What is the tick rule for union?

i) P(A∪B)

ii) P(A∪B’) =

iii) P(A’∪B) =

iv) P(A’∪B’) =

v) P(B∪B’) =

111

Statistics: Probability: Venn Diagrams: Intersection

P(A∩B) 5 min

What is the tick rule for intersection?

i) P(A∩B) =

ii) P(A∩B’) =

iii) P(A’∩B) =

iv) P(A’∩B’) =

v) P(B∩B’) =

112

Statistics: Probability: Venn Diagrams: Addition Rule 8 min

Venn Diagrams

Formulae to Learn:

Addition Rule

Mutually Exclusive:

Independent:

!(!′ ∪ !⬚ )∪

P(AUB)’

OR=

AND=

113

Statistics: Probability: Venn Diagrams: Given

P(A|B) 10 min Watch the examples then answer the questions, showing the formula used and your working… What is the “Given” Formula:

i) P(A|B) =

ii) P(B|A) =

iii) P(A’|B) =

iv) P(A|B’) =

v) P(A’|B’) =

vi) P(B’|A’) =

114

Statistics: Probability: Tree Diagrams Tree diagrams 6 min

Abagcontains3blueballsand5redballs.Twoareselectedatrandomwithoutreplacement.Findthe

probabilitythat

a)theyarebothblue

b)thereisoneofeachcolour

Pythagoras tree…look it up!

Thereare5blacksocksand3redsinabag.Ipick2sockswithoutreplacement.FindtheprobabilityIget

a)twoofthesamecolouredsock

b)atleastoneredsock

115

Statistics: Probability: Tree Diagrams: Given Interpreting conditional probability using tree diagrams 6 min

116

Statistics - Probability Mutually exclusive and independent events 5 min

117

Statistics - Statistical distributions Discrete Random Variables (DRVs) 7mins

FindtheprobabilitydistributionforSthescoreonadie.

AdieisthrownuntileitherasixappearsorI’vethrownitthreetimes.FindtheprobabilitydistributionforT

whereTisthenumberofthrows.

118

Statistics - Statistical distributions Discrete uniform distribution 3mins

Write down the rules you need to learn:

119

Statistics - Statistical distributions Binomial Distribution 10mins

Write down the four properties for a binomial distribution:

i)

ii)

iii)

iv)

Example,

A die is thrown three times and a success is defined as when a 6 is thrown:

120

Statistics - Hypothesis testing for Binomial Test for a Binomial distribution 10 min

A 6 sided die is thrown 30 times and the number of sixes recorded.

Let X be the r.v. number of6’s thrown in 30 throws, !~ !(30, !)

0 1 2 3 4 5 6 7 8 9 10 …

One Tail Tests

121

Statistics - Hypothesis testing Lower tails test 9 min

Is a normal six sided die fair when 1 six is thrown in 24 throws?

Let X be the r.v. number of6’s thrown in 24 throws, !~ !(24, !)

On the Casio fx-991EX

To find the probability that x = 1 in the above example, follow these instructions

Menu

7: Distribution

4: Binomial

2: Variable

X: 1

N: 24

P: 1 ÷ 6 get p = 0.06037975302

Can then find the probability x = 0 and add them.

(0.06037975302 + 0.01257911521 = 0.07295886823)

Alternatively, to find a cumulative probability x ≤ 1, which can be more useful in general, follow these.

Menu

7: Distribution

Scroll down to 1: Binomial CD

2: Variable

X: 1

N: 24

get 0.07295886823, as before

122

Statistics - Hypothesis testing Upper tail tests 7 min

In Luigi’s restaurant on average 1 in 10 people order a bottle of Chardonnay. Out if a sample of 50 people, 11 chose Chardonnay. Has the drink become more popular? Test at the 1% level of significance.

Let X be the r.v.’ number people ordering a bottle of Chardonnay out of a sample of 50, where !~ !(50 !)

Conclusion

Please note

Where he refers to tables, use Casio fx-991EX instead.

Adapting instructions for Binomial CD from before to find p(X ≤ 10),

X: 1

N: 50

P: 0.1 get 0.9906453984

Then required probability is 1 - 0.9906453984 = 0.0093546…

123

Statistics - Hypothesis testing Critical values – lower tail test 8 min

A manufacturer claims that 2 out of 5 people prefer Soapy Suds washing powder over any other brand. For a sample of 25 people only 4 people are found to prefer Soapy Suds. Is the manufacturers claim justified? Test at the 5% level of significance.

Conclusion

Please note


Adapting instructions for Binomial CD from before to find p(X ≤ x),

X: try x = 2, 3, 4 … until you reach a probability greater than 5% (0.05)

N: 25

P: 0.4

This occurs when p(x ≤ 6) = 0.0735…, so X = 5 is the critical , p(x ≤ 5) = 0.0294

124

Statistics - Hypothesis testing Critical regions – upper tail test 8 min

A particular drug has a 1 in 4 chance of curing a certain disease. A new drug is developed to cure the disease. How many people would need to be cured in a sample of 20 if the new drug was deemed more successful at curing the disease than the old drug to obtain a significant result at the 5% level?

Let x be the r.v. ‘Number of people cured by the new drug’, where !~ !(20 !)

Conclusion

Please note


Adapting instructions for Binomial CD from before to find p(X ≤ r – 1) ≥ 0.95

X: try x = 6, 7, 8, … until you reach a probability greater than 0.95

N: 20

P: 0.25

So r – 1 ≥ 8, r ≥ 9

125

Statistics - Hypothesis testing Critical regions – two tail test 16 min A person suggests that the proportion, p of red cars on a road is 0.3. In a random sample of 15 cars it is desired to test the null hypothesis against the alternative hypotheses p ≠0.3 of a nominal significance level of 10%. Determine the appropriate rejection region and the corresponding actual significance level.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Let X be the r.v. ‘Number of red cars in a sample of 15’, where !~ !(15 !)

Conclusion

Please note


Adapting instructions for Binomial CD from before to find

a) For lower tail p(X ≤ xL) ≥ 0.05

X: try x = 0, 1, … until you reach a probability greater than 0.05

N: 15

P: 0.3

xL = 1

b) For upper tail p(X ≤ xM- 1) ≥ 0.95

XM-1 = 7, XM = 8

126

127

Mechanics (Paper 3, Section B) ● Topic 6 – Quantities and units in mechanics ● Topic 7 – Kinematics ● Topic 8 – Forces and Newton’s laws ● Topic 9 – Moments

128

129

Quantities & Units in Mechanics

130

General features of the graphs (GCSE revision)

Upwards sloping line =

Horizontal line =

Downwards sloping line=

Acceleration =

Example: (show full working and units)

From 0 to 100 seconds the acceleration is:

From 200 to 500 seconds the acceleration is:

What is the significance of the minus sign?

What is the total distance travelled?

(show full working and units)

Mechanics - Constant acceleration VT graphs 7 min

131

Mechanics - Constant acceleration SUVAT 5 min

Write down what the letters stand for in the SUVAT equations

S

U

V

A

T

The four equations of motion are:

(These are the same equations you will use if you are doing physics)

V =

S =

S =

V! =

There is a fifth equation that is useful – you will learn this in class

V =

What is it about the acceleration that makes these equations work?

When don’t they work?

132

Mechanics - Projectiles Projectile motion 17 min

A particle is projected at an angle of elevation of 40o at a speed of 30ms-1

Time of flight T

Which direction of motion are you considering?

s=

u=

v=

a=

t=

Which equation are you using?

T=

Maximum height h

s=

u=

v=

a=

t=

Which equation are you using?

h=

Range R

R=

133

Mechanics - Forces and motion Free body (Forces) Diagrams 6 min

When drawing free body diagrams you should:

Include √ Don’t include X

How should you align your coordinate system?

Sketch the diagrams on the next page using the video for help, then sketch the free body diagram of this climber.

134

Sketch the free body diagram of:

Free body diagram (label the forces) Coordinate system

A stationary car resting on a platform

A car that is accelerating to the right

A car that is falling to earth

A car launched upwards at the top of its arc

A car accelerating down a ramp

A car held at rest on a slope

Fnormal

Fgravity

135

Newton’s first law states:

Newton’s second law states:

Newton’s third law:

When an object is in equilibrium, what does that tell us about the forces on it?

Why can the reindeer move the sleigh?

(You only need to watch the first 6 min of this video as the rest is a revision of force diagrams)

Mechanics - Forces and motion Newton’s Laws 6 min

136

Mechanics - Forces and motion Resolving forces on a slope 7 mins

Complete the forces diagram for a block on the slope.

Add the resolved components of the force due to gravity, and label the angles

The component of the force due to gravity parallel to the slope is:

Fparallel=

The component of the force due to gravity perpendicular to the slope is:

Fperpendicular=

137

Mechanics - Forces and motion Connected particles – pulleys 12 mins

Complete the forces diagram

of this pulley system:

Write out the steps needed to find the acceleration of the system, and the tension in the ropes:

Acceleration:

Tension in the rope connecting the masses:

Tension in the rope holding the pulley:

138

Mechanics - Forces and motion Connected particles – on a slope 16 min

A particle A of mass 2kg is attached by a light inextensible string, passing over a smooth pulley to a particle B of mass 4kg as shown in the diagram. A rests on a rough plane inclined at 30o to the horizontal.

If the particles are released from rest, and the coefficient of friction between A and the plane is 0.4 find: (i) the acceleration of A, (ii) the tension in the string.

139

Mechanics: Forces and Motion - Friction Friction (Fmax=µR) 8 min

Fmax = µ R Define the terms:

Fmax

µ

R

What does limiting equilibrium mean?

What are the two situations where the frictional force is at its maximum?

1)

2)

140

Mechanics - Forces and Motion Single particle problems including friction 7 min

A box weighing 100 N is at rest on a horizontal floor. The coefficient of friction between the box and the floor is 0.4. What is the smallest force F extended eastwards and upwards at an angle of 30o with the horizontal that can start the box moving?

AS Double Mathematics - ingridflynn.weebly.com · AS Double Mathematics Blue Block P1, ... (x, y) plane Topic 3 ... Probability: Tree Diagrams – Given (Conditional Probability)

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