BHASVIC Maths Linear A level Y1 Learning Pack

A LEVEL MATHEMATICS

YEAR 1 LEARNING PACK CONTENTS

To be watched before the lesson on week beginning:

Ref in spec

Videos Page Watched

0 9th Sept P2 Sketching factorised cubic functions 1P2 Sketching quartics 2P2 Sketching reciprocals 3P2 Graph Transformations 4-5

1 16th Sept

P3 Equation of a line 6P3 Equation of the perpendicular bisector of a line 72P2 Linear and quadratic simultaneous equations 7-8P3 Equation of a circle 73-74P3 Using circle properties to solve problems on

coordinate grids75

2 23rd Sept P2 Completing the square where a is not equal to 1 10P2 Discriminant 12P2 Sketching Quadratic Graphs 11P2 Solving Quadratic inequalities 9

3 30th Sept SM8 Newton’s laws 82SM8 Free body diagrams (also known as force diagrams) 83-84SM7 VT graphs 80SM7 SUVAT 81

4 7th Oct SM2 Mean and sd of a list 95-96SM2 Median and Interquartile Range (IQR) of a list 97-98SM2 Mean and sd with grouped data 99-

100SM2 Mean and sd of grouped data (by calculator) 101SM2 Interpolation Median and Interquartile Range (IQR)

of grouped data 105

SM2 Percentile Approximation 102SM2 Linear Coding 103SM2 Combined mean 104SM2 The three cases of skew 94

5 14th Oct SM8 Resolving forces 856 21st Oct P7 An introduction to differentiation 13

P7 Differentiation from 1st principles 15P7 Equations of Tangents and Normals 14P7 Increasing/decreasing functions 17

HALF TERMTo be watched Ref in Videos Page Watched

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BHASVIC Maths Linear A level Y1 Learning Pack

before the lesson on week beginning:

spec

7 4th Nov P7 2nd derivative and classification of turning points 16P7 Sketching gradient functions 18P7 Optimisation problems 19

READING WEEK9 18th Nov P6 Exponentials and logs – the basics 29

P6 Sketching ex and ln x 30-32P6 Log rules 33P6 Solving index form equations using logs and

powers34

P6 Solving log form equations using logs and powers 35-36P6 Modelling using logarithmic and power

relationships37

10 25th Nov P10 Defining and representing vectors in 2D 68-69SM8 Friction (FMAX=μR ¿ 86SM8 Single particle problems including friction 87

11 2nd Dec SM3 Venn diagrams 106SM3 Tree diagrams 108SM3 Sample space diagrams 109-

110SM3 Conditional probability 111SM3 Mutually exclusive and independent events 107

12 9th Dec P5 Radian measure 42P5 Graphs of standard trig functions and solving mini

trig equations45-46

13 16th Dec SM4 Discrete Random variables (DRVs) 112CHRISTMAS HOLIDAY

14 6th Jan P2 Factor and remainder theorems 66-67P2 Algebraic division 63-64P2 Factorising cubics 65P8 Introduction to indefinite integration 53P8 Definite Integration 54-55

15 13th Jan P8 Integration as limit of a sum 56P8 Areas under a curve (easy) 57-58P8 Areas under a curve (harder) 59-60P8 Areas under a curve (even harder) 61-62

16 20th Jan SM8 Connected particles - pulleys 88SM8 Connected particles – on a slope 89SM8 Connected particles – a lift problem 90

17 27th Jan SM4 Binomial distribution 113SM4 Binomial distribution – cumulative probabilities 114

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BHASVIC Maths Linear A level Y1 Learning Pack

To be watched before the lesson on week beginning:

Ref in spec

Videos Page Watched

18 3rd Feb P10 Defining and representing vectors in 3D 70-71P10 Vector problems - ships 91-92

19 10th Feb P7 Projectile motion 93 HALF TERM

20 24th Feb P5 Reciprocal trig functions 47P5 Reciprocal trig graphs 48P5 Harder trig 49-50P5 Pythagorean trig identities - proof 51P5 Pythagorean trig identities -use in solving

equations52

21 2nd Mar P7 Differentiation of trig functions 20P7 Differentiating ex 21P7 Differentiating ln x 22P7 Chain rule 23-24P7 Product rule 25-26P7 Quotient rule 27-28

22 9th Mar P2 Functions and mappings 76P2 Domain and range 77P2 Composite functions 78P2 Inverse functions f-1(x) 79

23 16th Mar SM5 Test for a Binomial distribution 115SM5 Lower tail test -p value method 116SM5 Upper tail test- p value method 117SM5 Lower tail test- critical region method 118SM5 Upper tail test- critical region method 119SM5 Two tail test-critical region method 120

24 23rd Mar P5 Radian measure and its applications (TOOLS) 43-4425 30th Mar P1 Proof by counter example 39

P1 Proof by contradiction 38P1 Proof by deduction 40P1 Proof by exhaustion 41PROGRESSION EXAM REVISION- STUDY LEAVE- HALF TERM

Ref to spec Topic Ref to spec TopicP1 Proof SM1 SamplingP2 Algebra and functions SM2 Data Presentation and interpretationP3 Coordinate geometry in the (x,y) plane SM3 ProbabilityP4 Sequences and series SM4 Statistical distributionsP5 Trigonometry SM5 Statistical hypothesis testingP6 Exponentials and logarithms SM6 Quantities and units in mechanicsP7 Differentiation SM7 KinematicsP8 Integration SM8 Forces and newton’s lawsP9 Numerical methods SM9 Moments

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BHASVIC Maths Linear A level Y1 Learning Pack

P10 Vectors

Pure - Graphs and transformationsSketching factorised cubic functions

https://youtu.be/7DtbMGBO-vk 14 min

Graph of = x 3 Graph of = - x 3

y= ( x−2)( x+3)( x−1 ) y=x (4−x ) (2+ x )

Crosses x axis at: Crosses x axis at:

Crosses y axis at: Crosses y axis at:

y= (2x+1)(2−x )( x−3 ) y= (x−1 ) ( x−2 )2

Crosses x axis at: Crosses x axis at:

Crosses y axis at: Crosses y axis at:

1

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BHASVIC Maths Linear A level Y1 Learning Pack

Pure - Graphs and transformationsSketching quartics

https://www.youtube.com/watch?v=wruKcSlo0N4 5 min

A quartic function has the form f ( x )=a x4+b x3+c x2+dx+e

Sketch the graphs of:

1. y=(x+2)(x+1)(x−1)(x−3) 2. y=−x(x+3)(x+1)( x−3)

3. y=(x+1)2(x−1)(x−3) 4. y=−(x+3)(x+1)¿

5. y=¿ 6. y=−¿

7. y= (x+1 ) ¿ 8. y=−¿

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BHASVIC Maths Linear A level Y1 Learning Pack

Pure - Graphs and transformationsSketching reciprocals No video to watch here. Use graphing software to check your answers

Using your knowledge of the shape of the graph of y=1x to sketch graphs of these

equations.

Mark the asymptotes with a dotted line and label them with their equation.

i) y= 1x2

ii) y= 1x2 +3

iii) y= 1x2−2

iv) y= 3x2

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BHASVIC Maths Linear A level Y1 Learning Pack

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BHASVIC Maths Linear A level Y1 Learning Pack

Pure - Graphs Transformations Translations, stretches and reflections https://www.youtube.com/watch?v=RBYzZ1UhghE 10 minGraph Transformations: Shifts

ActivitySketch on this graph a) f(x) + 2 b) f(x + 2)

Shifts Notation The plot (2.-3) moves to?

Which co-ordinate always changes

Outside f(x) + 2

f(x) – 2

Inside f(x+2)

f(x-2)

Graph Transformations: Stretches

ActivitySketch on this graph a) 2f(x) b) f(2x)

Stretches Notation The plot (2.-3) moves to?

Which co-ordinate always changes

Outside 2f(x)

Inside f(2x)

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BHASVIC Maths Linear A level Y1 Learning Pack

Graphs and Transformations (continued)Graphs Transformations: Reflections

Activity Sketch on this graph a) - f (x) b) f(-x)

Shifts Notation The plot (2.-3) moves to?

Which co-ordinate always changes

Outside -f(x)

Inside f(-x)

Summary of Graph Transformations

Type Outside Description Inside DescriptionShifts f(x) + a f(x+a)

f(x) – a f(x-a)

Stretches af(x) f(ax)

Reflections

-f(x) f(-x)

Pure: Co-ordinate Geometry

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BHASVIC Maths Linear A level Y1 Learning Pack

Equation of a line*One of these questions is not answered in the video. Try it and check in class. https://www.youtube.com/watch?v=qyUHHxthtcw&feature=youtu.be 7 min

What is the equation of the line through (3, 7) with gradient equal to 4?

What is the equation of a straight line which passes through (x1, y1) with gradient m

*What is the equation of a straight line which passes through (1,4) with gradient 7

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BHASVIC Maths Linear A level Y1 Learning Pack

What is the equation of a straight line which passes through (x1, y1) and (x2, y2)

What is the equation of a straight line which passes through (1,-2) and (-3,-7). Show your working.

Pure - Equations and inequalitiesLinear and quadratic simultaneous equations

https://www.youtube.com/watch?v=bToJwG9Snmk 20 min

This is GCSE Revision, if you can answer the questions on your own, you don’t need to watch the video. But it is there if you need it. You can scroll to the end of the video to check your answers.

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)

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BHASVIC Maths Linear A level Y1 Learning Pack

Level 1

2 x+3 y=104 x+ y=20 Little sketch of what you’re finding:

Level 2

2 x+3 y=104 x+ y2=20 Little sketch of what you’re finding:

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BHASVIC Maths Linear A level Y1 Learning Pack

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

2 x+3 y=104 x2+ y2=20 Little sketch of what you’re finding:

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BHASVIC Maths Linear A level Y1 Learning Pack

Pure - Equations and inequalities Solving Quadratic inequalities

https://youtu.be/2QrU94W72Rc 6 min

1) x2−7 x+12≥ 2

We also need to write solutions using set notation.

For this example, {x : x ≥ 5 U x ≤ 2}

X Is the set of number such that x is greeater or equal to 5 or less than or equal to 2.

2) x2+7x<4 x+5

Write this solution using set notation.

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BHASVIC Maths Linear A level Y1 Learning Pack

Pure – Quadratics Completing the square where a is not equal to 1

https://www.youtube.com/watch?v=TE0ISEBm9a0 8 min

Example 1:

Complete the square in order to solve 2 x2−8 x+7=0

Example 2:

Complete the square in order to solve 5 x2+8x−2=0

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BHASVIC Maths Linear A level Y1 Learning Pack

Pure – Quadratics Sketching Quadratic graphs

https://www.youtube.com/watch?v=8-LrQmpU8Uc 5 min

Sketching A Quadratic Graph

Example: Sketch the graph of y=x2+8 x+10

Sketching Checklist:

Crosses y axis when x = 0. Therefore y =

Crosses x axis when y = 0. Solve x2+8 x+10=0 (Can factorise, use quadratic formula or complete the square)

Co-ordinates of Turning/Stationary Point. Use the completed square form.

Pure – Quadratics16

BHASVIC Maths Linear A level Y1 Learning Pack

Discriminant

https://www.youtube.com/watch?v=YFUj375ubl8 6 min

Quadratic Function

Value of the Discriminant

Corresponding graph of Quadratic function

Number and Type of Roots

Question: Determine the number of the roots of these quadratic equations

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BHASVIC Maths Linear A level Y1 Learning Pack

Pure – Differentiation An introduction to differentiation

https://www.youtube.com/watch?v=VDLnI9dOuTc 13 min

18

The gradient function gives the gradient of a tangent to the curve at a specific point on the curve.

When the gradient function at a specific point is ZERO. This means that the tangent is a horizontal line and at that point the curve has a turning point (stationary point/max or min point).

y =

dydx

=¿

BHASVIC Maths Linear A level Y1 Learning Pack

Pure - DifferentiationEquations of tangents and normals https://www.youtube.com/watch?v=FNPdO9VyCRw 10 min

Watch the first example.

y=2 x3−x2+2 x−1What is the equation of the tangent at x = -1?

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

y=3 x2−2 x+5What is the equation of the tangent at x = 2?

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

Pure - Differentiation

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BHASVIC Maths Linear A level Y1 Learning Pack

Differentiation from 1st principles https://youtu.be/IGTcIUriGBs 3 min

Write out the steps of the proof of the derivative of f(x) = x2 + 3x - 4 from first principles.

Notation: f’(x) is the same as dydx

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BHASVIC Maths Linear A level Y1 Learning Pack

Pure - Differentiation2nd derivative and classification of turning points

https://www.youtube.com/watch?v=7TBhZliDH2g 13 min

y=2 x3+ 72

x2−5 x+10 DO THE BIRD!Find the coordinates of the stationary points

y=2 x3+ 72

x2−5 x+10

Find and classify the stationary points

Watch this! A little sonG

Pure - Differentiation

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BHASVIC Maths Linear A level Y1 Learning Pack

Increasing/decreasing functions https://www.youtube.com/watch?v=KxOp3s9ottg&feature=youtu.be 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.

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BHASVIC Maths Linear A level Y1 Learning Pack

Pure - DifferentiationSketching gradient functions https://youtu.be/XK9H2IPgMNM 5 min

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

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Copy this one yourself

BHASVIC Maths Linear A level Y1 Learning Pack

Pure - Differentiation

Optimisation problems

https://www.youtube.com/watch?v=aGAfMwg4JlY&list=UUhk8LXCP7lzhEK2lvyaQ7JQ&index=3612 min

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BHASVIC Maths Linear A level Y1 Learning Pack

Pure - DifferentiationDifferentiation of trig functions

https://www.youtube.com/watch?v=Ysv6neJO6CM 12 min

Please watch the whole of the video, but all we need are these STANDARD DERIVATIVES.

xdxd sin

x

dxd

cos

xdxd

tan xdxd

sec

xdxd

cosec ddx

cot x=¿¿

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BHASVIC Maths Linear A level Y1 Learning Pack

Pure - DifferentiationDifferentiating ex

https://youtu.be/1vdc8egFNKE 9 min

y=ex y=A ex y=5+ex y=3−5 ex s=3e t

4

dydx

=¿ dydx

=¿ dydx

=¿ dydx

=¿ dsdt

=¿

Find the equation of the tangent to the curve y=3−2ex

5 at the point where x=0.

Give your answer in the form ax+by+c=0 where a , b∧c are integers.

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BHASVIC Maths Linear A level Y1 Learning Pack

Pure - DifferentiationDifferentiating ln x

https://youtu.be/FO3D9AW2ZQY 7 min

y=lnx y=2+lnx y=5−2 lnx s=2 lnt3

dydx

=¿ dydx

=¿ dydx

=¿ dsdt

=¿

Find the coordinates of the stationary point on the curve y=34

x−3 lnx4

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BHASVIC Maths Linear A level Y1 Learning Pack

Pure - DifferentiationChain rule

https://www.youtube.com/watch?v=l8mVQ9FZe-k 10 min

Differentiate y= (4 x+1 )3

Examples

y=3 (1−x2)6 y= 13 x2+4

Chain Rule Song ⇓

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BHASVIC Maths Linear A level Y1 Learning Pack

Another Chain Rule Song ⇓

y=sin (4 x+10 ) y=cos( x2+1 )

y=sin ( x2 ) y=sin2 x

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

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BHASVIC Maths Linear A level Y1 Learning Pack

Pure - DifferentiationProduct rule

https://www.youtube.com/watch?v=rHNrXmLdphY 12 min

What is the product rule? If y = uv then dydx =

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

Examples Product Rule

Song ⇓

y=x2 (1+2 x )

y=( x+1)2(2 x−1)

y=x2cos 3 x

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BHASVIC Maths Linear A level Y1 Learning Pack

Exam type questionddx

((x+1)¿¿3(2 x−1)4)=( x+1 )2 (2 x−1 )3(ax+b)¿

a) Find the values a and bb) Find the x coordinates of the stationary points of y= (x+1 )3 (2 x−1 )4

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

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BHASVIC Maths Linear A level Y1 Learning Pack

Pure - DifferentiationQuotient rule (the sound goes a bit at the end of the simplifying in one question)https://www.youtube.com/watch?v=hJTS3zsl0SM 6 min

What is the quotient rule? If y = uv then

dydx =

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

Examples Differentiate

Quotient

Rule Song ⇓

y= x

x2+1

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BHASVIC Maths Linear A level Y1 Learning Pack

y= 2 x+1

( x+2 )2 Quotient Rule

Song ⇓

y=3 sin 3 xx3

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BHASVIC Maths Linear A level Y1 Learning Pack

Pure - Exponentials and logsThe basics

https://www.youtube.com/watch?v=dZybQLZg6Tc 4 min

Index form BP = A Log form logBA = P

23 = 8

32 = 9

log525 = 2

log101000 = 3

42 = 16

7-1 = 1/7

30 = 1

log5125 = 3

log100.1 = -1

log101000 = 3

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BHASVIC Maths Linear A level Y1 Learning Pack

Pure - Exponentials and logsSketching ex and lnx https://www.youtube.com/watch?v=EWVGIvhutVs&feature=youtu.be 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!

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BHASVIC Maths Linear A level Y1 Learning Pack

y = lnx Check on your

calculator!

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

y = lnx+1 y=-lnxRemember to show the x-intercept and/or y-intercept

Always label the equation of the asymptote!

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BHASVIC Maths Linear A level Y1 Learning Pack

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BHASVIC Maths Linear A level Y1 Learning Pack

Pure - Exponentials and logsLog rules

https://www.youtube.com/watch?v=UPakc_o8cO0 7 min

Addition rule

Subtraction rule

Power rule

Change of base rule

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BHASVIC Maths Linear A level Y1 Learning Pack

Pure - Exponentials and logsSolving index form equations using logs and powers https://www.youtube.com/watch?v=QC1Nv2_gJTc 10 min

Solving easy index equations

Solve

1. 2x = 3 2. 3x = 10

3. 4x = 5 4. 6x = 7

Solving harder index equations

72 x +7x=56

9x−3x+1−3x+3=0

39

Is it in the form BP= A ?

BHASVIC Maths Linear A level Y1 Learning Pack

Pure - Exponentials and logsSolving log form equations using logs and powers https://www.youtube.com/watch?v=ARhQrb5pphY 16 min

Solve

1. log3 x=2 2. log x 16=4 3. x= log4 2

4. log 9 x=1

2 5. log x 2=−1

3 6. 3=log3 x

7.

13=log x 2

8. log5 25=x

40

Is it in the form

LogBA = P?

BHASVIC Maths Linear A level Y1 Learning Pack

Solving harder log equations

log10( x+21)=2−log10 x

log4 (x2 )=1+ log4 3 x

2 log3 x− log3 ( x+4 )=2

log 4 x−8 logx 4−2=0

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BHASVIC Maths Linear A level Y1 Learning Pack

Pure - Exponentials and logsModelling using logarithmic and power relationships

https://www.youtube.com/watch?v=2iZXFd_ZImA 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?

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BHASVIC Maths Linear A level Y1 Learning Pack

Pure - Algebraic methods - ProofProof by contradiction

8 minhttps://www.youtube.com/watch?v=VNZoB0qao1U&feature=youtu.be

Definition of proof by contradiction:

Write down the proof that √2 is irrational

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BHASVIC Maths Linear A level Y1 Learning Pack

Pure - Algebraic methods - Proof

Proof by Counter Example

https://youtu.be/mxcGpNji4i k 6 min

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BHASVIC Maths Linear A level Y1 Learning Pack

Pure - Algebraic methods - ProofProof by deduction

https://youtu.be/_REBwAie_Tk 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

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BHASVIC Maths Linear A level Y1 Learning Pack

Odd numbersPure - Algebraic methods - ProofProof by exhaustion

https://youtu.be/1amtOenZEwU 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 ………

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BHASVIC Maths Linear A level Y1 Learning Pack

(Make sure you conclude your proof)

Pure - TrigonometryRadian Measure

https://youtu.be/iH0lK3Iy6pw 13 min

270° = 225° =

210° = 120° =

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BHASVIC Maths Linear A level Y1 Learning Pack

Pure - TrigonometryRadian measure and its applications (TOOLS)

https://www.youtube.com/watch?v=pYhHuIlRi5o 17 min

T Triangle Area

O Sector Area

O Arc Length

L Cosine Rule

S Sine Rule

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BHASVIC Maths Linear A level Y1 Learning Pack

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BHASVIC Maths Linear A level Y1 Learning Pack

Pure - TrigonometryGraphs of standard trig functions and solving mini trig equationshttps://www.youtube.com/watch?v=tdyUxjjQ1OY 13 min

sin x=√32 0≤x≤360

sin x=√32 0≤x≤2 π

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BHASVIC Maths Linear A level Y1 Learning Pack

cos x=−√32 0≤x≤360

2 tan x = - 2 0≤x≤360

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BHASVIC Maths Linear A level Y1 Learning Pack

Pure – TrigonometryReciprocal trig functions https://www.youtube.com/watch?v=ERogWYvsCts 3 min

Write down the three reciprocal trig functions

Secant θ (sec θ) =

Cosecant θ (cosec θ) =

Cotangent (cot θ) =

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

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BHASVIC Maths Linear A level Y1 Learning Pack

Pure – TrigonometryReciprocal trig graphs https://www.youtube.com/watch?v=c-Av9XSzQo8 13 min

y = cosec θ

y = sec θ

y = cot θ

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BHASVIC Maths Linear A level Y1 Learning Pack

Pure -harder trig equations Harder trig identities

https://www.youtube.com/watch?v=2idsvY-fUDY 8 min

LEARN: sin2 x+cos2 x=1

ALSO LEARN:

sin xcos x

=tan x

sin2 x−2sin x−3=0

sin2 x−sin x=0

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BHASVIC Maths Linear A level Y1 Learning Pack

cos2 x−sin x+1=0

sin x−4 cos x=0

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BHASVIC Maths Linear A level Y1 Learning Pack

Pure - TrigonometryPythagorean trig identities – proof

https://www.youtube.com/watch?v=bhQ5CXjxttI 6 min

56

Prove that: 1 + cot2 θ = cosec2θ

Divide through by sin2 θ

Prove that: tan2 θ + 1 = sec2 θ

Divide through by cos2 θ

BHASVIC Maths Linear A level Y1 Learning Pack

Ask me about the beach and the baby

Pure - TrigonometryPythagorean trig identities – use in solving equations

https://www.youtube.com/watch?time_continue=86&v=zk8Bled7bFc 2 min

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

Solve 4 cosec2θ−9=cotθ for 0≤ θ ≤ 360

57

Prove that: tan2 θ + 1 = sec2 θ

Divide through by cos2 θ

BHASVIC Maths Linear A level Y1 Learning Pack

Pure - IntegrationIntroduction to indefinite integration https://www.youtube.com/watch?v=oqXo-ezFzUU 6 min

Example:

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BHASVIC Maths Linear A level Y1 Learning Pack

Pure - IntegrationDefinite integration

https://www.youtube.com/watch?v=62Zf5HgUQhc 13 min

∫4

6

3+4 x3− 2x2 dx

∫1

2

(2√ x+3 )2dx

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BHASVIC Maths Linear A level Y1 Learning Pack

∫−3

−1

x3+3x2+5 dx

60

Pure – IntegrationIntegration as limit of a sum https://www.youtube.com/watch?v=YtKEd1xmf7U&feature=youtu.be 12 min

BHASVIC Maths Linear A level Y1 Learning Pack

Pure - IntegrationArea under a curve (easy)

https://www.youtube.com/watch?v=xlyVrUqtJ5A 16 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

Find the shaded area61

x

y

C

O

P

A

R

BHASVIC Maths Linear A level Y1 Learning Pack

Pure - Integration62

y

O x2

C

1 5

BHASVIC Maths Linear A level Y1 Learning Pack

Area under a curve (harder) https://www.youtube.com/watch?v=K-0kmt7Ubok 11 min

Find the area shaded between the curves

and

63

A

R

O

B

x

y

BHASVIC Maths Linear A level Y1 Learning Pack

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BHASVIC Maths Linear A level Y1 Learning Pack

R is the

region bounded by and Find the area of R

65

y

x

C

L

R

O

BHASVIC Maths Linear A level Y1 Learning Pack

Pure - IntegrationArea under a curve (even harder)

https://www.youtube.com/watch?v=y6PUuKgus-I 13 min

The curve passes through the point A(1, 6) and has a minimum turning point at B. Find the shaded area.

66

x

y

O

A

B

C

BHASVIC Maths Linear A level Y1 Learning Pack

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

20x)

67

y

x

AB

NO

y x x x= – 8 + 2 03 2

R

BHASVIC Maths Linear A level Y1 Learning Pack

Pure - AlgebraAlgebraic division

https://www.youtube.com/watch?v=ufZEGRo9l0I 16 min

Understanding the structure of what you’re doing

Numbers Algebra

73= x3+x2+2 x−1

x2−1=

7= x3+x2+2 x−1=

General idea of the method: |

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DMS (Doc Marten Shoes) to remember to divide, multiply, subtract.

Don’t forget to Edward Scissorhands back down!

BHASVIC Maths Linear A level Y1 Learning Pack

Example 1 x3+x2+2 x−1

x2−1=

Silly long division song

Example 2 (pause the video!) x3+3 x2+x+2

x+1=

IMPORTANT You’re in an exam (not really, just pretend that you are) and you’ve just done this question. You’re going to use your answer in part (b) of the question so if you’ve made a mistake then you’re going to lose a LOT of accuracy marks. How can you check your answer? Write your idea here and we will discuss in class.

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BHASVIC Maths Linear A level Y1 Learning Pack

Pure - AlgebraFactorising cubics

https://www.youtube.com/watch?v=DJU7U9x71sM 5 min

Factorise 2 x3+7 x2−14 x+5 using the fact that 2 x−1 is a factor

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BHASVIC Maths Linear A level Y1 Learning Pack

Pure - AlgebraFactor and remainder theorems

https://www.youtube.com/watch?v=456VjfXmtiA 10 min

The Remainder theorem

133 =

2 x3−4 x+1x−2 =

Find the remainder of

x3+x2+x+4x+1

Find the remainder when x3+2x5+6x is divided by (x+1)

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BHASVIC Maths Linear A level Y1 Learning Pack

Find the remainder when x4 - 2x3 + 4x + 5 is divided by (x-2)

The Factor theoremShow that (x+2) is a factor of x3 + 3x2 + 5x + 6

Show that (x-3) is a factor of x3 - 3x2 + 4x -12

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BHASVIC Maths Linear A level Y1 Learning Pack

Pure - VectorsDefining and representing vectors in 2D https://www.youtube.com/watch?v=Gw4nPxef48g 18 min

Vectors can be written in component form O⃗A = 5i + 7j = (57)

or in column vector form O⃗A = (57) Both have advantages but almost everyone prefers working with column vectors.

Write out the component and column vector form of the following vectors (reminder - the first 3 are called ‘position vectors’ but the 4th one is just called a ‘vector’):

O⃗B= O⃗C=

O⃗D= A⃗C=

What is the difference between (5 , 7 ) and (57 )?

The points A and B in the diagram have coordinates (3, 4) and (11, 2) respectively.

Find the position vector of A.

Find the position vector of B.

Find the vector A⃗B

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BHASVIC Maths Linear A level Y1 Learning Pack

Adding and subtracting vectors in component form (hint – write them in column vector notation!)

a + b = a + b + c =

a – b = b – c =

Given that a = 2i + 5j, b = 12i – 10j and c = –3i + 9j, find a + b + c, using column matrix notation (column vector notation) in your working

How would you find the modulus (magnitude) of xi+ yj ?a = i + j b = 2i + 3j

Find |a| Find |b|

Find |a+b| Find |b−a|

The vector a is equal to 5i – 12j. Find |a| and find a unit vector in the same direction as a.

The vector a is equal to 5i + j and b = –2i – 4j. Find the exact value of |2 a+b|

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BHASVIC Maths Linear A level Y1 Learning Pack

Pure - VectorsDefining and representing vectors in 3D

https://www.youtube.com/watch?v=Gw4nPxef48g 15 min

i is the unit vector in the x directionj 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 A⃗B

Read this ⇑

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

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BHASVIC Maths Linear A level Y1 Learning Pack

The distance between A ( x1 , y1 , z1)

and B ( x2 , y2 , z2)

is

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.

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BHASVIC Maths Linear A level Y1 Learning Pack

Pure - CirclesEquation of the perpendicular bisector of a line

https://www.youtube.com/watch?v=yAKWp480CzU 8 min

Find the equation of the perpendicular bisector of the line joining the points A (3, 5) and B (-2, -1) giving your answer in the form ax + by + c = 0 where a, b and c are integers.

Midpoint of AB =

Gradient of AB =

Perp. Gradient =

Equation of perpendicular bisector is:

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BHASVIC Maths Linear A level Y1 Learning Pack

Pure - CirclesEquation of a circle

https://www.youtube.com/watch?v=EN3F1KWgd6w 11 min

Most important top tip:

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BHASVIC Maths Linear A level Y1 Learning Pack

Pure - CirclesUsing circle properties to solve problems on coordinate grids

https://youtu.be/Joxsb1VOBZ4 10 min

i) Centre

ii) Gradient, m =

iii) CP =

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BHASVIC Maths Linear A level Y1 Learning Pack

Pure - FunctionsFunctions and mappings https://youtu.be/3BmYm5lbzkk 5 min

List of ordered pairs Mapping Table of values Graph

Determine if each mapping is a function, or not a function.

Domain Range Domain Range Domain Range

You can use the vertical line test for graphs of equations not just individual points. Which graphs are function?

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1

2

3

-5

3

9

-3

2

3

-2

6

-1

5

-4

-1

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BHASVIC Maths Linear A level Y1 Learning Pack

Pure - FunctionsDomain and Range

https://youtu.be/uz0fgAbLLjg?t=7 10 min

Complete the sketches for the functions on the video and complete the domain and range analyses.

f(x) = 2x – 1 Domain -1≤ x ≤ 1 f(x) = 2x – 1

g(x) = x2 Domain x ε R g(x) = x2 Domain -1≤ x < 3

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BHASVIC Maths Linear A level Y1 Learning Pack

Pure - FunctionsComposite functions

https://www.youtube.com/watch?v=nISOXLGwk-c 6 min

f ( x )=x2 , g ( x )=x−3

3 f f(3) = 9 g g(9) = 6

What do we mean when we write gf(x)?

Examples

gf (2 )=¿ f 2 (2 )=¿

fg ( x )=¿ g f ( x )=¿

Try

f ( x )=x2+5 , g (x )=x−1

fg (2 )=¿ gf (x )=¿ g2 (1 )=¿

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BHASVIC Maths Linear A level Y1 Learning Pack

Pure - FunctionsInverse functions f -1(x)

https://youtu.be/vgwPqKkZd_0?t=7 8 min

Watch the video and complete the notes for finding the inverse of a function written as f-1(x).

Let y = f(x) where f(x) = x + 2 Let x =

x y = f(x) make y the subject

f( ) = y

5 7 y =

f-1( ) f-1( x ) =

f-1( x ) =

If f ( x )=3 x−28 . Find f -1( x )

Let x =

=

y =

f-1( x ) =

x = 6

f(6) = 2 f-1( ) =

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BHASVIC Maths Linear A level Y1 Learning Pack

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