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

Mechanics - Constant accelerationVT graphs

https://youtu.be/bNyMI-859Lc 7 min

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)

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BHASVIC Maths Linear A level Y1 Learning Pack

Mechanics - Constant accelerationSUVAT

https://youtu.be/rCi3exlvdo0 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 2 =

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?

Mechanics - Forces and motion 5

V ms-1

t/s s

u

BHASVIC Maths Linear A level Y1 Learning Pack

Newton’s Laws https://www.youtube.com/watch?v=Z03y6SYhnjg 4 min

Newton’s first law states:

Newton’s second law states: (We tend to write the equation as F=ma)

Newton’s third law:

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BHASVIC Maths Linear A level Y1 Learning Pack

Mechanics - Forces and motionFree body Diagrams (also known as Force Diagrams) https://youtu.be/nDis6HbXxjg 6 min

When drawing free body diagrams (force 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 (force) diagram of this climber.

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BHASVIC Maths Linear A level Y1 Learning Pack

Sketch the free body (force) diagram of:

Free body (force) 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

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Fnormal

Fgravity

BHASVIC Maths Linear A level Y1 Learning Pack

Mechanics - ForcesResolving forces https://www.examsolutions.net/tutorials/resolving-forces/?level=A-Level&board=Edexcel&module=M1&topic=1632 12 min

From the video resolve these forces into perpendicular components (complete the diagrams)

Resolving a force means replacing a force by two forces at right angles to one another that would have the same effect as the single force. The forces that replace the single force are called the components of the force.

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BHASVIC Maths Linear A level Y1 Learning Pack

Mechanics - Forces and motionFriction (Fmax=R) https://youtu.be/OR4CjWKAKAU 8 min

Fmax = RDefine the terms:

Fmax

R

What does limiting equilibrium mean?

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

1)

2)

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BHASVIC Maths Linear A level Y1 Learning Pack

Mechanics - Forces and motionSingle particle problems including friction

https://youtu.be/_qmDu3kkdCU?t=8 8 min

A small box of mass 15kg rests on a rough horizontal plane. The coefficient of friction between the box and the plane is 0.2. The force of magnitude P Newtons is applied to the box at 50° to the horizontal as shown. The box is on the point of sliding down the plane.

Find the value of P, giving your answer to 2 significant figures.

Complete the diagram above with the other forces acting on the box and below form two equations resolving forces horizontally and vertically. Use these equations to solve the problem.

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50°15 kg

P N

BHASVIC Maths Linear A level Y1 Learning Pack

Mechanics - Forces and motionConnected particles – pulleys

https://youtu.be/SGEscXwQUFw 12 min

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A light inextensible string passes over a fixed smooth pulley. Particles of masses 2 kg and 3 kg are attached at opposite end of the string. Each part of the string hangs vertically. Find the acceleration of each particle, the tension in each string and the force required to hold the pulley in equilibrium

BHASVIC Maths Linear A level Y1 Learning Pack

Mechanics - Forces and motionConnected particles – on a slope https://youtu.be/RSIwqNonZL0 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.

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BHASVIC Maths Linear A level Y1 Learning Pack

Mechanics - Forces and motionConnected particles – a lift problem

https://www.youtube.com/watch?v=hU2705F9HS4 7 min

A woman travels in a lift. The mass of the woman is 50kg and the mass of the lift is 950kg. The lift is being raised vertically by a vertical cable, which is attached to the top of the lift. The lift is moving upwards and has constant deceleration of 2ms-2. By modelling the cable as being light and inextensible, find

a) the tension in the cable

b) the magnitude of the force exerted on the woman by the floor of the lift

Mechanics - Forces and motion14

BHASVIC Maths Linear A level Y1 Learning Pack

Vector problems – ships

https://www.youtube.com/watch?v=OVyfgc8AwOU 12 min

A coastguard station O monitors the movements of ships in a channel. At noon, the station’s radar records two ships moving with constant speed.

Ship A is at the point with position vector (-5i + 10j) km relative to O and has velocity (2i + 2j) kmh−1.

Ship B is at the pint with positon vector (3i + 4j) km and has velocity (-2i + 5j) kmh−1.

a) Given that the two ships maintain these velocities, show that they collide.

The coast guard radios ship A and orders it to reduce its speed to move with velocity ( i + j) km/h

Given that A obeys this order and maintains the new constant velocity.

b) Find an expression for the vector A⃗B at time, t hours after noon.

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BHASVIC Maths Linear A level Y1 Learning Pack

c)Find to 3 significant figures, the distance between A and B at 1400 hours

d)Find the time at which B will de due north of A

Mechanics - Projectiles

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BHASVIC Maths Linear A level Y1 Learning Pack

Projectile motion

https://youtu.be/umJ3zAIF5JE 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=

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BHASVIC Maths Linear A level Y1 Learning Pack

Statistics -Measures of location and spreadThe three cases of skew.

https://www.youtube.com/watch?v=nM-ftP4l-IE 10 min

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BHASVIC Maths Linear A level Y1 Learning Pack

Statistics -Measures of location and spreadMean and standard deviation of a list

https://www.youtube.com/watch?v=DKpXijtS8zA 8 min

Calculate the mean and standard deviation of the no of goals Palace scored in their last 6 matches

6 2 3 3 5 4

What is the variance?

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BHASVIC Maths Linear A level Y1 Learning Pack

Calculate the mean and standard deviation of the no of goals Brighton scored in their last 6 matches

0 2 1 2 0 1

What is the variance?

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BHASVIC Maths Linear A level Y1 Learning Pack

Statistics - Measures of location and spreadMedian and Interquartile Range (IQR) of a list(there maybe an error in this video?) https://www.youtube.com/watch?v=ERAUsoIH_pM 9 min

2|5 means 25

0 6 7 81 0 2 3 4 7 7 7 8 92 1 3 4 5 5 73 1 1 2 6 6 94 1 5 5 6 9567 9

Find the lower quartile, median, upper quartile, IQR and decide if there are outliers.

If it is a whole number…………………………………………………………………….

If it is not a whole number…………………………………………………………………

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BHASVIC Maths Linear A level Y1 Learning Pack

Key stem = 10s

9 0 68 3 5 77 1 6 66 0 2 2 45 1 1 24 3 4 7 83 5 721 6

Find the lower quartile, median, upper quartile, IQR and decide if there are outliers

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BHASVIC Maths Linear A level Y1 Learning Pack

Statistics - Measures of location and spreadMean and S.D with grouped data

https://www.youtube.com/watch?v=Qxam3TBVA1o 10 min

Ungrouped frequency

Maths test mark

No of people

1 62 53 74 4

Find the mean and standard deviation

Grouped frequency

height frequency0-4 25-10 411-16 617-20 521-30 5

Find the mean and standard deviation

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BHASVIC Maths Linear A level Y1 Learning Pack

Ungrouped frequency

No of pets owned

No of people

1 42 63 24 2

Find the mean and standard deviation

Grouped frequency

English mark

Frequency

5-14 315-19 420-29 530-34 2

Calculate the mean and standard deviation

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BHASVIC Maths Linear A level Y1 Learning Pack

Statistics - Measures of location and spreadMean and S.D of grouped data (by calculator)

https://www.youtube.com/watch?v=Y1gsxlSbojs 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:

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BHASVIC Maths Linear A level Y1 Learning Pack

Statistics - Measures of location and spreadPercentile Approximation

https://www.youtube.com/watch?v=gu7qakehDm8 7 min

1. Calculate the 50th percentile, P50 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

2. Calculate the 40th percentile P40 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

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BHASVIC Maths Linear A level Y1 Learning Pack

Statistics - Measures of location and spreadLinear Coding https://www.youtube.com/watch?v=hH1HAzPFM4U 6 min

Statistics - Measures of location and spread

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Mean x = 4

Standard deviation σ = 2.3

BHASVIC Maths Linear A level Y1 Learning Pack

Combined mean

https://www.youtube.com/watch?v=bb4PggxcYHU 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.

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BHASVIC Maths Linear A level Y1 Learning Pack

Statistics - Measures of location and spreadInterpolation Median and Interquartile Range (IQR) of grouped data https://www.youtube.com/watch?v=arSw96Wrgg8 13 min

height frequency0-4 25-10 411-16 617-20 521-30 5

gw( n−tgf )+gb

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BHASVIC Maths Linear A level Y1 Learning Pack

Statistics - ProbabilityVenn Diagrams https://www.youtube.com/watch?v=HEjKmmYKc4A 8 min

Venn Diagrams

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P(A '∪B) P(A U B)’ or

TICK METHOD:

OR = TWO TICKS

AND = ONE OR TWO TICKS

A BS

A BS

A BS

A BS

A BS

A BS

A BS

A BS

P( B’ )

P( A’ ∩ B )

ADDITION RULE:

= P (A )+P (B )--

BHASVIC Maths Linear A level Y1 Learning Pack

Statistics - ProbabilityMutually exclusive and independent events

https://youtu.be/LWVw9SI8WDI 5 minMutually exclusiveIf A and B are mutually exclusive they cannot happen together

IndependentIf A and B are independent, the probability of A happening is the same whether or not B happens.

P(A) x p(B) = P( A ∩B ¿

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BHASVIC Maths Linear A level Y1 Learning Pack

Statistics - ProbabilityTree diagrams

https://www.youtube.com/watch?v=tzXTJZI9dIQ 6 min

A bag contains 3 blue balls and 5 red balls. Two are selected at random without replacement. Find the probability that

a) they are both blue

b) there is one of each colour

Pythagoras tree…look it up!

There are 5 black socks and 3 reds in a bag. I pick 2 socks without replacement. Find the probability I get

a) two of the same coloured sock

b) at least one red sock

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BHASVIC Maths Linear A level Y1 Learning Pack

Statistics - ProbabilitySample space diagrams

https://www.youtube.com/watch?v=n-xM2K7E9TU 4 min

Two dice are thrown and their scores added. Find the probability of getting

a) a total of 7

b) a total of 8 or more

c) a score of 5 or more given the first score was even

A dice house

Two coins are thrown. What is the probability of getting two tails?

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BHASVIC Maths Linear A level Y1 Learning Pack

Spinner A has 2,3,4 and spinner B has 1, 2,3,4,5. They are spun and their scores are added. Find the probability of

a) a total of 6

b) a total of 7 or more

c) a total of 4 or more given that spinner A was an even number

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BHASVIC Maths Linear A level Y1 Learning Pack

Statistics - ProbabilityConditional probability

https://youtu.be/gEyIZ5Tlokg 6 min

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Can be rearranged and used like this. Look out for the word “given” to show dependence on another event.

P (A /B )= P(A ∩B)P(B)

Conditional Probability:

Known as Multiplication Law

P (A ∩B )=P (A /B ) . P(B)

BHASVIC Maths Linear A level Y1 Learning Pack

Statistics - Statistical distributionsDiscrete Random Variables (DRVs)

https://youtu.be/kuBf2dOVW0o 7 min

A discrete random variable must be ‘the number of …’

It is represented by a capital letter.

The probability distribution is a table of possible values and their assocaited probabilites.

Find the probability distribution for S the score on a die.

A die is thrown until either a six appears or I’ve thrown it three times.

Find the probability distribution for T where T is the number of throws.

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BHASVIC Maths Linear A level Y1 Learning Pack

Statistics - Statistical distributionsBinomial distribution https://youtu.be/NaDZ0zVTyXQ 10 min

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:

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BHASVIC Maths Linear A level Y1 Learning Pack

Statistics - Statistical distributionsBinomial distribution – cumulative probabilities

https://youtu.be/MwNuZgZYBDA 10 min

The Binomial CD (Cumulative Distribution) function gives P(X ≤ x) for X~B(n, p).

If X~B( 12, 0.2)EvaluateP(X ≤ 6)

P(X < 7)

P(X ≥ 7)

P(X > 6)

P(3 ≤ X ≤ 5)

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BHASVIC Maths Linear A level Y1 Learning Pack

Statistics - Hypothesis testing for Binomial Test for a Binomial distribution

https://www.youtube.com/watch?v=61Wi04SqF34&feature=youtu.be 10 min

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

Let X be the r.v. number of 6’s thrown in 30 throws, X B(30 , p)

0 1 2 3 4 5 6 7 8 9 10 …

One Tail Tests

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BHASVIC Maths Linear A level Y1 Learning Pack

Statistics - Hypothesis testingLower tail test – p value method

https://www.youtube.com/watch?v=RLoQH9O2gAE&feature=youtu.be 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, X B(24 , p)

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

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BHASVIC Maths Linear A level Y1 Learning Pack

Statistics - Hypothesis testingUpper tail test – p value method

https://www.youtube.com/watch?v=JpiaMkERYC8&feature=youtu.be 7 min

In Luigi’s restaurant on average 1 in 10 people order a bottle of Chardonnay. Out of 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 X B(50 p)

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…

Statistics - Hypothesis testing

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BHASVIC Maths Linear A level Y1 Learning Pack

Lower tail test – critical region method

https://www.youtube.com/watch?v=6CXptC72vyw&feature=youtu.be 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

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

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

Statistics - Hypothesis testing

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BHASVIC Maths Linear A level Y1 Learning Pack

Upper tail test – critical region method

https://www.youtube.com/watch?v=RKb7WHwIDxA&feature=youtu.be 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 X B(20 p)

Conclusion

Please note

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

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

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BHASVIC Maths Linear A level Y1 Learning Pack

Statistics - Hypothesis testingTwo tail test- critical region method

https://www.youtube.com/watch?v=ZJDfLb2zZCg&feature=youtu.be 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 X B(15 p)

Conclusion

Please note

Where he refers to tables, use Casio fx-991EX instead.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 0.05N: 15P: 0.3

xL = 1

b) For upper tail p(X ≤ xM- 1) ≥ 0.95XM-1 = 7, XM = 8

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BHASVIC Maths Linear A level Y1 Learning Pack

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