Paul Keogan

Paper 1 – Revision

Paul Keogan

LCOL Distance, Speed and Time

LCOL Distance, Speed & Time

Your progress through this topic0 of 1 lessons

Lessons in this topic

All 1 lessons.

1LCOL Distance, Speed and Time

Key ideas in this topic

The DST triangle — The DST triangle

Put D on top, S and T underneath, then cover the one you want: D = S × T, S = D/T, T = D/S. Check the units match first (km with km/h).

LCOL Distance, Speed & Time

Paul Keogan

LCHL Maths – Functions and Graphs

LCHL Functions and Graphs

Your progress through this topic0 of 11 lessons

Lessons in this topic

All 11 lessons, in order — each one ticks off as you open it.

1JC Revision 2Shapes of Quadratic Functions 3Vertex form/Completing the square 4Composite functions 5Injective Functions 6Surjective Functions 7Bijective Functions 8Inverse functions 9Drawing Graphs 10Limits, convergence, divergence, continuity 11Functions and Graphs 10

Definitions you must know

Definition of a function — Definition of a functionevery input → exactly one output

Bijective ⇒ inverse is a function — Bijective ⇒ inverse is a functionFunctions & Graphs 8

The horizontal line test

Injective test — Injective testno horizontal line meets the graph more than once

Surjective test — Surjective testevery horizontal line meets the graph at least once

Injective and surjective mappings

Injective (one-to-one) mapping — Injective (one-to-one) mappingdistinct inputs → distinct outputs

Surjective (onto) mapping — Surjective (onto) mappingevery output is used at least once

Key ideas in this topic

Vertex form — Vertex formFunctions & Graphs 3

Composite functions — Composite functionsFunctions & Graphs

Inverse functions — Inverse functionsFunctions & Graphs 8

Check it against the exam

Jump straight to the fully worked solution for that paper.

2023 Paper 1 Q2 →2023 Paper 1 Q5 →2023 Paper 1 Q9 →2022 Paper 1 Q5 →2022 Paper 1 Q6 →

Paul Keogan

LCHL Maths – Paper 1 Proofs

LCHL Paper 1 Proofs

Your progress through this topic0 of 6 lessons

Proofs in this section

All 6 proofs, in order — each one ticks off as you open it.

1Prove root 2 is irrational (Number theory)★ Learn off 2Prove by induction the sum of a geometric series★ Learn off 3Prove De Moivres theorem by induction★ Learn off 4Prove the Amortization formula (Financial maths)★ Learn off 5Prove Sum to Infinity formula (Geometric Sequences)★ Learn off 6Proof of Sin(3A) and Cos(3A)★ Learn off

Check it against the exam

Jump straight to the fully worked solution for that paper.

2023 Paper 1 Q3 →2021 Paper 1 Q7 →2020 Paper 1 Q5 →2019 Paper 1 Q6 →2018 Paper 1 Q4 →

Paul Keogan

LCHL Maths – Induction

LCHL Induction

Your progress through this topic0 of 7 lessons

Lessons in this topic

All 7 lessons, in order — each one ticks off as you open it.

1Introduction 2Inequalities 3Factorials 4Divisibility 5Series 6Proof of De Moivre’s Theorem by induction 7Induction 7 Proof by induction of the Sum of a geometric series

✍️ Learning work — the four-step method

Learn this skeleton off — exam answers lose marks without it.

1Prove P(n) is true for n = (the first value)

2Assume P(n) is true for n = k

3Hence, prove P(n) is true for n = k + 1

4State the conclusion

The four types of induction

Series (summation)Prove a sum formula — append the (k+1)th term, then factor.DivisibilityProve aⁿ ± c is divisible by D — substitute and factor.InequalitiesProve e.g. 3ⁿ > 2n — split into pairs with the coefficient.FactorialsProve factorial-based statements — the same four-step skeleton.

A rule you must know

Simplifying factorials — Simplifying factorials(n+1)! = (n+1)·n!

Check it against the exam

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2023 Paper 1 Q4 →2023 Paper 1 Q5 →2022 Paper 1 Q2 →2021 Paper 1 Q4 →2020 Paper 1 Q7 →

Paul Keogan

LCHL Maths – Complex Numbers

LCHL Complex Numbers

Your progress through this topic0 of 17 lessons

Lessons in this topic

All 17 lessons, in order — each one ticks off as you open it.

1Introduction 2Addition, Subtraction, Multiplication 3Division and Conjugates 4Argand Plane and Modulus 5Modulus and conjugates 6Equating Co-efficients 7Rotations 8Quadratic equations 9Cubic equations 10Introduction to Polar Form 11Rules for Polar Form 12Introduction to De Moivres theorem 13Converting to polar form and applying De Moivres theorem 14Using De Moivres theorem to find more than one answer 15Polar form, De Moivres theorem and fraction in the power 16Complex Numbers 16 Proof of De Moivres Theorem by Induction 17Proof of Sin(3A) and Cos(3A)

Key ideas in this topic

The imaginary unit — The imaginary unitComplex Numbers

Modulus — ModulusComplex Numbers

The argument arg(z) — The argument arg(z)Complex Numbers

Polar form — Polar formComplex Numbers

De Moivre's theorem — De Moivre’s theoremin the log tables

Rules of polar form you must know

Multiplication — multiply the moduli, ADD the arguments — Multiplication — multiply the moduli, ADD the argumentsComplex Numbers 11

Division — divide the moduli, SUBTRACT the arguments — Division — divide the moduli, SUBTRACT the argumentsComplex Numbers 11

Reciprocal — the modulus inverts, the i-sign flips — Reciprocal — the modulus inverts, the i-sign flipsComplex Numbers 11

Transformations on the Argand plane

Scalar multiples — scaling along the line through O and z — Scalar multiples — scaling along the line through O and zk·z is collinear with z; |kz| = |k||z|

Rotations — Rotations×i = 90° anticlockwise, ×(−i) = 90° clockwise, ×(−1) = 180°

Addition — the parallelogram rule — Addition — the parallelogram ruleorigin, z₁, z₃, z₂ form a parallelogram

Conjugation — axial symmetry in the real axis — Conjugation — axial symmetry in the real axisz̄ is the mirror image of z across the real axis

Check it against the exam

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2023 Paper 1 Q3 →2023 Paper 1 Q4 →2022 Paper 1 Q1 →2022 Paper 1 Q3 →2021 Paper 1 Q1 →

Paul Keogan

LCHL Maths – Financial Maths

LCHL Financial Maths

Your progress through this topic0 of 8 lessons

Lessons in this topic

All 8 lessons, in order — each one ticks off as you open it.

1Simple Interest 2Compound Interest 3Time value of money 4Converting interest rates from monthly to yearly etc.5Annuities ( Saving Money)6Annuities (Paying Money Back)7Proof of Amortization formula★ Learn off 8Pensions

Key ideas in this topic

Future value (compound interest) — Future value (compound interest)Financial Maths

Present value — Present valueFinancial Maths

Amortisation (loan repayment) — Amortisation (loan repayment)Financial Maths 7

Depreciation — DepreciationFinancial Maths

Check it against the exam

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2023 Paper 1 Q7 →2023 Paper 1 Q8 →2022 Paper 1 Q7 →2020 Paper 1 Q5 →2018 Paper 2 Q8 →

Paul Keogan

LCHL Maths – Integration

LCHL Integration

Your progress through this topic0 of 13 lessons

Lessons in this topic

All 13 lessons, in order — each one ticks off as you open it.

1Algebra 2Integrating 1/x 3Exponentials 4Integrating a constant to the power of an unknown 5Trigonometry 1 6Trigonometry (II)7Integration with limits 8Finding f(x) when given the slope 9Distance, Speed, Acceleration 10Anti- Derivatives 11Average value 12Area between a curve and the x- axis 13Area between a curve and the y-axis

Key ideas in this topic

Power rule — Power ruleIntegration 1

Definite integral — Definite integralIntegration

Integral of 1/x — Integral of 1/xIntegration 2

Integral of eˣ — Integral of eˣIntegration 3

Applications you must know

Average value of a function — Average value of a functionIntegration

Acceleration → velocity → displacement — Acceleration → velocity → displacementintegrate with respect to time

Check it against the exam

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2023 Paper 1 Q10 →2023 Paper 1 Q6 →2023 Paper 1 Q7 →2022 Paper 1 Q2 →2022 Paper 1 Q5 →

Paul Keogan

LCHL Maths – Differentiation

LCHL Differentiation

Your progress through this topic0 of 21 lessons

Lessons in this topic

All 21 lessons, in order — each one ticks off as you open it.

Key ideas in this topic

From first principles — From first principlesDifferentiation 2

Power rule — Power ruleDifferentiation

Product rule — Product ruleDifferentiation 6

Quotient rule — Quotient ruleDifferentiation 7

Chain rule — Chain ruleDifferentiation 8

Applications you must know

The first derivative = tan of the angle the tangent makes with the horizontal — The first derivative = tan of the angle the tangent makes with the horizontaldy/dx = tan θ (angle with the positive x-axis)

Turning point — Turning pointdy/dx = 0

Point of inflection — Point of inflectiond²y/dx² = 0

Increasing / decreasing — Increasing / decreasingsign of dy/dx

Displacement → velocity → acceleration — Displacement → velocity → accelerationdifferentiate with respect to time

Check it against the exam

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2023 Paper 1 Q5 →2023 Paper 1 Q6 →2023 Paper 1 Q9 →2022 Paper 1 Q10 →2022 Paper 1 Q6 →

Paul Keogan

LCHL Maths – Sequences and Series, Patterns and Limits

LCHL Sequences and Series

Your progress through this topic0 of 12 lessons

Lessons in this topic

All 12 lessons, in order — each one ticks off as you open it.

1Arithmetic Sequences 1 2Arithmetic Sequences 2 3Arithmetic sequences 3 4Arithmetic Sequences 4 5Geometric Sequences 1 6Geometric Sequences 2 7Geometric Sequences 3 8Geometric Sequences 4 9Geometric Sequences 5 10Prove the Sum of Geometric Series by induction 11Quadratic Patterns 12Cubic patterns

Common differences

Linear pattern — common first difference — Linear pattern — common first differencethe 1st differences are constant

Quadratic pattern — common second difference — Quadratic pattern — common second differencethe 2nd differences are constant

Cubic pattern — common third difference — Cubic pattern — common third differencethe 3rd differences are constant

Patterns & tests you must know

Quadratic pattern — Quadratic pattern2a = second difference

Cubic pattern — Cubic pattern6a = third difference

Three terms in an arithmetic sequence — Three terms in an arithmetic sequencemiddle term is the average

Three terms in a geometric sequence — Three terms in a geometric sequenceT₃/T₂ = T₂/T₁

Prove a sequence is arithmetic — Prove a sequence is arithmeticconstant first difference

Prove a sequence is geometric — Prove a sequence is geometricconstant ratio

The rules of limits — The rules of limitssum, product & quotient rules

Key ideas in this topic

Arithmetic — nth term — Arithmetic — nth termSequences & Series

Arithmetic — sum of n terms — Arithmetic — sum of n termsSequences & Series

Geometric — nth term — Geometric — nth termSequences & Series

Geometric — sum of n terms — Geometric — sum of n termsSequences & Series

Sum to infinity — Sum to infinitySequences & Series

Check it against the exam

Jump straight to the fully worked solution for that paper.

2023 Paper 1 Q2 →2023 Paper 1 Q9 →2022 Paper 1 Q9 →2021 Paper 1 Q7 →2020 Paper 1 Q2 →