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Master A Level Maths Step by Step

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Begin with a strong foundation in algebra and functions - they're crucial for understanding more advanced topics. Master calculus concepts step-by-step, seeing how derivatives and integrals connect. Explore statistics and probability to round out your mathematical toolkit, preparing you for real-world applications and exam success.

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Maths was hard for me. Luckily I got StudyPug. When I don't understand the maths taught in class, I am able to watch the video lessons in StudyPug as many times as I want. Besides, the practice questions are very useful for preparing my exams.

Millie Lawson - Year 7 maths student, Birmingham, UK

Year 12 Maths Topics

Linear Equations

Introduction to linear equations

Introduction to nonlinear equations

Special case of linear equations: Horizontal lines

Special case of linear equations: Vertical lines

Parallel line equation

Perpendicular line equation

Combination of both parallel and perpendicular line equations

Applications of linear equations

Linear Inequalities

Express linear inequalities graphically and algebraically

Solving one-step linear inequalities

Solving multi-step linear inequalities

Compound inequalities

Inequalities in Two Variables

Graphing linear inequalities in two variables

Graphing simultaneous linear inequalities

Graphing quadratic inequalities in two variables

Graphing simultaneous quadratic inequalities

Applications of inequalities

What is linear programming?

Linear programming word problems

Introduction to Relations and Functions

Relationship between two variables

Understand relations between x- and y-intercepts

Domain and range of a function

Identifying functions

Function notation

Linear Functions

Distance formula: \(d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)

Midpoint formula: \(M = ( \frac{x_1+x_2}2 ,\frac{y_1+y_2}2)\)

Gradient equation: \(m = \frac{y_2-y_1}{x_2- x_1}\)

Gradient intercept form: y = mx + b

General form: Ax + By + C = 0

Gradient-point form: \(y - y_1 = m (x - x_1)\)

Rate of change

Graphing linear functions using table of values

Graphing linear functions using x- and y-intercepts

Graphing from gradient-intercept form y=mx+b

Graphing linear functions using a single point and gradient

Word problems of graphing linear functions

Parallel and perpendicular lines in linear functions

Applications of linear relations

Absolute Value Functions

Absolute value functions

Solving absolute value equations

Solving absolute value inequalities

Solving Simultaneous Equations

Determining number of solutions to linear equations

Solving simultaneous equations by graphing

Solving simultaneous equations by elimination

Solving simultaneous equations by substitution

Money related questions in linear equations

Unknown number related questions in linear equations

Distance and time related questions in linear equations

Rectangular shape related questions in linear equations

Simultaneous linear-quadratic equations

Factorising Polynomial Expressions

Common factors of polynomials

Factorising polynomials by grouping

Solving polynomials with the unknown "b" from x^2 + bx + c

Solving polynomials with the unknown "c" from x^2 + bx + c

Factorising polynomials: x^2 + bx + c

Applications of polynomials: x^2 + bx + c

Solving polynomials with the unknown "b" from \(ax^2 + bx + c\)

Factorising polynomials: \(ax^2 + bx + c\)

Factorising perfect square trinomials: (a + b)^2 = a^2 + 2ab + b^2 or (a - b)^2 = a^2 - 2ab + b^2

Find the difference of squares: (a - b)(a + b) = (a^2 - b^2)

Evaluating polynomials

Using algebra tiles to factorise polynomials

Solving polynomial equations

Word problems of polynomials

Quadratic Functions

Characteristics of quadratic functions

Transformations of quadratic functions

Quadratic function in general form: y = ax^2 + bx + c

Quadratic function in vertex form: y = a(x-p)^2 + q

Completing the square

Converting from general to vertex form by completing the square

Shortcut: Vertex formula

Graphing parabolas for given quadratic functions

Finding the quadratic functions for given parabolas

Applications of quadratic functions

Polynomial Functions

What is a polynomial function?

Polynomial long division

Polynomial synthetic division

Remainder theorem

Factor theorem

Rational zero theorem

Characteristics of polynomial graphs

Multiplicities of polynomials

Imaginary zeros of polynomials

Determining the equation of a polynomial function

Applications of polynomial functions

Solving polynomial inequalities

Solving polynomial equations by iteration

Fundamental theorem of algebra

Descartes' rule of signs

Surds

Operations with surds

Conversion between entire radicals and mixed surds

Adding and subtracting surds

Multiplying surds

Solving surd equations

Laws of indices

Product rule of exponents

Quotient rule of exponents

Power of a product rule

Power of a quotient rule

Power of a power rule

Negative exponent rule

Combining the exponent rules

Standard form

Convert between radicals and rational exponents

Solving for exponents

Algebraic Fractions

Simplifying algebraic fractions and restrictions

Adding and subtracting algebraic fractions

Multiplying algebraic fractions

Dividing algebraic fractions

Solving equations with algebraic fractions

Applications of equations with algebraic fractions

Simplifying complex fractions

Partial fraction decomposition

Functions

Function notation

Operations with functions

Adding functions

Subtracting functions

Multiplying functions

Dividing functions

Composite functions

Inequalities of combined functions

Inverse functions

One to one functions

Difference quotient: applications of functions

Transformations of Functions

Transformations of functions: Horizontal translations

Transformations of functions: Vertical translations

Reflection across the y-axis: y = f(-x)

Reflection across the x-axis: y = -f(x)

Transformations of functions: Horizontal stretches

Transformations of functions: Vertical stretches

Combining transformations of functions

Even and odd functions

Exponential Functions

Exponents: Product rule (a^x)(a^y) = a^(x+y)

Exponents: Division rule (a^x / a^y) = a^(x-y)

Exponents: Power rule (a^x)^y = a^(x * y)

Exponents: Negative exponents

Exponents: Zero exponent: a^0 = 1

Exponents: Rational exponents

Solving exponential equations using exponent rules

Graphing exponential functions

Graphing transformations of exponential functions

Finding an exponential function given its graph

Logarithmic Functions

What is a logarithm?

Converting from logarithmic form to exponential form

Evaluating logarithms without a calculator

Common logarithms

Natural log: ln

Evaluating logarithms using change-of-base formula

Converting from exponential form to logarithmic form

Solving exponential equations with logarithms

Product rule of logarithms

Quotient rule of logarithms

Combining product rule and quotient rule in logarithms

Evaluating logarithms using logarithm rules

Solving logarithmic equations

Graphing logarithmic functions

Finding a logarithmic function given its graph

Applications of Exponential and Logarithmic Functions

Exponential growth and decay by a factor

Exponential decay: Half-life

Exponential growth and decay by percentage

Finance: Compound interest

Continuous growth and decay

Logarithmic scale: Richter scale (earthquake)

Logarithmic scale: pH scale

Logarithmic scale: dB scale

Finance: Future value and present value

Circles and Parabolas

Angles in a circle

Chord properties

Tangent properties

Circles and circumference

Arcs of a circle

Areas and sectors of circles

Inscribed quadrilaterlas in circles

Central and inscribed angles in circles

Circles in coordinate plane

Parabola

Introduction to Trigonometry

Use sine ratio to calculate angles and side (Sin = \( \frac{o}{h}\) )

Use cosine ratio to calculate angles and side (Cos = \( \frac{a}{h}\) )

Use tangent ratio to calculate angles and side (Tan = \( \frac{o}{a}\) )

Combination of SohCahToa questions

Solving expressions using 45-45-90 special right triangles

Solving expressions using 30-60-90 special right triangles

Word problems relating ladder in trigonometry

Word problems relating guy wire in trigonometry

Other word problems relating angles in trigonometry

Trigonometry

Angle in standard position

Coterminal angles

Reference angle

Find the exact value of trigonometric ratios

ASTC rule in trigonometry (All Students Take Calculus)

Unit circle

Converting between degrees and radians

Trigonometric ratios of angles in radians

Radian measure and arc length

Graphing Trigonometric Functions

Sine graph: y = sin x

Cosine graph: y = cos x

Tangent graph: y = tan x

Cotangent graph: y = cot x

Secant graph: y = sec x

Cosecant graph: y = csc x

Graphing transformations of trigonometric functions

Determining trigonometric functions given their graphs

Trigonometric Identities

Quotient identities and reciprocal identities

Pythagorean identities

Sum and difference identities

Cofunction identities

Double-angle identities

Sequences and Series

Arithmetic sequences

Arithmetic series

Geometric sequences

Geometric series

Infinite geometric series

Sigma notation

Arithmetic mean vs. Geometric mean

Probability

Determining probabilities using tree diagrams and tables

Probability of independent events

Finding probabilities using two-way frequency tables

Probability with Venn diagrams

Permutations and Combinations

Fundamental counting principle

Factorial notation

Path counting problems

Permutation vs. Combination

Permutations

Combinations

Problems involving both permutations and combinations

Pascal's triangle

Binomial theorem

Statistics

Median and mode

Mean

Range and outliers

Application of averages

Influencing factors in data collection

Data collection

Data and Graphs

Reading and drawing bar graphs

Reading and drawing histograms

Reading and drawing line graphs

Box-and-whisker plots and scatter plots

Stem-and-leaf plots

Reading and drawing Venn diagrams

Sine Rule and Cosine Rule

Sine rule

Cosine rule

Applications of the sine rule and cosine rule

Limits

Finding limits from graphs

Continuity

Finding limits algebraically - direct substitution

Finding limits algebraically - when direct substitution is not possible

Infinite limits - vertical asymptotes

Limits at infinity - horizontal asymptotes

Intermediate value theorem

Squeeze theorem

Differentiation

Definition of derivative

Power rule

Slope and equation of tangent line

Chain rule

Derivative of trigonometric functions

Derivative of exponential functions

Product rule

Quotient rule

Implicit differentiation

Derivative of inverse trigonometric functions

Derivative of logarithmic functions

Higher order derivatives

Critical number & maximum and minimum values

Integration

Antiderivatives

Riemann sum

Definite integral

Fundamental theorem of calculus

Integration Applications

Areas between curves

Volumes of solids with known cross-sections

Volumes of solids of revolution - Disc method

Volumes of solids of revolution - Shell method

Average value of a function

Arc length

Quadratic Equations and Inequalities

Solving quadratic equations by factoring

Solving quadratic equations by completing the square

Using quadratic formula to solve quadratic equations

Nature of roots of quadratic equations: The discriminant

Applications of quadratic equations

Solving quadratic inequalities

Reciprocal functions

Graphing reciprocals of linear functions

Graphing reciprocals of quadratic functions

Parametric Equations and Polar Coordinates

Defining curves with parametric equations

Polar coordinates

Set Theory

Set notation

Set builder notation

Intersection and union of 2 sets

Intersection and union of 3 sets

Bearings

Introduction to bearings

Bearings and direction word problems

Angle of elevation and depression

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How Can I Pass A Year 12 Maths Exam?

To help you obtain a passing grade in your year 12 maths exam, you should be doing maths revision on a regular basis. Try and take the time to get yourself into a weekly study routine that involves reviewing your class notes (which if you're not taking, you should be!), working on homework assignments, and testing your knowledge via online maths aids like worksheets and past year 12 maths exam papers.

Use these past papers to build your confidence and improve your time management ahead of your actual exams. Assess how long it takes for you to answer questions and see how long you should realistic spend on each section. If a section proves to be too difficult, skip it and come back to it later. It's more important to get through the paper, answering as many questions as possible, so don't waste time on one problem.

Using these past papers, you can also single out any topics that are giving you trouble. By Highlighting these weaker areas, you will able able to build much more effective revision strategies.

Once you know where you need to improve, use StudyPug for some additional year 12 maths help. You can find the lesson you need help with via the easy to use search function and from there, we'll help you tackle the problems by introducing step-by-step examples that work to build your understanding.

Our year 12 maths tutors will cover all the complicated A level maths questions and we've made sure that our content reflects the current Year 12 maths curriculum and the A level maths syllabus. You'll also have access to your very own study planner that you can use to track your progress and keep you moving towards your goals.

Once you're comfortable using our videos to study maths, try sitting additional past papers and attempt more worksheets to chart your progress. You'll soon see improvements in your performance and you'll be in a much better position to pass your upcoming exams.

Finally, when you're actually in your exam, please keep in mind that the external markers will be looking for proof that you understood the problem. They want to see that you knew how to solve the problems beyond memorizing formulae and other tools. Where possible, show your working out! This will show them your thought process and your mathematical reasoning abilities. In doing so, you may earn additional marks and receive a bump in your final grade.

Should I Try to Take an A-level Exam Early?

If you feel confident enough to take the test ahead of time, you can, but please be aware that it's not necessarily something that will help you in your university applications. The UCAS points you earn will still count towards your overall UCAS score, but keep in mind that universities are looking for more than just UCAS points.

The Russell Group, an association of United Kingdom-based universities, founded in 1994, had this to say in regards to taking A-levels early and how it may impact your university applications.

Some universities or their individual subject departments may want to see that you have taken a number of Advanced level qualifications all at the same time; for example, they may want to see three A-levels taken in Year 13.

This can be because they want to know that you can comfortably manage a workload of this size… Admissions policies may therefore differ in relation to A-levels taken early, and whether these are included in offers made or not.

For example, some courses that typically make a conditional offer of AAB may take account of an A-level A grade achieved at the end of Year 12 and, as a result, make a conditional offer of AB for A-levels taken in Year 13. Others may still make a conditional offer of AAB on subjects taken at the end of Year 13 and will not include the A-level already taken in their conditional offer.

What Units Will I Study in A-Level Maths?

Choosing to study A level maths will not only benefit your university applications, but a qualification in A-Level maths, will help you in your future career too. It demonstrates to potential employers that you have the ability to apply analytical thinking, finding logical solutions to solve a variety of problems.

We're not suggesting that employers are specifically looking for a candidate that can solve maths equations or knows how to calculate angels (unless it's an architectural role), but they'll certainly see the value in the various transferable skills that are associated with solving those problems.

Once you've decided to study A/AS-level maths, you'll need to narrow down that actual units you'd like to cover throughout the program. Within A-Level mathematics, there are two different programs of study.

Firstly, there's AS Level maths, this is a one year course that will cover 2 core maths units plus one additional unit from either mechanics, decisions, or statistics. If you decide to study a full 2 year A-level course, you'll cover all 4 core maths units plus two additional units.

With this in mind, you should select the optional units that will benefit you best in your future university applications and career. For example, if you're looking to enter into engineering, you may want to consider studying mechanics.

To give you an idea of what to expect, here is a breakdown of topics you'll likely cover in the three optional units (not a complete list).

Statistics

Statistical models
Summarising data
Probability
Correlation and Regression
Discrete Random Variables
Continuous Random Variables
Continuous Distributions
Hypothesis tests
Combinations of random variables

Mechanics

Vectors
Kinematics
Dynamics
Statics
Moments
Centres of mass
Work and energy
Collisions
Statics of rigid bodies

Decision

Algorithms
Algorithms on graphs
Route inspection
Critical path analysis
Linear programming
Matchings
Flows in a network

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