A Level ExamSolutions Maths Edexcel
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Content Overview
227 topics in 3 modules
βοΈ Pure 159 topics
- Algebra and Functions β Rational Expressions: Simplifying β Simplifying algebraic fractions
- Algebra and Functions β Rational Expressions: Simplifying β Exam Questions - Simplifying a rational expression
- Algebra and Functions β Rational Expressions: Simplifying β Addition and subtraction of algebraic fractions
- Algebra and Functions β Rational Expressions: Simplifying β Exam Questions - Addition & subtraction
- Algebra and Functions β Rational Expressions: Simplifying β Multiplication of algebraic fractions
- Algebra and Functions β Rational Expressions: Simplifying β Further simplifying of 'stacked fractions'
- Algebra and Functions β Rational Expressions: Simplifying β Exam Questions - Algebraic long division
- Coordinate Geometry β Parametric equations
- Coordinate Geometry β Converting to Cartesian form
- Coordinate Geometry β Exam Questions - Parametric to Cartesian equations
- Coordinate Geometry β Sketching parametric graphs
- Coordinate Geometry β Parametric equations of circle, ellipse, parabola and hyperbola
- Coordinate Geometry β Finding Points of Intersection between a Parametric and Cartesian Equation
- Coordinate Geometry β Exam Questions - Parametric equations
- Algebra and Functions β Modulus Functions, Equations and Inequalities β The modulus function
- Algebra and Functions β Modulus Functions, Equations and Inequalities β Graphing y=|f(x)|
- Algebra and Functions β Modulus Functions, Equations and Inequalities β Modulus equations
- Algebra and Functions β Modulus Functions, Equations and Inequalities β Exam Questions - Modulus equations
- Algebra and Functions β Modulus Functions, Equations and Inequalities β Modulus inequalities
- Algebra and Functions β Modulus Functions, Equations and Inequalities β Exam Questions - Modulus inequalities
- Algebra and Functions β Working with Functions β Mappings
- Algebra and Functions β Working with Functions β Mappings - More examples
- Algebra and Functions β Working with Functions β Mappings, functions or both?
- Algebra and Functions β Working with Functions β f(x) notation
- Algebra and Functions β Working with Functions β Domain and range
- Algebra and Functions β Working with Functions β Exam Questions - Domain and range
- Algebra and Functions β Working with Functions β Combination of functions
- Algebra and Functions β Working with Functions β The inverse of a function
- Algebra and Functions β Working with Functions β Graphical relationship between f(x) and its inverse
- Algebra and Functions β Working with Functions β Exam Questions - Inverse functions
- Algebra and Functions β Working with Functions β Exam Questions - Functions
- Algebra and Functions β Partial Fractions β Partial fractions
- Algebra and Functions β Partial Fractions β Denominator contains 2 or 3 linear factors
- Algebra and Functions β Partial Fractions β Denominator contains repeated factors
- Algebra and Functions β Partial Fractions β Exam Questions - Partial fractions
- Binomial Expansion β Exam Questions - Binomial expansion for rational and negative powers
- Binomial Expansion β Exam Questions - Partial fractions with the binomial expansion
- Definition and finding the nth term
- Increasing and decreasing sequences
- Recurrence relationships
- Sigma notation
- Recurrence relationships - Exam Questions
- Arithmetic progressions
- Sum of the first n terms
- Finding a and d given two terms
- Working with consecutive terms
- Arithmetic sequences and series - Exam Questions
- Examsolutions Beastie - Arithmetic progressions
- Geometric series
- Proof of sum of first n terms, Sn
- Sum to infinity
- Geometric Series and Progressions - Exam Style Questions
- Geometric series - Exam Style Questions
- Radians
- Arcs, sectors and segments
- Arcs, sectors and segments - Exam Questions
- Trig functions sec ΞΈ, cosec ΞΈ and cot ΞΈ
- Graphs of sec ΞΈ, cosec ΞΈ and cot ΞΈ
- Inverse trigonometric functions - arcsin x, arccos x, arctan x
- Examples using Inverse trigonometric functions
- sinΒ²x + cosΒ²x β‘1 , 1 + tanΒ²x β‘ secΒ²x , 1 + cotΒ²x β‘ cosecΒ²x
- Solving equations using Pythagorean identities
- Small-angle approximations
- Small Angle Approximations - Exam Questions
- sin(AΒ±B), cos(AΒ±B) and tan(AΒ±B)
- Using the Addition formulae to get exact values
- Proving identities using the addition formulae
- Identities - Addition type - Equations
- Identities for sin2A, cos2A and tan2A
- Examples using double angle identities
- Solving equations using double angle identities
- Double Angles - Exam Questions
- Examples using half angle identities
- Identity for cos 3ΞΈ and sin 3ΞΈ
- Harmonic Identities Rsin(x Β± Ξ±), Rcos(x Β± Ξ±)
- Equations using harmonic identities
- Harmonic identities - Max and Min
- Harmonic identities and equations - Exam Questions
- Mixed trigonometry - Exam Questions
- Exponential function ex
- The natural log function, ln(x)
- The trig functions sin(x), cos(x) and tan(x)
- Chain rule: Polynomial to a rational power
- Chain rule: Exponential types
- Chain rule: Natural log types
- Chain rule: Trigonometric types
- The product rule
- The quotient rule
- Quotient Rule - Exam Questions
- The trig functions, sec(x), cosec(x) and cot(x)
- The reciprocal function of dy/dx
- Differentiation methods - Exam Questions
- Differentiation: tangents, normals and stationary points - Exam Questions
- Exponential rates of change - Exam Questions
- Differentiation: Exponential functions of the form y=ax
- Differentiation: Parametric functions
- Implicit functions
- Tangents and normals
- Implicit functions - Exam Questions
- Connected rates of change
- Using three connected rates of change
- Connected rates of change cone type problems
- Connected rates of change - Exam Questions
- Area bound by a curve and x-axis
- Area bound by a curve and x-axis - Exam Questions
- Area under a graph : parametric type
- Integration:(ax+b)n types
- Integration:(ax+b)n types - Exam Questions
- Integrating exponential functions ex, eax and e(ax+b)
- Integrating exponential functions ex, eax and e(ax+b) - Exam Questions
- Integrating reciprocal functions 1/x and 1/(ax+b)
- Integrating reciprocal functions 1/x and 1/(ax+b) - Exam Questions
- Integrals of the form : f '(x)/f(x)
- Integrals of the form : f'(x)ef(x)
- Integrals of sin x, cos x, secΒ² x
- Integrals of the form sin(ax+b), cos(ax+b), secΒ² (ax+b) types
- Integrals Using Trigonometric Identities
- sin2x and cos2x types
- Trigonometric types - Exam Questions
- Integrals involving partial fractions
- Integrals involving partial fractions - Exam Questions
- Integration by substitution
- Square root types
- Integration of trigonometric functions by substitution
- Integration of exponential types by substitution
- Integration by substitution using limits
- Integration of trigonometric functions by substitution with limits
- Integration by substitution - Exam Questions
- Integrating products of the form f[g(x)]g'(x) by inspection
- Integration by parts
- Integration by parts (ln types)
- eaxsin(bx) and eaxcos(bx) types
- Integration by parts using limits
- Proof of the formula - Integration by parts
- Exam Questions - Integration by parts
- Integration - General Methods
- Mixed Examples - Integration
- Integration - Exam Questions
- Differential Equations - Finding a general and a particular solution
- Working with constants in log types
- Differential Equations - Exponential and trig type
- Differential equations β Forming differential equations β Direct proportion type
- Differential equations β Forming differential equations β Inverse proportion type
- Differential equations β Forming differential equations β Newton's law of cooling
- Differential equations β Forming differential equations β Forming differential equations - Exam Questions
- Solution of Equations by Numerical methods β Graphical methods
- Solution of Equations by Numerical methods β Change of sign
- Solution of Equations by Numerical methods β Iteration
- Solution of Equations by Numerical methods β Iteration - Exam Questions
- Solution of Equations by Numerical methods β Newton-Raphson method for locating a root in a given interval
- Solution of Equations by Numerical methods β Newton-Raphson - Exam Questions
- Trapezium rule
- Trapezium rule - Exam Questions
- Vector notation
- Position vectors
- Multiplying a vector by a scalar
- Addition and subtraction of vectors
- Magnitude of a 3 dimensional vector
- Vectors - Exam Questions
βοΈ Statistics 27 topics
- Conditional probability in tree diagrams
- Exam Questions - Tree diagrams
- Conditional probability in Venn diagrams
- Exam Questions - Venn diagrams
- Probability in two way tables
- Independent, dependent and mutually exclusive events
- Mean and variance
- The normal distribution
- Normal Cumulative Distribution Function
- Inverse Normal Function to find observed values
- The standard normal distribution Z~N(0,1) - Using tables or calculator
- Probabilities from a normal distribution using tables
- Exam Questions - Normal distribution, finding a probability
- Calculating the mean ΞΌ and standard deviation Ο
- Exam Questions - Calculating the mean and standard deviation
- Exam Questions - Finding an observed value
- Sampling distribution of the sample means (Normal distribution)
- Sampling distribution of the sample means (Normal distribution) proof
- Introduction to Hypothesis testing for Normal distribution
- Hypothesis testing for Normal distribution (One tailed and 2 tailed test)
- Understanding Critical values (Hypothesis testing for Normal Distribution)
- Hypothesis testing for Normal Distribution - Critical values method (3 examples)
- Measuring linear correlation using a calculator
- Hypothesis testing for zero correlation
- Continuity corrections
- The normal approximation to the binomial distribution
- Exam Questions - Normal approximation to the binomial distribution
βοΈ Mechanics 41 topics
- Resolving forces
- Resultant forces - two forces at an angle
- Exam Questions - Resultant forces: two forces at an angle
- Resultant forces - three or more forces at an angle
- Equilibrium of a particle
- Exam Questions - Equilibrium
- What is friction, limiting equilibrium and the coefficient of friction?
- Exam Questions - Friction
- What is the moment of a force?
- Horizontal beams in equilibrium resting on supports
- Exam Questions - Moments horizontal beams
- Tilting beam
- Exam Questions - Moments tilting beam
- Moment of a non-perpendicular force
- Exam Questions - Moment of a non-perpendicular force
- Moments - Ladder problems
- Exam Questions - Moments ladder problems
- Exam Questions - Horizontal rough plane
- Motion on a smooth inclined plane
- Motion on a rough inclined plane
- Exam Questions - Rough inclined plane
- Inclined planes
- Exam Questions - Inclined planes
- Exam Questions - Force on a pulley
- Projectiles
- Particle projected at an upward angle from a height above ground
- Particle projected at a downward angle from a height above ground
- Particle projected horizontally from a height
- Finding the speed and direction at a given time
- Equation of the trajectory of a projectile
- Exam Questions - Projectiles
- Projectile Motion (hitting targets)
- Projectile motion 'Show that' questions (part 1)
- Projectile motion 'Show that' questions (part 2)
- Projectiles - "Show that" questions (part 3) - Particles colliding
- Calculating the position vector after time t
- Constant velocity as the rate of change of position vectors
- Exam Questions - Velocity vectors
- Position, velocity and variable acceleration vectors
- Working backwards using integration methods
- Exam Questions - Variable acceleration using vectors
A Level ExamSolutions Maths Edexcel Revision Content
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A Level ExamSolutions Maths Edexcel - Pure - Algebra and Functions β Rational Expressions: Simplifying β Simplifying algebraic fractions Content Preview
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Algebra and Functions β Rational Expressions: Simplifying β Simplifying algebraic fractions
Algebra and Functions β Rational Expressions: Simplifying β Simplifying Algebraic Fractions
Key Concepts
- Algebraic fractions, also known as rational expressions, are fractions in which the numerator and the denominator are both polynomials.
- These fractions can often be simplified by factoring and cancelling common factors in the numerator and the denominator.
- The process of simplifying an algebraic fraction is similar to that of simplifying a numeric fraction: you divide the numerator and the denominator by their highest common factor.
Factoring
- The first step in simplifying an algebraic fraction is typically to factor both the numerator and the denominator.
- Factoring breaks down the expressions into the product of their factors, which can simplify the process of identifying and cancelling common factors.
- Remember that a difference of squares, such as a^2 - b^2, can be factored into (a - b)(a + b).
Cancelling Common Factors
- Once the numerator and denominator have been factored, the next step is to identify and cancel any common factors.
- Common factors are expressions that appear in both the numerator and the denominator. By definition, any expression divided by itself is 1, so common factors can be cancelled out.
- This step is crucial: cancelling common factors can greatly simplify the fraction and highlight its important features.
Multiplying and Dividing Algebraic Fractions
- When multiplying algebraic fractions, you simply multiply the numerators together and the denominators together. Simplify if possible.
- When dividing algebraic fractions, you multiply by the reciprocal of the divisor. Realise that this is the same as multiplying by the fraction upside-down. Simplify if possible.
Adding and Subtracting Algebraic Fractions
- Like numeric fractions, algebraic fractions can only be added or subtracted if they have the same denominator (i.e., they are like fractions).
- If the fractions have different denominators, you will need to find a common denominator before you can add or subtract them.
Useful Strategies
- Practice makes perfect. Spend time working on a variety of problems involving algebraic fractions to solidify your understanding.
- Watch out for complex fractions - fractions where the numerator and/or denominator itself contains fractions. You'll need to simplify these before you can proceed.
- Stick with it. Simplifying algebraic fractions can be a complex process, but it's also a foundational skill in much of algebra. Mastering this will set you up for success in more complex topics.
Question: Simplify the following algebraic fraction: (6x^3 - 18x^2 + 12x) / (3x^2)
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