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Further Maths ExamSolutions Maths Edexcel
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Further Maths ExamSolutions Maths Edexcel
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Further Maths ExamSolutions Maths Edexcel
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Core Pure
Exam Questions - Particular solutions using boundary conditions
Exam Questions - Trig Type
Exam Questions - Exponential Type kepx (exponential types)
Particular solutions using boundary conditions to solve differential equations
Exam Questions - General solutions where f(x) = kx (linear types)
Special types of particular integrals
General solutions where f(x) = λ cosωx + µ sinωx (trig types)
General solutions where f(x) = kepx (exponential types)
General solutions where f(x) = kx2 (quadratic types)
General solutions where f(x) = kx (linear types)
General solutions where f(x) = k (constant types)
How to solve second order linear differential equations that equal zero
Exam Questions - Exact equations (integrating factors)
Solving equations of the form dy/dx + Py = Q using an integrating factor
Exact equations where one side is the exact derivative of a product
Integrating expressions involving hyperbolic functions
Differentiation of inverse hyperbolic functions
Differentiation of hyperbolic functions
Solving hyperbolic equations using hyperbolic identities
Expressing inverse hyperbolic functions as natural logarithms
Inverse hyperbolic functions and their graphs
Osborn's rule
Hyperbolic identities
Solving equations using inverse and exponential functions
Graphs of sech(x), cosech(x) and coth(x)
Graphs of sinh(x), cosh(x) and tanh(x)
Definitions
Exam Questions - Tangents
Finding equations of tangents parallel to and perpendicular to the initial line
Exam Questions - Area bounded by a polar curve
Area bounded by a cardioid and a loop
Area bounded by a polar curve
Sketching curves the curve r = asin 2θ
Sketching curves the cardioid r = a (1+cosθ)
Sketching curves a circle and arc
Sketching polar curves a half-line
Sketching polar curves a spiral
Converting the equation of a Cartesian curve to polar form
Converting the equation of a polar curve to Cartesian form
Converting polar coordinates to Cartesian coordinates
Converting Cartesian coordinates to polar coordinates
Defining the position of a point
Reflection of a line in a plane
The angle between two planes
Angle between a line and a plane
Shortest distance from a point to a plane
Finding the point of intersection between a line and a plane
Cartesian form of a plane
Scalar product forms of a plane
Parametric vector form of a plane
Cartesian form of a line
Exam Questions - Vectors
Shortest distance of a point to a line
Closest point to a line and shortest distance from the origin
Exam Questions - Parallel intersecting and skew lines
Intersecting and skew lines
Parallel lines
Angle between two lines
Vector equation of a line
Exam Questions - Scalar product
Perpendicular vectors
Scalar product
Integrals of the form 1/(a2+x2) and 1/√(a2-x2)
Mixed examples : Differentiating inverse trig functions
Derivatives of sin-1(x), cos-1(x) and tan-1(x)
Exam Questions - Integrals involving partial fractions
Integrals involving partial fractions
The mean value of a function
Convergent and Divergent Integrals
Exam Questions - Volume of revolution: parametric form
Volume of revolution for a curve given in parametric form
Volume of Revolution about the y-axis generated between curves
Exam Questions - Volume of Revolution about the y-axis
Volume of Revolution about the y-axis
Volume of revolution about the x-axis generated between curves
Exam Questions - Volume of revolution about the x-axis
Volume of revolution about the x-axis
Exam Questions - Matrix proofs
Matrix proofs
Exam Questions - Divisibility and multiple tests
Divisibility and multiple test proofs
Exam Questions - Sum of series
Proof for other series
Proof of the sum of the series ∑r³
Proof of the sum of the series ∑r²
Proof of the sum of the series ∑r
Exam Questions - Maclaurin’s series
Further series
Series expansion for ln(1+x)
Series expansion for sin(x) and cos(x)
Series expansion for ex
Maclaurin's series expansion
Exam Questions - Method of differences
Method of differences
Exam Questions - Series
Using known formulae to sum more complex series
Sum of the cubes of the first n natural numbers ∑r3
Sum of the squares of the first n natural numbers ∑r2
Sum of the first n natural numbers ∑r and the results for ∑a and ∑(ar+b)
Relationships between the roots and coefficients of a quartic equation
Relationships between the roots and coefficients of a cubic equation
Relationships between the roots and coefficients of a quadratic equation
Exam Questions - Matrix transformations
Determinant as the area scale factor of a transformation
Inverse matrices to reverse linear transformations
Combinations of transformations
How well do you know your transformations?
Linear transformations - enlargement
Linear transformations - reflections
Linear transformations - rotations
Solving three linear simultaneous equations
Solving simultaneous linear equations in 2 unknowns
Finding the inverse of a 3x3 matrix where it exists
Finding the determinant of a 3x3 matrix
Transposed and symmetric matrices
Exam Questions - Identity and inverse of a 2x2 matrix
Identity and Inverse of a 2x2 matrix
Matrix multiplication
Addition and subtraction and multiplying a matrix by a scalar
Introduction and dimension of a matrix
Exam Questions - Loci in the complex plane
Using complex numbers to represent regions on an Argand diagram
The locus of a point moving along a half-line
The locus of a point moving along a perpendicular bisector
The locus of a point moving in a circle
Exam Questions - nth roots of a complex number
nth roots of a complex number
Expressing sinnθ and cosnθ in terms of sin(kθ) and cos(kθ)
Exam Questions - Expressing sin(nθ) and cos(nθ) in terms of sinθ and cosθ
Expressing sin(nθ) and cos(nθ) in terms of sinθ and cosθ
De Moivre's theorem
Exam Questions - Further complex numbers
Multiplication and division rules for mod and argument of two complex numbers
Exponential Form (Euler's relation)
Modulus-argument form of a complex number
Exam Questions - Finding roots
Geometrical effects of conjugating a complex number
Exam Questions - Complex numbers
Solving quartic equations
Solving cubic equations
Solving quadratic equations with complex roots
Square roots of a complex number
Solving problems with complex numbers
Modulus and argument of a complex number
Argand diagrams
Division of a complex number by a complex number
Complex conjugates
Addition, subtraction and multiplying complex numbers and simplifying powers of i
Real and imaginary numbers
Further Mechanics 1
Successive oblique impacts
Loss of kinetic energy due to oblique impact
Oblique impact between a sphere and wall
Oblique impact between two spheres
Exam Questions - Newton's law of restitution
Collision with a fixed surface
Collision between two particles
Newtons law of restitution (experimental law)
Energy stored in an elastic string or spring
Hooke's law
Exam Questions - Work, energy and power
Exam Questions - Power
Car accelerating along the horizontal and then up a hill
Calculating the power developed by an engine
Exam Questions - Work Energy Principle
Motion down an undulating slope
Motion down a rough plane
Motion down a smooth plane
Conservation of energy / work energy principle
Kinetic energy
Gravitational potential energy
Moving up a rough inclined plane against friction and against gravity
Moving in the vertical plane against gravity
Moving in the horizontal plane with and without friction
Exam Questions - Impulse (vector form)
Impulse in vector form
Exam Questions - Momentum and impulse
Impulse
Exam Questions - Momentum
A bullet being fired from a gun
Two particles colliding and moving off as one particle
Two particles colliding and then separating
Collisions and the conservation of linear momentum
Further Mechanics 2
Exam Questions - Linear motion with variable acceleration
Linear motion with variable acceleration
Vertical elastic strings and springs
Horizontal elastic strings and springs
Simple harmonic motion
Newton's law of gravitation
Particle moving in a straight line under a variable forces
Centre of mass of a lamina by integration
Exam Questions - Centre of mass of wire frameworks
Exam Questions - Centre of mass of laminas
Lamina toppling on an inclined plane
Freely hanging suspended lamina
Wire framework consisting of a rectangle and triangle
Composite Laminas - Rectangle with circle removed
Composite Laminas - Rectangle + semi-circle
Composite Laminas - Rectangle + triangle
Composite Laminas - Rectangles
Uniform Laminas - Sectors and semi-circles
Triangular laminas
Centre of mass of a uniform lamina that has lines of symmetry
Centre of mass of a system of particles in a plane
Centre of mass of a system of particles in a line
Particle on the outer surface of a sphere
Motion of a particle on a light inextensible string
Car moving around a banked track
Particle moving on the inside of a hemispherical shell
Particle moving on the inside of a cone
Conical pendulum
Angular speed and acceleration
Further Pure 1
Exam Questions - Modulus inequalities fractional type
Modulus inequalities fractional type
Exam Questions - Modulus inequalities
Modulus inequalities
Fractional inequalities
Exam Questions - Simpson's Rule
Simpson's Rule
Second Order Differential Equations - Still to be done
First Order Differential Equations - Still to be done
Intersection of three planes
The angle between two planes
Angle between a line and a plane
Shortest distance from a point to a plane
The equation of the line of intersection between two non parallel planes
Finding the point of intersection between a line and a plane
Cartesian form of a plane
Scalar product forms of a plane
Parametric vector form of a plane
Shortest distance between two skew lines
Vector product form of a line
Cartesian form of a line
Applications: volume
Triple scalar product
Applications : Area of a triangle and parallelogram
Vector product (cross product)
Finding equations of simple loci
Finding equations of tangents and normals to a hyperbola
Cartesian equation, focus and directrices
Cartesian and parametric equations and asymptotes for a hyperbola
Equation of the directrix and coordinates of the focus
Tangents and normals to an ellipse
The Ellipse - Cartesian and parametric forms
Exam Questions - Hyperbola (rectangular)
Tangents and normals: parametric type
Tangents and normals : Cartesian type
Cartesian and parametric type
Exam Questions - Parabola
Tangents and normals: parametric types
Tangents and normals: Cartesian types
Parabola parametric form
Coordinate geometry: Parabola directrix, focus, locus and equation
The conic sections of a circle, parabola, ellipse and hyperbola
Using substitution to reduce a differential equation to a known form
Exam Questions - Substitution types
The Weierstrass substitution
L'Hospital's Rule - Still to be made
Leibnitz’s theorem
Using Series Expansions to Find Limits : still to do
Exam Questions - Taylor’s series
Solution to differential equations using Taylor's series
Taylor's series
Solving trigonometric equations using the t-formulae
The t-formulae (Proof)
Further Pure 2
Exam Questions - Recurrence relations
Recurrence relationship proofs
Exam Questions - Divisibility and multiple tests
Divisibility and multiple test proofs
Exam Questions - Recurrence relationships
Recurrence relationships
Working with Sequences and Series - Definition and finding the nth term
Combining combinations
Combinations of n different items
Permutations with restrictions : items must not be together
Permutations with restrictions : items at the ends
Permutations with restrictions: letters / items together
Permutations of repeated letters in a word
Permutations of r items from n different items
Permutations of n different items
Exam Questions - Transformations of the complex plane
Further transformations of the complex plane
Transformations of the complex plane
Exam Questions - Loci in the complex plane
Using complex numbers to represent regions on an Argand diagram
The locus of a point moving on the arc of a circle
The locus of a point moving along a half-line
The locus of a point moving along a perpendicular bisector
The locus of a point moving in a circle
The Cayley-Hamilton theorem
Reducing a symmetrical matrix to diagonal form
Eigenvalues and eigenvectors
Area of surface of revolution about the x-axis
Applications of Integration - Length of an arc
Deriving and using reduction formulae
Isomorphism of groups
Lagrange’s theorem
Subgroups
Order of a group
Commutative groups
Different types of groups
What defines a group
Identity and inverse elements
Binary operations
Binary Operations - Definition
Further Stats 1
Probability Generating Functions
Chi squared test for contingency tables
Central Limit Theorem
Test for the parameter p of a Geometric Distribution
Exam Questions - Hypothesis tests: Poisson distribution
Hypothesis Tests – Poisson Distribution - Two tailed test
Hypothesis Tests – Poisson Distribution - Upper tail test
Hypothesis Tests – Poisson Distribution - Lower tail test
Test for a Poisson mean
Negative Binomial distribution
The variance Var(X) - Geometric Distribution
The expected value, E(X)
Calculating least trials before a success
Calculating probabilities
Geometric distribution
Exam Questions - Poisson approximation to the binomial distribution
Poisson approximation to the binomial distribution
Mean and variance
Exam Questions - Poisson distribution
Sum of two Poisson distributions
Finding an observed value x
Changing the mean
Poisson distribution
Expected value and variance of a function of X
Exam Questions - Discrete random variables
Linear combinations of discrete random variables
Variance, Var(X)
Expected values E(X²)
Expected values E(X)
Probability distribution tables
Further Stats 2
Exam Questions - Continuous uniform / rectangular distribution
Expected mean E(X), the variance Var(X), cumulative distribution function F(X)
Defining the pdf, calculating probabilities and quartiles
Continuous uniform / rectangular distribution
Exam Questions - Probability density functions and cumulative distribution functions
E(X) and Var(X)
Finding the mode from a p.d.f.
Finding the median quartiles and percentiles
The cumulative distribution function, (c.d.f)
Calculating probability from a p.d.f.
Finding the constant k in a p.d.f
What is a probability density function (p.d.f.)?
Hypothesis testing for Spearman's correlation coefficient
Exam Questions - Spearman's Rank Correlation Coefficient
Spearman's rank correlation coefficient
Exam Questions - Regression
Making predictions and reliability of predicted values
Interpreting the regression line constants, explanatory and response variables
Regression
Hypothesis testing for zero correlation
Exam Questions - Correlation
Measuring linear correlation using a calculator
Product moment correlation coefficient (pmcc)
Correlation