Topic module: Further Pure Core Maths A-Level OCR

Complex numbers specimen questions (mixed exam boards)

Complex numbers In this tutorial, we solve problems from different exam boards involving complex numbers. The idea is to give you a more rounded approach to solving different topic problems. Edexcel A-Level Further Maths Specimen Paper 1 Q3 (Core Pure 1) 9 MARKS IB Maths AA HL Specimen Paper 1 (Section B) a) 5MARKS b) […]

Systems of differential equations

Damped Oscillations

Solving the differential equation: ẍ = – ω²x

The identity: cosh²x-sinh²x ≡ 1

Exam Questions – Partial Fractions (Improper type)

Invariant points

An invariant point is a point that does not move. Not all points in a plane are moved by a transformation. These points are said to be invariant.

3D matrix transformations

You will need to be familiar with matrix transformations in 3D (3 dimensions). These can be worked out in a similar way to the ones described in 2D.

Convergent and Divergent Integrals

Solving three linear simultaneous equations

Mixed examples : Differentiating inverse trig functions

Angle between a line and a plane

The mean value of a function

Exam Questions – Trig Type

Exam Questions – Exponential Type ke^px (exponential types)

Exam Questions – Matrix transformations

Exam Questions – Finding roots

Shortest distance of a point to a line

Shortest distance from a point to a plane

Geometrical effects of conjugating a complex number

Modulus-argument form of a complex number

Transposed and symmetric matrices

Shortest distance between two skew lines

Vector product (cross product)

Standard integrals relating to hyperbolic functions

Integrating inverse trigonometric and hyperbolic functions using integration by parts

Using trigonometric and hyperbolic substitutions

Integrating expressions involving hyperbolic functions

Finding the inverse of a 3×3 matrix where it exists

Finding the determinant of a 3×3 matrix

Derivatives of sin^-1(x), cos^-1(x) and tan^-1(x)

Denominator contains 1 linear and 1 quadratic factor

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 – Volume of Revolution about the y-axis

Volume of revolution about the x-axis generated between curves

Volume of Revolution about the y-axis generated between curves

Volume of Revolution about the y-axis

Exam Questions – Maclaurin’s series

Exam Questions – General solutions where f(x) = kx (linear types)

Exam Questions – Expressing sin(nθ) and cos(nθ) in terms of sinθ and cosθ

Exam Questions – Matrix proofs

Exam Questions – Vectors

Exam Questions – Parallel intersecting and skew lines

Exam Questions – Scalar product

Exam Questions – Volume of revolution about the x-axis

Integrals of the form 1/(a²+x²) and 1/√(a²-x²)

Solving quartic equations

Solving cubic equations

Solving quadratic equations with complex roots

Square roots of a complex number

Exam Questions – Area bounded by a polar curve

Exam Questions – Particular solutions using boundary conditions

Exam Questions – Substitution types

Exam Questions – Exact equations (integrating factors)

Exam Questions – nth roots of a complex number

Exam Questions – Further complex numbers

Exam Questions – Method of differences

Exam Questions – Divisibility and multiple tests

Exam Questions – Sum of series

Exam Questions – Series

Exam Questions – Identity and inverse of a 2×2 matrix

Exam Questions – Complex numbers

Exam Questions – Volume of revolution: parametric form

Exam Questions – Integrals involving partial fractions

The angle between two planes

Finding the point of intersection between a line and a plane

Scalar product forms of a plane

Cartesian form of a plane

Parametric vector form of a plane

Cartesian form of a line

Differentiation of inverse hyperbolic functions

Differentiation of hyperbolic functions

Solving hyperbolic equations using hyperbolic identities

Expressing inverse hyperbolic functions as natural logarithms

Solving equations using inverse and exponential functions

Graphs of sinh(x), cosh(x) and tanh(x)

Definitions

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

Further series

Series expansion for ln(1+x)

Series expansion for sin(x) and cos(x)

Series expansion for e^x

Maclaurin’s series expansion

Using substitution to reduce a differential equation to a known form

Particular solutions using boundary conditions to solve differential equations

Special types of particular integrals

General solutions where f(x) = λ cosωx + µ sinωx (trig types)

General solutions where f(x) = ke^px (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

Using substitution to reduce a differential equation to a known form

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

Separating the variables and sketching a family of curves

Working with constants in log types

Differential Equations – Finding a general and a particular solution

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

nth roots of a complex number

Expressing sinⁿθ and cosⁿθ in terms of sin(kθ) and cos(kθ)

Expressing sin(nθ) and cos(nθ) in terms of sinθ and cosθ

De Moivre’s theorem

Multiplication and division rules for mod and argument of two complex numbers

Exponential Form (Euler’s relation)

Method of differences

Matrix proofs

Divisibility and multiple test proofs

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

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

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 simultaneous linear equations in 2 unknowns

Sum of the first n natural numbers ∑r and the results for ∑a and ∑(ar+b)

Identity and Inverse of a 2×2 matrix

Matrix multiplication

Addition and subtraction and multiplying a matrix by a scalar

Introduction and dimension of a matrix

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

Closest point to a line and shortest distance from the origin

Intersecting and skew lines

Parallel lines

Angle between two lines

Vector equation of a line

Perpendicular vectors

Scalar product

Differential Equations – Exponential and trig type

Volume of revolution for a curve given in parametric form

Volume of revolution about the x-axis

Integrals involving partial fractions

Improper types

Sigma notation