This course is spread across three units :

Methods in Algebra and Calculus (MAC),Geometry, Proof and Systems of Equations (GPS) and Applications of Algebra and Calculus (Apps)

An external SQA exam is sat at the end of the course.

Advanced Higher Maths Past Papers

Apps 1.1 Applying algebraic skills to the binomial theorem and to complex numbers

Apps 1.2 Applying algebraic skills to sequences and series.

Apps 1.3 Applying algebraic skills to summation and mathematical proof.

Apps 1.4 Applying algebraic and calculus skills to properties of functions.

Apps 1.5 Applying algebraic and calculus skills to problems.

GPS 1.1 Applying algebraic skills to matrices and systems of equations.

GPS 1.2 Applying algebraic and geometric skills to vectors.

GPS 1.3 Applying geometric skills to complex numbers.

GPS 1.4 Applying algebraic skills to number theory.

GPS 1.5 Applying algebraic and geometric skills to methods of proof.

MAC 1.1 Applying algebraic skills to partial fractions.

MAC 1.2 Applying calculus skills through techniques of differentiation.

MAC 1.3 Applying calculus skills through techniques of integration.

MAC 1.4 Applying calculus skills to solving differential equations.

Prerequisite Knowledge

You must have a good grasp of the following :

- Differentiation.
- Integration.
- Manipulating Vectors.
- Algebraic manipulation.

Key subskills:

Expand expressions using the binomial theorem.

Performing algebraic operations on complex numbers.

Apps 1.1 Applying algebraic skills to the binomial theorem and to complex numbers.

Unit 1 LO1 Use algebraic skills

Key subskills:

Performing geometric operations on complex numbers
.

GPS 1.3 Applying geometric skills to complex numbers.

Apps 1.1 Applying algebraic skills to the binomial theorem and to complex numbers.

Unit 2 LO3 Understand and use complex numbers .

Key subskills:

Differentiating functions given in the form of a product and in the form of a quotient.

Differentiating exponential and logarithmic functions.

Differentiating inverse trigonometric functions.

Finding the derivative of functions defined implicitly.

Finding the derivative of functions defined parametrically.

First Principles

Differentiation Refresher

Rules

Parametric Differentiation

MAC 1.2
Applying calculus skills through techniques of differentiation.

Unit 1 LO2Use the rules of differentiation on elementary functions.

Unit 2 LO1Use matrix methods to solve systems of linear equations.

Key subskills:

Solving a first order differential equation with variables separable.

Solving a first order linear differential equation using the integrating factor.

Solving second order differential equations.

Applying differentiation to problems, in context where appropriate.

Applying integration to problems, in context where appropriate.

Applications of Calculus

MAC 1.4 Applying calculus skills to solving differential equations.

Apps 1.5 Applying algebraic and calculus skills to problems.

Unit 3 LO4 Solve further ordinary differential equations .

Key subskills:

Finding the asymptotes of rational functions.

Investigating features of graphs and sketching graphs of functions,
including appropriate analysis of stationary points.

Apps 1.4 Applying algebraic and calculus skills to properties of functions.

Unit 1 LO4Use properties of functions.

Key subskills:

Integrating expressions using standard results.

Integrating by substitution.

Integrating by parts.

MAC 1.3
Applying calculus skills through techniques of integration.

Unit 1 LO3 -Integrate using standard results and the substitution metho .

Unit 2 LO2Use further integration techniques.

Key subskills:

Using Gaussian elimination to solve a 3x3 system of linear equations.

Performing matrix operations of addition, subtraction and multiplication.

Calculating the determinant of a matrix.

Finding the inverse of a matrix.

Transpose of a matrix
Matrix multiplication
2 x 2 matrix
3 x 3 matrix
Cofactor of a 3 x 3 matrix
Adjoint of a 3 x 3 matrix

Matrices - Determinant

Matrices - Inverse

Using Gaussian elimination to find inverse of matrix
Special matrices
Diagonal matrix

Special Matrices

GPS 1.1 Applying algebraic skills to matrices and systems of equations.

Unit 1 LO5 Use matrix methods to solve systems of linear equations.

Unit 3 LO2Use matrix algebra.

Key subskills:

Using Euclid’s algorithm to find the greatest common divisor of two positive integers.

GPS 1.4Applying algebraic skills to number theory.

Unit 3 LO5 Use further number theory and direct methods of proof.

Key subskills:

Expressing proper rational functions as a sum of partial fractions.

MAC 1.1 Applying algebraic skills to partial fractions

Unit 1 LO 1Use algebraic skills.

Key subskills:

Disproving a conjecture by providing a counter-example.

Using direct and indirect proof in straightforward examples.

GPS 1.5Applying algebraic and geometric skills to methods of proof.

Apps 1.3 Applying algebraic skills to summation and mathematical proof .

Unit 2 LO5 Use standard methods to prove results in elementary number theory.

Key subskills :

Finding the general term and summing arithmetic and geometric sequences.

Using the Maclaurin series expansion to find
a stated number of terms of the power series for a simple function.

Power Series

Application of summation formulae

Apps 1.2 Applying algebraic skills to sequences and series.

Apps 1.3 Applying algebraic skills to summation and mathematical proof .

Unit 2 LO4 Understand and use sequences and series.

Unit 3 LO3 Understand and use further aspects of sequences and series.

Key subskills:

Calculating a vector product.

Finding the equation of a line in three dimensions.

Finding the equation of a plane.

Components
Scalar triple product
Intersection of two lines
Intersection of two planes

Vector equations of lines

The distance from a point to a line
Equations of a line
Planes
Planes in space

Vector equation of planes

GPS 1.2 Applying algebraic and geometric skills to vectors.

Unit 3 LO1 Use vectors in three dimensions.