Course

# Preliminary Maths Extension 1

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9 Lessons

## Preliminary Maths Extension 1 Course

This Preliminary Maths Extension 1 syllabus is constructed on the assumption that students have acquired competence in the various mathematical skills related to the content of the mathematics course for the School Certificate. In particular it is expected that some familiarity with the material specified in the first few topics will have been gained. Nevertheless, the
content of all topics listed in this syllabus is expected to be covered in the teaching of the course.

## Syllabus

The order of topics in this syllabus is an indication of the connections among them, but is not prescriptive. Teachers are advised to familiarise themselves with the syllabus as a whole before planning a teaching program.
Part B of this syllabus is written for teachers and is intended to clarify the mathematical ideas underlying the whole syllabus and the various topics and to indicate the depth of treatment required. The methods and examples contained in them are not intended as a paradigm; it is the responsibility of each teacher to decide on
matters such as the method of presentation of a topic and the setting out of examples.

## Proofs

All proofs given in the syllabus are expected to be discussed and treated as a normal part of the exposition, except where Part B indicates a lighter treatment. Students are not required to reproduce proofs of results contained in items preceded by the symbol † except where Part B indicates that Preliminary Maths Extension 1 students are expected to be able to do so. It is assumed that electronic calculators will be available and used throughout the course.

### Specific objectives of the course

to give an understanding of important mathematical ideas such as variable, function, limit, etc, and to introduce students to mathematical techniques which are relevant to the real world;
to understand the need to prove results, to appreciate the role of deductive reasoning in establishing such proofs, and to develop the ability to construct these proofs;
to enhance those mathematical skills required for further studies in mathematics, the physical sciences and the technological sciences.

### Statement of Syllabus Topics for Preliminary Maths Extension 1

#### Further Algebra

• Other Inequalities
• \begin{aligned} x &\le 1 \\ x+1 &\ge 2 \\ x+y+2 &\le 3 \\ x+y+z+3 &\gt 4 \\ \end{aligned} \\

\begin{aligned} \displaystyle \frac{x}{x-1} &> 2 \\ (x-1)^2 \times \frac{x}{x-1} &> 2 \times (x-1)^2 \\ (x-1)x &\gt 2(x-1)^2 \\ 2(x-1)^2 – (x-1)x &\lt 0 \\ (x-1)\big[2(x-1) – x\big] &\lt 0 \\ (x-1)(2x-2 – x) &\lt 0 \\ (x-1)(x-2) &\lt 0 \\ \therefore 1 &\lt x \lt 2 \\ \end{aligned} \\

#### Trigonometry

• Harder applications of the exact ratios.
• Harder applications of Bearings and angles of elevation.
• Harder applications of Sine and cosine rules for a triangle. Area of a triangle, given two sides and the included angle.
• Trigonometric functions of sums and differences of angles.
• Expressions for sin θ, cos θ and tan θ in terms of tan(θ/2).
• Simple trigonometric identities and equations.
• The general solution of trigonometric equations.

#### Angles and Divisions of Linear Functions

• The angle between two lines.
• Internal and external division of an interval in a given ratio.

#### Parametric Representation of Parabola

• Parametric representation.
• Applications to problems concerned with tangents, normals and other geometric properties.

#### Remainder and Factor Theorem

• Definitions of polynomial, degree, polynomial equation. Graph of simple polynomials.
• The remainder and factor theorems.
• The roots and coefficients of a polynomial equation.
• Iterative methods for numerical estimation of the roots of a polynomial equation.
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