## Statement of Syllabus Topics of Preliminary Maths

Preliminary Maths (commonly referred to as Year 11 2 Unit Mathematics) is the standard Year 11 course. The Preliminary Maths course is an absolute necessity for any student considering either the HSC Mathematics Extension 1 and 2 courses. The Preliminary Maths course covers:

#### 1. Basic arithmetic and algebra

- Review of arithmetical operations on rational numbers and quadratic surds.
- Inequalities and absolute values.
- Review of manipulation of and substitution in algebraic expressions, factorisation, and operations on simple algebraic fractions.
- Linear equations and inequalities. Quadratic equations. Simultaneous equations.

#### 2. Real functions

- Dependent and independent variables. Functional notation. Range and domain.
- The graph of a function. Simple examples.
- Algebraic representation of geometrical relationships. Locus problems.
- Region and inequality. Simple examples.

#### 3. Trigonometric ratios

- Review of the trigonometric ratios, using the unit circle.
- Trigonometric ratios of: \( \pm \theta, 90^{\circ} \pm \theta, 180^{\circ} \pm \theta, 360^{\circ} \pm \theta.\)
- The exact ratios.
- Bearings and angles of elevation.
- Sine and cosine rules for a triangle. Area of a triangle, given two sides and the included angle.

#### 4. Linear functions

- The linear function \(y = mx + b\) and its graph.
- The straight line: equation of a line passing through a given point with given slope; equation of a line passing through two given points; the general equation \(ax + by + c = 0\); parallel lines; perpendicular lines.
- Intersection of lines: intersection of two lines and the solution of two linear equations in two unknowns; the equation of a line passing through the point of intersection of two given lines.
- Regions determined by lines: linear inequalities.
- Distance between two points and the (perpendicular) distance of a point from a line.
- The mid-point of an interval.
- Coordinate methods in geometry.

#### 5. The quadratic polynomial and the parabola

- The quadratic polynomial \(ax^{2} + bx + c\). Graph of a quadratic function. Roots of a quadratic equation. Quadratic inequalities.
- General theory of quadratic equations, relation between roots and coefficients. The discriminant.
- Classification of quadratic expressions; identity of two quadratic expressions.
- Equations reducible to quadratics.
- The parabola defined as a locus. The equation \(x^{2} = 4Ay\). Use of change of origin when vertex is not at \((0, 0)\).

#### 6. Plane geometry – geometrical properties

- Preliminaries on diagrams, notation, symbols and conventions.
- Definitions of special plane figures.
- Properties of angles at a point and of angles formed by transversals to parallel lines.
- Tests for parallel lines.
- Angle sums of triangles, quadrilaterals and general polygons.
- Exterior angle properties.
- Congruence of triangles.
- Tests for congruence.
- Properties of special triangles and quadrilaterals.
- Tests for special quadrilaterals.
- Properties of transversals to parallel lines. Similarity of triangles. Tests for similarity. Pythagoras’ theorem and its converse.
- Area formulae.

- Application of above properties to the solution of numerical exercises requiring one or more steps of reasoning.
- Application of above properties to simple theoretical problems requiring one or more steps of reasoning.

#### 7. Tangent to a curve and derivative of a function

- Informal discussion of continuity.
- The notion of the limit of a function and the definition of continuity in terms of this notion. Continuity of \(f + g\), \(f-g\), \(fg\) in terms of continuity of \(f\) and \(g\).
- Gradient of a secant to the curve \(y = f(x)\).
- Tangent as the limiting position of a secant. The gradient of the tangent. Equations of tangent and normal at a given point of the curve \(y = f(x)\).
- Formal definition of the gradient of \(y = f(x)\) at the point where \(x = c\). Notations \(f'(c)\), at \(x = c\).
- Differentiation of: general polynomial, \(x^{n}\) for
*n*rational, and functions of the form \({f(x)}^{n}\) or \( \displaystyle \frac{f(x)}{g(x)}\), where \(f(x)\), \(g(x)\) are polynomials.

#### 8. Series and Applications

- Arithmetic series. Formulae for the
*n*th term and sum of n terms. - Geometric series. Formulae for the
*n*th term and sum of n terms. - Geometric series with a ratio between –1 and 1. The limit of \(x^{n}\), as \(n \rightarrow \infty\), for \(| x | < 1\), and the concept of limiting sum for a geometric series.
- Applications of arithmetic series. Applications of geometric series: compound interest, simplified hire purchase and repayment problems. Applications to recurring decimals.

#### Course Materials

Lessons

#### Basic Arithmetic and Algebra > Equations

- Basic Linear Equations
- Linear Equations With Pronumerals On Both Sides
- Linear Equations With Grouping Symbols
- Linear Equations with One Fraction
- Linear Equations With More Than One Fractions
- Linear Equations With Pronumerals In The Denominator
- Quadratic Equations
- Equations with Absolute Value
- Miscellaneous Linear Equations

#### Basic Arithmetic and Algebra > Exam Preparation Papers

- Basic Arithmetic and Algebra > Exam Preparation Paper 1
- Basic Arithmetic and Algebra > Exam Preparation Paper 2
- Basic Arithmetic and Algebra > Exam Preparation Paper 3
- Basic Arithmetic and Algebra > Exam Preparation Paper 4
- Basic Arithmetic and Algebra > Exam Preparation Paper 5
- Basic Arithmetic and Algebra > Exam Preparation Paper 6
- Basic Arithmetic and Algebra > Exam Preparation Paper 7
- Basic Arithmetic and Algebra > Exam Preparation Paper 8
- Basic Arithmetic and Algebra > Exam Preparation Paper 9
- Basic Arithmetic and Algebra > Exam Preparation Paper 10
- Basic Arithmetic and Algebra > Exam Preparation Paper 11
- Basic Arithmetic and Algebra > Exam Preparation Paper 12

#### Sequences and Series > Geometric Sequences

- Identifying Geometric Sequences
- First Term and Common Ratio of Geometric Sequences
- Finding a Term from a Rule
- Set up a Rule
- Finding the number of Terms of Geometric Series and Series
- Applications of Geometric Sequences and Series
- Compound Interest using Geometric Sequences and Series
- Finding Unknown Values of Geometric Sequences and Series