Course

Preliminary Maths

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40 Lessons

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 nth term and sum of n terms.
  • Geometric series. Formulae for the nth 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.

 

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Course Materials

Basic Arithmetic and Algebra > Arithmetic

Basic Arithmetic and Algebra > Laws of Surds

Basic Arithmetic and Algebra > Laws of Indices

Basic Arithmetic and Algebra > Index Equations

Basic Arithmetic and Algebra > Expansion

Basic Arithmetic and Algebra > Factorisation

Basic Arithmetic and Algebra > Equations

Basic Arithmetic and Algebra > Inequalities

Basic Arithmetic and Algebra > Simultaneous Equations

Basic Arithmetic and Algebra > Complete the Square

Basic Arithmetic and Algebra > Exam Preparation Papers

Real Functions > Domain and Range

Real Functions > Graphs

Real Functions > Limits Regions

Real Functions > Exam Preparation Papers

Trigonometry > Bearing and Exact Ratios

Trigonometry > Trigonometric Equations

Trigonometry > Trigonometric Graphs

Trigonometry > Trigonometric Identities and Rules

Linear Functions > Distance and Midpoint

Linear Functions > Equation of a Straight Line

Linear Functions > Parallel Lines

Linear Functions > Perpendicular Lines

Linear Functions > Exam Preparation Papers

Differentiation > Basic Rules

Differentiation > Further Differentiation Rules (Free)

Differentiation > Tangents and Normals

Differentiation > Exam Preparation Papers

Quadratic Functions > Quadratic Graphs

Quadratic Functions > Properties

Quadratic Functions > Equations

Quadratic Functions > Reducible to Quadratics

Quadratic Functions > Exam Preparation Papers

Locus and Parabola > Definitions

Locus and Parabola > Tangents and Normals

Sequences and Series > Arithmetic Sequences

Sequences and Series > Arithmetic Series

Sequences and Series > Geometric Sequences

Sequences and Series > Geometric Series

Sequences and Series > Infinite Geometric Series