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.
• 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.
• 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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