Algebra and Inequalities?
Examples of solving algebraic inequalities will be discussed and how to graphically represent these solutions. Following this, discriminants will be reviewed and how to analyse the value obtained from them. Finally, the absolute modulus will be discussed and how to solve equations with absolute modulus. Exam questions will be answered and explained.
Sequences and Series and Financial Maths?
The difference between a sequence and a series is explained. A method of determining if a series is arithmetic or geometric is carried out as well as an investigation into both. Limits are introduced and how to solve such problems. Finally, infinite series and sequences are introduced and their applications such as reoccurring decimals.
Sequences and Series and Financial Maths?
Percentage profit and loss is discussed. Different types of income tax are also explained, such as PRSI and USC. An example of present and future value analyzed. Compound interest examples are then carried out. Depreciation examples are completed as well as different interest period and rates questions. Finally, savings, investments, loans and mortgages questions are carried out using both the amortization formula and sequences and series.
Complex Numbers?
An explanation of why complex number is needed. The Argand Diagram is introduced and how to analyze the modulus and the argument from the diagram as well as from their equations. Operations on complex numbers are carried out, such as their sum, products, addition and multiplication. Complex equations are also introduced and how to analyze their solutions.
Complex Numbers?
Focusing on Complex Equations and the Conjugate Root Theorem. An explanation of the polar form is given and a step by step analysis is provided on transforming from the cartesian to polar coordinates. Operations on the polar form of the complex number will be carried out. Finally, De Moivre’s Theorem will be introduced as well as several examples of its uses.
Trigonometry?
We review Pythagorus’ Theorem and different equations such as the arc of a circle to solve several problems. The unit circle and its properties will be presented. Several examples on solving trigonometric functions are carried out. A step-by-step guide to drawing trigonometric lines is introduced and their properties such as range, period etc.
Trigonometry?
Different geometric formulas using trigonometry are given and several examples of using such formulae are carried out. Several examples are calculated to demonstrate where trigonometry can be used to solve 2D and 3D problems. Derivations for trigonometric formula are given that are required for the exam and several exam questions will be completed.
The Line and the Circle?
Different formulae for properties of a line are analyzed, slope, perpendicular distance, midpoint etc. Problems including areas of triangles, dividing lines in ratios and perpendicular distance are solved demonstrating uses of these formulae. The circle formula is also introduced and how to determine the radius and centre from the equation of a circle.
The Line and the Circle?
The circle is introduced and how to write the equation of a circle given certain information. Several examples are then carried out involving a line and a circle, touching circles, using algebraic and geometric approaches.
Arrangements and Combinations, Probability?
An introduction to Permutations and Combinatorics. Several example questions are conducted to demonstrate applications of the above. Probability definitions are introduced and discussed, following a review on how permutations and combinatorics can be applied to probability questions. Probability formulae are introduced and are used to solve problems related to probability theory.
Probability and Expected Values?
The concept of a probability distribution and expected values is examined. Examples of where one can use the expected value is also considered. The Bernoulli Trials is introduced, and several problems are solved using this method. Questions on the margin of error, confidence levels, p values and hypothesis testing are discussed. These concepts will be introduced in statistics.
Statistics?
Different types of data and sampling are explained, and statistical definitions are also given. Methods of recognizing bias and limits are explained, such as mean, median mode, skewed data etc. The standard deviation is introduced and how to calculate its value from a data set. Finally, percentiles, histograms, stem and leaf plots, bar charts and scatter plots are discussed where the correlation fit, line of best fit and clusters are explained for the latter.
Inferencial Statistics?
The concept of a probability distribution is explained. Bernoulli trials and their uses are also introduced, and several examples are carried out. The empirical rule is stated and definitions such as the margin of error, confidence intervals and constructing these are explained. Finally, hypothesis tests are completed using both p values and confidence intervals.
Induction?
The structure of an answer using proof by induction is introduced. A step-by-step guide to solutions of the three different types of induction questions (Summation, division and inequality) is provided and example questions will be answered. Common questions and De Moivres proof are also provided.