# Kilroy's College

## Online & Home Study Courses   # Leaving Certificate Maths course

### Course Contents

Remember, you can complete this Leaving Certificate Maths course within 6 to 24 Months of enrolling. So take your first step by contacting us today !

## Higher Level

See Ordinary level >>

## Detailed Content

Algebra

• Algebraic operations on polynomials and rational functions.
• Addition, subtraction, multiplication and division and the use of brackets and surds
• Laws of indices and logarithms
• The Factor Theorem for polynomials of degree two or three.
• Factorisation of such polynomials (the linear and quadratic factors having integer coefficients).
• Solution of cubic equations with at least one integer root.
• Solution of equations e.g. f(x) = g(x)
• Sums and products of roots of quadratic equations.
• Unique solution of simultaneous linear equations with two or more unknowns.
• Inequalities: solution of inequalities of the form g(x) < k, where g(x) = ax2+bx+c
• Use of the notation |x|; solution of |x-a| < b, |x – a| >b and combinations of these

Euclidean geometry

• Geometrical constructions
• Proof of theorems
• Enlargements

Probability

• Fundamental principle of counting
• Discrete probability – simple cases – tossing coins, dice throwing, birthday distribution etc.
• Outcome space, events
• Addition of probabilities, conditional probability, independent events
• Binomial and normal distributions
• Populations and samples

Coordinate geometry of the straight line and circle

• General equation of the line in form ax+by+c = 0
• Length of perpendicular from (x1, y1) to ax+by+c = 0
• Angle between two lines with slopes m1 and m2
• Equation of line through the intersection of two lines a1x+b1y+c1 = 0 and a2x+b2y+c2 = 0 {form L(a1x+b1y+c1)+M(a2x+b2y+c2) = 0, L, M constant}
• Division of a line segment in ratio m:n
• Addition, substraction of vectors, multiplication by a scalar. Unit vectors i and j
• Equation of circle centre (0,0) and radius r (x2 + y2 = r2)
• General equation of circle centre (- g, - f) and radius r (x2 + y2 + 2gx + 2fy + c = 0)
• Equation of tangent at (x1, y1) to x2 + y2 = r2
• Intersection of line and specific circle

Arithmetics

• Percentage error and tolerance
• Rates, income tax, PRSI and USC
• Costing (materials, labour and wastage)
• Area and perimeter of 2D shapes
• Area and circumference of circles and sectors of circles
• Problems involving area
• The nets of rectangular solids
• Surface area and volume of various 3D shapes
• Problems involving surface area and volume
• Trapezoid rule to approximate area

Functions and Differential Calculus

• Finding the period and range of a continuous periodic function, given its graph on scaled and labelled axis
• Informal treatment of limits of functions: Rules for sums, products and quotients
• Derivations from first principles of x2, x3, sin x, cos x, 1/x
• First derivatives of polynomials, rational, power and trigonometric functions
• First derivatives of sums
• First derivatives of differences
• First derivatives of products
• First derivatives of quotients
• Application to finding tangents to curves
• Simple second derivatives
• First derivatives of implicit and parametric functions
• Rates of change
• Maxima and minima
• Curve sketching

Statistics

• Calculation of mean and standard deviation
• Centres of measure – mean, median and mode
• Line of best fit
• Make predictions based on the line of best fit
• Percentiles
• Interquartile range

Number Systems

• Review of numbers systems
• Complex numbers: Argand diagram; addition, subtraction, multiplication, division; modulus; conjugate; conjugates of sums and products; conjugate root theorem
• De Moivre’s theorem: proof by induction, applications such as nth roots of unity and identities

Series and Induction

• Geometric sequences and series
• Sum of finite and infinite geometric series
• Sum to infinity of geometric series
• Applications of above e.g. derive the formula for a mortgage repayment
• Recurring decimals as infinite GPs
• Induction

Trigonometry

• Use the radian measure of angles
• Calculate the area of a sector of a circle and the length of an arc and solve problems involving these calculations
• Use trigonometry to calculate the area of a triangle
• Use the sine and cosine rules to solve problems 2D and 3D
• Define sine A, cos A and tan A for all values of A
• Graph the trigonometric function sine, cosine and tangent
• Graph trigonometric functions of type a sinnA, a cosnA for an n Îµ N
• Solve trigonometric equations such as sinnA = 0 and cosnA = ½ giving all solutions
• Calculate the area of a sector of a circle and the length of an arc and solve problems involving these calculations
• Derive the trigonometric formulae 1, 2, 3, 4, 5, 6, 7, 9
• Apply the trigonometric formulae 1 - 24