# 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 !

Please note the content shown below is 2023 syllabus. This course is also available for 2024 syllabus. Please contact college for details.

## Higher Level

## Detailed Content

- 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

- Geometrical constructions
- Proof of theorems
- Enlargements

- 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

- 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

- Percentage error and tolerance
- Rates, income tax, PRSI and USC
- Value Added Tax (VAT)
- 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

- 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

- 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

- 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

- 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

- 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

- Integration techniques (integrals of sums, multiplying constants, and substitution)
- Definite integrals with applications to areas and volumes of revolution (confined to cones and spheres)

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