Maths Paper 1 and 2 Layout and Question Breakdown for Leaving Cert Higher Level

Maths Proofs Predictions 2025

Algebra (manipulating formulae)

Algebra (factorisation of polynomials)

Sequences and series (limits)

Functions and graphs (transformations)

Logs and indices

Algebra (forming equation from given roots)

Complex numbers (Argand diagram)

Complex numbers (complex conjugates)

Functions and graphs (transformations)

Functions (composite functions)

Functions (injective/surjective/bijective)

Functions and graphs

Integration

Differentiation

Integration

Speed calculations ()

Probability (expected value)

Price calculations (cost price, selling price, profit)

Functions and graphs

Differentiation (tangents)

Geometry (area of a triangle)

Sequences and series

Integration

Differentiation (related rates of change)

Probability (expected values)

Trigonometric equations

Coordinate geometry (midpoint)

Coordinate geometry (perpendicular bisectors)

Perpendicular distance

Basic area and volume concepts

Trigonometric graphs

Differentiation

Statistics (hypothesis testing)

Probability

Probability (tree diagrams)

Geometry

Trigonometry

Circles

Area and volume

Area and volume (constructing a net)

Permutations and combinations

Finding where a line cuts the y-axis
Finding the slope of a line given the angle it forms with another line

Square roots of complex numbers

De Moivre’s theorem

Roots of cubic functions

Multiplication of surds

Roots of quadratic functions

Logs and indices

Arithmetic sequences

Co-ordinate geometry of the line

Differentiation

Trigonometry

Differentiation

Integration

Differentiation

Differentiation

Trigonometry

Volume

Perpendicular distance of a point from a line

Distance between two points

Similar triangles

Circles that touch externally

Trigonometric equations

Volume of a sphere

Distance, velocity, time

Co-ordinate geometry of the line – axioms and corollaries

Distance, velocity, time

Area of a triangle

Perpendicular distance from a point to a line

Z-scores

P-tests

Probability – Bernoulli trials

Distance, velocity, time

Trigonometric functions

Bernoulli trials

Z-scores

Expected value

Roots of a function/finding missing terms of a function

Functions with imaginary roots

Modulus

Complex numbers/sequences and series – finding common ratio of a geometric sequence (dividing complex numbers)

De Moivre’s Theorem

Logs

Functions – where a line intersects a curve

Differentiation – finding the point of inflection

Financial maths – present value and total mortgage owed

Differentiation of an equation containing the natural log

Integration

Substituting into the formula for a sequence

Algebraic fractions

Simplifying algebraic expressions

Algebra – solving for n

Substitution into a formula

Proof by induction

Area

Differentiation to find maximum area

Differentiation – rates of change

Logs

Integration – average value

Differentiation

Differentiation/inequality

Algebra/ logs

Perpendicular distance from a point to a line

Finding the slope of a line given the angle it makes with the x-axis

Angle between two lines

Distance between two points

Trigonometry

Circles with centres on the axes

Finding the equation of a circle given the radius and a point on the circle

Sine rule

Area of a circle

Ratios

Area of a sector

Area of a triangle

Independent events

Probability of an outcome in a game

Bernoulli trials

Surface area of a cone

Sectors

Volume of a cone

Flow rate – cross sectional area multiplied by rate

Z-scores

95% confidence interval

Sample proportion

Hypothesis testing

Measures of spread and central tendency

Functions

Pythagoras’ Theorem

Differentiation – minimum value

1. Algebra – undetermined coefficients
2. Algebraic fractions

1. Algebra – Plotting a linear graph given the equation/ substitution
2. Proof by induction

1. Factorising by grouping
2. Substitution
3. Differentiation

1. Integration
2. Finding the equation of a function from a graph/ integration/ solving a cubic function

1. Complex numbers – cubic equations
2. Writing complex numbers in polar forms/ De Moivre’s Theorem

1. Algebra – irrational equations/ surds/ Statistics – mean, median
2. Proof by contradiction

1. Sequences and series – geometric progressions/ finite series
2. Geometric progressions/ limits of a series

1. Trigonometric functions
2. Trigonometric functions
3. Differentiation of trigonometric functions
4. Differentiation – show a function is increasing
5. Differentiation – minimum point, point of inflection

1. Perimeter/ The Circle/ Algebra
2. Algebra/ Graphing a linear function/ Finding the slope of the graph/ Interpreting the slope
3. Algebra/ Functions/ Area/ Differentiation of a function/ Maximum point of a function

1. Probability – combinations/selections
2. Probability – combinations/selections

1. The Line – equation of a line
2. The Line – Finding the equation of a line given the slope and a co-ordinate/ intersection of two lines

1. The Circle – finding a point on the circle given the equation of the circle
2. The Circle – circle with the axes as tangents/ finding the equation of a circle

1. Trigonometric proof: cos2Θ = 1 – 2sin²Θ
2. Trigonometry/ angle between two lines

1. Construction – orthocentre of a triangle
2. Trigonometry/ The Circle/ The Line/ Properties of triangles/ circles/ tangents

1. Probability – independent events
2. Probability

1. Pythagoras’ Theorem/ Trigonometry/ Area/ Volume
2. Area/ Volume/ 3D shapes

1. Statistics – hypothesis testing/ interpreting normal distributions/ z-scores
2. Statistics – Interpreting p-values/ sample mean

1. Trigonometry – cosine rule
2. Sine rule
3. Area of a triangle
4. Area/ radius of a circle
5. Trigonometry – angle of elevation

1. Simultaneous equations
2. Inequalities

1. Expressing a function as a perfect square
2. Finding the minimum point of a function
3. Roots of a function

1. Complex roots of a function
2. De Moivre’s Theorem
3. Complex roots of a function

1. Geometric progression
2. Sum to finite geometric series
3. Sum to infinity

1. Roots of a function
2. Finding points of intersection of two functions using algebra
3. Graphing a function

1. Sequences and series – geometric progression and common ratio
2. Roots of a function
3. Sum to infinity

1. De Moivre’s Theorem
2. Complex numbers

1. Modulus inequalities
2. Simultaneous equations

1. Solving cubic functions

1. Complex roots of a function
2. Plotting complex numbers on the Argand diagram
3. Trigonometry – finding an angle in a triangle

1. Differentiation to find the slope of a tangent and the angle it makes with the x-axis
2. Integration – average value

1. Differentiation from first principles
2. Compound functions/ differentiation

1. Plotting natural/exponential functions
2. Finding points of intersection of two functions using algebra

1. Functions/ trapezoidal rule
2. Integration – definite integral / percentage error

1. Proof by induction
2. Sum of a finite arithmetic series
3. Sequences and series/ algebra – constructing an equation

1. Proving De Moivre’s Theorem by induction
2. Using De Moivre’s theorem

1. Geometric progression/ inequalities
2. Sum to infinity

1. Proof by induction
2. Logs

1. Simplifying complex numbers by multiplying by the conjugate
2. Sum of a finite geometric series

1. Differentiation from first principles
2. Differentiation/ slope of a tangent to the curve

1. Arithmetic sequence
2. Arithmetic sequence

1. Finding the roots of a function
2. Finding local maximum and minimum points of a function
3. Finding missing terms of a function

1. Pythagoras’ Theorem and Pythagorean triples
2. Prove a function is injective/ find the inverse function

1. Solving equations with surds
2. Differentiation
3. Differentiation – finding the turning points of an equation

1. Integration of a trigonometric function
2. Integrating a derivative to find the original function/ integration – average value

1. Points of intersection of two functions
2. Integration – area between two functions
3. Inverse functions – plot on graph using symmetry

1. Graphing an exponential function
2. Integration – finding the area bound by two functions and a line

1. Differentiation from first principles
2. Differentiation of trigonometric functions
3. Using differentiation to find the slope of the tangent to the curve at a given point

1. Compound interest
2. Financial maths – amortisation

1. Proving that a sequence is arithmetic/ finding common difference
2. Sum of an arithmetic sequence
3. Sequences and series/ algebra

1. Functions – finding missing terms
2. Functions – filling into a function
3. Graphing a function
4. Graphing a function
5. Estimations using the graphs of functions
6. Differentiation – finding maximum value

1. Substitution/ algebra
2. Substitution/ algebra
3. Substitution/ algebra
4. Using logs to find missing terms
5. Logs/ algebra
6. Integration – average value
7. Differentiation – rates of change

1. Differentiation – rates of change
2. Functions/ Integration (average value)

1. Functions – finding missing terms
2. Differentiation and evaluating the derivative at a certain value of x
3. Differentiation – finding the acute angle to the horizontal
4. Differentiation – point of inflection
5. Verifying points are on a curve/ finding images under symmetry in the point of inflection

1. Pythagoras’ Theorem/ Pythagorean triples/ algebra
2. Functions/ trigonometry/ Differentiation to find minimum value/ Algebra

1. Co-ordinate geometry – where graph intersects the y-axis
2. Area
3. Differentiation- show a graph is decreasing at a certain point
4. Differentiation – point of inflection

1. Financial maths/ Sequences & Series -derivation of amortisation formula
2. Financial maths – monthly repayments, APR, compound interest

1. Differentiation to find maximum points of a function and acute angle to horizontal/ Translations/ Simultaneous equations/ Functions
2. Algebra/ Logs

1. Functions – plotting a function/ constructing an equation from a graph
2. Volume of a cylinder/ differentiation – rates of change
3. Differentiation – rates of change
4. Area/ velocity/ time

1. Differentiation – maximum value on a curve
2. Functions/ algebra – finding x at a given value of y
3. Integration – finding area
4. Expressing a function as a perfect square
5. Identifying a pattern/ constructing an equation

1. Sequences and series
2. Constructing an expression for a geometric sequence/ inequalities
3. Inequalities/ sequences and series/ limits
4. Geometric sequence/ limits

1. Graphing trigonometric functions
2. Trigonometric functions – missing terms
3. Interpreting graph of a trigonometric function

1. Sequences and series – sum of a finite geometric series/ sum to infinity
2. Substituting into a given formula/ algebra

1. Trigonometric functions (filling in)
2. Trigonometric functions (solving for an unknown term)
3. Differentiation of trigonometric functions
4. Finding the maximum value of a trigonometric function
5. Integration – average value

1. Exponential equations – finding missing terms
2. Exponential equations – finding missing terms
3. Exponential equations – finding missing terms
4. Sketching an exponential function
5. Sketching another exponential function/ comparing the two sketches
6. Differentiation – rates of change

1. Probability – expected value
2. Probability – expected value/ Algebra

1. Probability – fundamental principle of counting
2. Bernoulli trials
3. Probability
4. Exponential terms/ inequalities/ algebra

1. Co-ordinate geometry – finding line perpendicular to a given line
2. Finding co-ordinates of the orthocentre

1. Probability – completing sample space diagram
2. Probability of winning/ interpreting sample space
3. Bernoulli trials

1. Trigonometry – cosine rule/ area of a triangle
2. Trigonometry – circumcentre

1. Probability/ Statistics – Normal distribution + z-scores
2. Probability/ Statistics -Standardising formula/ Algebra/ Empirical rule

1. Correlation coefficient
2. Drawing line of best fit
3. Interpreting/explaining slope of line of best fit
4. Distance, speed, time/ money

1. Co-ordinate geometry – equation of a line
2. Line/ circle – tangents, equation of a circle

1. Statistics – confidence intervals
2. Hypothesis testing
3. P-values

1. Trigonometry proof: cos2A = cos²A – sin²A
2. Trigonometry – sectors/ angle/ radius

1. Permutations
2. Factorials/ Algebra

1. Co-ordinate geometry – translations/ dividing a line in a given ratio
2. Co-ordinate geometry – midpoint of a line
3. Finding the orthocentre of a triangle

1. Trigonometry – proving by substitution
2. Solving a trigonometric equation

1. Co-ordinate geometry – perpendicular lines/ equation of a line
2. Perpendicular distance
3. Perpendicular distance/ Algebra
4. Perpendicular distance/ Algebra

1. Probability
2. Probability – expected value
3. Bernoulli trials

1. Trigonometry – solving trigonometric equations
2. Trigonometry/ Algebra/ Substitution

1. Finding the equation of a circle given 3 points on the circle
2. Trigonometry – finding an angle in a triangle

1. Similar triangles
2. Construction of a line segment equal in length to the square root of a given line segment

1. The Circle – circles touching internally/ centre/ radius
2. Ratios/ Translations/ Finding the equation of a circle
3. Common tangent/ Equation of a line

1. Trigonometry – trigonometric functions/ period/ range
2. Trigonometry – filling into trigonometric function/ Calculating standard deviation
3. Standard deviation/ Algebra/ Trigonometric functions

1. Co-ordinate geometry – verify a point is on a line
2. Co-ordinate geometry – perpendicular distance/ Algebra
3. The Circle – equation of a circle/ circles touching externally/ dividing a line in a given ratio

1. Proving that triangle are similar
2. Trigonometry – finding the length of a side
3. Trigonometry – finding the length of a side
4. Trigonometry – finding the area of a quadrilateral

1. Probability – filling in table/ fundamental principle of counting
2. Probability/ Sets – independent events

1. Trigonometry proof: tan(A+B) =
2. Solving trigonometric equations

1. Co-ordinate geometry/ Area of a triangle
2. Co-ordinate geometry – equation of a line/ showing a point is on a line
3. Co-ordinate geometry/ Algebra/ Area of a triangle

1. Proof – Theorem 12
2. Trigonometry – ratios/ Pythagoras’ Theorem

1. Pythagoras’ Theorem
2. Trigonometry/ circumference of a circle/ radius

1. Probability – fundamental principle of counting
2. Probability – expected value
3. Profit/ Algebra

1. Construction – centroid of a triangle
2. Proof – Theorem 11

1. Proof – Theorem 13
2. Construction of an angle of 60ᵒ without using a protractor/ set square
3. Similar triangles
4. Similar triangles
5. Similar triangles/ Algebra

1. Volume/ geometric progression
2. Surface area/ algebra/ arithmetic sequence
3. Length/ radius
4. Volume/ trigonometry/ algebra

1. Volume/ Algebra
2. Pythagoras’ Theorem/ similar triangles/ surface area/ Algebra
3. Volume/ Algebra

1. Trigonometry – Pythagoras’ Theorem/ finding an angle or length of a side using trig ratios/ Area
2. Trigonometry – angle of elevation/ trig ratios/ Pythagoras’ Theorem

1. The Circle/ Trigonometry
2. Area
3. Trigonometry – trig ratios to find an angle/ Area

1. Interpreting data presented in a table
2. Statistics – median/ interquartile range
3. Interpreting data in a table/ completing a table
4. Comparing different types of charts/ interpreting charts

1. Probability/ z-scores/ one-sample t-test/ confidence interval
2. Statistics – sample size/ sample proportion/ Differentiation – maximum value
3. Financial maths/ geometric progression

1. Statistics – normal distribution/ z-scores/ hypothesis testing
2. Probability – filling in tree diagram/ interpreting tree diagram/ conditional probability

1. Trigonometric function – period/ range
2. Maximum value of a trigonometric function
3. Differentiation – trigonometric functions/ rates of change
4. Graphing a trigonometric function
5. Interpreting graph of trigonometric function
6. Interpreting graph of trigonometric function

1. Probability – fundamental principle of counting
2. Probability
3. Probability
4. Probability/ Algebra
5. Sequences and series – geometric progression/ Inequalities
6. Probability – independent events

1. Probability – completing tree diagram/ interpreting tree diagram/ conditional probability
2. Hypothesis testing

1. Trigonometry – sine rule
2. Trigonometry – trigonometric functions/ range/ period/ graph
3. Trigonometry – sine rule/ cosine rule

1. Trigonometry – angles in a triangle/ Algebra
2. Trigonometry – angles in a triangle/ Algebra
3. Trigonometry/ Algebra
4. Trigonometry/ Algebra
5. Trigonometry – using trig ratios/ Algebra
6. Probability

1. Probability/ Statistics – z-scores/ standardising formula/ tax/ hypothesis testing
2. Statistics – confidence intervals
3. Statistics – Central Limit Theorem
4. Margin of error/ sample error

1. Trigonometry
2. Trigonometry – cosine rule/ Pythagoras’ Theorem
3. Trigonometry/ Functions/ Distance, velocity and time
4. Trigonometry

1. The Circle – centre/ radius/ Co-ordinate geometry – distance formula/ perpendicular distance/ mid-point of a line/ Translations
2. Construction of a circle given diameter/ Trigonometry/ The Circle/ Area