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Key Facts: LC Higher Maths Exam

2 papers

Higher Level Maths is assessed by two written papers, each 2 hours 30 minutes

State Examinations Commission

300 marks

Each paper is worth 300 marks, giving 600 marks in total

State Examinations Commission examination papers

5 strands

The Project Maths syllabus has five strands across Papers 1 and 2

NCCA Leaving Certificate Mathematics syllabus

25 bonus points

A grade of H6 or higher in Higher Level Maths earns 25 bonus CAO points

Central Applications Office (CAO)

H1-H8

Higher Level grades range from H1 (90-100%) to H8 (below 10%)

State Examinations Commission grading scale

2 sections

Each paper has Section A (Concepts and Skills) and Section B (Contexts and Applications)

State Examinations Commission examination papers

June sitting

Higher Level Maths papers are sat during the annual June Leaving Certificate examinations

State Examinations Commission timetable

100

Free original single-best-answer practice questions here

OpenExamPrep

Leaving Certificate Higher Level Mathematics is the top of three levels in the Irish State exam, set by the State Examinations Commission and built on the five-strand Project Maths syllabus. It is assessed by two written papers, each 2 hours 30 minutes and 300 marks (600 marks total), each split into Section A (Concepts and Skills) and Section B (Contexts and Applications). Paper 1 covers Number, Algebra and Functions including calculus; Paper 2 covers Geometry and Trigonometry and Statistics and Probability. Results use the H1-H8 scale, and a grade of H6 (40-49%) or higher earns 25 bonus CAO points. This 100-question bank provides original single-best-answer concept practice across all five strands.

Sample LC Higher Maths Practice Questions

Try these sample questions to review concepts for the LC Higher Maths exam. Each question includes a detailed explanation. Start the interactive quiz above for the full 100+ question experience with AI tutoring.

1Solve the equation 2x^2 - 7x + 3 = 0.
A.x = 3 or x = 1/2
B.x = -3 or x = -1/2
C.x = 3 or x = 2
D.x = 7 or x = 3
Explanation: Factorise: 2x^2 - 7x + 3 = (2x - 1)(x - 3) = 0, so x = 1/2 or x = 3. You can confirm by the quadratic formula with a = 2, b = -7, c = 3, giving x = (7 ± 5)/4.
2For the quadratic 3x^2 - 4x + 2 = 0, what does the discriminant tell you about the roots?
A.Two distinct real roots
B.One repeated real root
C.No real roots (two complex roots)
D.Exactly one real root and one complex root
Explanation: The discriminant is b^2 - 4ac = (-4)^2 - 4(3)(2) = 16 - 24 = -8. A negative discriminant means there are no real roots; the two roots are complex conjugates.
3Simplify (x^2 - 9)/(x^2 - x - 6) for x not equal to 3 or -2.
A.(x - 3)/(x - 2)
B.(x + 3)/(x + 2)
C.(x - 3)/(x + 2)
D.(x + 3)/(x - 2)
Explanation: Factorise numerator and denominator: (x^2 - 9) = (x - 3)(x + 3) and (x^2 - x - 6) = (x - 3)(x + 2). Cancelling the common factor (x - 3) leaves (x + 3)/(x + 2).
4Solve the inequality x^2 - 5x + 6 < 0.
A.x < 2 or x > 3
B.2 < x < 3
C.x < -3 or x > -2
D.-3 < x < -2
Explanation: Factorise: (x - 2)(x - 3) < 0. The product is negative between the roots, so the solution is 2 < x < 3. The parabola opens upward and dips below zero between its roots.
5Simplify (2 + 3i)(4 - i), where i^2 = -1.
A.11 + 10i
B.8 - 3i
C.5 + 14i
D.11 - 10i
Explanation: Expand: (2 + 3i)(4 - i) = 8 - 2i + 12i - 3i^2 = 8 + 10i - 3(-1) = 8 + 3 + 10i = 11 + 10i.
6What is the modulus of the complex number z = 5 - 12i?
A.7
B.13
C.17
D.169
Explanation: The modulus is |z| = sqrt(a^2 + b^2) = sqrt(5^2 + (-12)^2) = sqrt(25 + 144) = sqrt(169) = 13.
7Using De Moivre's theorem, evaluate (cos 30 degrees + i sin 30 degrees)^3.
A.cos 90 degrees + i sin 90 degrees
B.cos 10 degrees + i sin 10 degrees
C.cos 33 degrees + i sin 33 degrees
D.3 cos 30 degrees + 3i sin 30 degrees
Explanation: De Moivre's theorem gives (cos theta + i sin theta)^n = cos(n theta) + i sin(n theta). With theta = 30 degrees and n = 3, the result is cos 90 degrees + i sin 90 degrees, which equals i.
8Solve the simultaneous equations 2x + y = 7 and x - y = 2.
A.x = 3, y = 1
B.x = 1, y = 5
C.x = 2, y = 3
D.x = 4, y = -1
Explanation: Adding the equations: (2x + y) + (x - y) = 7 + 2 gives 3x = 9, so x = 3. Substituting into x - y = 2 gives 3 - y = 2, so y = 1.
9Simplify the surd expression sqrt(50) + sqrt(8).
A.7 sqrt(2)
B.sqrt(58)
C.10 sqrt(2)
D.5 sqrt(10)
Explanation: sqrt(50) = sqrt(25 x 2) = 5 sqrt(2) and sqrt(8) = sqrt(4 x 2) = 2 sqrt(2). Adding gives 5 sqrt(2) + 2 sqrt(2) = 7 sqrt(2).
10If log_2(x) = 5, what is the value of x?
A.10
B.25
C.32
D.7
Explanation: By the definition of a logarithm, log_2(x) = 5 means 2^5 = x. Since 2^5 = 32, x = 32.

About the LC Higher Maths Exam

Leaving Certificate Higher Level Mathematics is the most advanced of the three levels (Higher, Ordinary and Foundation) of the Irish State examination in mathematics, sat at the end of the two-year senior cycle and administered by the State Examinations Commission. The Project Maths syllabus is built around five strands: Statistics and Probability; Geometry and Trigonometry; Number; Algebra; and Functions (including differential and integral calculus). It is assessed by two written papers of 2 hours 30 minutes each, worth 300 marks per paper, with each paper split into Section A (Concepts and Skills) and Section B (Contexts and Applications). Paper 1 focuses on Number, Algebra and Functions/calculus, while Paper 2 focuses on Geometry and Trigonometry and on Statistics and Probability. Results are reported on the H1-H8 grade scale, and a grade of H6 or above earns 25 bonus CAO points used in third-level admission.

Exam sponsor: State Examinations Commission (SEC), Ireland. The requirements and fees below concern the certification or admission exam, separate from our free practice resources.

Assessment

Two written papers, each 300 marks. Each paper has Section A (Concepts and Skills) and Section B (Contexts and Applications). Paper 1 covers Number, Algebra and Functions (with calculus); Paper 2 covers Geometry and Trigonometry, and Statistics and Probability.

Time Limit

Each paper lasts 2 hours 30 minutes and is sat on a separate day during the June examination period (600 marks across the two papers).

Passing Score

No fixed pass mark. Results are reported on the H1-H8 scale (H1 = 90-100% down to H8 = below 10%). A grade of H6 (40-49%) or higher earns 25 bonus CAO points for Higher Level Mathematics.

Exam / Certification Fees

Candidates in recognised schools do not pay a separate sitting fee; external and repeat candidates pay an examination fee set annually by the State Examinations Commission.

Exam sponsor website

Fees, eligibility, and exam policies can change. Confirm them with the exam sponsor before applying or paying.

Our practice resources: topics covered

We aim to reflect publicly available exam outlines and topic information in our study resources. Coverage, format, and difficulty may differ from the actual exam, and we cannot guarantee that every detail is accurate or current. Confirm exam requirements, fees, and policies with the official exam sponsor.

20%

Algebra (Strand 4)

Practice covers manipulation of algebraic expressions, linear, quadratic and simultaneous equations, inequalities, surds, indices and logarithms, factor and remainder theorems, and complex numbers including modulus, argument and De Moivre's theorem. These skills underpin much of Paper 1.

25%

Functions and Calculus (Strand 5)

Practice covers domain, range and graphs of linear, quadratic, cubic, exponential, logarithmic and trigonometric functions, limits and continuity, differentiation rules and applications (rates of change, maxima and minima, tangents), and integration including the area under a curve and definite integrals.

20%

Geometry and Trigonometry (Strand 2)

Practice covers coordinate geometry of the line and the circle, synthetic geometry, theorems and enlargements, the unit circle, trigonometric ratios and identities, solving trigonometric equations, radian measure, and the sine and cosine rules with area of a triangle.

20%

Statistics and Probability (Strand 1)

Practice covers data presentation, mean, median, mode, range, standard deviation, the fundamental principle of counting, permutations and combinations, probability rules, the normal distribution, the empirical (68-95-99.7) rule, and inferential statistics including confidence intervals and hypothesis testing.

15%

Number (Strand 3)

Practice covers number systems, ratio and proportion, arithmetic and geometric sequences and series, financial mathematics including compound interest, present value, loans and amortisation, indices and logarithms in context, and quantitative reasoning.

Preparing for the LC Higher Maths Exam

What You Need to Know

  • Passing score: No fixed pass mark. Results are reported on the H1-H8 scale (H1 = 90-100% down to H8 = below 10%). A grade of H6 (40-49%) or higher earns 25 bonus CAO points for Higher Level Mathematics.
  • Assessment: Two written papers, each 300 marks. Each paper has Section A (Concepts and Skills) and Section B (Contexts and Applications). Paper 1 covers Number, Algebra and Functions (with calculus); Paper 2 covers Geometry and Trigonometry, and Statistics and Probability.
  • Time limit: Each paper lasts 2 hours 30 minutes and is sat on a separate day during the June examination period (600 marks across the two papers).
  • Exam / certification fees: Candidates in recognised schools do not pay a separate sitting fee; external and repeat candidates pay an examination fee set annually by the State Examinations Commission. Official sources

Using Our Practice Resources

  • Work through all 100 available questions
  • Review every answer and explanation
  • Track weak areas and revisit them
  • Use our AI tutor for tough concepts

LC Higher Maths: Suggested Study Strategy

1Work through past SEC papers and marking schemes for both Paper 1 and Paper 2, since the marking scheme shows exactly how method and accuracy marks are awarded.
2Learn the formulae that are NOT in the official tables booklet (Formulae and Tables) and practise locating the ones that are, so you do not waste time in the exam.
3For calculus, master the rules first (product, quotient and chain) before applied questions on rates of change, maxima and minima, and area under curves.
4In Statistics and Probability, be fluent with the empirical rule and the normal distribution, and practise full hypothesis-testing and confidence-interval questions, which appear regularly on Paper 2.
5Show full working: Higher Level marking awards partial credit, so a clearly set-out method can earn marks even when the final answer is wrong.
6Practise Section B context questions under time pressure, as these longer applied problems combine several strands and reward careful reading of the scenario.

Frequently Asked Questions

How is Leaving Certificate Higher Level Maths examined?

It is examined by two written papers, each lasting 2 hours 30 minutes and worth 300 marks (600 marks in total). Each paper has Section A (Concepts and Skills) and Section B (Contexts and Applications).

What is the difference between Paper 1 and Paper 2?

Paper 1 mainly covers Number, Algebra and Functions including calculus. Paper 2 mainly covers Geometry and Trigonometry, and Statistics and Probability. Both papers include short-answer and longer applied questions.

What are the five strands of the syllabus?

The Project Maths syllabus has five strands: Strand 1 Statistics and Probability, Strand 2 Geometry and Trigonometry, Strand 3 Number, Strand 4 Algebra, and Strand 5 Functions (including differential and integral calculus).

What are the bonus points for Higher Level Maths?

A grade of H6 (40-49%) or higher in Higher Level Mathematics earns 25 bonus CAO points, which are added to the points from the candidate's best six subjects for third-level admission.

How are the grades reported?

Higher Level results use the H1-H8 scale: H1 is 90-100%, H2 is 80 to under 90%, and so on in 10% bands down to H8 (below 10%). H7 (30-39%) is the lowest grade that still awards CAO points at Higher Level.

Are these official State Examinations Commission questions?

No. These are original OpenExamPrep single-best-answer concept questions modelled on the syllabus strands. The State Examinations Commission publishes official past papers and marking schemes separately, and you should also practise full long-form questions.