LEAVING CERT HIGHER LEVEL · MATHS
2023 — Paper 2
300 marks · fully worked video solutions
Section A · Concepts & Skills
Q130 marks
Probability
Q230 marks
Trigonometry
Q330 marks
Coordinate Geometry
Q430 marks
Coordinate GeometryAlgebra
📚 2 relevant video tutorials
Q530 marks
StatisticsArea & Volume
Q630 marks
GeometryTrigonometry
📚 3 relevant video tutorials
Geometry 1.1 — Axioms, Theorems, Corollaries (🏁 §10 Synthetic Geometry strand-opener: a 50-minute MAP of the entire LCHL geometry course — 5 axioms, 21 theorems, 6 corollaries, 22 constructions, 8 proofs to know (5 JC + 3 LC). 🚨 LOAD-BEARING — the three LC proofs you actually need are **Theorem 11** (three parallel lines cut equal segments on any transversal), **Theorem 12** (line parallel to a side of a triangle cuts the other two sides in the same ratio) and **Theorem 13** (similar triangles ⇒ sides proportional in order). The five JC proofs (Theorems 4, 6, 9, 14, 19) are technically on the LC course but Paul flags they are 'very unlikely' to be asked. Plus key Junior-Cert recall: 🎯 vertically-opposite / alternate / corresponding angles (Theorems 1, 3, 5 — always grouped together in exam diagrams); 🎯 cyclic-quadrilateral opposite-angles-sum-to-180° (Corollary 5); 🎯 the canonical 'converse-not-true' example: *all rectangles are parallelograms* but NOT all parallelograms are rectangles)Geometry 1.13 — Congruent Triangles (the FOUR congruence criteria for triangles + how to PROVE two triangles are congruent in exam questions. 🚨 LOAD-BEARING — congruent triangles have 3 sets of equal sides AND 3 sets of equal angles (same triangle, same area, written twice). Four criteria: SSS (three sides), SAS (two sides + included angle), ASA (two angles + included side), RHS (right angle + hypotenuse + one other side). 🚨 CRITICAL — AAA does NOT prove congruence; only similarity. The four criteria each require THREE 'things in common' but not just three angles. Exam-question template: (1) draw left/right two-column structure for the two triangles; (2) state three pairs of equal sides/angles with justification for each; (3) name the criterion (SSS/SAS/ASA/RHS); (4) conclude congruence. Justifications draw on Theorem 2 (isosceles base-angles), Theorem 5 (alternate/corresponding angles), Theorem 9 (parallelogram opposite-sides), and 'common side' / 'given'. The third in Paul's worked-examples trio after Theorem 19 questions and Similar Triangles)Trigonometry 1.2 — Sin, Cos and Tan
Section B · Contexts & Applications
Q750 marks
Trigonometry
📚 9 relevant video tutorials
Trigonometry 3.4 — Changing The MidlineDifferentiation 16 — Increasing / Decreasing CurvesDifferentiation 3 — Trigonometric FunctionsTrigonometry 1.1 — Pythagoras TheoremTrigonometry 1.2 — Sin, Cos and TanTrigonometry 1.5 — Sine RuleTrigonometry 3.2 — Changing the AmplitudeTrigonometry 3.3 — Changing the PeriodTrigonometry 3.5 — Identifying Functions From Their Graph
Q850 marks
StatisticsProbability
📚 7 relevant video tutorials
Probability 1 — JC RevisionStatistics 17 — Confidence IntervalsProbability 3 — Independent Events And Conditional ProbabilityStatistics 13 — z-scores 5 (The standardising formula z = (x − μ)/σ; practical-applications pivot for the entire arc)Statistics 16 — Margin of ErrorStatistics 19 — Hypothesis TestingStatistics 21 — Hypothesis Testing and Confidence Intervals
Q950 marks
TrigonometryCoordinate Geometry
📚 5 relevant video tutorials
AVM 1.1 — Area, Volume & Measurement JC Revision (🏁 §10 Measurement strand-opener: comprehensive shape-formula recap for the LCHL course. 🚨 LOAD-BEARING — Paul opens with the framing that AVM is NOT a stand-alone topic at LCHL: 'it pops up in differentiation, it pops up in integration, it pops up in different parts of the course.' Treat this lesson as a FORMULA TOOLKIT to be applied wherever shape calculations appear elsewhere. Covers the full inventory: 🎯 2D shapes (square, rectangle, triangle, circle — page 8 log tables) and 🎯 3D shapes (cube, rectangular block, cylinder, cone, sphere — page 10 log tables), with 🚨 LOAD-BEARING insights: (1) 'consistent-cross-section' rule — Volume = (area of base) × height works for cylinder/rectangular block/triangular prism BUT NOT for cone (because the cross-section shrinks); (2) cone has THREE dimensions (radius R, perpendicular height h, slant height L) related by Pythagoras (R² + h² = L²) — if you know two you can find the third; (3) sphere has only ONE dimension (R). Three worked questions: Q1 cylinder (V → R → CSA), Q2 cone (CSA + L → R → h via Pythagoras → V), Q3 sphere (A → R → V). All three reduce to the same skeleton: use one formula to find a missing dimension, then sub into another formula.)The Circle 7 — Equation of a Tangent to a Circle at a Point, or Parallel to a Given Line (the two tangent techniques: perpendicular-to-radius for a point on the circle; perpendicular-distance-equals-radius for tangents parallel to a known line)Trigonometry 1.2 — Sin, Cos and TanTrigonometry 1.6 — Cosine RuleTrigonometry 1.7 — Area of a Non-Right-Angled Triangle (`A = ½ ab sin C` on page 16; the third of the three non-right-angled-triangle formulae; the 🚨 LOAD-BEARING two-sides-plus-the-angle-between-them setup; the 🎯 sine-rule-or-cosine-rule-as-a-setup-step pattern when the question doesn't give those three pieces directly)
Q1050 marks
GeometryArea & VolumeProbability
📚 2 relevant video tutorials
Probability 5 — PermutationsGeometry 1.16 — Similar Triangles (applications: the 'write out three fractions, then eliminate the irrelevant one' workflow, the colour-coded opposite-angle technique for matching corresponding sides, and the semicircle 'bridge through a third triangle' pattern that's the most common exam setup)