Proofs, Derivations and Constructions for Leaving Cert Higher Level Maths
This page gives a comprehensive list of all proofs, derivations and constructions that are on the leaving cert higher level maths syllabus. I have taken this information directly from the syllabus to be as accurate as possible. The complete syllabus can be found here.
Where a video solution of the proof is available simply click on the Video button to go to the video
You can download this list here.
Proofs

prove theorems 11,12,13,

prove that √2 is not rational Video

prove De Moivre’s Theorem by induction for n ∈ N Video

prove by induction

simple identities such as the sum of the first n natural numbers and the sum of a finite geometric series

simple inequalities such as n! > 2n , 2n ≥ n2 (n ≥ 4) (1+ x) n ≥ 1+nx (x > –1)

factorisation results such as 3 is a factor of 4n –1

Derivations

derive the trigonometric formulae 1 Video, 2, 3, 4 Video, 5 Video, 6, 7, 9

derive the Sine Rule Formula (trig formula 2) Video

derive the Cosine Rule Formula (trig formula 3)Video

derive the formula for the sum to infinity of geometric series by considering the limit of a sequence of partial sums

solve problems involving finite and infinite geometric series including applications such as recurring decimals and financial applications, e.g. deriving the formula for a mortgage repayment
Constructions

geometrically construct √2 and √3

perform constructions 1622 (115 known from JC  A knowledge of the constructions prescribed for JCHL will be assumed, and may be examined)

The prescribed constructions are:

1. Bisector of a given angle, using only compass and straight edge. Video

2. Perpendicular bisector of a segment, using only compass and straight edge. Video

3. Line perpendicular to a given line l, passing through a given point not on l. Video

4. Line perpendicular to a given line l, passing through a given point on l. Video

5. Line parallel to given line, through given point.

6. Division of a segment into 2, 3 equal segments, without measuring it.

7. Division of a segment into any number of equal segments, without measuring it.

8. Line segment of given length on a given ray.

9. Angle of given number of degrees with a given ray as one arm.

10. Triangle, given lengths of three sides.

11. Triangle, given SAS data.

12. Triangle, given ASA data.

13. Rightangled triangle, given the length of the hypotenuse and one other side.

14. Rightangled triangle, given one side and one of the acute angles (several cases).

15. Rectangle, given side lengths.

16. Circumcentre and circumcircle of a given triangle, using only straightedge and compass. Video

17. Incentre and incircle of a given triangle, using only straightedge and compass. Video

18. Angle of 60◦ , without using a protractor or set square. Video

19. Tangent to a given circle at a given point on it. Video

20. Parallelogram, given the length of the sides and the measure of the angles.

21. Centroid of a triangle. Video

22. Orthocentre of a triangle.

Source of material: https://www.curriculumonline.ie/getmedia/f6f2e8222b0c461ebcd4dfcde6decc0c/SCSEC25_Maths_syllabus_examination2015_English.pdf
Accessed on 20/04/2020