Here is a breakdown of the Papers on the leaving Cert higher maths;





Paper 1 Essentials!

Question 1 : Algebra :

(a) Linear Algebra : You must be able to solve linear equations in ,one ,two and three variables (Simultaneous equations)(b)You must be able to solve inequalities . (c)You must be able to prove and use the factor theorem . (d) You must be able to use the remainder theorem .

Question 2: Algebra :

Quadratic Equations :(a) You must be able to solve a quadratic equation .(b)You must know the rules which connect the roots to the coefficients of a quadratic equation . (c) You must know the conditions for which a quadratic equation has (i)Real,(ii)Unreal,(iii)equal roots .(d)You must be able to solve a difference equation . (e)You must be able to solve an inequality involving indices .

Question 3:Matrices /Complex Numbers :

(a)You must be able to add,subtract,multiply,2x2 matrices . (b)You must be able to find the inverse of a 2x2 matrix and be able to solve a matrix equation . (c)You must be able to deal with complex numbers both in Cartesian (x + iy) form and in Polar form (Cosx + i Sinx). (d)You must be able to solve equations of the form .Conjugate roots theorem appeared in ,99 might be worth a look at

Question 4 :Sequences and Series :

(a)Must know the formulae for Un and Sn of an AP and a GP . (b) Must know Sáof a GP . (b)To do this question you must be very familiar with all aspects of the properties of AP's,and GP's.Question 4 is the main sequences and series question it often contains an equation in see 99.98,997,96 this can be very easy

Question 5: Series /Induction/Logs :

(a)You must be able to prove all of the following by Induction (i)Sums of Series (ii)Inequalities (iii)That a given number is a factor of a given expression ,(b)Must be able to solve equations involving Logs .(c)You must be able to use the Binomial expansion,you must be able to use the general term to find specific terms ,you must know the properties of Binomial Coefficients .(d)You must be able to find Un ,Sn and S of a telescopic type series ,You must be able to find Un,Sn,S of an APGP .

Question 6 :Differential Calculus :

(a)You must be able to differentiate functions using the product,quotient,and chain rules (b)You must be able to differentiate implicit functions . (c)You must be able to use calculus to find the turning points on a curve .(d)You must be able to find the Asymptotes of a curve .

Question 7 : Differential Calculus :

(a)You must be able to differentiate specific functions from first principals (can appear in Q6orQ7) ;(b)You must be able to use calculus to solve problems involving distance,speed ,time and rate of change problems in general,(c)You must be able to differentiate functions in parametric form .Newton Raphson has been a particular favourite in this question in recent years.

Question 8 : Integral Calculus :

(a)You must be able to integrate standard integrals (b)You must be able to use substitutions in particular the udu substitution . (c)You must be able to use integration to find (i)the Area under a curve (ii)the volume formed by rotating a function about an axis (objects formed can only be cones or spheres) .(d)The last part of this question may be quite difficult be well prepared .

General Comments :

(a)The marking scheme is as follows (i)Each question is broken into three parts a = 10 marks,b = 20 marks,c = 20 marks .(ii) Attempt marks will be awarded for any step in the right direction (iii)Errors are marked as follows, - 1 for a slip a small arithmetical error ,-3 for a blunder a technical error . (iv) The same mistake is never punished more than once ie you cannot lose marks more that once for a repeated error . (v)The order in which the questions are attempted is not important . For some students the best advice is to do all the part a and b's first then come back and do the more difficult part c's.

Proofs on this paper (1)Factor theorem(2)Induction(3)The Calculus proofs from first principals


Leaving Cert Higher Maths Paper 2 Essentials:

Question 1: The Circle :

(a)You must be able to change the equation of a circle from Polar Form to Carthesian Form .(b)You must be able to find the centre and radius of given circles,(c)You must be able to find the equation of a tangent from a point to a circle , and must be able to find the equation of a tangent at a point on a circle .(d)You must know the conditions that apply when two circles touch ,(although orthogonal circles ,and coaxial circles are not on the course knowledge of their properties can be useful ) Parts a and b very easy the part c,s can give trouble don’t forget the geometry of the circle.

Question 2: Vectors :

(a)Must know how to add two vectors using the triangle /parallelogram rule . (b)Must be able to write a vector in terms of two given vectors .(c)Must be able to use the dot product to find the measure of the angles of a triangle . this is an easy question put it on your list

Question 3:The line /Transformations :

(a)Must know the two coordinate geometry proofs . (b)Must be able to use the Tan x rule for the angle between two lines .(c)You must be able to use the perpendicular distance formula *(remember the bisectors of angles are not on the course . (d)You must be able to find the image of a line by a transformation .(e)You must be able to prove that the image of a specific line is parallel to the original line or if two lines are perpendicular their images are/are not perpendicular .

Question 4 : Trigonometry :

(a)Must be familiar with all the identities (only the first 12) on page 9 in the tables (10 marks) .(b)Must be able to solve triangles using the Sine and Cosine Rules (20 marks)

Question 5 : Trigonometry:

(a)Must be able to find the area of a sector and the lenght of an arc . (b)You must be able to solve Trig equations of the form SinX = 0,Cos2x + Sinx = -1, Sinx + Sin3x = 0 .(c)You must be able to solve a problem which involves three dimensions (a vertical pole on a horizontal plane) and possible use of a compound angle formula (Sin(X +Y).

Question 6 :Probability /Statistics:

(a)Must be able to find the mean and standard deviation of a frequency table (b)you must be know the what happens to the mean and standard deviation when a constant is added to the data ,or the data is multiplied /divided by a constant (properties of mean and standard deviation) .(c)You must know all the rules for probability ie the and or rules . (d)Be careful with this question as it is difficult to get attempt marks ,to get the best value out of this question tell the examiner exactly what you are doing at each stage so that if there are any attempt marks going you will get them .

Question 7: Probability /Permutations and Combinations /Difference Equations :

(a)You must be able to use the rules of permutations and combinations (note repetitions are out except in the cases of telephone numbers/ Licence plate numbers ). (b)Must be familiar with the ways of selecting committees . (c)Must be able to solve a difference equation ,(d)Must be able to verify that a given Un is a root of a given difference equation .

The Options :

I will only deal with question 8 the further Calculus Option .

(a)You must be able to integrate by parts ,(b)You must be able to integrate by parts a function which involves at most two steps( ). (c)You must be able to find the max or min of a given object ,remember the method (i)find the equation of the problem ,(ii)reduce the equation to an equation in one variable .(iii)find dy/dx and set equal to zero, and solve for x . (iv)differentiate again to establish max/min .(v)use the value found to find the max/min .(d)You must be able to use the Maclaurin expansion to write a given function as a power series.The following are the required functions xxSinxexln,2xex,1+ ,ln(1 + x) ,Cos x, Sin x,Tan x . You must also be able to find ? using the sum of two inverse tans ..(e)You must be able to use the ratio test to establish if a series converges or diverges the Un of this series can only be of the form SAnxn . x1tan-

Be careful with this question marks lost in this question cannot be replaced by doing another question !