# Leaving Certificate Mathematics/Algebra

**Algebra** is a branch of mathematics concerning the study of structure, relation and quantity. The name is derived from the treatise written by the Persian mathematician, astronomer, astrologer and geographer, Muhammad bin Mūsā al-Khwārizmī titled Kitab al-Jabr wa-l-Muqabala meaning "The Compendious Book on Calculation by Completion and Balancing", which provided symbolic operations for the systematic solution of linear and quadratic equations. Al-Khwarizimi's book made its way to Europe and was translated into Latin as Liber algebrae et almucabala.

Together with geometry, analysis, combinatorics, and number theory, algebra is one of the main branches of mathematics. Elementary algebra is often part of the curriculum in secondary education and provides an introduction to the basic ideas of algebra, including effects of adding and multiplying numbers, the concept of variables, definition of polynomials, along with factorization and determining their roots.

Algebra is much broader than elementary algebra and can be generalized. In addition to working directly with numbers, algebra covers working with symbols, variables, and set elements. Addition and multiplication are viewed as general operations, and their precise definitions lead to structures such as groups, rings and fields.

## Contents

## Exam Questions[edit]

### 2003[edit]

#### Paper 1 Question 1[edit]

1. (a) Express the following as a single fraction in its simplest form:

(b) (i) where

Given that is a real number such that , prove that is a factor of .

(ii) Show that is a factor of and find the other factor.

(c) The real roots of differ by where and .

(i) Show that .

(ii) Given that one root is greater than zero and the other root is less than zero, find the range of possible values of .

#### Paper 1 Question2[edit]

2. (a) Solve the simultaneous equations:

(b) (i) Solve for x:

(ii) Given that is a factor of where

find the value of and the value of .

(c) (i) Solve for y:

(ii) Given that {math>\ x = \alpha</math> and are the solutions of the quadratic equation

where and

show that is independent of and .

### 2004[edit]

#### Paper 1 Question 1[edit]

(a) Express in the form where and .

(b)

(i) Let where is a constant Given that is a factor of find the value of

(ii) Show that simplifies to a constant.

(c)

(i) Show that .

(ii) Hence, or otherwise, find, in terms of and , the three values of for which .

#### Paper 1 Question 2[edit]

(a) Solve without using a calculator, the following simultaneous equations:

(b)

(i)

Solve the inequality where and

(ii)

the roots of are and where .

Find the quadratic equation whose roots are and .

(c)

(i)

for

Show that there exists a real number such that for all

(ii)

Show that for any real values of the quadratic equation

has real roots.

### 2005[edit]

#### Paper 1 Question 1[edit]

(a) Solve the simultaneous equations:

(b)

(i) Exspress in the form where

(ii) Let .

Show that is a factor of .

(c) is a factor of

Show that

Exspress the roots of in terms of p

#### Paper 1 Question 2[edit]

(a) Solve for x where

(b) The cubic equation has one integer root and two irrational roots. Exspess the rational roots in simplest surd form.

(c) Let wher and are constants and

(i) show that .

(ii) and are real numbers such that and . Show that if , then

### 2006[edit]

#### Paper 1 Question 1[edit]

(a) Find the real number a such that for all ,

(b) , where and are constants. Given that and are factors of , find the value of and the value of .

(c) is a factor of .

(i) Show that .

(ii) Express the roots of in terms of and .

#### Paper 1 Question 2[edit]

(a) Solve the simultaneous equations

(b)

(i) Find the range of values of for which the quadratic equation

(ii) Explain why the roots are real when t is an integer.

(c) and , where is a positive real number. Find, in terms of , the value of for which .