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.

Exam Questions

2003

Paper 1 Question 1

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

$\ {\frac {6y}{x(x+4y)}}-{\frac {3}{2x}}$$\ {\frac {6y}{x(x+4y)}}-{\frac {3}{2x}}$

(b) (i) $\ f(x)=ax^{2}+bx+c$$\ f(x)=ax^{2}+bx+c$ where $\ a,b,c\epsilon R$$\ a,b,c\epsilon R$

Given that $\ k$$\ k$ is a real number such that $\ f(k)=0$$\ f(k)=0$, prove that $\ x-k$$\ x-k$ is a factor of $\ f(x)$$\ f(x)$.

(ii) Show that $\ 2x-{\sqrt {3}}$$\ 2x-{\sqrt {3}}$ is a factor of $\ 4x^{2}-2(1+{\sqrt {3}})+{\sqrt {3}}$$\ 4x^{2}-2(1+{\sqrt {3}})+{\sqrt {3}}$ and find the other factor.

(c) The real roots of $\ x^{2}+10x+c=0$$\ x^{2}+10x+c=0$ differ by $\ 2p$$\ 2p$ where $\ c,p\epsilon R$$\ c,p\epsilon R$ and $\ p>0$$\ p>0$.

(i) Show that $\ p^{2}=25-c$$\ p^{2}=25-c$.

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

Paper 1 Question2

2. (a) Solve the simultaneous equations:

$\ 3x-y=8$$\ 3x-y=8$

$\ x^{2}+y^{2}=10$$\ x^{2}+y^{2}=10$

(b) (i) Solve for x:

$\ |4x+7|<1$$\ |4x+7|<1$

(ii) Given that $\ x^{2}-ax-3$$\ x^{2}-ax-3$ is a factor of $\ x^{3}-5x^{2}+bx+9$$\ x^{3}-5x^{2}+bx+9$ where $\ a,b\epsilon R$$\ a,b\epsilon R$

find the value of $\ a$$\ a$ and the value of $\ b$$\ b$.

(c) (i) Solve for y: $\ 2^{2y+1}-5(2^{y}+2=0$$\ 2^{2y+1}-5(2^{y}+2=0$

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

$\ 2k^{2}x^{2}+2ktx+t^{2}-3k^{2}=0$$\ 2k^{2}x^{2}+2ktx+t^{2}-3k^{2}=0$ where $\ k,t,\epsilon R$$\ k,t,\epsilon R$ and $\ k\neq 0$$\ k\neq 0$

show that $\ \alpha ^{2}+\beta ^{2}$$\ \alpha ^{2}+\beta ^{2}$ is independent of $\ k$$\ k$ and $\ t$$\ t$.

2004

Paper 1 Question 1

(a) Express $\ {\frac {1-{\sqrt {3}}}{1+{\sqrt {3}}}}$$\ {\frac {1-{\sqrt {3}}}{1+{\sqrt {3}}}}$ in the form $\ a{\sqrt {3}}-b$$\ a{\sqrt {3}}-b$ where $\ a$$\ a$ and $\ b\epsilon N$$\ b\epsilon N$.

(b)

(i) Let $\ f(x)=x^{3}+kx^{2}-4x-12$$\ f(x)=x^{3}+kx^{2}-4x-12$ where $\ k$$\ k$ is a constant Given that $\ x+3$$\ x+3$ is a factor of $\ f(x)$$\ f(x)$ find the value of $\ k$$\ k$

(ii) Show that $\ {\frac {3}{1+x^{p}}}+{\frac {3}{1+x^{-}p}}$$\ {\frac {3}{1+x^{p}}}+{\frac {3}{1+x^{-}p}}$ simplifies to a constant.

(c)

(i) Show that $\ p^{3}+q^{3}-(p+q)^{3}=-3pq(p+q)$$\ p^{3}+q^{3}-(p+q)^{3}=-3pq(p+q)$.

(ii) Hence, or otherwise, find, in terms of $\ a$$\ a$ and $\ b$$\ b$, the three values of $\ x$$\ x$ for which $\ (a-x)^{3}+(b-x)^{3}-(a+b-2x)^{3}=0$$\ (a-x)^{3}+(b-x)^{3}-(a+b-2x)^{3}=0$.

Paper 1 Question 2

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

$\ 3x+y+z=0$$\ 3x+y+z=0$

$\ x-y+z=0$$\ x-y+z=0$

$\ 2x-3y-z=9$$\ 2x-3y-z=9$

(b)

(i)

Solve the inequality $\ {\frac {x+1}{x-1}}<4$$\ {\frac {x+1}{x-1}}<4$ where $\ x\epsilon R$$\ x\epsilon R$ and $\ x\neq 0$$\ x\neq 0$

(ii)

the roots of $\ x^{2}+px+q=0$$\ x^{2}+px+q=0$ are $\ alpha$$\ alpha$ and $\beta$$\beta$ where $\ p,q\epsilon R$$\ p,q\epsilon R$.

Find the quadratic equation whose roots are $\ \alpha ^{2}\beta$$\ \alpha ^{2}\beta$ and $\ \alpha \beta ^{2}$$\ \alpha \beta ^{2}$.

(c)

(i)

$\ f(x)=2x+1$$\ f(x)=2x+1$ for $\ x\epsilon R$$\ x\epsilon R$

Show that there exists a real number $\ k$$\ k$ such that for all $\ x$$\ x$

$\ f(x+f(x))=kf(x)$$\ f(x+f(x))=kf(x)$

(ii)

Show that for any real values of $\ a,b,h$$\ a,b,h$ the quadratic equation

$\ (x-a)(x-b)-h^{2}=0$$\ (x-a)(x-b)-h^{2}=0$

has real roots.

2005

Paper 1 Question 1

(a) Solve the simultaneous equations:

${\frac {x}{5}}-{\frac {y}{4}}=0$${\frac {x}{5}}-{\frac {y}{4}}=0$

$\ 3x+{\frac {y}{2}}=17$$\ 3x+{\frac {y}{2}}=17$

(b)

(i) Exspress $\ 2^{1/4}+2^{1/4}+2^{1/4}+2^{1/4}$$\ 2^{1/4}+2^{1/4}+2^{1/4}+2^{1/4}$ in the form $\ 2^{p/q}$$\ 2^{p/q}$ where $\ p,q\epsilon Z$$\ p,q\epsilon Z$

(ii) Let $\ f(x)=ax^{3}+bx^{2}+cx+d$$\ f(x)=ax^{3}+bx^{2}+cx+d$.

Show that $\ (x-t)$$\ (x-t)$ is a factor of $\ f(x)-f(t)$$\ f(x)-f(t)$.

(c)$\ (x-p)^{2}$$\ (x-p)^{2}$ is a factor of $\ x^{3}+qx+r$$\ x^{3}+qx+r$

Show that $\ 27r^{2}+4q^{3}=0$$\ 27r^{2}+4q^{3}=0$

Exspress the roots of $\ 3x^{3}+q=0$$\ 3x^{3}+q=0$ in terms of p

Paper 1 Question 2

(a) Solve for x $\ |x-1|<7$$\ |x-1|<7$ where $\ x\epsilon R$$\ x\epsilon R$

(b) The cubic equation $\ 4x^{3}+10x^{2}-7x-3=0$$\ 4x^{3}+10x^{2}-7x-3=0$ has one integer root and two irrational roots. Exspess the rational roots in simplest surd form.

(c) Let $\ f(x)={\frac {x^{2}+k^{2}}{mx}}$$\ f(x)={\frac {x^{2}+k^{2}}{mx}}$ wher $\ k$$\ k$ and $/m$$/m$ are constants and $\ m\neq 0$$\ m\neq 0$

(i) show that $\ f(km)=f({\frac {k}{m}})$$\ f(km)=f({\frac {k}{m}})$.

(ii) $\ a$$\ a$ and $\ b$$\ b$ are real numbers such that $\ a\neq 0,b\neq 0$$\ a\neq 0,b\neq 0$ and $\ a\neq b$$\ a\neq b$. Show that if $\ f(a)=f(b)$$\ f(a)=f(b)$, then $\ ab=k^{2}$$\ ab=k^{2}$

2006

Paper 1 Question 1

(a) Find the real number a such that for all $\ x\neq 9$$\ x\neq 9$,

${\frac {x-9}{{\sqrt {x}}-3}}$${\frac {x-9}{{\sqrt {x}}-3}}$

(b)$\ f(x)=3x^{3}+mx^{2}-17x+n$$\ f(x)=3x^{3}+mx^{2}-17x+n$ , where $\ m$$\ m$ and $\ n$$\ n$ are constants. Given that $\ x-3$$\ x-3$ and $\ x+2$$\ x+2$ are factors of $\ f(x)$$\ f(x)$, find the value of $\ m$$\ m$ and the value of $\ n$$\ n$.

(c)$\ x^{2}-t$$\ x^{2}-t$ is a factor of $\ x^{3}-px^{2}+r$$\ x^{3}-px^{2}+r$.

(i) Show that $\ {pq=r}$$\ {pq=r}$.

(ii) Express the roots of $\ x^{3}-px^{2}+r=0$$\ x^{3}-px^{2}+r=0$ in terms of $\ p$$\ p$ and $\ q$$\ q$.

Paper 1 Question 2

(a) Solve the simultaneous equations

$\ y=2x-5$$\ y=2x-5$

$\ x^{2}+xy=2$$\ x^{2}+xy=2$

(b)

(i) Find the range of values of $\ t\epsilon R$$\ t\epsilon R$ for which the quadratic equation

$\ (2t-1)x^{2}+5tx+2t=0$$\ (2t-1)x^{2}+5tx+2t=0$

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

(c) $\ f(x)=1-b^{2x}$$\ f(x)=1-b^{2x}$ and $\ g(x)=1-b^{1+2x}$$\ g(x)=1-b^{1+2x}$, where $\ b$$\ b$ is a positive real number. Find, in terms of $\ b$$\ b$, the value of $\ x$$\ x$ for which $\ f(x)=g(x)$$\ f(x)=g(x)$.