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 [ edit ] Paper 1 Question 1 [ edit ] 1. (a) Express the following as a single fraction in its simplest form:
6
y
x
(
x
+
4
y
)
−
3
2
x
{\displaystyle \ {\frac {6y}{x(x+4y)}}-{\frac {3}{2x}}}
f
(
x
)
=
a
x
2
+
b
x
+
c
{\displaystyle \ f(x)=ax^{2}+bx+c}
a
,
b
,
c
ϵ
R
{\displaystyle \ a,b,c\epsilon R}
Given that
k
{\displaystyle \ k}
f
(
k
)
=
0
{\displaystyle \ f(k)=0}
x
−
k
{\displaystyle \ x-k}
f
(
x
)
{\displaystyle \ f(x)}
(ii) Show that
2
x
−
3
{\displaystyle \ 2x-{\sqrt {3}}}
4
x
2
−
2
(
1
+
3
)
+
3
{\displaystyle \ 4x^{2}-2(1+{\sqrt {3}})+{\sqrt {3}}}
(c) The real roots of
x
2
+
10
x
+
c
=
0
{\displaystyle \ x^{2}+10x+c=0}
2
p
{\displaystyle \ 2p}
c
,
p
ϵ
R
{\displaystyle \ c,p\epsilon R}
p
>
0
{\displaystyle \ p>0}
(i) Show that
p
2
=
25
−
c
{\displaystyle \ 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
{\displaystyle \ p}
Paper 1 Question2 [ edit ] 2. (a) Solve the simultaneous equations:
3
x
−
y
=
8
{\displaystyle \ 3x-y=8}
x
2
+
y
2
=
10
{\displaystyle \ x^{2}+y^{2}=10}
(b) (i) Solve for x:
|
4
x
+
7
|
<
1
{\displaystyle \ |4x+7|<1}
x
2
−
a
x
−
3
{\displaystyle \ x^{2}-ax-3}
x
3
−
5
x
2
+
b
x
+
9
{\displaystyle \ x^{3}-5x^{2}+bx+9}
a
,
b
ϵ
R
{\displaystyle \ a,b\epsilon R}
find the value of
a
{\displaystyle \ a}
b
{\displaystyle \ b}
(c) (i) Solve for y:
2
2
y
+
1
−
5
(
2
y
+
2
=
0
{\displaystyle \ 2^{2y+1}-5(2^{y}+2=0}
x
=
β
{\displaystyle \ x=\beta }
2
k
2
x
2
+
2
k
t
x
+
t
2
−
3
k
2
=
0
{\displaystyle \ 2k^{2}x^{2}+2ktx+t^{2}-3k^{2}=0}
k
,
t
,
ϵ
R
{\displaystyle \ k,t,\epsilon R}
k
≠
0
{\displaystyle \ k\neq 0}
show that
α
2
+
β
2
{\displaystyle \ \alpha ^{2}+\beta ^{2}}
k
{\displaystyle \ k}
t
{\displaystyle \ t}
Paper 1 Question 1 [ edit ] (a) Express
1
−
3
1
+
3
{\displaystyle \ {\frac {1-{\sqrt {3}}}{1+{\sqrt {3}}}}}
a
3
−
b
{\displaystyle \ a{\sqrt {3}}-b}
a
{\displaystyle \ a}
b
ϵ
N
{\displaystyle \ b\epsilon N}
(b)
(i) Let
f
(
x
)
=
x
3
+
k
x
2
−
4
x
−
12
{\displaystyle \ f(x)=x^{3}+kx^{2}-4x-12}
k
{\displaystyle \ k}
x
+
3
{\displaystyle \ x+3}
f
(
x
)
{\displaystyle \ f(x)}
k
{\displaystyle \ k}
(ii) Show that
3
1
+
x
p
+
3
1
+
x
−
p
{\displaystyle \ {\frac {3}{1+x^{p}}}+{\frac {3}{1+x^{-}p}}}
(c)
(i) Show that
p
3
+
q
3
−
(
p
+
q
)
3
=
−
3
p
q
(
p
+
q
)
{\displaystyle \ p^{3}+q^{3}-(p+q)^{3}=-3pq(p+q)}
(ii) Hence, or otherwise, find, in terms of
a
{\displaystyle \ a}
b
{\displaystyle \ b}
x
{\displaystyle \ x}
(
a
−
x
)
3
+
(
b
−
x
)
3
−
(
a
+
b
−
2
x
)
3
=
0
{\displaystyle \ (a-x)^{3}+(b-x)^{3}-(a+b-2x)^{3}=0}
Paper 1 Question 2 [ edit ] (a) Solve without using a calculator, the following simultaneous equations:
3
x
+
y
+
z
=
0
{\displaystyle \ 3x+y+z=0}
x
−
y
+
z
=
0
{\displaystyle \ x-y+z=0}
2
x
−
3
y
−
z
=
9
{\displaystyle \ 2x-3y-z=9}
(b)
(i)
Solve the inequality
x
+
1
x
−
1
<
4
{\displaystyle \ {\frac {x+1}{x-1}}<4}
x
ϵ
R
{\displaystyle \ x\epsilon R}
x
≠
0
{\displaystyle \ x\neq 0}
(ii)
the roots of
x
2
+
p
x
+
q
=
0
{\displaystyle \ x^{2}+px+q=0}
a
l
p
h
a
{\displaystyle \ alpha}
β
{\displaystyle \beta }
p
,
q
ϵ
R
{\displaystyle \ p,q\epsilon R}
Find the quadratic equation whose roots are
α
2
β
{\displaystyle \ \alpha ^{2}\beta }
α
β
2
{\displaystyle \ \alpha \beta ^{2}}
(c)
(i)
f
(
x
)
=
2
x
+
1
{\displaystyle \ f(x)=2x+1}
x
ϵ
R
{\displaystyle \ x\epsilon R}
Show that there exists a real number
k
{\displaystyle \ k}
x
{\displaystyle \ x}
f
(
x
+
f
(
x
)
)
=
k
f
(
x
)
{\displaystyle \ f(x+f(x))=kf(x)}
(ii)
Show that for any real values of
a
,
b
,
h
{\displaystyle \ a,b,h}
(
x
−
a
)
(
x
−
b
)
−
h
2
=
0
{\displaystyle \ (x-a)(x-b)-h^{2}=0}
has real roots.
Paper 1 Question 1 [ edit ] (a) Solve the simultaneous equations:
x
5
−
y
4
=
0
{\displaystyle {\frac {x}{5}}-{\frac {y}{4}}=0}
3
x
+
y
2
=
17
{\displaystyle \ 3x+{\frac {y}{2}}=17}
(i) Exspress
2
1
/
4
+
2
1
/
4
+
2
1
/
4
+
2
1
/
4
{\displaystyle \ 2^{1/4}+2^{1/4}+2^{1/4}+2^{1/4}}
2
p
/
q
{\displaystyle \ 2^{p/q}}
p
,
q
ϵ
Z
{\displaystyle \ p,q\epsilon Z}
(ii) Let
f
(
x
)
=
a
x
3
+
b
x
2
+
c
x
+
d
{\displaystyle \ f(x)=ax^{3}+bx^{2}+cx+d}
Show that
(
x
−
t
)
{\displaystyle \ (x-t)}
f
(
x
)
−
f
(
t
)
{\displaystyle \ f(x)-f(t)}
(
x
−
p
)
2
{\displaystyle \ (x-p)^{2}}
x
3
+
q
x
+
r
{\displaystyle \ x^{3}+qx+r}
Show that
27
r
2
+
4
q
3
=
0
{\displaystyle \ 27r^{2}+4q^{3}=0}
Exspress the roots of
3
x
3
+
q
=
0
{\displaystyle \ 3x^{3}+q=0}
Paper 1 Question 2 [ edit ] (a) Solve for x
|
x
−
1
|
<
7
{\displaystyle \ |x-1|<7}
x
ϵ
R
{\displaystyle \ x\epsilon R}
4
x
3
+
10
x
2
−
7
x
−
3
=
0
{\displaystyle \ 4x^{3}+10x^{2}-7x-3=0}
f
(
x
)
=
x
2
+
k
2
m
x
{\displaystyle \ f(x)={\frac {x^{2}+k^{2}}{mx}}}
k
{\displaystyle \ k}
/
m
{\displaystyle /m}
m
≠
0
{\displaystyle \ m\neq 0}
(i) show that
f
(
k
m
)
=
f
(
k
m
)
{\displaystyle \ f(km)=f({\frac {k}{m}})}
(ii)
a
{\displaystyle \ a}
b
{\displaystyle \ b}
a
≠
0
,
b
≠
0
{\displaystyle \ a\neq 0,b\neq 0}
a
≠
b
{\displaystyle \ a\neq b}
f
(
a
)
=
f
(
b
)
{\displaystyle \ f(a)=f(b)}
a
b
=
k
2
{\displaystyle \ ab=k^{2}}
Paper 1 Question 1 [ edit ] (a) Find the real number a such that for all
x
≠
9
{\displaystyle \ x\neq 9}
x
−
9
x
−
3
{\displaystyle {\frac {x-9}{{\sqrt {x}}-3}}}
f
(
x
)
=
3
x
3
+
m
x
2
−
17
x
+
n
{\displaystyle \ f(x)=3x^{3}+mx^{2}-17x+n}
m
{\displaystyle \ m}
n
{\displaystyle \ n}
x
−
3
{\displaystyle \ x-3}
x
+
2
{\displaystyle \ x+2}
f
(
x
)
{\displaystyle \ f(x)}
m
{\displaystyle \ m}
n
{\displaystyle \ n}
(c)
x
2
−
t
{\displaystyle \ x^{2}-t}
x
3
−
p
x
2
+
r
{\displaystyle \ x^{3}-px^{2}+r}
(i) Show that
p
q
=
r
{\displaystyle \ {pq=r}}
(ii) Express the roots of
x
3
−
p
x
2
+
r
=
0
{\displaystyle \ x^{3}-px^{2}+r=0}
p
{\displaystyle \ p}
q
{\displaystyle \ q}
Paper 1 Question 2 [ edit ] (a) Solve the simultaneous equations
y
=
2
x
−
5
{\displaystyle \ y=2x-5}
x
2
+
x
y
=
2
{\displaystyle \ x^{2}+xy=2}
(b)
(i) Find the range of values of
t
ϵ
R
{\displaystyle \ t\epsilon R}
(
2
t
−
1
)
x
2
+
5
t
x
+
2
t
=
0
{\displaystyle \ (2t-1)x^{2}+5tx+2t=0}
(ii) Explain why the roots are real when t is an integer.
(c)
f
(
x
)
=
1
−
b
2
x
{\displaystyle \ f(x)=1-b^{2x}}
g
(
x
)
=
1
−
b
1
+
2
x
{\displaystyle \ g(x)=1-b^{1+2x}}
b
{\displaystyle \ b}
b
{\displaystyle \ b}
x
{\displaystyle \ x}
f
(
x
)
=
g
(
x
)
{\displaystyle \ f(x)=g(x)}
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