Chapter 1
REAL NUMBER SYSTEM
Real number system topics. Properties of real numbers- Transitive, commutative,
associative, distributive, and inverse. Binary relations - Reflexivity, symmetry and
transitivity. Laws of indices and simple examples. Simplification of Algebraic Expressions
and Expansion.
CONTENTS
1. Number .......................................................................................................................... 2
1.1. Natural numbers or Positive integers (N) ............................................................ 2
1.2. Integers (Z) ............................................................................................................. 2
1.3. Prime numbers ....................................................................................................... 2
1.4. Rational numbers (Q) ............................................................................................ 2
1.5. Irrational number (T) ............................................................................................ 2
1.6. Real numbers (R) ................................................................................................... 3
1.7. Real line................................................................................................................... 3
1.8. Intervals .................................................................................................................. 3
1.9 Complex numbers (C) ............................................................................................ 3
2. Basic Rules of Algebra .................................................................................................. 4
2.1. Closure Property .................................................................................................... 4
2.2. Commutative Property .......................................................................................... 4
2.3 Associative law. ....................................................................................................... 4
2.4 The Distributive law ............................................................................................... 5
2.5 The Inverse law ......................................................................................................... 5
2.6 Identity Element ........................................................................................................ 5
3. Cartesian product of Two Sets ........................................................................................ 5
4. Relations or binary relations ........................................................................................... 6
4.1. Properties of a Relation ............................................................................................ 8
(a) Reflexive property ................................................................................................. 8
(b) Symmetric property ............................................................................................... 8
(c) Transitive Property ................................................................................................ 9
(d) Equivalence Relation............................................................................................. 9
5. Laws of Indices ............................................................................................................. 10
Law 1 ........................................................................................................................ 11
Law 2 ........................................................................................................................ 11
Law 3 ........................................................................................................................ 11
Law 4 ........................................................................................................................ 11
Law 5 ........................................................................................................................ 11
Law 6 ........................................................................................................................ 11
6. ALGEBRAIC EXPRESSIONS AND EXPANSION ................................................... 12
6.1 Priority of Mathematical Operations ...................................................................... 12
6.2 Some basic algebraic expansion formulae .............................................................. 12
6.2.1 Factorizing algebraic expressions .................................................................... 12
6.3 Simplifying rational expressions............................................................................. 13
6.4 Adding and subtracting rational expressions .......................................................... 14
EXERCISES ..................................................................................................................... 14
APPENDIX-I: PAST TEST 1 AND EXAM QUESTIONS ............................................. 16
1
1. NUMBER
Definition: A number is a mathematical entity used to count, label, and measure of
objects or items.
In mathematics, the definition of number has been extended over the years to include
such numbers as (i) 0, (ii) negative numbers, (iii) rational numbers,(iv) irrational
numbers, (v) real numbers, and (vi) complex numbers.
1.1. Natural numbers or Positive integers (N)
The most familiar numbers are the natural numbers or counting numbers or positive
integers: 1, 2, 3, and so on. Traditionally, the sequence of natural numbers started with 1
(0 was not even considered a number). The mathematical symbol for the set of all natural
numbers is N, also written as .
Remark: In the 19th century, set theorists and other mathematicians started including 0 in
the set of natural numbers. Today, different mathematicians use the term to describe both
sets, including 0 or not.
1.2. Integers (Z)
These consist of the set of natural numbers (or positive integers) , their negatives i.e
negative integers and zero, e.g. …-4,-3,-2,-1,0,1,2,3,4,… Mathematically the set of
integers is denoted by the symbol Z.
Remark: Natural numbers (positive integers) are a subset of the set of integers.
1.3. Prime numbers
A prime number is a number that is divisible only by itself only, e.g. 2,3,7,11,…
Remark: A natural number can be a prime or a nonprime number.
1.4. Rational numbers (Q)
Rational number consists of numbers as a ratio of two integers;
a
,b0
b
a
1 3
, e.g. , ,  5, 0, 1.38, . . .
1
2 4
Mathematically, the set of rational numbers is denoted by the symbol Q.
Every integer is a rational number since a 
Remark 1: Any terminating decimal is a rational number e.g.0.721 is a rational number,
721
.
since 0.721 
1000
Remark 2: Also any repeating decimal is also a rational number, e.g.
1
2
 0.333... ,  0.6666... etc.
3
3
1.5. Irrational number (T)
A number that cannot be represented by the ratio of two integers is called an irrational
number.
2
Mathematically, the set of irrational numbers is denoted by the symbol T.
Remark: An irrational number is a non-terminating and non-repeating decimal number.
For example √2 =1.414213… is a irrational number because it is non-terminating and
non-repeating decimal number.
1.6. Real numbers (R)
Definition: Real numbers consist of the set of all rational and irrational numbers.
Mathematically the set of real number is denoted by R=(-∞, +∞).
Remark: The set of natural numbers (positive integers) is a subset of set of integers,
which in turn is a subset of integers, which in turn is a subset of rational numbers.
Remark: We observe that   Z  Q and QUT   R
N
1.7. Real line
Real numbers can be represented on the points on a straight line. This line is called a real
number line or real line.
∞
-∞
-3
-2
-1
0
1
2
A real line is denoted by the notation   ,   .
3
4
1.8. Intervals
Open Interval: A collection of real numbers, say, x, between two numbers say a and b,
where but not including a or b is called an Open interval, written as a<x<b or (a<b).


a
x
b
Closed Interval: If the end points a and b are to be included in the interval, then the
collection of real numbers, x, is called a Closed interval, written as a≤x≤b or [a,b].


a
x
b
We may also define half open/closed (semi closed) intervals such as
(i) (a,b)]
left open right closed interval.
(ii) [a,b)
left closed right open interval.
1.9 Complex numbers (C)
An imaginary number
i 2  i  i   1  1 

is
 1 
denoted
2
by
i,
where
i   1.
Note
that
 1. The set of imaginary numbers is denoted by I .
Definition: A complex numbers is denoted by z  a  ib, where a , b are any real
numbers and i   1. Thus the complex number is combination of a real and an
imaginary number.
3
The set of complex numbers is denoted by C .
2. BASIC RULES OF ALGEBRA
2.1. Closure Property
Definition: An operation is said to demonstrate closure property, if for all real numbers
a and b , the result a  b ( can be , , / and x ) is also a real number.
Remark: A set has closure property under an operation if the outcome of that operation
on members of the set always produces a member of the same set; in this case we also say
that the set is closed under the operation.
For example:
5  R, 7  R, then
5  7  12  R
5-7  -2  R,
5
 R,
7
5  7  35  R.
 That the operations +, -, / and  demonstrate closure property.
Remark: In general, the real numbers are closed under addition, subtraction and
multiplication. However it is not closed under division as 0/0 is not a real number.
Remark: The set of positive integers is not closed under subtraction for, 3 and 8 are both
positive integers, but 3 − 8 = −5, is not a positive integer.
Remark: Another example is the set containing only the number zero, which is closed
under addition, subtraction and multiplication but not division.
2.2. Commutative Property
A binary operation * on a set S is called commutative if:
x*y = y*x
for all x, y ϵ S
Remark: An operation that does not satisfy the above property is called noncommutative.
(i) The Commutative law of Addition
Let a and b be any two real numbers, then a+b = b+a
e.g. 2+5 = 5+2
(ii) The Commutative Law for Multiplication
Let a and b be any two real numbers, then a x b = b x a
e.g. 2 x 5= 5x2
(  is also denoted by  )
2.3 Associative law.
(i) Associative law of Addition
Let a, b and c be any three real numbers, then (a+b)+c = a+(b+c)
e.g. (2+5)+3 = 2+(5+3) = 10
4
(ii) The Associative law of Multiplication
Let, a, b and c be any three real numbers, then (a.b).c = a.(b.c)
e.g. (2.5).3 = 2.(5.3) =30
2.4 The Distributive law
Distributive law of addition and Multiplication
Let a, b and c be any three real numbers, then a.(b+c)= (a.b)+(a.c).
e.g., 2.(5+3)= (2.5)+(2.3) 10+6= 16.
2.5 The Inverse law
(i) Inverse Law of Addition
For any real number a, a+(-a)=a - a=-a+a=0.
Hence ‘-a’ is the inverse element of ‘a’ for addition.
(ii) Inverse law of Multiplication
For any real number a such that a≠0 there exists a number which, when
multiplied by a, we have a product equals to 1 (one). This number is called the
inverse of a and is denoted by a -1, that is, a.a -1=1. Note that a -1=1/a.
e.g. 12.12 -1 = 12. 1/12 = 1, here a=12 and a -1 =1/12
2.6 Identity Element
(i) Additive Identity: If a number I is added to another real number a and if the result is
a itself, then the number I is said to be additive identity.
Remark: Zero is the additive identity of the real number system. This is because,
a+0=0+a=a. Hence 0 is the identity element for addition.
(ii) Multiplicative Identity: If a real number a is multiplied by another real number I and
if the result is a, then I is said to be the multiplicative identity of the real number system.
Remark: One is the multiplicative identity of the real number system. This is because,.
ax1=1xa=a
3. CARTESIAN PRODUCT OF TWO SETS
(i) The Cartesian product of two sets A  a , b and B  x , y , z denoted by A  B (A
cross B) is defined as the set of all ordered pairs. That is,
A  B  a , b   x , y , z   a , x  ,  a , y  ,  a , z  ,  b , x  ,  b , y  ,  b , z  .
(ii) While the Cartesian product of two sets B  x , y , z and A  a , b denoted by
B  A (B cross A) is defined as the set of all ordered pairs. That is,
B  A  x , y , z  a , b   x , a  ,  x , b  ,  y , a  ,  y , b  ,  z , a  ,  z , b 
Notice that there are six ordered pairs in both cases (i) & (ii) and A x B ≠ B x A.
5
Example 3.1
Let A  4,8, 2 and B  10,12 .
Solution 3.1
Then
A  B  4,8, 2  10,12   4,10  ,  4,12  , 8,10  , 8,12  ,  2,10  ,  2,12  .
B  A  10 ,12  4 ,8, 2  10 , 4  , 10,8  , 10, 2  , 12, 4  , 12,8  , 12 , 2 
AxB≠BxA
4. RELATIONS OR BINARY RELATIONS
Definition: Let A and B be sets. A binary relation from A to B is a subset of AB. In
other words, for a binary relation R we have R  AB.
When (x, y) belongs to R, x is said to be related to y by R.
Note: However we will deal with relations on a set.
Definition: Here we relate one member x (say) of a set A with one member y (say) of the
set B (B may be same as A) through some rule (relation) R.
We denote such relation by R as a set of co-ordinates (x, y) where x is a member of the
set A which relates to y through such a rule R. This is written as
R   x, y  x is related to y , x  A, y  B.


Example 4.1: Let A = {1, 2, 3, 4}. Find ordered pairs which are in the following
relations;
(i) R = {(x, y) | x < y}
(ii) R = {(x, y) | x ≥ y}
Represent above relationships using arrow diagrams.
Solution 4.1
y
x
A
1
2
3
4
1
11
21
31
41
2
12
22
32
42
3
13
23
33
43
4
14
24
34
44
(i) R = {(x, y) | x < y}
R = {(1,2), (1,3), (1,4), (2,3), (2,4), (3,4)}
Arrow diagrams may be used to represent such relationship as follows.
6
x
y
1
1
2
2
3
3
4
4
(ii) R = {(x, y) | x ≥ y}
R= {(1,1),(2,1), (3,1), (4,1), (2,2),(3,2), (4,2), (3,3),(4,3),(4,4)}
Arrow diagrams may be used to represent such relationship as follows.
x
y
1
1
2
2
3
3
4
4
Example 4.2
(i) Let R be the ‘is a multiple of’ relation on the set A={2,4,6,8}.
i.e. R = {(x, y) | x is multiple of y}
(ii) Let R be the ‘is a factor of’ relation on the set A={2,4,6,8}
i.e. R = {(x, y) | x is a factor of y}
Represent the relation by ordered pairs of the relation and represent such relationship
using arrow diagrams.
Solution 4.2
There are sixteen (16) ordered pairs
y
x
2
4
6
8
2
22
42
62
82
4
24
44
64
84
6
26
46
66
86
8
28
48
68
88
7
(i)
R = {(x, y) | x is multiple of y}
Thus we have,
R = {(2,2), (4,2), (6,2), (8,2), (4,4), (8,4), (6,6), (8,8)}
Arrow diagrams may be used to represent such relationship as follows
x
y
2
2
4
4
6
6
8
8
(ii) R = {(x, y) | x is factor of y}
Thus we have,
R = {(2,2), (2,4), (2,6), (2,8), (4,4), (4,8), (6,6), (8,8)}
x
y
2
2
4
4
6
6
8
8
4.1. Properties of a Relation
(a) Reflexive property
A relation R on a set A is said to be reflexive if (x,x) ∈ R for all x ∈ R. That is, we say that
R is reflexive if every element in A is related to itself.
In Example 4.2 above, the relation R is reflexive on the set A = {2,4,6,8} because (2,2),
(4,4), (6,6) and (8,8) are in the set of the relation R.
(b) Symmetric property
A relation R on a set A is said to be symmetric if for every (x,y) ∈ R, (y,x) ∈ R. In other
words, if x is related to y, then y is related to x.
8
Example 4.1.1
Let R be the relation ‘-x is the opposite of x’ on the set A = {1,-1,2,-2}. Represent the
relation by ordered pairs of the relation. Also show that relation R is symmetric.
Solution 4.1.1
The relation is defined as R= {(x, y)| y is opposite of x }, x ∈ A, y ∈ A
Thus we have,
R= {(1,-1), (-1,1), (2,-2), (-2,2)}
The relation R (-x is the opposite of x) is symmetric on the set A = {1,-1,2,-2} because
(1,-1), (-1,1), (2,-2) and (-2,2) are in the set of the relation R.
(c) Transitive Property
A relation R on a set A is transitive if (x,y) ∈ R and (y,z) ∈ R, then (x,z) ∈ R.
Example 4.1.2
Consider the relation R ‘is a factor of’ on the set A = {2,4,6,8}. Represent the relation by
ordered pairs of the relation. Also show that relation R is transitive.
Solution 4.1.2
Set A for R gives
R = {(x, y) | x is factor of y)}
Thus we have,
R = {(2,2), (2,4), (2,6), (2,8), (4,4), (4,8), (6,6), (8,8)}
Note:
 2, 4  R and  4, 8  R   2,8  R ,
(2,6)  R and (6,6)  R  (2,6)  R so on.
Remark: Note that to show that the transitive property is satisfied, once needs to establish
the fact if (x,y) ∈ R and (y,z) ∈ R, then (x,z) ∈ R for all such pairs of sets in R. However,
the existence of one such pair such that if (x,y) ∈ R and (y,z) ∈ R, but (x,z) ∉ R, is
sufficient to show that a relation is not transitive.
(d) Equivalence Relation
Any relation on a set that is reflexive, symmetric and transitive is called an Equivalence
Relation.
Example 4.1.3
Let A   1,0,1,2and a relation on A be given by


 1, 1 ,  1,0  ,  1,1 ,  1, 2  ,  0,0  ,  0, 1 , 0,1 ,

R


 0, 2  , 1, 1 , 1,0  , 1,1 , 1, 2  ,  2, 1 ,  2,0  ,  2,1 ,  2, 2  

Note that
(i) R is a relation on A since R  A  A.
9
(ii) R is reflexive, since  a, a   R for all a  A i.e.,
(ii)
 1, 1 , 0,0 , 1,1 ,, 2,2   R ,
 1,0 ,  0, 1  R,
 1,1 , 1, 1  R,
 0,1 , 1,0  R,
R is symmetric, since for every  a, b   R ,  b, a   R . i.e.,
 0, 2  ,  2,0   R,
 1, 2  ,  2, 1  R and
1, 2  ,  2,1  R
(iv) R is transitive since,  1,2, 2,1  R, then  1,1  R , 0,1, 1,2  R, then 0,2  R
Since R satisfies the reflexive, symmetry and transitive properties it is an equivalence
relation.
Example 4.1.4
Let R be the ‘is a factor of’ relation on the set A  3,6,9, 27 . Examine, giving reasons,
if R is not an equivalence relation.
Solution 4.1.4
The ordered pair of the relation
R = {(x, y) | x is a factor of y} is given below
R= {(3,3), (3,6), (3,9), (3,27), (6,6), (9,9), (9,27), (27,27)}
To determine giving reasons if R is an equivalence relation
Any relation on a set that is reflexive, symmetric and transitive is called an Equivalence
Relation. So we will check these three properties on R.
Reflexive
Reflexive property is satisfied because
(3,3),(6,6),(9,9),(27,27) ϵ R
Symmetry
Symmetry property is not satisfied since  3,6  R, but  6,3  R
Transitive: Transitive property is satisfied, because of (a,b) ϵ R, (b,c) ϵ R then (a,c) ϵ R:
(i)
For each (3,b) and (b, c) in R we see that (3,c) is in R
(ii)
For each (6,b) and (b, c) in R we see that (6,c) is in R
(iii)
For (9,9) and (9, 27) in R we see that (9,27) is in R
Since R does not satisfy symmetry, it is not an equivalence relation.
5. LAWS OF INDICES
Definition of an
In general let a and n be non-zero real number (positive, negative or fraction), then
a n  a  a  ...  a ( n factors)
10
In the definition, a is called the base and n is called the exponent.
We always take a 0  1 (by definition).
Thus,
10  1,
 5  1,
0
1.35  1, and
0
so on.
Law of Indices
Law 1
a m  a n  a m n , where m and n are real numbers.
e.g. 23  25   2  2  2   2  2  2  2  2   235  28  256
Law 2
a 
n m
 a nm ,
e. g.  32   323  36  729
3
Law 3
 a  b   a n  b n , where m and n are real numbers.
4
 3  2    3  2    3  2   3  2   3  2   34  24  64  6  6  6  6  1296 .
n
Law 4
m
am
a

, b0
 
bm
b
4
16
 2   2   2   2   2  2222 2
e. g.               
 4 
81
 3   3   3   3   3  3333 3
4
Law 5
am
 a m n , a  0
n
a
53 5  5  5
e. g. 2 
 532  5
5
55
Law 6
1
(i)
(ii)
an  n a
m
n
a  n am
1
Thus,
3
1
3
27   27   3,
3
125   125
11
1
3
 27  3 3
.
 5, 
 
4
 64 
6. ALGEBRAIC EXPRESSIONS AND EXPANSION
6.1 Priority of Mathematical Operations
For all numerical or algebraic expressions, the order of evaluation is (PEDMAS):
Parentheses and Brackets (innermost first)
First Priority
Exponents or powers
Second Priority
Division (proceed from left to right)
Third Priority
Multiplication (proceed from left to right)
Third Priority
Addition (proceed from left to right)
Fourth Priority
Subtraction (proceed from left to right)
Fourth Priority
Example:
4 + [–1(–2 – 1)]2 = 4 + [–1(–3)]2 = 4 + [3]2 = 4 + 9 = 13
6.2 Some basic algebraic expansion formulae
We may use of the following standard results for simplifying algebraic expressions.
(i) (a  b)2  a 2  2ab  b2
(ii) (a  b)2  a 2  2ab  b2
(iii ) (a  b)3  a 3  3a 2b  3ab2  b3
(iv) (a  b)3  a3  3a 2b  3ab2  b3
(v) a 2  b2  (a  b)(a  b)
Example 6.2.1
Expand and simplify the following:
3  5 x  2   2  x 2  2 x  1
Solution 6.2.1
3  5 x  2   2  x 2  2 x  1  15 x  6  2 x 2  4 x  2  2 x 2  19 x  4
6.2.1 Factorizing algebraic expressions
Example 6.2.1.1
Factorize the following algebraic expressions:
(a) 3x2  5x  2  6 x 5x  2  3 5x  2  ,
(b) 4 x 2  4 x  1 and
(c) 2ab 4  4ab 2 c 2  2ac 4 .
Solution 6.2.1.1
(a) 3x2  5x  2  6 x 5x  2  3 5x  2 
  5 x  2   3x 2  6 x  3  3 5 x  2   x 2  2 x  1 ;
12
since 5x  2  is common
(b) 4 x 2  4 x  1  (2 x) 2  2(2 x).1  12 u sin g the formula ( a  b) 2  a 2  2ab  b2
where, a  2 x and b  1
;
 (2 x  1)
Alternative method
( b) 4 x 2  4 x  1  4 x 2  2 x  2 x  1
2
 2 x(2 x  1)  1(2 x  1)
 (2 x  1) (2 x  1)
sin ce (2 x  1) is common
 (2 x  1)2
(c) 2ab 4  4ab 2 c 2  2ac 4



 2a b 4  2b 2c 2  c 4  2a b 2  c 2

2
Hint: use formula (a  b)2  a 2  2ab  b2 then
 2a  b4  2b2 c 2  c 4   2a  b2  c 2 
2
 2a  b  c  b  c   2a  b  c   b  c  .
2
2
2
6.3 Simplifying rational expressions
A rational expression is a ratio (fraction) in which both numerator and denominator is a
polynomial.
P x 
Thus Rx  
, where Px  and Qx  are plynomial functions
Q x 
Example 6.3.1
Simplify the following algebraic expressions:
4a 2 x  8abx  60b2 x
(a)
,
5a 2 y  30aby  45b2 y
x 2  2 xy  y 2 x 2  2 xy  y 2
(b)
and

x y
6x  6 y
x2  4x  5
3x  15
(c)
 2
3x  6
x  2x 1
Solution 6.3.1
4a 2 x  8abx  60b2 x
N
(a)
(1)

2
2
5a y  30aby  45b y D
N  4a 2 x  8abx  60b2 x  4 x[a 2  2ab  15b2 ]  4 x[a 2  5ab  3ab  15b2 ]
 4 x[a ( a  5b)  3b( a  5b)]  4 x( a  5b)( a  3b)
(2)
Similarly,
(3)
D  5a 2 y  30aby  45b2 y  5 y  a  3b  a  3b 
Substituting (2) and (3) in (1) we have,
4 x  a  5b  a  3b  4 x  a  5b 
4a 2 x  8abx  60b2 x


2
2
5a y  30aby  45b y 5 y  a  3b  a  3b  5 y  a  3b 
(b)
2
2
2
2
x 2  2 xy  y 2 x 2  2 xy  y 2  x  2 xy  y  x  2 xy  y 
.

x y
6x  6 y
 x  y  6 x  6 y 
13
 x  y   x  y    x  y  x  y  .

6  x  y  x  y 
6
2
2
Note: In the above example following formulae are used
(a  b)2  a 2  2ab  b2 and (a  b)2  a 2  2ab  b2
(c)



x2  4x  5 x2  2x  1
x2  4 x  5
3x  15
 2

3x  6
x  2x 1
3x  6 3x  15
 x  5  x  1  x  1

3  x  2  3  x  5
2
 x  1

.
9  x  2
3
6.4 Adding and subtracting rational expressions
Example 6.4.1
Perform the indicated operations and simplify, if possible:
x  4 x 1

(a)
and
x 1 x 1
2x
x
1
 2

(b)
x3 x  x6 x 2
Solution 6.4.1
x  4 x  1  x  4  x  1   x  1 x  1


(a)
x 1 x 1
 x  1 x  1
x

(b)
2
 

 3x  4  x 2  2 x  1
 x  1 x  1
2 x 2  5 x  3  2 x  1 x  3
.

 x  1 x  1  x  1 x  1
2x
x
1
2x
x
1
 2




x  3 x  x  6 x  2 x  3  x  3 x  2  x  2

2 x  x  2   x   x  3 2 x 2  4 x  x  x  3

 x  3 x  2 
 x  3 x  2 

2x2  4x  3
 x  3 x  2 
EXERCISES
1. Let R be the relation "  a is the opposite of a " on the set A  4,  4,7,  7  .
Write ordered pairs of the relation and represent the relation by an arrow diagram.
2. Let the relation R be “is less than or equal to” on the set A={11,22,35}.
(i)
Write the ordered pairs of the relation.
(ii)
Represent the relation by an arrow diagram.
(iii) Determine which of the reflexive, symmetric and transitive properties are
satisfied by the relation.
3. Simplify the following algebraic expression:
 2x 4 y 
 2 3 
 8x y 
4
14
x8
Ans
256 y 8
4. Perform the indicated operations and simplify the following:
(i) 3 x  6 x  2  5 x  2  

 7 x  4 Ans
(ii)  9 y  14 yz  8 z 2   3z  4 y  5 z   2 y  3 y  2 z 
2
 15 y 2  22 yz  7 z 2 Ans
(iii) 2 x  4 x 2  x  3x  1   x 3x 2  2 x  3x  2  
 11x 3  2 x 2 Ans.
(iv)  x 2  2  2 x 2  3
 2 x4  x2  6
Ans.
(v) 3xyz  2 x  7 yz   3 y  2 xz  
  60 x 2 y 2 z 2 Ans.
5. Factorize the following:
(i) x 2  2 x  1
( x  1)2 Ans.
(ii)  a  3 a  6   a  3 2a  8
  a  3 a  14 
Ans.
(iii) 16 x3  16 x 2  4 x
 4 x(2 x  1)2 Ans.
(iv) 25a 2b 2  100a 2 c 2
 25a 2 (b  2c)(b  2c) Ans.
6. Perform the following operations and simplify:
y 1 y  2
2y  7
(i)

 2
y  2 y 3 y  y 6
2 y2
Ans.
( y  2)( y  3)
4
x
4 
x 2  12 

x
OR
x


3

(ii)

x 2  12
x  
x 2  4 
3 2
x 4
( x  2)2 ( x  2)2

Ans.
4 x3
x
3

1
(iii)
 x  3  x  2 x  3
=

3  x  1
 x  2  x  3
Ans
15
4
3
x 1
3x  1
Ans
x 1
6
3
(v)
x2
3x
Ans
x2
x  4 x 1

(vi)
x 1 x 1
 2 x  1 x  3 Ans .

 x  1 x  1
(iv)
APPENDIX-I: PAST TEST 1 AND EXAM QUESTIONS
TEST 1-2008
Question 1
(a) Perform the indicated operations and simplify completely.
p 2 q  4q3
pq  2q 2
(i)

4 p  8q 12 p  4 pq
 p(3  q)
Ans.
4
3
(ii)
x 1
3x  1

Ans
x 1
[5+3 Marks]
Question 2
Let the relation R be ‘is greater than or equal to’ on the set A  5,  1, 0, 2
i.e. R = {(x, y) | x ≥ y}
(a) Write the ordered pairs of the relation and represent the relation by an arrow
diagram.
(b) Are the reflexive, symmetric and transitive properties satisfied by the relation?
Give reasons.
[7+6 Marks]
TEST 1-2010
Question 1 (same as in 2008)
(a) Perform the indicated operations and simplify completely.
p 2 q  4q3
pq  2q 2
(i)

4 p  8q 12 p  4 pq
 p(3  q)
Ans.
6
3
(ii)
x2
3x  1

Ans
x 1
[5+3 Marks]
16
Question 2
Let the relation R be ‘is greater than or equal to’ on the set A={-3,-2,0,1,2}
i.e. R = {(x, y) | x ≥ y}
(a) Write the ordered pairs of the relation and represent the relation by an arrow
diagram.
(b) Are the reflexive, symmetric and transitive properties satisfied by the relation?
Give reasons.
[7+6 Marks]
TEST 1-2012
Question 1
(a) Define an equivalence relation.
(b) Let the relation R be defined as “is greater than or equal to” on the set
A  0.01,  0.1, 0,1 .
(i)
Write down the ordered pairs of the relation.
(ii)
Represent the relation by an arrow diagram.
(iii) Determine, with reasons, whether the reflexive, symmetric and transitive
properties are satisfied by the relation.
[3+(3+3+6)=15 Marks]
Question 2
(a) Perform the indicated operation and simplify completely:
2a 2  3ax  x 2 2a  x
 2
x y
x  xy
 x(a  x )
Ans.
[5 Marks]
TEST 1-2016
Question 1
a) Identify the rules of algebra associated with the number system in each of the
following numeric statements :
(i)
(3+7)+2 = 3+(7+2)
(ii)
2  (4+6) = (2  4)+( 2  6)
1
2  21  2   1
(iii)
2
1
1 1
0  0 
(iv)
5
5 5
b) Let R be the ‘is a factor of’ relation on the set A  3,6,9, 27 i.e. R = {(x, y) | x is a
factor of y}. Examine, giving reasons, if R is an equivalence relation
[4+11=15 Marks]
Question 2
a) Perform the indicated operation and simplify completely:
(i) 3xyz  2 x  7 yz   3 y  2 xz  
  60 x 2 y 2 z 2
1
4a  6b
 2
(ii)
2a  3b 4a  9b 2
Ans.
17
1
Ans.
2
b) Factorize the following:
5 p3q  10 p 2 q 2  5 pq3

 5 pq( p  q)2
Ans.
[5+8+7=20 Marks]
EXAM-2012
1a) Simplify the following expression completely;
x
3

1
 x  3  x  2 x  3
5 Marks
3c) Suppose that the set A  {2,  1, 0, 1, 2}, the set B  {0, 1, 2, 3, 4, 5} and R is
the relation ‘the square is’ on the set A to set B R = {(x, y) | y is the square of x}
(i) Write the ordered pairs of the relation.
(ii) Represent R by an arrow diagram or on a graph.
6+4 = 10 Marks
SOLUTION-EXAM-2012
1a)
x
3

1
 x  3  x  2 x  3


x  x  2   3   x  2  x  3
 x  2  x  3
x 2  2 x  3   x 2  3x  2 x  6 
 x  2  x  3

x 2  2 x  3  x 2  3x  2 x  6
 x  2  x  3

3  x  1
 x  2  x  3
3c)
(i) Ordered pairs of relation (x, y) = {(-2,4), (-1, 1) (0, 0) (1,1) (2,4)
(ii) Represent R by an arrow diagram or on a graph.
EXAM-2013
1a) Let R be the relation defined as “ a is the additive inverse of a ” on the
set A  2,  2,3,  3,1, 1 . Write the ordered pairs of the relation and represent the
relation by an arrow diagram.
5 Marks
2
2
2
2
x  4 y x  4 xy  4 y

2a) Simplify completely:
.
4x  8 y
x  2y
8 Marks
3a) All roads linking different neighborhoods in a certain city are one-way. The following
arrow diagram shows how 4 such localities are linked.
18
(i) Write down the ordered pairs for the relations
(ii) Verify which of the properties of relations, namely: Reflexive, Symmetric,
Transitive and equivalence are satisfied.
(4+11 = 15 Marks)
SOLUTION-EXAM-2013
1 a) R = {(x, y) | y is the additive inverse of x}
Ordered pairs of relation (x, y) = {(2,-2), (-2, 2), (3,-3), (-3, 3), (1,-1) (-1,1)
2a)
( x  2 y)( x  2 y) ( x  2 y) 2 ( x  2 y )( x  2 y) ( x  2 y )
( x  2 y) 2
.

/

4( x  2 y )
( x  2 y ) 2 4( x  2 y ) 2
4( x  2 y)
( x  2 y)
3a)
(a) Ordered pairs are
{(1,1), (2,2), (2,4), (3,2), (3,3), (3,4), (4,4)}
Reflective property: for every a in the relation, (a, a) belongs to relation. Hence
Reflective property is satisfied.
Symmetric; for every a and b in relation, if (a, b) exists, then (b, a) exists (2, 3) and (3, 2)
exist but (2,4) exists and (4,2) does not exist hence not reflective
Transitive: If (a, b) exists and (a, c) exists, then (b, c) must exit. (2, 3) exists and (2,4)
exist, and (3, 4) exists. Hence Transitive.
EXAM-2014
1a) Let the relation R be ‘is less than ’on the set A  3,6,9, 27 .
(i)
Write down the ordered pairs of the relation.
(ii)
Represent the relation by an arrow diagram.
(iii) Determine, with reasons, whether the reflexive, symmetric and transitive
properties are satisfied by the relation.
5 Marks
2
1   1  x  
1 2
2a) Simplify completely:   2  3   
.
x   x3 
x x
5 Marks
3a) Let the relation R be “is a factor of” on the set A   2,2,4,6,8.
19
i  Write the Cartesian product of the set above.
 ii  Represent the relation by an arrow diagram and ordered pairs.
 iii  Determine, giving reasons, which of the reflexive, symmetric and transitive
properties are satisfied.
(2+4+6 = 12 Marks)
SOLUTION-EXAM-2014
1a) see Test 1-2016
2a)
2
2

1   1  x    x 2  2 x  1 
x3
1  x
x3
1 2


  2  3 

  1  x 2  x 3  1  x 2  1
x   x 3  
x3
x x

3a)
 2, 2  ,  2, 2  ,  2, 4  ,  2, 6  ,  2,8  ,  2, 2  ,



 2, 2  ,  2, 4  ,  2, 6  ,  2,8  ,  4, 2  ,  4, 2  ,  4, 4  ,  4, 6  
 i  AxA  2, 2, 4, 6,8 x 2, 2, 4, 6,8  

,  4,8  ,  6, 2  ,  6, 2  ,  6, 4  ,  6, 6  ,  6,8  , 8, 2  , 8, 2  
, 8, 4 , 8, 6 , 8,8

  


(iii)
 2, 2  ,  2, 2  ,  2, 4  ,  2, 6  ,  2,8  ,  2, 2  ,  2, 2  ,  2, 4  ,  2, 6  , 
R

 2,8 ,  4, 4  ,  4,8  ,  6, 6  , 8,8 

The reflexive property is satisfied by the relation
 2, 2  R,  2, 2  R,  4, 4  R,  6,6  R, 8,8
The symmetric property is not satisfied by the relation  2, 4  R but  4, 2  R .
The transitive property is satisfied by the relation  2, 2  R,  2, 4  R,  2, 4   R and so
on.
EXAM-2015
Section A
(vi) Perform the indicated operation and simplify the following:
x 2  5x  6 2 x  4
 2
x2  9
x  3x
(viii) Let the relation R be ‘is a factor of’ on the set A  3,6,9, 27 . Represent the
relation by an arrow diagram and write the ordered pairs of the relation. Determine,
giving reasons, which of the reflexive, symmetric and transitive properties are satisfied
by the relation.
10 Marks
20
SOLUTION-EXAM-2015
Section A
(vi)
x 2  5 x  6 2 x  4 x 2  5 x  6 x 2  3x
 2


x2  9
x  3x
x2  9
2x  4
( x  3)( x  2) x( x  3) x( x  2)



( x  3)( x  3) 2( x  2) 2( x  2)
(viii) see Test 1-2016
R  {(3,3), (3, 6), (3,9), (3, 27), (6, 6), (9,9), (9, 27), (27, 27)}
Reflexive: yes, (3,3)  R, (6, 6)  R, (9,9)  R, (27, 27)  R
Symmetric: no, (3, 6)  R, (6,3)  R
21