Sanjay_Mishra_Mathematics_At_A_Glance_for_Class_XI__XII,_Engineering

Prévia do material em texto

for Class XI & XII, Engineering Entrance
and other Competitive Exams
Mathematics 
at a Glance
Sanjay Mishra
B. Tech (IIT-Varanasi)
ISBN: 978-93-325-2206-0
 
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Contents
Preface iv
Acknowledgements v
 1. Foundation of Mathematics ����������������������������������������������������������������������������������������������������������������� 1�1-1�28
 2. Exponential Logarithm ����������������������������������������������������������������������������������������������������������������������� 2�29-2�36
 3. Sequence and Progression ������������������������������������������������������������������������������������������������������������������ 3�37-3�47
 4. Inequality ���������������������������������������������������������������������������������������������������������������������������������������������� 4�48-4�54
 5. Theory of Equation ������������������������������������������������������������������������������������������������������������������������������ 5�55-5�63
 6. Permutation and Combination ���������������������������������������������������������������������������������������������������������� 6�64-6�78
 7. Binomial Theorem ������������������������������������������������������������������������������������������������������������������������������� 7�79-7�83
 8. Infinite Series���������������������������������������������������������������������������������������������������������������������������������������� 8�84-8�86
 9. Trigonometric Ratios and Identities�������������������������������������������������������������������������������������������������� 9�87-9�97
 10. Trigonometric Equation ��������������������������������������������������������������������������������������������������������������� 10�98-10�109
 11. Properties of Triangle ������������������������������������������������������������������������������������������������������������������ 11�110-11�120
 12. Inverse Trigonometric Function ������������������������������������������������������������������������������������������������ 12�121-12�131
 13. Properties of Triangle ������������������������������������������������������������������������������������������������������������������ 13�132-13�139
 14. Straight Line and Pair of Straight Line �������������������������������������������������������������������������������������� 14�140-14�151
 15. Circle and Family of Circle ��������������������������������������������������������������������������������������������������������� 15�152-15�161
 16. Parabola ����������������������������������������������������������������������������������������������������������������������������������������� 16�162-16�172
 17. Ellipse �������������������������������������������������������������������������������������������������������������������������������������������� 17�173-17�179
 18. Hyperbola ������������������������������������������������������������������������������������������������������������������������������������� 18�180-18�188
 19. Complex Number ������������������������������������������������������������������������������������������������������������������������ 19�189-19�211
 20. Sets and Relation �������������������������������������������������������������������������������������������������������������������������� 20�212-20�225
 21. Functions �������������������������������������������������������������������������������������������������������������������������������������� 21�226-21�254
 22. Limit, Continuity and Differentiability ������������������������������������������������������������������������������������� 22�255-22�272
 23. Method of Differentiation ����������������������������������������������������������������������������������������������������������� 23�273-23�277
 24. Application of Derivatives ���������������������������������������������������������������������������������������������������������� 24�278-24�304
 25. Indefinite Integration ������������������������������������������������������������������������������������������������������������������ 25�305-25�321
 26. Definite Integration and Area Under the Curve ��������������������������������������������������������������������� 26�322-26�336
 27. Differential Equation ������������������������������������������������������������������������������������������������������������������� 27�337-27�350
 28. Vectors ������������������������������������������������������������������������������������������������������������������������������������������� 28�351-28�365
 29. Three Dimensional Geometry ��������������������������������������������������������������������������������������������������� 29�366-29�381
 30. Probability ������������������������������������������������������������������������������������������������������������������������������������� 30�382-30�391
 31. Matrices and Determinants �������������������������������������������������������������������������������������������������������� 31�392-31�411
 32. Statistics ����������������������������������������������������������������������������������������������������������������������������������������� 32�412-32�419
Any presentation or work on Mathematics must be conceived as an art rather than a text� This is where 
this work holds it differently� During my school days and throughout my long teaching career, I realized 
that most of the JEE aspirants feel the need of a book that may provide them with rapid revision of all 
the concepts they learned and their important applications, throughout their two years long time of 
preparation� I prefer to call it Mathematics at a Glance� The present book is written with sole objective 
of that� The entire syllabus of Mathematics for AIEEE, JEE Mains and JEE Advanced has been presented 
in an unprecedented format� The reader ought to have the following pre requisites before going 
through it:
 (i) He/She must have ample knowledge of high school Mathematics
 (ii) Must have conceptual/theoretical knowledge behind the various mathematical thoughts presented�
 (iii) Must be confident enough that he/she is not the father of Mathematics and, if not comfortable with 
any concept or text, we shall be thankful to have your valuable advice� 
As the name of this work suggests that it has been designed to help during revision� It must be kept in mind 
that the motive of the text is to provide a recapitulation of the entire mathematics that you have studied in 
your mainstream syllabus� While going through the book if you want detailed analysis of any thought or 
idea you must go for:
“Fundamentals of Mathematics---By Sanjay Mishra”�
All the suggestions for improvement are welcome and shall be greatfully acknowledged�
—Sanjay Mishra
Preface
I am really grateful to “Pearson Education”, for showing their faith in me and for providing me an 
opportunity to transform my yearning, my years-long teaching experience and knowledge into the present 
rapid revision book, “Mathematics at a Glance”� I would like to thank all teachers and my friends, for 
their valuable criticism, support and advicethat was really helpful to carve out this work� I wish to thank 
my parents and all my family members, for their patience and support in bringing out this book and 
contributing their valuable share of time for this cause� I extend my special thanks to my team, including 
my assistant teachers Rakesh Gupta, Parinika Mishra, managers and computer operators, for their hard 
work and dedication in completing this task�
—Sanjay Mishra
Acknowledgements
Chapter 1
Foundation oF 
MatheMatiCs
MatheMatical Reasoning
1.1 INTRODUCTION
Mathematics is a pure application of brains. To crack mathematical problems an analytical approach is 
required.
1.2 PRE-REQUISITES
		Flush out your thoughts of maintaining algorithms for mathematical problems.
		Try to connect the text and work in this chapter from high-school mathematics and make conclusive 
analysis of applying basic principles of mathematics. 
1.2.1 Greek Words (Symbols)
Symbol Meaning Symbol Meaning Symbol Meaning
α Alpha β Beta g Gamma
δ, D Delta ∈, ε Epsilon ξ Zeta
η Eta θ Theta i Iota
κ Kappa λ Lambda µ Mu
v Nu ξ Xi o Omicron
π Pi ρ Rho σ, ∑ Sigma
τ Tau υ Upsilon f Phi
χ Chi ψ Psi ω Omega
1.3 UNDERSTANDING THE LANGUAGE OF MATHEMATICS
Well! Obviously mathematics is no language by itself but as remarked by Albert Einstein 
“Mathematics is the language in which god has written the universe.”
1.2 Mathematics at a Glance
1.3.1 Mathematical Symbols
Symbol Meaning Symbol Meaning Symbol Meaning
\ Therefore ∫ Single Integration D Triangle
∵ Because, Since ∫ ∫ Double Integration ⇒ Implies
: Such that Σ Sigma N The set of natural numbers
:: So as a Proportionate to ⇔ Implies and is implied by
: Ratio f Function Z or I The set of integers
:: Proportion ∞ Infinity Q The set of rational numbers
= Equal to _ Line bracket ℝ The set of real numbers
≠ Not equal to () Small bracket |x| Absolute value of x
> Greater than {} Mid bracket i.e. i.e., (that is)
< Less than [] Large bracket e.g. example gratia (for example)
≥ Greater than or 
equal to
∈ Belongs to Q.E.D. Quod erat demonstrandum
≤ Less than or 
equal to
∉ Does not belong to ⊄ Is not a subset of
∢ Not less than ⊂ Is a subset of ∪ Universal set
nth root ∪ Union of sets ~ Similar to
Cube root ∩ Intersection of sets iff If and only if
∠ Angle A × B Cartesian product 
of A and B
|| Parallel
^ Perpendicular A – B Difference of two 
sets A and B
f Null Set (phi)
@ Congruent to ∀ For all ∩ Arc
$ There exists
1.4 STATEMENTS AND MATHEMATICAL STATEMENTS
1.4.1 Statement
It is a sentence which is complete in itself and explains its meanings completely, e.g., Delhi is the 
capital of India.
1.4.2 Mathematical Statements
A given statement is mathematical, if either it is true or it is false but not both.
1.4.3 Scientific Statement
A given sentence will qualify as a scientific statement even if it may be true conditionally, e.g., mass can be 
neither created nor destroyed.
Foundation of Mathematics 1.3
1.5 CLASSIFICATION OF MATHEMATICAL STATEMENTS
 1. axiom: Mathematical statements which are accepted as truth without any formal proof given for 
it. e.g., Equals added to equals are equals.
 2. Definition: Mathematical statement which is used to explain the meaning of certain words used in 
the subject.
 E.g. “The integers other than ±1 and 0 which are divisible by either one or by themselves are called 
prime integers”.
 3. Theorems: A mathematical statement which is accepted as ‘truth’ only when a formal proof is 
given for it like summation of interior angles of a triangle is 180 degree, is a theorem.
1.5.1 Conjectures
In mathematics, a conjecture is an unproven proposition that appears correct. For example, every even 
integer greater than two, can be expressed as a sum of two primes.
1.5.2 Mathematical Reasoning
Reasoning is a process of logical steps that enables us to arrive at a conclusion. In mathematics there are 
two types of reasoning. These are as follows:
 1. inductive Reasoning: Like that in mathematical induction.
 2. Deductive Reasoning: Series of steps to deduct one mathematical statement from the other and 
their proof, which will be discussed in the text.
1.6 WORKING ON MATHEMATICAL STATEMENTS
1.6.1 Negation of a Statement
The denial of a statement is called its negation. To negate a statement we can use phrases like “It is false” 
“is not”. Rita is at home ⇒ Rita is not at home.
1.6.2 Compounding of Statements
Compounding of statements is defined as combining two or more component statements using the 
connecting words like ‘and’ and ‘or’, etc. The new statement formed is called a compound statement.
Compounding 
with OR
p : x is odd prime number.
q : x is perfect square of integer.
x is a odd prime or a perfect 
square integer.
Compounding 
with AND
p : 2 is a prime number.
q : 2 is an even number.
2 is a prime and even natural 
number.
Note:
OR be inclusive or exclusive depending both conditions are simultaneously possible or not 
respectively.
http://en.wikipedia.org/wiki/Mathematics
http://en.wikipedia.org/wiki/Proposition
1.4 Mathematics at a Glance
1.7 IMPLICATION OF A STATEMENT
If two statement p and q are connected by the group of words, ‘if…, then…’ the resulting compound 
statement:
‘if p then q’ is called ‘conditional implications’ of p and q is written in symbolic form as ‘p → q’ 
(read as ‘p implies q’) 
e.g., p: the pressure increases 
q: the volume increases
Then, implication of the statements p and q is given by p → q: if the pressure increases, then the 
volume decreases.
1.7.1 Converse of a Statement
it is given by p ⇒ q means q ⇒ p.
If a integer n is even then n2 is divisible by 4. Converse is “If n2 is divisible by 4, then n must be even”.
1.7.2 Contra Positive of a Statement p ⇒ q is ~q ⇒ ~p
If a triangle has two equal sides, then it is isosceles triangle. Its contrapositive is: ‘if a 
triangle is not isosceles then it has no two sides equal’.
1.8 TRUTH VALUE
The truth (T) or falsity (F) of any statement is called its truth value. E.g., every mathematical statement is 
either true or false. Truth value of a true statement is (T), and that of a false statement is (F).
Given below in the table are Venn Diagrams and truth tables of various mathematical and logical 
operations:
Operation Venn Diagram Truth Table 
And p q p and q/p ^ q
T T T
T F F
F T F
F F F
Or p q p or q/p ∨ q
T T T
T F T
F T T
F F F
Foundation of Mathematics 1.5
Operation Venn Diagram Truth Table 
Negation p ~p
T F
F T
Implies and is 
Implied by
p q p → q q → p (p→q) ^ (q→p)
T T T T T
T F F T F
F T T F F
F F T T T
Implication p q p → q
T T T
T F F
F T T
F F T
1.9 QUANTIFIERS
These are phrases like “there exists $”, “for all ∀”, less than, greater than, etc. For example, there exist a 
polygon having its all sides equal is known as a regular polygon.
1.9.1 Proofs in Mathematics
We can prove a mathematical statement in various ways which are categorized as: straightforward, Mathod 
of exhaustion, Mathematical induction, Using counter example, Contradiction and Contrapositive statements.
1.9.2 What is a Mathematical Assumption?
A mathematical statement which is assumed to be true until a contradiction is achieved. An assumed 
statement may prove to be false at a later stage of mathematical analysis.
nuMbeR systeM
Well! Life without numbers is unpredictable. Numbers have been used since ages to facilitate our transac-
tions regarding trade exchange or other mathematical purposes. Number system has successfully replaced 
the Bartar system of exchange. In this text, we will discuss the number system followed by mathematical 
analysis of real world problems. Our present number system is known as Indo-Arabic number system.
1.10 SET OF NATURAL NUMBERS
ℕ = {x: x is counting number). Counting numbers are called, ‘natural numbers’ and their set is denoted 
as = {1, 2, 3, 4, 5,....}.
1.6 Mathematics at a Glance
If 0 is not included in the set of natural numbers, then we obtain whole numbers (W). 
W = {0, 1, 2, 3....}.1.10.1 Algebraic Properties of Natural Numbers
  They are associative and commutative, i.e., for all a, b, c in the set of natural numbers.
associative law: a + (b + c) = (a + b) + c; a(bc) = (ab)c
commutative law: a + b = b + a; ab = ba
  The cancellation law holds for natural numbers. If a, b, c are natural numbers a + c = b + c
⇒ a = b; ac = bc ⇒ a = b (c is not equal to zero)
  Distribution of multiplication over addition a(b + c) = ac + bc
  Order properties:
 (i) law of trichomy: Given any two natural numbers a and b, exactly one of the following 
holds: a > b or a < b or a = b.
 (ii) transitivity: For each triplet of natural numbers a, b, c; a > b and b > c implies that a > c.
 (iii) Monotone Property for addition and Multiplication: For each triplet of natural 
numbers, a > b ⇒ a + c > b + c and ac > bc.
  existence of additive and multiplicative identity: Zero is an additive identity element and 1 is 
a multiplicative identity element.
  existence of additive and multiplicative inverse: For every integer x, there always exists its 
negative –x which, when added to x makes additive identity. Multiplicative inverse of x is an element 
which, when multiplied to x, makes multiplicative identity 1.
1.11 SET OF INTEGERS
When negatives of natural numbers are included in a set of whole numbers then a set of integers is formed: 
ℤ = {–4, –3, –2, –1, 0, 1, 2, 3, 4,...}.
1.12 GEOMETRICAL REPRESENTATION OF INTEGERS
Greek Mathematicians invented Geometrical method of representing numbers on a line known as 
‘number line’. In this method, a point is marked as zero (0) and, with respect to zero, the numbers are 
located in order of their magnitude. The distance of number (x), from zero represents its magnitude (|x|).
1.12.1 Properties of Integers
 (a) It is closed, commutative, associative and distributive for addition, subtraction and multiplication.
 (b) Zero is the identity element for addition and 1 is the identity for multiplication.
Foundation of Mathematics 1.7
 (c) Additive inverse of x is equal to –x Q x+ (–x) = 0.
 (d) Multiplicative inverse of x is 1/x provided x ≠ 0 as x. 1/x = 1.
 (e) Cancellation law holds for addition as well as multiplication.
 (f) Property of order: ∀ x, y ∈ℤ either x > y or x = y or x < y. Also known as law of trichomy.
1.13 DIVISION ALGORITHM
Given are two integers a and b, such that a > b and b > 0; then there exist two integers q and r such that 
a = bq + r; where a: dividend; b: divisor; q: quotient; r: remainder.
Properties:
	 The remainder r is a non-negative integer which is less than the divisor b.
	 0 ≤ r < b; where r = 0, 1, 2, 3, 4,..., b – 1.
	 If the remainder r = 0, then a = b.q.
	 Then a is called completely divisible by b (i.e., multiple of b) and, b and q are called factors of a.
1.13.1 Even and Odd Integers
 (a) Set of even integer = {x: x = 2k; where k ∈ ℤ}
 (b) Set of odd integers = {x: x = 2k + 1; where k ∈ ℤ}
1.13.2 Prime Integer
An integer x (other than 0, –1 and 1) is called prime iff it has only positive divisors as 1 and itself. 
e.g., 2, 3, 5, 7, etc.
1.13.2.1 Properties
	 An integer other than 0, –1 and 1, which are non-primes are called composite numbers. A composite 
integer has atleast three factors.
	 1, –1, 0 are neither prime nor composite.
	 Twin Primes: A pair of primes is said to be twin primes if they differ by 2. i.e., 3, 5 and 11, 13 etc.
	 Co-Primes: A pair of integers is said to be co-primes if they have no common positive divisor except 
1. e.g. 8, 5 and 12, 35.
	 If p is prime and greater than or equal to 5, then p is either 6k + 1, or 6k – 1, but converse is not 
necessarily true.
	 If p is prime and greater than 5, then p2 – 1 is always divisible by 24.
1.14 FACTORIAL NOTATION
Factorial of r is denoted as r! and is defined as product of first r natural numbers. i.e., r! = 1. 2. 3. 4….. (r – 1).r
e.g.: 1! = 1 ; 2! = 2 ; 3! = 6 ; 4! = 24
 5! = 120 6! = 720 7! = 5040
	 Product of any r consecutive integer is always divisible by r!.
1.8 Mathematics at a Glance
1.14.1 Related Theorems
Theorem 1: xn – yn is divisible by (x – y) ∀ x ∈ ℕ, {since putting x = y makes expression xn – yn = yn – yn = 0. 
Therefore, x – y must be factor in the above expression.
Theorem 2: xn – yn is divisible by (x + y) ∀ odd natural numbers n. Since putting x = –y makes 
expression xn + yn = yn + (–y)n = yn + (–1)n yn = yn – yn = 0. Thus, x + y must be factor in the above 
expression (xn + yn) = (x + y)(xn–1 – xn–2y + xn–3y2 –…+ (–1)n–1 yn–1).
Theorem 3: Given n ∈ ℕ and p and p is prime such that “n is co-prime to p” then np –n is always divisible by p.
Fermat’s Theorem: n = 2 and p = 5 ⇒ 5|25 – 2 ⇒ 5|24 – 1.
corollary 1: np – n is also divisible by n and (n – 1).
corollary 2: np – n is divisible by n(n – 1). Since n and (n – 1) are always co-prime.
corollary 3: np–1 – 1 is always divisible by p.
Theorem 4: (fundamental theorem of arithmetic) A natural number N can be expressed as product of 
non-negative exponent of primes. N = pa. qb. rc. sd … where p, q, r, s are primes and a, b, c, d are whole 
numbers. e.g., 1800 = 23.32.52.70....
Theorem 5: (Wilson’s theorem) if p is a prime number, then 1 + (p – 1)! is divisible by p. i.e., 16! + 1 is 
divisible b.
1.14.2 Divisors and Their Property
A natural number x = pa qb rg is called divisor of N = pa qb rc ⇒ N, is completely divisible by x.
⇔ all the prime factors of x are present in N.
⇔ 0 ≤ α ≤ a; 0 ≤ b ≤ b; 0 ≤ g ≤ c where a, b, g are whole numbers.
 Set of all divisors of N is given as: {x : x = pα. qb. rg where 0 ≤ α ≤ a, 0 ≤ b ≤ b, 0 ≤ g ≤ c}
1.14.3 Number of Divisors
n {(α,b,g) : 0 ≤ α ≤ a, 0 ≤ b ≤ b, 0 ≤ g ≤ c} = number of ways the integers a, b, g can take values applying the 
above restrictions = (a + 1).(b + 1).(c + 1).
sum of Divisor of n = pa qb rc (1 + p + p2 +...+ pa). (1 + q + q2 +...+ qb). (1 + r +r2 +...+ rc)
improper Divisors of N = pa qb rc when a = b = g = 0 ⇒ x = 1 this is divisor of every integer and a = a, 
b = b and g = c, then x becomes number N itself. These two are called ‘improper divisor.’ The number of 
proper divisors of N = (a + 1).(b + 1).(c + 1) – 2.
If p = 2, then number of even divisors = a(b + 1)(c + 1). Number of odd divisors = (b + 1)(c + 1).
Number and sum of divisors of N divisible by a natural number 1 1 1a b cy p .q .r= :
Let x = pa. qb. rg be such divisors. ∵ 1 1 1a b cy | x p .q .r | p .q .rα β γ⇒
⇒ a1 ≤ α ≤ a and b2 ≤ b ≤ b and c1 ≤ g ≤ c 
⇒ Number of such divisors = (a – a1 + 1) (b – b1 + 1) (c – c1 + 1)
 Sum of such divisors Sy = 
1 1 1 1 1 1a a 1 b b 1 c c 1a b c
yS (p p .... p ).(q q .... q ).(r r .... r )
+ + += + + + + + + + + +
 = 1 1 1a a b b c c2 2y(1 p p ... p )(1 q q ... q )(1 r ... r )− − −+ + + + + + + + + + + 
 = 
1 1 1a a 1 b b 1 c c 1p 1 q 1 r 1y
p 1 q 1 r 1
− + − + − +   − − −
   
− − −   
Foundation of Mathematics 1.9
Notes:
 1. The number of ways of resolving n into two factors is + + +
1
( a 1)( b 1)( c 1)...,
2
 when n is not a perfect 
square and + + + +
1
{( a 1)( b 1)( c 1) 1},
2
 when n is a perfect square.
 2. Every number n has two improper divisors 1 and n itself and the remaining divisors are called proper 
divisors. E.g., number of proper divisors of 108 is 10.
1.14.4 Least Common Multiple (LCM)
LCM of set of numbers is the smallest number (integer/rational) which is completely divisible by each of 
them. i.e. x is said to be LCM of y and z iff y divides x, z also divides x and x is least positive of all such 
numbers. E.g., LCM of 6, 4, 9, is 36.
Let x and y be two given integer x = pa.qβ.rg.sd and 1 1 1 1y p .q .r .sα β γ δ= where p, q, r, are primes.
If z is LCM of x and y then 1 1 1 1max( , ) max( , ) max( , ) max( , )z p .q .r .sα α β β γ γ δ δ=
LCM 
LCM (a and c)a cand
b d HCF (b and d)
=
1.14.5 Greatest Common Divisor (GCD)/Highest Common Factor (HCF)
HCF of a given set of numbers is the largest number which divides each of the given numbers HCF of y and 
z is also denoted as (y, z). Therefore,x is said to be GCD of y and z if x divides both y and z and x is largest 
of such numbers. So, clearly every common divisor of y and z also divides x and x ≠ 0.
E.g., HCF of 12 and 64 is 4. GCD of 6 and 35 is 1 (co-prime).
HFC 
HCF (a and c)a cand
b d LCM (b and d)
= .
Method to find hcF: For two given integers x and y.
Method 1: Consider their prime factors 1 1 1 2 2 2x p .q .r and y p .q .r ....α β γ α β γ= =
If z is HCF of x and y ⇒ z/x and z/y
\ z contains the least power for each corresponding prime factor. 
⇒ { } { } { }1 2 1 2 1 2min , min , min ,z (x, y) p .q .r ...α α β β γ γ= =
1.14.6 Decimal Representation of Number 
given a natural number, x abcde= ; where e, d, c, b, a are respectively digits occupying unit, ten’s, 
hundredth, thousandth, ten thousandth places. So the numerical value of x is defined as, ‘sum of products of 
digits multiplied by their corresponding place values’.
    
th th th
4 3 2 1 0
ten's place unit placetenthousand thousand hundred
value valueplace value place value place value
x = a 10 + b 10 + c 10 + d 10 + e 10
− − −
× × × × ×
Theorem: If an integer x is divided by 10, the reminder is a digit at the unit place of x.
Proof: = = + + + + = +4 3 2x abcde a(10 ) b(10 ) c(10 ) d(10) e 10m e ⇒ Remainder is e.
1.10 Mathematics at a Glance
Theorem: The remainder, if an integer x is divided by 5, is e 0 e 4
e 5 5 e 9
≤ ≤
 − ≤ ≤
; where e is are unit place 
digit of the number 4 3 2x abcde a(10 ) b(10 ) c(10 ) d(10) e= = + + + +
= a(104) + b(103) + c(102) + d(10) + e = 5m + e; 0 ≤ e ≤ 9
0
5m e 0 e 4 5m e 0 e 4
5m 5 e 5 5 e 9 5m (e 5) 5 e 9
+ ≤ ≤ + ≤ ≤  = = + + − ≤ ≤ + − ≤ ≤  
1.14.7 Periodic Properties of Integers
Theorem 1: Unit digit of nth power of an integer having zero at its unit place is zero.
⇒ n 1 1 1(abc...0) (a b c ...0)=
Theorem 2: Unit digit of nth power of an integer having one at its unit place is one.
⇒ n 1 1 1(abc...1) (a b c ...1)=
Theorem 3: Unit digit of nth power of an integer having two at its unit place is described as follows:
⇒ cn 1 1 1(abc...2) (a b c ...2)= if n = 4k + 1, i.e., 
n
1 1 1(abc...2) (a b c ...4)= if n = 4k + 2
 i.e., n 1 1 1(abc...2) (a b c ...8)= if n = 4k + 3, i.e., 
n
1 1 1(abc...2) (a b c ...6)= if n = 4k
Theorem 4: Unit digit of nth power of an integer having three at its unit place is described as follows:
⇒ n 1 1 1(abc...3) (a b c ...3)= if n = 4k + 1, i.e., 
n
1 1 1(abc...3) (a b c ...9)= if n = 4k + 2
⇒ i.e., n 1 1 1(abc...3) (a b c ...7)= if n = 4k + 3, i.e., 
n
1 1 1(abc...3) (a b c ...1)= if n = 4k
Theorem 5: Unit digit of nth power of an integer having four at its unit place is described as follows:
⇒ n 1 1 1(abc...4) (a b c ...4)= if n = 2k + 1, i.e., 
n
1 1 1(abc...4) (a b c ...6)= if n = 2k
Theorem 6: Unit digit of nth power of an integer having five at its unit place has five at unit place.
⇒ n 1 1 1(abc...5) (a b c ...5)= if n ∈ ℕ
Theorem 7: Unit digit of nth power of an integer having six at its unit place has six at unit place.
⇒ n 1 1 1(abc...6) (a b c ...6)= if n ∈ ℕ
Theorem 8: Unit digit of nth power of an integer having seven at its unit place is described as follows:
i.e., n 1 1 1(abc...7) (a b c ...7)= if n = 4k + 1, i.e., 
n
1 1 1(abc...7) (a b c ...9)= if n = 4k + 2
i.e., n 1 1 1(abc...7) (a b c ...3)= if n = 4k + 3, i.e., 
n
1 1 1(abc...7) (a b c ...1)= if n = 4k
Theorem 9: Unit digit of nth power of an integer having eight at its unit place is described as follows:
i.e., n 1 1 1(abc...8) (a b c ...8)= if n = 4k + 1, i.e., 
n
1 1 1(abc...8) (a b c ...4)= if n = 4k + 2
i.e., n 1 1 1(abc...8) (a b c ...2)= if 4k + 3, 
n
1 1 1i.e. (abc...8) (a b c ...6) if n 4k= =
Theorem 10: Unit digit of nth power of an integer having nine at its unit place is described as follows:
n
1 1 1i.e. (abc...9) (a b c ...9) if n 2k 1= = + , 
n
1 1 1i.e. (abc...9) (a b c ...1) if n 2k= =
Foundation of Mathematics 1.11
1.15 TESTS OF DIVISIBILITy
 1. Divisibility by 2: A number N is divisible by 2 if and only if its last digit is divisible by 2. (i.e., even)
 2. Divisibility by 3: A number N is divisible by 3 if and only if the sums of all digits are divisible by 3.
 3. Divisibility by 4: A number N is divisible by 4 if its units digit plus twice its ten’s digit is divisible by 4.
 4. Divisibility by 5: A number N is divisible by 5 if and only if its last digit is divisible by 5 (i.e., if it 
ends in 0 or 5).
 5. Divisibility by 6: A number N is divisible by 6 if and only if its units’s digit is even and the sum of 
its digits are divisible by 3
 6. Divisibility by 7: A number N is divisible by 7 if and only if 3 × unit’s digit + 2 × ten’s digit – 1 
× hundred’s digit – 3 × thousand’s digit -2 × ten thousand’s digit + 1 × hundred thousand’s digit is 
divisible by 7. i.e., 3(a0) + 2(a1) – 1(a2) – 3(a3) – 2(a4) + 1(a5) + 3(a6) +... is divisible by 7.
 i.e., If there are more digits present in the sequence of multipliers 3, 2, – 1, – 3, – 2, 1 is repeated as 
often necessary.
 7. Divisibility by 8: A number N is divisible by 8 if and only if its unit’s digit + 2× ten’s digit + 4 × 
hundred’s digit is divisible
 8. Divisibility by 9: A number N is divisible by 9 if and only if the sum of its digits is divisible by 9
 9. Divisibility by 10: A number N is divisible by 10 if and only if the last digit is 0
 10. Divisibility by 11: N is divisible by 11 if and only if the difference between the sum of the digits in 
the odd places (starting from the right) and the sum of the digits in the even places (starting from the 
right) is a multiple of 11, e.g., 1221, 123321, 2783 etc. 
 12. Divisibility by 13: A number N is divisible by 13 if and only if 10 × units’s digit – 4 × ten’s 
digit – 1 × hundred’s digit + 3 × thousand’s digit + 4 × ten thousand’s digit + 1 × hundred thousand’s 
digit is divisible by 13. (If there are more digits present, the sequence of multipliers 10, –4, –1, 3, 4, 1 
is repeated as often as necessary).
1.16 RATIONAL (ℚ) AND IRRATIONAL NUMBERS (ℚ′)
A number x in the form p/q where p and q are integers and q is not equal to 0, is called rational and 
otherwise it is called irrational numbers  ( or ). e.g. 1, 3, 5, 0, 2/5, 10/16,....., 10/7, are rational while 
√2, √3, √5,…., √x : x is not a perfect square of rational are irrationals. Pie (p) : is ratio of circumference 
of any circle to the diameter of the same circle. It is an irrational number approximately equal to rational 
numbers 22/7 or 3.14.
euler number (e): 1 1 1e 1 ... 2.7 e 8
1! 2! 3!
= + + + + ∞⇒ < <
1.16.1 Properties of Rational and Irrational Numbers
	 If a number x in decimal form is written as x cde.pqr= , then
     
th
2 1 0 1 2 3
ten's place unit place first decimal Second decimal third decimalhundred
value value place value place value place valueplace value
x c 10 d 10 e 10 p 10 q 10 r 10− − −
− − −−
= × + × + × + × + × + ×
	 All terminating decimals are rational. e.g. 4
abcdea.bcde
10
= ; = = 1 2 n1 2 3 n n
ax x ......xx a.x x x ...., x
10
1.12 Mathematics at a Glance
	 If a rational p/q (in lowest term) is terminating decimal, then q = 2m.5n, i.e., q must not contain any 
prime factor other than 2 or 5.
	 Non-terminating, but repeating decimals are also rationals, e.g., y = x.ab ab ab …… y x.ab⇒ =
 ……..(i)
	 If number of repeating digits be n, then multiply both side by 10n, i.e., 210 y xab.ab= …….(ii)
Subtracting (i) from (ii), we get xab xy
99
−
= (which is a rational number).
	 Non-terminating and non-repeating decimals are irrationals. 2.71354921275718 ….. (no periodic 
re-occurrence up to µ).
	 Set of rational numbers is countable while set of irrational numbers is uncountable.
1.16.2 nth Root of a Number
A real number y is called nth root of real number x, where n is a natural number (n ≥ 2). Iff yn = x. 
When n = 2, then it is called as square root and for n = 3, known as cube root. All the numbers other 
than zero have more than one nth roots. e.g., both 2 and –2 are square root of4.
1.16.3 Principal nth Root
The principal nth root of a real number x (having atleast one n-th root) is that nth root which has its sign 
same as that of x. It is denoted by a radical symbol n x .
The positive integer n is known as the index of the radical symbol. Usually we omit the index from the 
radical sign if index n = 2, and write as x
e.g., 27 336 6,
8 2
= = and 5 4243 ( 3), 16 2− = − = whereas 4 16− is a non-real number since fourth 
power of no real number can be –16 which is negative.
1.16.4 Properties of nth Root
 (i) Every positive real number x has exactly two real nth roots when n is a positive even natural number 
(n = 2m) denoted by 2m 2mx and x− are two real fourth roots of 256. e.g., 4 4256 4; 256 4= − = −
 (ii) Every real number x has only one real nth roots when n is a positive odd natural number 
(n= 2m + 1) denoted by 2m 1 x+ , e.g., 3 3125 5; 125 5= = − .
 (iii) nth root of a negative real number x is non-real when n is an even integer. E.g., 424, 16− − has 
no real values 1− is a non-real number symbolized as i (iota).
 (iv) Zero is only real number which has only one nth root and n 0 0=
 (v) Integers, such as 1, 4, 9, 16, 25 and 49 are called perfect squares because they have integer 
square roots.
 (vi) Integers such as 1, 8, 27, 64 are called perfect cubes as they have integer cube roots.
square roots: If b is the square root of a where a is the non-negative real number then b when squared 
must become equal to a.
⇒ b2 = a ⇒ b2 – a = 0 ⇒ (b – √a) (b + √a) = 0
⇒ b – √a = 0 or b + √a = 0 ⇒ b = √a (positive) or b = –√a (negative)
Foundation of Mathematics 1.13
1.16.4.1 Properties of Square Roots
 (i) Zero has only one square root, i.e., zero.
 (ii) Every positive real number (except zero) has two square roots. One of them is positive (called as 
principle square root denoted as √a) and the other is negative, denoted as (–√a).
 (iii) Magnitude of real number x, denoted as |x| and defined as the quantity of x is 
2
x if x 0
| x | x 0 if x 0
x if x 0
 >
= = =
− <
.
1.16.5 Algebraic Structure of 
 and
	 closure law: For addition and subtraction, multiplication.
	 commutative law: For addition and multiplication.
	 associative law: For both addition and multiplication.
	 Distributive law: For addition and subtraction operation.
Notes:
 (i) Zero is the identity element for addition and 1 is the identity for multiplication.
 Q x + 0 = x and 
1
x 1;
x
 × = 
 
 ∀ x ∈ ℚ x ≠ 0.
 (ii) Additive inverse of x = p | q is equal to –x Q x + (–x) = 0
 (iii) Multiplicative inverse of = =
p 1
x
q x
provided x ≠ 0 as 1x. 1.
x
=
  cancellation law holds for addition as well as multiplication. 1 2 1 3 2 3
1 2 1 3 2 3
x x x x x x
x .x x x x x
+ = + ⇒ =
 = ⇒ =
 
provided x1 ≠ 0.
  Property of order: ∀ x, y ∈ ℚ either x > y or x = y or x < y. Also known as law of trichotomy.
  Union of set of rationals and set of irrationals is called set of real numbers ℝ.
1.17 SURDS AND THEIR CONJUGATES
Sum of a rational and an irrational number is always irrational and called as surd; denoted by s.


rational irrational
part of s part of s
s a b= + ; where b is not a perfect square of the rational number.
	 For every surd s there exist element s : s a b= − ; where s a b= + , called as conjugate of s.
	 Rationalization of denominator of an irrational number:
2
2 2
s a b (a b)(a b) a b 2a b
s a b a b(a b) (a b)(a b)
+ + + +
= = = +
− −− − +
1.14 Mathematics at a Glance
1.18 REAL NUMBERS SySTEM
Union of set of Rationales and set of Irrationals is called set of Real numbers (ℝ).   = ∪
Properties:
 (i) Square of real numbers is always non-negative. If x ∈ ℝ ⇒ x2 ≥ 0.
 (ii) Between any two real numbers, there are infinitely many real numbers.
 (iii) Magnitude of real number x is denoted as |x| and defined as the quantity of x.
i.e., 2
x if x 0
| x | x 0 if x 0
x if x 0
 >
= = =
− <
.
 (iv) They are represented on a straight line called as real number line in order of their magnitude, 
such that distance of the number of x from zero is equal to magnitude of x (|x|).
 (v) A real number line is infinitely dense and continuous line. i.e., between two any two number (how 
so ever closed they are) there lies infinitely real number.
1.18.1 Concept of Interval
As the set of all real numbers lying between two unequal real numbers a and b can never be expressed in 
roster form, therefore these are expressed in set builder form using the concept of intervals.
open interval: Denoted as (a, b) x ∈ (a, b) = {x : a < x < b, x ∈ ℝ} 
i.e., end points are not included. 
closed interval [a, b]: x ∈ [a, b] = {x : a ≤ x ≤ b, x∈ ℝ} the end points 
are included.
semi-open interval: x ∈ (a, b] ⇒ a < x ≤ b and x ∈ [a, b) 
⇒ a < x ≤ b.
1.18.2 Intersection and Union of Two or More Intervals 
To find the intersection or union of two or more intervals, locate each interval over the same real number 
line and for intersection take the interval, which is common to both and for union locate the interval 
which includes the numbers of all the interval considered.
1.19 MATHEMATICAL INDUCTION
Mathematical induction is a mathematical tool by which we can prove the correctness of any 
mathematical statement or proposition. It works on the principle that results for higher integers 
are induced from the results for lower integers. 
Foundation of Mathematics 1.15
Statement Working Rule
First principle 
of mathematical 
induction
The set of statements {P(n): 
n ∈ N} is true for each natural 
number n ≥ m, is provided that:
P(m) is true,
P(k) is true for n = k, 
(where k ≥ m)
⇒ P(n) is true for 
n = k + 1
Let there be a proposition or a mathematical 
statement, namely P(n), involving a natural num-
ber n. In order to prove that P(n) is true for all 
natural numbers n ≥ m, we proceed as follows:
Verify that P(m) is true.
Assume that P(k) is true (where k ≥ m).
Prove that P(k + 1) is true.
Once step – (c) is completed after (a) and (b), 
we are through. i.e., P(n) is true for all natural 
numbers n ≥ m.
Second 
principle of 
mathematical 
induction
The set of statements {P(n): n 
∈ N} is true for each natural 
number n ≥ m, provided that:
P(m) and P(m + 1) are true; 
P(n) is true for n ≤ k, 
(where k ≥ m)
⇒ P(n) is true for 
n = k + 1
This is also called extended 
principle of Mathematical 
Induction.
Verify that P(n) is true for n = m, n = m + 1.
Assume that P(n) is true for n ≤ k (where k ≥ m)
Prove that P(n) is true for n = k + 1.
Once rule (c) is completed after (a) and (b), we 
are through. That is P(n) is true for all natural 
numbers n ≥ m. This method is to be used when 
P(n) can be expressed as a combination of 
P(n -1) and P(n - 2). In case P(n) turns out to be 
a combination of P(n -1), P(n - 2) and 
P(n -3), we can verify for n = m + 2 also in 
Rule(a).
1.19.1 Ratio and Proportion
Ratio and proportions are algebraic operations which are operated on one or more variables as:
Ratio: It is a rational between two quantities that tells us what multiple/part; one quantity is of the other. 
Therefore if x and y are two quantities of the same kind, then their ratio is x : y which may be denoted 
by x/y (This may be an integer or fraction.)
 1. A ratio may be represented in a number of ways, e.g., x mx nx
y my ny
= = = .....; where m, n.....are 
non-zero numbers.
 2. To compare two or more ratios, always reduce them to a common denominator.
 3. Ratio of two fractions may be represented as the ratio of two integers, e.g., x z x/y xu: :
y u z/u yz
= or xu : yz.
 4. Ratios are compounded by taking their product, i.e., x z v x.z.v. . .... .
y u w y.u.w
=
 5. Duplicate/triplicate ratio: If x : y is any ratio, then its duplicate ratio is x2: y2 ; triplicate ratio is 
x3: y3...., etc. If x:y is any ratio, then its sub-duplicate ratio is x1/2: y1/2; sub triplicate ratio is x1/3: y1/3, etc.
Proportion:
When two ratios a/b and c/d are equal, then the four quantities composing them are said to be propor-tional. If a,b,c,d are proportional, then a/b = c/d, and it is written as a:b = c:d or a:b :: c: d.
 1. ‘a’ and ‘d’ are known as extremes whereas ‘b and c’ are called as means.
 2. Product of extremes = product of means.
1.16 Mathematics at a Glance
1.19.2 Some Important Applications of Proportion
If four a, b, c, d are proportional, then many other useful proportions can be derived using various laws of 
fraction which are extremely useful in mathematical calculations and simplifications. 
invertando: If a : b = c : d, then b : a = d : c.
alternando: If a : b = c : d, then a : c = b : d.
componendo: If a:b = c:d, then 
a b c d
b d
+ +
= .
∵ a c
b d
= adding 1 from both sides a c1 1
b d
+ = + ⇒ a b c d
b d
+ +
= .
Dividendo: If a : b = c : d, then a b c d
b d
− −
=
∵ a c
b d
= subtracting one to both sides a c1 1
b d
− = − ⇒ a b c d
b d
− −
=
componendo and dividendo: If a : b = c : d, then applying both componendo and dividendo operations 
together we get, a b c d
a b c d
+ +
=
− −
. If a c e
b d f
= = (say = l), then 
1/nn n n
n n n
xa yc ze
xb yd zf
 + +
 + + 
.
1.19.3 Linear Equalities
An expression of the form y = ax + b where a and b ∈ ℝ is called a linear polynomial function of x, y 
and set of points (x, y) satisfying the above relations if plotted on the xy plane, a straight line is obtained. 
An equation of the form ax + by + c = 0 is termed as linear equation in x and y.
solving simultaneous linear equations in two unknowns: 
To solve a pair of linear equation a1x + b1y = c1 .... (i)
a2x + b2y = c2 .... (ii)
The following three approaches are adopted:
1.19.4 Method of Comparison
From both equations find the value of any one variable, say y in terms of other, i.e., x. 
1 1 2 2
1 2
c a x c a xy
b b
− −
= = ⇒ 1 2 1 2
1 2 1 2
c c a a x
b b b b
 
− = − 
 
 
⇒ 2 1 1 2
1 2 2 1
b c b c
a b a b
−
−
 and similarly, we get: 2 1 1 2
1 2 1 2
a c a cy
b a a b
−
=
−
.
1.19.5 Method of Substitution
To solve equations (i) and (ii) substitute the value of y from equation (i) to (ii) get x and y then can also be 
obtained. 1 12 2 2
1
c a xa x b c
b
 −
+ = 
 
; a2b1x + b2c1 – b2a1x = b1c2
⇒ (a2b1 – a1b2)x = b1c2 – b2c1 ⇒ 
1 2 2 1
2 1 1 2
b c b cx
a b a b
 −
=  
− 
, and so, we get: 1 2 2 1
1 2 2 1
a c a cy
a b a b
−
=
−
.
Foundation of Mathematics 1.17
1.19.6 Method of Elimination
a1x + b1y = c1 .... (i)
a2x + b2y = c2 .... (ii) 
Multiplying equation (i) by a2 and equation (ii) by a1 and subtracting, x gets eliminated
a1a2x + b1a2y = a1c1 ...(iii)
a1a2x + a1b2y = a1c2 ...(iv)
Subtracting equation (iii) and (iv); 2 1 1 2
2 1 1 2
a c a cy
a b a b
 −
=  
− 
 and thus 1 2 2 1
2 1 1 2
b c b cx
a b a b
−
=
−
.
1.19.6.1 Method of cross-multiplication
It is a very useful method for solving pair of linear equations in two or three variables. 
Given two equations a1x + b1y + c1 ……. (i)
a2x + b2y + c2z …….. (ii) 
Dividing both equations by z and replacing 0
x x
z
= and 0
y y
z
= , we get:
a1x0 + b1y0 + c1 ……. (iii)
a2x0 + b2y0 + c2z …….. (iv) 
Solving by any of the above mentioned three elementary methods, we get:
2 1 1 2 2 1 1 2
0 0
2 1 1 2 2 1 1 2
b c b c b c b cx xx ; x
a b a b z a b a b z
− −
= = = =
− −
that can be symmetrically expressed as 
1 2 2 1 1 2 2 1 1 2 2 1
x y z
b c b c c a c a a b a b
= =
− − −
.
Thus, we can conclude that the set of solution of above pair of equation can always be expressed by the ratio 
x : y : z in terms of coefficients of the equations.
step (1): Express the coefficients of x, y, z beginning with y in cyclic 
order as shown in the figure, and take the product of the coefficients 
indicated by arrows.
step (2): The product formed by descending arrows is considered positive and those by ascending 
arrows is taken negative.
step (3): So, we get x : y : z : : (b1c2 – b2c1) : (c1a2 – c2a1) : (a1b2 – a2b1).
FunDaMentals oF inequality
1.20 INTRODUCTION
The concept of inequality finds its origin from the property of order of real numbers. An inequation is 
marked by the use of logical operations, such as <, >, ≤, ≥, ≠, etc. An inequation can have one or more 
than one variables ax + by + c ≥ 0.
inequation: An inequation is a statement involving sign of inequality, i.e., <, >, ≤, ≥, ≮, ≯, ≠.
1.18 Mathematics at a Glance
1.20.1 Classification of Inequality
Inequalities are of four types.
If a – b > 0 ⇒ a > b (read a greater than b)
If a – b ≥ 0 ⇒ a ≥ b (read a greater than or equal to b)
If a – b < 0 ⇒ a < b (read a is less than b)
If a – b ≤ 0 ⇒ a ≤ b (read a is less than or equal to b)
linear inequality: Inequality having variables in one degree, e.g., 2x + 3y > 5, x – 2y + 3z = 5, etc.
solution of inequality: The values of unknown variable which satisfies the given inequation are called 
solutions of inequality, e.g., x = 2, y = 4 is a particular solution of inequality 2x + 3y > 5.
1.20.1.1 Basic properties of inequality and laws
 (i) transition property: If a > b and b > c ⇒ a > c.
 (ii) law of trichotomy: If x and y are two real numbers, then exactly one of the three statements 
hold, i.e., x > y or x < y or x = y.
 (iii) If a > b, then a + c > b + c and a – c > b – c ∀ c ∈ ℝ.
 (iv) If x < y < 0 ⇒ |x| > |y| (Larger the number smaller the magnitude).
 (v) If x > y > 0 ⇒ |x| > |y| (Larger the number larger the magnitude).
 (vi) If a > b, then a.c > b.c ∀ c > 0 (sign of inequality does not change on multiplying by positive 
real number).
 (vii) If a > b; then a.c < b.c ∀ c < 0 (sign of inequality gets reverse when multiplied both sides by negative 
real number).
 (viii) If a > b, then, a b for c 0
c c
> > and a b for c 0
c c
< < .
 (ix) If a c
b d
≥ , then ad ≥ bc, if b and d same sign.
 (x) If a c
b d
≥ , then ad ≤ bc, if b and d are opposite signs.
 (xi) law of addition: If a1 > b1 and a2 > b2… and an > bn ⇒ (a1+a2+…+ an) > (b1+ b2 +…+ bn)
 (xii) law of Multiplication: If a1 > b1 > 0 and a2 > b2 > 0… and an > bn > 0 ⇒ (a1.a2.a3…an) > (b1.b2.b3….bn)
 (xiii) laws of reciprocal:
(a) If 0 < a < b, then 1 1
a b
> (b) If b < a < 0, then 1 1
b a
>
(c) If x ∈ [a, b], then 
  > <  
   −∞ ∪ ∞ < > =     ∈
  ∞ = > = 

  −∞ = < =  
1 1, for a, b 0, or a, b 0.
b a
1 1, , for a 0; b 0; not defined at x 0.
a b1
x 1 , for a 0, b 0; not defined at x 0.
b
1, for b 0, a 0;not defined at x 0.
a
Foundation of Mathematics 1.19
 (xiv) laws of squares or positive even powers: 
2 2
2 2
2 2
a b if both a,b 0
a>b a b If |a| = |b|
a b If a,b < 0
 > ≥

⇒ =
 <
 
If a and b have opposite sign and a > b, i.e., a > 0 and b < 0, then 
2 2
2 2
2 2
a b iff |a | | b |
a b a b iff |a | | b |
a b iff |a | | b |
 > >

> ⇒ = =
 < <
.
This law can be extended for any even natural power (2n)
If x ∈ [a, b], then 
2 2
2 2
2 2
2
2
[a ,b ] for a, b 0
[b ,a ] for a, b 0
x x
[0,a ] for a b and a.b 0
[0,b ] for b a anda.b 0
 >

<∈ ∈
> <
 > <
; similar is the case for x2n, n ∈ℕ.
 (xv) law of square root: If a and b both are non-negative and 
2n 2n
a b
a b
a b
 >> ⇒ 
>
.
 (xvi) laws of cubes or positive odd powers: If x ∈ [a, b], then x3 ∈ [a3, b3], similarly x2n+1 ∈ [a2n+1, b2n+1] 
for n ∈ ℕ.
 (xvii) law of cube root: a > b ⇒ a3 > b3 and a1/3 > b1/3 ∀ a, b ∈ℝ a < b ⇒ a3 < b3 and a1/3 < b1/3 ∀ a, b ∈ ℝ, 
this law can be extended for any odd natural power (2n+ 1) and odd root.
 (xviii) laws of exponential inequality: 
(a) If 0 < a < 1 and r ∈ ℝ+, then 0 < ar < 1 < a–r .
(b) If a > 1 and r ∈ ℝ+, then ar > 1 > a–r > 0.
(c) For a > 1; ax > ay for x > y and x, y ∈ ℝ.
(d) For 0 < a < 1; ax < ay for x > y and x, y ∈ ℝ.
(e) For a, b∈ (0, 1) or a, b ∈ (1,∞); if a > b, then ax < bx for x < 0 and ax > bx for x > 0.
(f) For a ∈ (0, 1) and b ∈ (1, ∞); ax > bx for x < 0 and ax < bx for x > 0.
 (xix) laws of logarithmic inequality: 
(a) x ≥ y ⇔ logax ≥ logay for a > 1 (b) x ≥ y ⇔ logax ≤ logay for 0 < a < 1
(c) ax ≥ y ⇒ x ≥ logay for a > 1 (d) a
x ≥ y ⇒ x ≤ logay for 0 < a < 1Remark:
Above two results follow from the fact that logarithmic and exponential function to the base a > 1 are increasing 
function and when base lies between 0 and 1 then they become decreasing function.
 (xx) inequalities containing modulus functions:
(a) |x| < a ⇔ –a < x < a; where a > 0, i.e., x ∈ (-a, a)
(b) |x| ≤ a ⇔ –a ≤ x ≤ a; where a > 0, i.e., x ∈ [-a, a]
(c) |x| > a ⇔ x < –a or x > a, i.e., x ∈ (–∞, –a] ∪ (a, ∞) 
(d) |x| ≥ a ⇔ x ≤ –a or x ≥ a, i.e., x ∈ (–∞,–a] ∪ [a, ∞)
(e) a < |x| < b ⇔ x ∈ (–b, b) for a ≤ 0
(f) a < |x| < b ⇔ x ∈ (–b, –a] ∪ [a, b) for a > 0
1.20 Mathematics at a Glance
 (xxi) triangle inequality: | |x| – |y| | ≤ |x ± y| ≤ |x| + |y| ∀ x, y ∈ ℝ. Further:
(a) |x + y| = |x| + |y| for x.y ≥ 0 (b) |x + y| < |x| + |y| for x.y < 0
(c) |x – y| = |x| + |y| for x.y ≤ 0 (d) |x – y| < |x| + |y| for x.y > 0
(e) | |x| – |y| | = |x + y| for x.y ≤ 0 (f) | |x| – |y| | < |x + y| for x.y > 0
(g) | |x| – |y| | < |x –y| for x.y < 0 (h) | |x| – |y| | = |x –y| for x.y ≥ 0
1.20.1.2 Solutions of linear in equations in two variables
 1. by graphical method:
 Let L ≡ ax + by + c = 0 be a line, then by = –ax – c.
 Since the P point satisfies the equation of the line.
 \ aa + bb + c = 0
 From the given diagram, we interpret that g > b.
 \ bg > bb for b > 0
 ⇒ aa + bg > aa + bb ⇒ aa + bg + c > aa + bb + c
 ⇒ aa + bg + c > 0.
 Thus, all the points lying in the half plane II above the line ax + by + c = 0 satisfies the 
inequality ax + by + c > 0.
 Similarly in case b < 0, we can prove that the point satisfying ax + by + c > 0 lies in the 
half plane I.
 Hence, we infer that all points satisfying ax + by + c > 0 lies in one of the half plane II or I 
according as b > 0 or b < 0 and conversely.
 Thus, the straight line ax + by + c = 0, divides the whole x –y plane into three regions.
 (a) For b < 0
 (i) R1 = {(a, b); aa + bb + c = 0}
 (ii) R2 = {(a, b); aa + bb + c < 0}
 (iii) R3 = {(a, b); aa + bb + c > 0}
 (b) For b > 0
 2. short-cut method
 step i: Consider the equation from the Inequality.
 step ii: Draw the straight line representing the Equation.
 step iii: Consider a Point P (a, b) (not on the line) and find the sign of 
linear expression for P (a,b).
 step iV: Check whether it satisfies the inequality or not. If it satisfies, 
then the inequality represents the half plane which contains the point and 
shade the region.
 step V: Otherwise, the inequality represents that half plane which does not contain the point 
within it. 
  For convenience, the point (0, 0) is preferred.
 step Vi: The set R1 is a straight line while the sets R2 and R3 are called open half planes. The set 
R1 ∪ R3 represent the points, whose co-ordinates satisfy ax + by + c ≥ 0 (b > 0) and 
R1 ∪ R2 represent the points whose co-ordinates satisfying ax + by + c ≤ 0 (b > 0). Here R2 is 
the solution region of inequality ax + by + c < 0; b > 0 and R3 is the solution region of inequality 
ax + by + c > 0; b > 0.
+
–––– – –– – –– – – – – –– –
++++++++++++++++
– – –– – –– – – – – – – – – – –
+++++++++++++++++
–––– –
–– – –– – – – – –– –
+++++++++++++++++
Foundation of Mathematics 1.21
1.20.2 Rational Algebraic Inequalities:
type 1: P(x) P(x) P(x) P(x)0, 0, 0, 0
Q(x) Q(x) Q(x) Q(x)
> < ≥ ≤ , P(x), Q(x) are polynomials.
step: 1: Factor P(x) and Q(x) into linear factors.
step 2:
 (i) P(x) 0 P(x).Q(x) 0 P(x) 0,Q(x) 0 or P(x) 0,Q(x) 0
Q(x)
> ⇒ > ⇒ > > < < .
 (ii) P(x) 0 P(x).Q(x) 0 P(x) 0, Q(x) 0 or P(x) 0,Q(x) 0
Q(x)
< ⇒ < ⇒ > < < > .
 (iii) 
P(x)Q(x) 0P(x) 0 P(x) 0, Q(x) 0 or P(x) 0, Q(x) 0
Q(x) 0Q(x)
≥≥ ⇒ ⇒ ≥ > ≤ < ≠
.
 (iv) 
P(x)Q(x) 0P(x) 0 P(x) 0, Q(x) 0or P(x) 0, Q(x) 0
Q(x) 0Q(x)
≤≤ ⇒ ⇒ ≥ < ≤ > ≠
.
step 3: For solving the above inequalities formed, e.g., P(x). Q(x) > 0 use wavy curve method or solution 
set is given by {x: P(x) > 0, Q(x) > 0} ∪ {x: P(x) < 0, Q(x) < 0}
type ii: For solving inequality of the form P(x) R(x)
Q(x) S(x)
<
step 1: P(x) R(x) R(x) P(x)0or 0
Q(x) S(x) S(x) Q(x)
− < − >
⇒ P(x).S(x) R(x).Q(x) R(x).Q(x) P(x).S(x)0 or 0
Q(x).S(x) S(x).Q(x)
− −   < >   
   
. Now, solve as in Type 1
type iii: For solving inequality of the form 
P(x) R(x) T(x)
Q(x) S(x) M(x)
< < .
step 1: Solve the inequalities P(x) R(x) 0
Q(x) S(x)
− < and R(x) T(x) 0
S(x) M(x)
− < .
⇒ 
P(x).S(x) R(x)Q(x) 0
Q(x).S(x)
−  < 
 
 ……(i) and R(x).M(x) T(x).S(x) 0
S(x).M(x)
−  < 
 
 ……(ii)
 Intersection of solution set of equations (i) and (ii) gives the solution set of the given inequality.
Remarks:
 (i) If we have inequality of form >
P( x )
0
Q( x )
 and Q(x) > 0 ∀ x ∈ ℝ, then P(x) > 0.Q(x) ⇒ P(x) > 0
 (ii) If P( x ) 0
Q( x )
> and Q(x) < 0 ∀ x ∈ ℝ, then P(x).Q(x) < 0 is multiplying by +ve real number does not 
change the sign of inequality where as multiplying by –ve real number reverses the sign of inequality.
 (iii) For all positive a, b, x;
a x a if a b
b x b
a x a if a b
b x b
+ > < + + < >
 +
.
1.22 Mathematics at a Glance
1.21 POLyNOMIALS
An algebraic expression involving one or more variable that contains two mathematical operations, 
multiplication and raising to a natural exponent (power) with respect to the variable/variables involved is 
called, ‘mono-nomial’.
1.21.1 Leading Terms/Leading Coefficient 
The term containing highest power of variable x is called ‘leading term’ and its coefficient is called leading 
coefficient. Because it governs the value of f(x); where x → ∞) 
∵ n n 1 n 2 nn 2 n
a a af(x) x a ...
x x x
− − = + + + + 
 
1.21.2 Degree of Polynomials
Highest power of x in the polynomial expression, is called ‘degree of polynomial’ (i.e., power of x in 
leading term). Based on degree, polynomials can be classfied as:
 0. (Constant) ax0 1. (linear) ax + b
 2. (quadratic) ax2 + bx + c 3. (cubic) ax3 + ax2 + cx + d
 4. (bi-quadratic) ax4 + bx3 + cx2 + dx + e
1.21.2.1 Rational function and rational equation
An equation of the form f(x)/g(x) where f(x) and g(x) are polynomials in x is known as rational function 
of x and when equated to zero it generates a rational equation.
solving rational inequality: While solving rational inequality the following facts must be always 
bear in mind:
 
> ⇒ > <
 
f (x) and g(x) have f (x) and g(x) have
same sign opposite sign
f (x) f(x)0 f(x).g(x) 0 0
g(x) g(x)
 
⇒ 

f (x) and g(x) of
samesign or f (x) 0
f (x)f(x).g(x) 0 0
g(x)
=
< ≥
⇒ 
=
 >
 ≤
 = ≠ f (x) g(x) ofand
opposite sign or
f (x) 0
f (x).g(x) 0
f(x)or 0
g(x)
f(x) 0 andg(x) 0
 ⇒ 
 <


 = ≠
f(x).g(x) 0
or
f(x) 0 and g(x) 0
1.21.3 Wavy-curve Method
To find the set of solution for inequality f(x) > 0 (f(x) is polynomial):
Factorize the polynomial and find all the roots e.g., f(x) = (x – a)3 (x – b)2 (x – d) (x – g)5; say a > b > d > g.
Locate the roots (with their multiplicity) on the real number line. Keep the sign expression in the 
right-most interval same as that of the leading coefficient.
Foundation of Mathematics 1.23
Moving towards left change the sign of expression across the root with multiplicity odd and, retain 
the same sign across the root with multiplicity even.
∴ f(x) > 0 ⇒ (a, b) ∪ (b, g) ∪ (d, ∞). Also, 
f(x) ≥ 0 ⇒ (a, b) ∪ (b, g) ∪ (d, ∞) ∪ (a, b,g,d)
⇒ x ∈ [a, g] ∪ [d, ∞). Similarly f(x) < 0 
⇒ (–∞, a) ∪ (g, d) and f(x) ≤ 0 
⇒ (–∞, a) ∪ (g, d) ∪ {a, b, g, d} f(x) ≤ 0 (–∞, a] 
∪ [g, d] ∪ {b}
1.21.3.1 Concept of continued sums and products
continued sum (∑): Sigma (Σ) stands for sum of indexed terms. e.g., 
n
k
k 1
a
=
∑ = a1 + a2 + a3 +....+ an. 
In the above symbol, ak is called, ‘general term’ and k is known as index.
Properties:
 1. 
n
k 1
a
=
∑ = a + a + a +....+ a (n terms) = na
 2. Sigma distributes on addition and subtraction 
n
k k
k 1
(a b )
=
±∑ = (a1 ± b1) + (a2 ± b2) +....+ (an ± bn)
 3. Sigma does not distribute on product and ratio of terms, i.e., 
n
k k
k 1
(a b )
=
×∑ = (a1 × b1) + (a2 × b2) 
+....+ (an × bn) ≠ 
n n
k k
k 1 k 1
ab
= =
  
  
  
∑ ∑ and 
n
k k
k 1
(a /b )
=
∑ = (a1/b1) + (a2/b2) +....+ (an/bn) ≠
n
k
k 1
n
k
k 1
a
b
=
=
 
 
 
 
 
 
∑
∑
 4. A constant factor can be taken out of sigma notation. i.e.,
n n
k k
k 1 k 1
m.a m a
= =
=∑ ∑ = m (a1 + a2 + a3 
+.... + an)
 cyclic and symmetric expressions: 
 An expression is called symmetric in variable x and y iff interchanging x and y does not changes the 
expression x2 + y2, x2 + y2 – xy; x3 + y3 + x2y + y2x. x3 – y3 is not symmetric.
 An expression is called cyclic in x, y, z iff cyclic replacement of variables does not change the 
expression, e.g. x + y + z, xy + yz + zx, etc. Such expression can be abbreviated by cyclic sigma 
notation as follows: Σx2 = x2 + y2 + z2, Σxy = xy + yz + zx
 Σ(x – y) = 0 ⇒ x + y + z + x2 + y2 + z2 = Σx + Σx2
 5. If sigma is defined for three variables say a, b, c occurring cyclically, then it is evaluated as 
follows Σa = a + b + c = a + b + c, Σ a. b = ab + bc + ca, Σa2 = a2 + b2 + c2.
 continued Products (π): Continued product of indexed terms
n
k
k 1
a
=
∏ is defined as product 
of n number of indexed terms as 
n
k 1 2 3 n
k 1
a a .a .a .......a
=
=∏ .
1.24 Mathematics at a Glance
 Properties:
 1. 
=
=∏
n
k 1
a a.a.a...a (n times) = an
 2. 
= =
λ = λ λ λ = λ = λ∏ ∏
n n
n
n 1 2 n 1 2 n k
k 1 k 1
a ( a )( a )...( a ) (a .a ...a ) a
 3. π distributes over product and ratio of indexed terms but not over sum and difference of terms. 
i.e., 
= = =
= =∏ ∏ ∏
n n n
k k k k 1 2 n 1 2 n
k 1 k 1 k 1
a .b a b (a a ...a )(b b ...b )
 =
=
=
  
= =   
   
∏
∏
∏
n
kn
1 2 3 nk k 1
n
k 1 k 1 2 3 n
k
k 1
a
a .a .a ...aa
b b .b .b ...bb
n n n
k k k k
k 1 k 1 k 1
(a b ) a b
= = =
± ≠ ±∏ ∏ ∏
1.22 PARTIAL FRACTIONS
1.22.3.1 Linear and non-repeating
Let D(x) = (x - a1) (x - a2) (x - a3). Then, = + + + +− − − −
31 2 n
1 2 3 n
AA A AN(x) Q(x) ...
D(x) x a x a x a x a
1.22.3.2 Linear and repeated roots
Let D(x) = (x - a)K (x - a1) (x - a2)...(x - an)
Then, = + + + + + + + +
− − − − − −
1 2 k 1 2
2 k
1 2 n
A A A B BN(x) BnQ(x) ... ...
D(x) x a (x a) (x a) x a x a x a
1.22.3.3 Quadratic and non-repeated roots
Let D(x) = (x2 + ax + b) (x – a1) (x – a2)...(x – an), then
+
= + + + + +
+ + − − −
1 2 1 2 n
2 1 2 n
A x A B B BN(x) Q(x) ...
D(x) (x ax b) x a x a x a
1.22.3.4 Quadratic and repeated
Let D(x) = (x2 + a1 x + b1) (x
2 + a2x + b2)...(x
2 + anx + bn)
type V: When both N(x) and D(x) contain only the even powers of x. To solve these types of integrals, 
follow the steps given below:
step 1. Put x2 = t in both N(x) and D(x). step 2. Make partial fractions of N(t)/D(t).
step 3. Put back t = x2 and solve the simplified integral now.
1.23 THEOREMS RELATED TO TRIANGLES
Theorem 1: If two straight lines cut each other, the vertically opposite angles are equal.
Theorem 2: If two triangles have two sides of the one equal to two sides of the other, each to each, and 
the angles included by those sides are equal, then the triangles are equal in all respects.
Theorem 3: If two angles of a triangle are equal to one another, then the sides which are opposite to the 
equal angles are equal to one another.
Foundation of Mathematics 1.25
Theorem 4: If two triangles have the three sides of which one side is equal to three sides of another, then 
they are equal in all respects.
Theorem 5: If one side of a triangle is greater than other, then the angle opposite to the greater side is 
greater then the angle opposite to the smaller side.
Theorem 6: If one angle of a triangle is a greater than another then the side opposite to greater angle is 
greater than the side opposite to less.
Theorem 7: Any two sides of a triangle are together greater they third side.
Theorem 8: If all straight lines drawn from a given point to a given point on a given straight line, then 
the perpendicular is the least.
Theorem 9: If a straight line cuts two straight lines to make:
 (i) The alternate angles equal, or
 (ii) Exterior angles equal to the interior opposite angles on the same side of the cutting line, or
 (iii) The interior angles on the same is side equal to two right angles, then in each case, the two straight 
lines are parallel.
Theorem 10: If a straight line cuts two parallel lines, it makes:
 (i) The alternate angles equal to one another.
 (ii) The exterior angle equal to the interior opposite angle on the same side of the cutting line.
 (iii) The two interior angles on the same side together equal to two right angles.
Theorem 11: The three angles of a triangle are together equal to two right angles.
Theorem 12: If two triangles have two angles of one equal to two angles of the other, each to each, and any side 
of the first equal to the corresponding side of the other, the triangles are equal in all respects called, ‘conjugate’.
Theorem 13: Two right angled triangles which have their hypotenuses equal, and one side of one equal 
to one side of the other, are equal in all respects.
Theorem 14: If two triangles have two sides of the one equal to two sides of the other, each to each, but 
the angles included by the two sides of one greater than the angle included by the corresponding sides of 
the other, then the base of that, which has the greater angle is greater than the base of the other.
1.23.3.1 Theorems related to parallelograms
Theorem 15: The straight lines which join the extremities of two equal and parallel straight lines towards 
the same parts are themselves equal and parallel.
Theorem 16: The opposite sides and angles of a parallelogram are equal to one another, and each 
diagonal bisects the parallelogram.
Theorem 17: If there are three or more parallel straight lines, and the intersepts made by them on any 
transversal are equal, then the corresponding intercept on any other transversal are also equal.
Theorem 18: Parallelograms on the same base and between the same parallels are equal in terms of area.
1.23.3.2 Theorems related to intersection of loci
The concurrence of straight lines in a triangle.
 (i) The perpendiculars drawn to the sides of a triangle from their middle points are concurrent.
 (ii) The bisectors of the angles of a triangles are concurrent.
 (iii) The medians of a triangle are concurrent.
Theorem 19: Triangles on the same base and between the same parallel line are equal in area.
1.26 Mathematics at a Glance
Theorem 20: If two triangles are equal in area, and stand on the same base and on the same side of it, 
they are between the same parallel line.
Theorem 21: Pythagoras’s theorem. In any right-angled triangle, the area of the square on the hypotenuse 
equals to the sum of the area of the squares on the other two sides.
1.23.1 Theorems Related to the Circle, Definitions and First Principles
1.23.1.1 Chords
Theorem 22: If a straight line drawn from the centre of a circle bisects a chord, which does not pass 
through the centre, it cuts the chord at right angles. Conversely, if it cuts the chords at right angles, the 
straight line bisects it.
Theorem 23: One circle, and only one, can pass through any three points not in the same straight line.
Theorem 24: If from a point within a circle, more than two equal straight lines can be drawn to the 
circumference, that point is the centre of the circle.
Theorem 25: Equal chords of a circle are equidistant from the centre. Conversely, chords which are 
equidistant from the centre than the equal.
Theorem 26: Of any two chords of a circle, which is nearer to the centre is greater than one more remote. 
Conversely, the greater of two chords is nearer to the centre than the less.
Theorem 27: If from any external point, straight lines are drawn to the circumference of a circle, the great-
est is that which passes through the centre, and the least is that which, when produced, passes through the 
centre. And of any other two such lines, the greater is that which subtends the greater angle at the centre.
1.23.1.2 Angles in a circle
Theorem 28: The angle at the centreof a circle is double of an angle at the circumference standing on 
the same arc.
Theorem 29: Angles in the same segment of a circle are equal. Coverse of this theorem states, “equal 
angles standing on the same base, and on the same side of it, have their vertices on an arc of a circle, of 
which the given base is the chord.”
Theorem 30: The opposite angles of quadrilateral inscribed in a circle are together equal to two right 
angles coverse of this theorem is also true.
Theorem 31: The angle in a semi-circle is a right angle.
Theorem 32: In equal circles arcs, which subtend equal angles, either at the centres or at the 
circumferences, are equal.
Theorem 33: In equal circles, arcs which are cut-off by equal chords are equal, the major arc equal to the 
major arc, and the minor to the minor.
Theorem 34: In equal circles chords which cut-off equal arcs are equal.
1.23.2 Tangency
Theorem 35: The tangent at any point of a circle, is perpendicular to the radius 
drawn to the point of contact.
Theorem 36: Two tangent can be drawn to a circle from an external point.
Theorem 37: If two circles touch one another, the centres and the point of 
contact are in one straight line.
Foundation of Mathematics 1.27
Theorem 38: The angles made by a tangent to a circle with a chord drawn from the point of contact are 
respectively equal to the angles in the alternate segments of the circle. 
Theorem 39: If two of straight lines, one is divided into any number of parts, the rectangle contained 
between the two lines is equal to the sum of the rectangles contained by the undivided line and the several 
parts of the divided line.
Theorem 40: If a straight line is divided internally at any point, the square on the given line is equal to 
the sum of the squares on the squares on the two segments together with twice the rectangle contained by 
the segments.
Theorem 41: If a straight line is divided externally at any point, the square on the given line is equal to 
the sum of the squares on the two segments diminished by twice the rectangle contained by the segments.
Theorem 42: The difference of the squares on the two straight lines is equal to the rectangle contained 
by their sum and difference.
Theorem 43: In an obtuse-angled triangle, the square on the side subtending the obtuse angle is equal to 
the sum of the squares on the sides containing the obtuse angle together with twice the rectangle contained 
by one of those sides and the projection of the other side upon it.
Theorem 44: In every triangle the square on the side subtending an acute angle is equal to the sum of 
the squares on the sides containing that angle diminished by twice the rectangle contained by one of these 
sides and the projection of the other side upon it.
Theorem 45: steward’s theorem: If D is any point on the side BC of a, then AB2.DC + AC2. BD = AC. 
(AD2 + BD. DC)
Theorem 46: In any triangle, the sum of the squares on two sides is equal to twice the square 
on half the third side together with twice the square on the median which bisects the third side. 
(Appolonius theorem, which is a special case of Steward’s theorem).
1.23.3 Rectangles in Connection with Circles
Theorem 47: If two chords of a circle cut a point within it, the rectangle contained by their segments are equal.
Theorem 48: If two chords of a circle, when produced, cut at a point outside it, the rectangles contained by 
their segments are equal. And each rectangle is equal to the square on the tangent from the point of intersection.
Theorem 49: If from a point outside a circle two straight lines are drawn, one of which cuts the circle, 
and the other meets it and if the rectangle contained by the whole line which cuts the circle and the part 
of it outside the circle is equal to the square on the line which meets the circle, then the line which meets 
the circle is a tangent to it.
1.23.4 Proportional Division of Straight Lines
Theorem 50: A straight-line drawn parallel to one side of a triangle cuts the other two sides, or those 
sides produced proportionally.
Theorem 51: If the vertical angle of a triangle is bisected internally into segments which have the same 
ratio as the other sides of the triangle. Conversely, if the base is divided internally or externally into segments 
proportional to the other sides of the triangle, the line joining the point of section to the vertex bisects the 
vertical angle internally or externally. AD and AD’ are internal and external angle bisectors of the triangle.
1.23.5 Equiangular Triangles
Theorem 52: I f two triangles are equiangular to each other, their corresponding sides are proportional.
Theorem 53: If two triangles have their sides proportional when taken in order, the triangles are 
equiangular to one another, and those angles are equal which are opposite to the corresponding sides.
1.28 Mathematics at a Glance
Theorem 54: If two triangles have one angle of which one is equal to one angle of the other, and the sides 
about the equal angles are proportionals, then the triangles are similar.
Theorem 55: If two triangles have one angles of which one is equal to one angle of the other, and the sides 
about another angle in one proportional to the corresponding sides of the other, then the third angles are 
either equal or supplementary; and in the former case the triangles are similar.
Theorem 56: In a right-angled triangle, if a perpendicular is drawn from the right angle to the hypotenuse, 
the triangles on each side of it are similar to the whole triangles and to each other.
1.23.5.1 Similar Figures
Theorem 57: Similar polygons can be divided into the same number of similar triangles; and the lines 
joining the corresponding vertices, in each figure, are proportional.
Theorem 58: Any two similar rectilinear figures may be placed in a way that the lines joining corre-
sponding the vertices are concurrent.
Theorem 59: In equal circles, angles, whether at the centres or circumferences, have the same ratio as the 
arcs on which they stand.
1.23.5.2 Proportion applied to area
Theorem 60: The areas of similar triangles are proportional to the squares on there corresponding sides.
Theorem 61: The area of similar polygons are proportional to the squares on there corresponding sides.
1.23.6 Some Important Formulae
 1. (a + b)2 = z2 + 2ab + b2 = (a – b)2 + 4ab 2. (a + b)2 = a2 – 2ab + b2 = (a – b)2 + 4ab 
 3. a2 – b2 = (a + b) (a – b) 4. (a + b)3 = a3 + b3 + 3ab (a + b) 
 5. (a – b)3 = a3 + b3 – 3ab(a – b) 6. a3 + b3 = (a + b)3 – 3ab(a + b) = (a + b) (a2 + b2 – ab) 
 7. a3 – b3 = (a – b)3 + 3ab (a – b) = (a – b) (a2 + b2 + ab) 
 8. 
2 2 2 2 2 2 2 1 1 1(a b c) a b c 1ab 2bc 2ca a b c 2abc
a b c
 + + = + + + + + = + + + + + 
 
 9. 3 3 3 2 2 21a b c ab bc ca (a b) (b c) (c a)
2
 + + − − − = − + − + − 
 10. ( )( )3 3 2 2 2 2a b c 3abc a b c a b c ab bc ca+ + − = + + + + − − − = ( ) ( )2 2 21 a b c (a b) (b c) (c a)2  + + − + − + − 
 11. a4 – b4 = (a + b) (a – b) (a2 – b2) 
 
12. a4 + a2 + 1 = (a2 + 1)2 – a2 = (1 + a + a2) (1 – a + a2) 
 13. 
2 2a b a b
ab
2 2
+ −   
= −   
   
 14. a b (a b)(a b)− = − +
 15. a2 + b2 + c2 – ab – bc – ca = {(a – b2) + (b – c)2 + (c – a)2}
 16. (x + a) (x + b) = x2 + (a + b)x + ab 17. (a + b + c)3 = a3 +| b3 |+c3 + 3 (a + b) (b + c) (c + a)
 18. a3 + b3 + c3 –3abc = (a + b + c) (a2 + b2 + c2 – ab – bc – ca)
 19. (a + b)4 = (a +| b)2 |(a +| b)2 = a4 + b4 + 4a3b + 6a2b2 + 4ab3
 20. (a – b)4 = (a –| b)2 |(a –| b)2 = a4 + b4 – 4a3b + 6a2b2 – 4ab3
 21. (a + b)5 = (a +| b)3 |(a +| b)2 = a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5
Chapter 2
eXpONeNtIaL 
LOGarIthM
2.1 ExponEntial Function
If a is a positive real number then ax; (a ≠ 1) is always positive and it is called ‘exponential function of x’. 
Here a is called ‘base’ and x is called index.
2.1.1 Properties of Exponential Functions
 (i) As we know that = × × × ∀ ∈

n
n times
a a a .... a n where a is called ‘base’ and n is index or exponent. 
Exponential function f(x) = ax is generalisationof this law to facilitate some useful applications with 
some imposed functional restrictions i.e., a > 0 and a ≠ 1.
 (ii) Domain of f(x) is set of real number and range of f(x) is (0, ∞), i.e., ∀ x ∈ℝ, f(x) = ax associates 
x to some positive real number, uniquely, i.e., exponential function f(x) is defined, such that it is 
invertible.
 (iii) For a < 0 and a = 0 the function f(x) = ax loses its meaning for some values of x ∈ ℝ. For instance 
for a = –1, –1/2, –3 etc.
f(x) = ax becomes non-real ∀ = px
q
 where p and q are co-prime and q is an even integer, 
e.g., (–3)3/2, (–1)1/4 etc.
Similarly, when base a = 0, then f(x) = 0x does not remain an one-to-one function which is required 
for invariability, same restriction also holds for a = 1. Since then f(x) = 1x, again becomes many one 
function as all inputs x get associated to single output 1.
Therefore, we conclude that for f(x) = ax, the base a > 0, a ≠ 1 and x ∈ ℝ, thus y ∈ (0, ∞)
 (iv) If the base a is Euler number ‘e’, then the exponential function ex is known as natural exponential 
function.
2.1.2 Laws of Indices
 (i) ax is defined and ax > 0 ∀ x ∈ ℝ.
 (ii) a0 = 1. We can observe that →n a 1 as n assumes very large value (n → ∞), and it is true for both 
cases, i.e., a > 1 or a ∈ (0, 1), therefore when n → ∞. = = =1/n 0n a a a 1 .
2.30 Mathematics at a Glance
 (iii) ax×ay = ax+y
 (iv) −=
x
x y
y
a a
a
 (v) (ax)y = axy = (ay)x
 (vi) = qp/q pa a ; where ∈q and q ≠ 1.
 (vii) ax = ay ⇒ x = y or a = 1.
 (viii) ax = bx ⇒ either x = 0 or a = b
 (ix) ax.bx = (ab)x and  =  
 
xx
x
a a
b b
.
 (x) ax ≥ ay ⇒ 
≥ >
 ≤ ∈
x y if a 1
x y if a (0,1)
.
2.1.3 Graphical Representation of an Exponential Function
 1. ax; where a > 1 behaves as an increasing nature function.
 For example, when a = 2, the value of function 2x increases as the 
input x increases. It can be understood from the table given below.
x –3 –2 –1 0 1 2 3 4 2
ax 1/8 1/4 1/2 1 2 4 8 16 32
 2. If 0 < a < 1 behaves like a decreasing nature function.
 For example, when a = 1/2, the value of function 2–x decreases as the 
input x increases which can be observed in the following table.
x –5 –4 –3 –2 –1 0 1 2 3
ax 32 16 8 4 2 1 1/2 1/4 1/8
 3. If the base a > 1, then ax ≥ 1 for all x ≥ 0 and ax < 1 when x < 0, if 0 < a < 1, then 0 < ax < 1 for x > 0 
and ax > 1 for x < 0. The above fact as well as the relative position of graphs of exponenital 
functions with different bases can be understood with the help of following figure.
 If the base a > 1, then ax ≥ 1 for all x ≥ 0.
Exponential Logarithm 2.31
2.1.4 Composite Exponential Functions
A composite function is a function in which both the base and the exponent are the functions of x. 
Generally, any function of this form is a composite exponential function. This function is also called an 
exponential power function or a power exponential function. i.e., y = [u(x)]v(x) = uv. In calculus, the domain 
consists of such values of x for which u(x) and v(x) are defined and u(x) > 0.
2.1.5 Methods of Solving Exponential Equation
To solve an exponential equation, we make use of the following facts.
 (i) If the equation is of the form ax = ay(a > 0) ⇒ x = y or a = 1.
 (ii) If the equation is of the form ax = bx (a, b > 0) ⇒ either x = 0 or a = b.
 (iii) If the equation is of the form ax = k (a > 0), then
Case I: If b ≥ 0 ⇒ x ∈ { }
Case II: If b > 0, k ≠ 1 ⇒ x = logak 
Case III: If a = 1, k ≠ 1 ⇒ x ∈ { }
Case IV: If a = 1, k = 1 ⇒ x ∈ ℝ
(Since 1x = 1 ⇒ 1 = 1, x ∈ ℝ)
 (iii) If the equation is of the form af(x) = ag(x) where a > 0 and a ≠ 1, then the equation will be equivalent 
to the equation f(x) = g(x).
Remarks
ax = 1 ⇒ x = 0 is an incomplete conclusion; it is only true if the base a ≠ 0; ± 1.
if a = 0 so equality does not holds as 00 is meaningless.
Where as when a = 1, then an = 1 ⇒ 1x = 1. Thus, x ∈ ℝ.
In case a = –1, then (–1)x = 1 is true for x = p/q when p is even and GCD of p and q = 1.
2.2 Solving ExponEntial inEquality
 (i) The value of ax increases as the value of x increases when base a ∈ (1, ∞), but the value of ax de-
creases as the value of x increases when base a ∈ (0, 1), 
≥ >≥ ⇒  ≤ ∈
x y x y if a 1a a
x y if a (0,1) .
 (ii) The elementary exponential inequalities are inequalities of form ax > k, ax < k, where a and k are 
certain numbers (a > 0, a ≠ 1). Depending on the values of the parameters a and k, the set of 
solutions of the inequality ax > k can be in the following forms:
1. x ∈ (logak, ∞) for a > 1, k > 0
2. x ∈ (–∞, logak) for 0 < a < 1, k > 0
3. x ∈ ℝ for a > 0, k < 0
Depending on the values of a and k, the set of solutions of the inequality ax < k can be in the fol-
lowing forms:
1. x ∈ (–∞, logak) for a > 1, k > 0;
2. x ∈ (logak, ∞ ) for 0 < a < 1, k > 0;
3. x ∈ { } for a > 0, k < 0; (i.e., the inequality has no solutions)
 (iii) 
+ =  ∀ ∈− = 

f(x y) f(x) . f(y)
x, y
f(x y) f(x) / f(y)
2.32 Mathematics at a Glance
2.3 logaRitHMic Function
The logarithm of any number N to the given base a is the exponent, or index, or the power to which the 
base must be raised to obtain the number N. Thus, if ax = N, x is called the logarithm of N to the base a. 
It is denoted as logaN.
\ loga N = x; ⇔ a
x = N, a > 0, a ≠ 1 and N > 0.
Notes:
 (a) The logarithm of a number is unique, i.e., no number can have two different logarithms to a given base.
 (b) The base ‘a’ is a positive real number, but excluding 1, i.e., a > 0, a ≠ 1. As a consequence of the 
definition of exponential function, we exclude a = 1.
 Since for a = 1, logax = y ⇒ x = a
y = 1y which has no relevance to the cases of logax when x ≠ 1, 
i.e., for all values of exponent, the value of x remains 1.
 (c) The number ‘x’ represents result of exponentiation, i.e., ay therefore, it is also a positive real number, 
i.e., x = ay > 0.
 (d) The exponent ‘y’, i.e., logarithm of ‘x’ is a real number and neither a nor x equals to zero.
 (d) Domain of function y = logax is (0, ∞) and the range (-∞, ∞).
 when x → 0, then logax →-∞ (for a > 1) and logax → ∞ (for 0 < a < 1) because y = logax ⇒ x = a
y 
which approaches to zero iff y →-∞ as a-∞ = 0 ∀ a > 1 and when a ∈ (0, 1), x = ay approaches to zero 
iff y → ∞ ∵ a∞ = 0 if 0 < a < 1.
 (e) Common Logarithms and Natural Logarithms: The base of logarithm can be any positive 
number other than 1, but basically two bases are mostly used. They are 10 and e (=2.718 
approximately). Logarithm of numbers to the base 10 are named as Common Logarithms; 
whereas the logarithms of the numbers to the base e are called as Natural or Napierian 
logarithms.
 If a = 10; then we write log b instead of log10b.
 If a = e; then we write ℓnb instead of logeb.
 \ We find logea = log10a. loge10 or 
e
10
e
log a
log a 0.434
log 10
= = logea (this transformation is used to convert 
natural logarithm to common logarithm).
2.3.1 Properties of Logarithm
 P 1. loga 1 = 0 because 0 is the power to which a must be raised to obtain 1.
 P 2. logaa = 1 since 1 is the power to which a must be raised to obtain a.
 P 3. alogaN = N and logaa
N = N as N is the power to which a must be raised to obtain aN.
 P 4. logm(a.b) = logm a
 + logmb (a > 0, b > 0). Logarithm of the product of two numbers to a certain base 
is equal to the sum of the logarithms of the numbers to the same base.
Exponential Logarithm 2.33
 P 5. logm (a/b) = logma-logmb: logarithm of the quotient of two numbers is equal to the difference of their 
logarithms, base remaining the same throughout.
 P 6. loga N
k = k logaN (k is any real number). Logarithm of the power of a number is equal to the product 
of the power and logarithm of the number (base remaining the same).
 P 7. logak N = (1/k).loga N.
Note:
 (1) The property 4, 5, 6, 7 are not applicable conditionally because logaM + logaN is defined only when 
M and N are both positive, whereas logaMN is defined even if M and N are both negative. Therefore 
logaMN cannot be always replaced by logaM + logaN. Therefore, such