Mathematics At A Glance for Class XI & XII, Engineering Entrance and other Competitive Exams

Cover
Contents
Preface
Acknowledgements
Chapter 1 : Foundation of Mathematics
	Mathematical Reasoning
		1.1 Introduction
		1.2 Pre-Requisites
			1.2.1 Greek Words (Symbols)
		1.3 Understanding the Language of Mathematics
			1.3.1 Mathematical Symbols
		1.4 Statements and Mathematical Statemens
			1.4.1 Statement
			1.4.2 Mathematical Statements
			1.4.3 Scientific Statement
		1.5 Classification of Mathematical Statements
			1.5.1 Conjectures
			1.5.2 Mathematical Reasoning
		1.6 Working on Mathematical Statements
			1.6.1 Negation of a Statement
			1.6.2 Compounding of Statements
		1.7 Implication of a Statement
			1.7.1 Converse of a Statement
			1.7.2 Contra Positive of a Statement p ⇒ q is ~q ⇒ ~p
		1.8 Truth Value
		1.9 Quantifiers
			1.9.1 Proofs in Mathematics
			1.9.2 What is a Mathematical Assumption?
	Number System
		1.10 Set of Natural Numbers
			1.10.1 Algebraic Properties of Natural Numbers
		1.11 Set of Integers
		1.12 Geometrical Representation of Integers
			1.12.1 Properties of Integers
		1.13 Division Algorithm
			1.13.1 Even and Odd Integers
			1.13.2 Prime Integer
		1.14 Factorial Notation
			1.14.1 Related Theorems
			1.14.2 Divisors and Their Property
			1.14.3 Number of Divisors
			1.14.4 Least Common Multiple (LCM)
			1.14.5 Greatest Common Divisor (GCD)/Highest Common Factor (HCF)
			1.14.6 Decimal Representation of Number
			1.14.7 Periodic Properties of Integers
		1.15 Tests of Divisibility
		1.16 Rrational (ℚ) and Irrational Numbers (ℚ′)
			1.16.1 Properties of Rational and Irrational Numbers
			1.16.2 nth Root of a Number
			1.16.3 Principal nth Root
			1.16.4 Properties of nth Root
			1.16.5 Algebraic Structure of ℚ and ℚ
		1.17 Surds and Their Conjugates
		1.18 Real Numbers System
			1.18.1 Concept of Interval
			1.18.2 Intersection and Union of Two or More Intervals
		1.19 Mathematical Induction
			1.19.1 Ratio and Proportion
			1.19.2 Some Important Applications of Proportion
			1.19.3 Linear Equalities
			1.19.4 Method of Comparison
			1.19.5 Method of Substitution
			1.19.6 Method of Elimination
	Fundamentals of Inequality
		1.20 Introduction
			1.20.1 Classification of Inequality
		1.21 Polynomials
			1.21.1 Leading Terms/Leading Coefficient
			1.21.2 Degree of Polynomials
			1.21.3 Wavy-curve Method
		1.22 Partial Fractions
		1.23 Theorems Related to Triangles
			1.23.1 Theorems Related to the Circle, Definitions and First Principles
			1.23.2 Tangency
			1.23.3 Rectangles in Connection with Circles
			1.23.4 Proportional Division of Straight Lines
			1.23.5 Equiangular Triangles
			1.23.6 Some Important Formulae
Chapter 2 : Exponential
Logarithm
	2.1 Exponential Function
		2.1.1 Properties of Exponential Functions
		2.1.2 Laws of Indices
		2.1.3 Graphical Representation of an Exponential Function
		2.1.4 Composite Exponential Functions
		2.1.5 Methods of Solving Exponential Equation
	2.2 Solving Exponential Inequality
	2.3 Logarithmic Function
		2.3.1 Properties of Logarithm
	2.4 Logarithmic Equations
		2.4.1 Some Standard Forms to Solve Logarithmic Equations
	2.5 Logarithmic Inequalities
		2.5.1 Characteristic and Mantissa
		2.5.2 Characteristic and Mantissa
Chapter 3 : Sequence and Progression
	3.1 Definition
		3.1.1 Types of Sequence
		3.1.2 Progression and Series
	3.2 Series
		3.2.1 Properties of Arithmetic Progression
	3.3 Arithmetic Mean
		3.3.1 Arithmetic Means of Numbers
		3.3.2 Insertion of n AM’s between Two Numbers
	3.4 Geometric Progression
		3.4.1 Properties of Geometric Progression
	3.5 Geometric Mean
		3.5.1 Geometric Means of Numbers
		3.5.2 Geometric Mean between Two Numbers
	3.6 Harmonic Progression
		3.6.1 Properties of Harmonic Progression
	3.7 Harmonic Mean
	3.8 Inequality of Means
	3.9 Arithmetic-Geometric Progression
		3.9.1 Standard Form
		3.9.2 Sum to Infinity Terms
	3.10 (Σ) Sigma Notation
		3.10.1 Concept of Continued Sum [Sigma (Σ) Notation]
	3.11 Properties
	3.12 Double Sigma Notation
		3.12.1 Representation
	3.13 Methods of Difference
	3.14 Vn Method
Chapter 4 : Inequality
	4.1 Inequality Containing Modulus Function
	4.2 Irrational Inequalities
		4.2.1 Exponential Inequalities
		4.2.2 Canonical Forms of Logarithmic Inequality
		4.2.3 Some Standard Forms to Solve Logarithmic Inequality
		4.2.4 Inequalities of Mean of Two Positive Real Numbers
	4.3 Theorem of Weighted Mean
		4.3.1 Theorem
		4.3.2 Weighted Power Mean Inequality
		4.3.3 Cauchy-Schwarz Inequality
		4.3.4 Tchebysheff’s Inequality
	4.4 Weierstrass Inequality
		4.4.1 Application to Problems of Maxima and Minima
	4.5 Use of Calculus In Proving Inequalities
		4.5.1 Monotonicity
		4.5.2 Test of Monotonicity
Chapter 5 : Theory of Equation
	5.1 Polynomial Expression
		5.1.1 Leading Terms/Leading Coefficient
	5.2 Classification of Polynomials
		5.2.1 Polynomial Equation
		5.2.2 Polynomials Identity
	5.3 Equation, Standard Equation and Quadratic
		5.3.1 Quadratic Equation
	5.4 Nature of Roots
		5.4.1 Formation of Quadratic Equation
		5.4.2 Sum and Product of the Roots
	5.5 Condition for Common Roots
	5.6 Symmetric Function of the Roots
		5.6.1 Maximum/Minimum Value and Sign of Quadratic Equation
	5.7 Location of Roots
	5.8 Descartes Rule
		5.8.1 Some Important Forms of Quadratic Equations
		5.8.2 Position of Roots of a Polynomial Eqution
	5.9 Equation of Higher Degree
Chapter 6 : Permutation and Combination
	6.1 introduction
	6.2 Fundamental Principles of Counting
		6.2.1 Addition Rule
		6.2.2 Multiplication Rule
		6.2.3 Complementation Rule
		6.2.4 Principles of Inclusion-Exclusion
		6.2.5 Injection and Bijection Principles
	6.3 Combinations and PermutationS
	6.4 Permutation of Different Objects
	6.5 Permutation of Identical Objects (Taking all of them at a Time)
	6.6 Rank of Words
	6.7 Circular Permutation
		6.7.1 Circular Permutation of n Objects
	6.8 Number of Numbers and their Sum
		6.8.1 Divisor of Composite Number
		6.8.2 Sum of Divisor
		6.8.3 Number/Sum of Divisors Divisible by a Given Number
		6.8.4 Factorizing a Number into Two Integer Factors
	6.9 Combination
		6.9.1 Properties of Combinations
		6.9.2 Restricted Combinations
		6.9.3 Combination of Objects Taking any Number of Them at a Time
		6.9.4 Combination When Some Objects are Identical(Taking any Number of Them at a Time)
		6.9.5 Combination When Some Objects are Identical(Taking specific number of them at a time)
	6.10 Distribution
		6.10.1 Distribution Among Unequal Groups
		6.10.2 To Find the Number of Ways in Which the m + n + p Things Can be Divided into Three Groups Containing m, n, p Things Separately
		6.10.3 Distribution Among Equal Groups
		6.10.4 When Name of Groups Specified
	6.11 Multinomial Theorem
		6.11.1 Number of Distinct Terms
	6.12 Dearrangements and Distribution in Parcels
	6.13 Distribution in Parcels
		6.13.1 Distribution in Parcels When Empty Parcels are Allowed
		6.13.2 When at Least One Parcel is Empty
	6.14 Exponent of a Prime in N!
		6.14.1 Exponent of Prime ‘P’ in n!
Chapter 7 : Binomial Theorem
	7.1 Introduction
	7.2 Binomial
		7.2.1 Binomial Expansion (Natural Index)
	7.3 General Term
		7.3.1 rth Term from Beginning
		7.3.2 kth Term from End
	7.4 Middle Term
	7.5 Number of Terms in Expansions
	7.6 Greatest Term
	7.7 Greatest Coefficient
	7.8 Properties of Binomial Coefficient
		7.8.1 Properties of nCr
	7.9 Properties of Coefficients
	7.10 Multinomial Theorem
	7.11 Tips and Tricks
Chapter 8 : Infinite Series
	8.1 Binomial theorem for any index (N)
	8.2 Greatest Term
	8.3 Taylor Expansion
		8.3.1 Maclaurins Expansions
		8.3.2 Euler’s Number
		8.3.3 Properties of e
		8.3.4 Expansion of ex
		8.3.5 Important Deduction from Exponential Series
	8.4 Logarithmic Series
		8.4.1 Important Deduction from Logarithmic Series
Chapter 9 : Trigonometric Ratios and Identities
	9.1 Introduction
	9.2 Angle
		9.2.1 Rules for Signs of Angles
		9.2.2 Measurement of Angle
	9.3 Polygon and its Properties
	9.4 Trigonometric Ratios
		9.4.1 Signs of Trigonometric Ratios
		9.4.2 Range of Trigonometric Ratios
		9.4.3 Trigonometric Ratios of Allied Angles
	9.5 Graphs of Different Trigonometric Ratios
		9.5.1 y = sin x
		9.5.2 y = cos x
		9.5.3 y = cot x
		9.5.4 y = cosec x
		9.5.5 y = sec x
		9.5.6 Trigonometric Identities
		9.5.7 Trigonometric Ratios of Compound Angles
		9.5.8 Trigonometric Ratios of Multiples of Angles
		9.5.9 Transformation Formulae
		9.5.10 Conditional Identities
	9.6 Some Other Useful Results
	9.7 Some Other Important Values
	9.8 Maximum and Minimum Values of a Cos θ + B Sin θ
	9.9 Tips and Trics
Chapter 10 : Trigonometric Equation
	10.1 Introduction
	10.2 Solution of Trigonometric Equation
	10.3 Particular Solution
	10.4 Principal Solution
	10.5 General Solution
	10.6 Summary of the above Results
	10.7 Type of Trigonometric Equations
	10.8 Homogeneous Equation in Sinx and Cosx
	10.9 Solving Simultaneous Equations
		10.9.1 More Than One Variable Problems
	10.10 Transcedental Equations
	10.11 Graphical Solutions of Equations
	10.12 Solving Inequalities
		10.12.1 Review of Some Important Trigonometric Values
Chapter 11 : Properties of Triangles
	11.1 Introduction
	11.2 Napier’s Analogy
		11.2.1 Solution of Triangle
	11.3 Geometric Discussion
	11.4 Area of Triangle ABC
	11.5 'M–N' Theorem
		11.5.1 Some Definitions
	11.6 Orthocentre and Pedal Triangle
		11.6.1 Sides and Angles of the Pedal Triangle
	11.7 In-Centre of Pedal Triangle
	11.8 Circumcircle of Pedal Triangle (Nine-Point Circle)
		11.8.1 Properties of Nine-point Circle
	11.9 The Ex-Central Triangle
	11.10 Centroid and Medians of Any Triangle
	11.11 Length of Medians
	11.12 Result Related To Cyclic Quadrilatral
Chapter 12 : Inverse Trigonometric Function
	12.1 Inverse Function
		12.1.1 Inverse Trigonometric Functions
	12.2 Domain and Range of Inverse Functions
	12.3 Graphs of Inverse Circular Functions and their Domain and Range
	12.4 Compositions of Trigonometric Functions and their Inverse Functions
		12.4.1 Trigonometric Functions of their Corresponding Circular Functions
	12.5 Inverse Circular Functions of their Corresponding Trigonometric Functions on Principal Domain
	12.6 Inverse Circular Functions of their Corresponding Trigonometric Functions on Domain
	12.7 Inverse Trigonometric Functions of Negative Inputs
	12.8 Inverse Trigonometric Functions of Reciprocal Inputs
	12.9 Inter Conversion of Inverse Trigonometric Functions
	12.10 Three Important Identities of Inverse Trigonometric Functions
	12.11 Multiples of Inverse Trigonometric Functions
	12.12 Sum and Difference of Inverse Trigonometric Functions
Chapter 13 : Point and Cartesian System
	13.1 Introduction
	13.2 Frame of Refrence
		13.2.1 Rectangular Co-ordinate System
		13.2.2 Polar Co-ordinate System
	13.3 Distance Formula
		13.3.1 Applications of Distance Formula
	13.4 Section Formula Internal Division
	13.5 Slope of Line Segment
		13.5.1 Area of Triangle
		13.5.2 Area of General Quadrilateral
		13.5.3 Area of Polygon
	13.6 Locus of Point and Equation of Locus
		13.6.1 Union of Loci
		13.6.2 Intersection of Loci
		13.6.3 Locus Passing Through Intersection of Two Locus
	13.7 Choice of Origin and Selection of Coordinate Axes
	13.8 Geometrical Transformations
		13.8.1 Transformations in Cartesian Plane
		13.8.2 Transformation of Coordinates Axis
		13.8.3 Rotation of the Axes (Without Changing Origin)
	13.9 Geometrical Tips and Tricks
		13.9.1 The Coordinates of Centroid
		13.9.2 Coordinates of Incentre
		13.9.3 Coordinates of Ex-centre
Chapter 14 : Straight Line and Pair of Straight Line
	14.1 Definition
		14.1.1 Equation of Straight Line
		14.1.2 Different Forms of the Equation of Straight Line
		14.1.3 Angle Between Two Lines
		14.1.4 Equation of a Line Perpendicular and Parallel to Given Line
		14.1.5 Straight Line Through (x1, y1) Making an Angle α with y = mx + c
		14.1.6 Position of Two Points w.r.t. a Straight Line
		14.1.7 Distance of a Point From a Line
		14.1.8 Distance Between Two Parallel Straight Lines
		14.1.9 Intersection of Two Lines
		14.1.10 Equation of the Bisectors of the Angles Between Lines
		14.1.11 Family of Straight Lines
	14.2 General Equation of Second Degreeand Pair of Straight Lines
		14.2.1 Pair of Straight Lines Through the Origin
		14.2.2 Angle Between the Pair of Straight Lines
Chapter 15 : Circle and Family of Circle
	15.1 Introduction
	15.2 Definiton of Circle
		15.2.1 Equation of a Circle in Various Forms
		15.2.2 General Equation
		15.2.3 Diametric Form
		15.2.4 Equation of Circle Thorugh Three Points
		15.2.5 The Carametric Coordinates of any Point on the Circle
		15.2.6 Position of a Point with Respect to a Circle
		15.2.7 Position of a Line with Respect to a Circle
	15.3 Equation of Tangent and Normal
		15.3.1 Tangents
		15.3.2 Parametric Form
		15.3.3 Pair of Tangents
		15.3.4 Normals
	15.4 Chord of Contact
		15.4.1 Relative Position of Two Circles
		15.4.2 Direct Common Tangent
	15.5 Intercept Made on Coordinate Axes by the Circle
	15.6 Family of Circles
	15.7 Radical Axes and Radical Centre
Chapter 16 : Parabola
	16.1 Introduction to Conic Sections
		16.1.1 Definition of Various Terms Related to Conics
	16.2 Parabola
		16.2.1 Standard Equation
		16.2.2 Position of Point w.r.t. Parabola
		16.2.3 Position of Line w.r.t. Parabola
	16.3 Chords of Parabola and Its Properties
		16.3.1 Chord of Parabola in Parametric Form
		16.3.2 Properties of Focal Chord
	16.4 Tangent of Parabola and Its ProPerties
		16.4.1 Properties of Tangents of a Parabola
	16.5 Normals and their Properties
		16.5.1 Properties
		16.5.2 Normals in Terms of Slope
Chapter 17 : Ellipse
	17.1 Definition
	17.2 Standard Equation of Ellipse
		17.2.1 Focal Distance
	17.3 Tracing of Ellipse
	17.4 Properties Related to Ellipse and Auxiliary Circle
		17.4.1 Position of a Point with Respect to Ellipse S : X2/a2 + y2/b2 −1 =0
		17.4.2 Position of a Line with Respect to Ellipse
	17.5 Properties of Tangents and Normals
Chapter 18 : Hyperbola
	18.1 Definition
		18.1.1 Standard Equation
		18.1.2 Tracing of Hyperbola
		18.1.3 Auxiliary Circle of Hyperbola
	18.2 Director Circle
		18.2.1 Position of a Point with Respect to Hyperbola
		18.2.2 Position of a Line with Respect to Hyperbola S: x2/a2−y2/b2-1=0
		18.2.3 Properties of Tangents and Normals
		18.2.4 Asymptote Hyperbola
	18.3 Rectangular Hyperbola
		18.3.1 Rectangular Hyperbola where Asymptote are Coordinate Axis
		18.3.2 Parametric Equations of Chord, Tangents and Normal
		18.3.3 Co-normal Points
		18.3.4 Properties of Conjugate Diameters
Chapter 19 : Complex Number
	19.1 Introduction
		19.1.1 Imaginary Numbers (Non-real Numbers)
		19.1.2 Purely Imaginary Numbers (I)
		19.1.3 Properties of Iota
	19.2 Complex Number
	19.3 Argand Plane
		19.3.1 Representation of Complex Numbers
		19.3.2 Properties of Complex Numbers
		19.3.3 Result
	19.4 Algebraic Structure of Set of Complex Numbers
		19.4.1 Conjugate of a Complex Number
		19.4.2 Properties of Conjugate of a Complex Number
		19.4.3 Modulus of a Complex Number
	19.5 De Moiver’s Theorem
		19.5.1 nth Root of Unity
		19.5.2 Properties of nth Root of Unity
	19.6 Geometry of Complex Number
		19.6.1 Line Segment in Argand’s Plane
		19.6.2 Application of the Rotation Theorem
		19.6.3 Loci in Argand Plane
	19.7 Theorem
	19.8 Complex Slope of the Line
		19.8.1 Circle in Argand Plane
	19.9 Appoloneous Circle
	19.10 Equation of Circular Arc
		19.10.1 Equation of Tangent to a Given Circle
		19.10.2 Explanation
		19.10.3 Equation of Parabola
		19.10.4 Equation of Ellipse
	19.11 Equation of Hyperbola
	19.12 Some Impotant Facts
		19.12.1 Dot and Cross Product
		19.12.2 Inverse Points w.r.t. a Circle
		19.12.3 Ptolemys Theorem’s
Chapter 20 : Sets and Relations
	20.1 Sets
	20.2 Representation of Sets
	20.3 Notation of Sets
	20.4 Notation for Some Special Sets
	20.5 Notation For Some Special Sets
	20.6 Method Representation of Sets
	20.7 Cardinal Number of a Sets
	20.8 Types of Sets
	20.9 Subsets
	20.10 Number of Subsets
	20.11 Types of Subsets
	20.12 Power Sets
	20.13 Disjoint Sets
	20.14 Universal Sets
	20.15 Complement Set of a Given Set
	20.16 Complementry Set of a Given Sets
	20.17 Comparable Sets
	20.18 Venn (Euler) Diagrams
	20.19 Operations on Sets
		20.19.1 Union of Two Sets
		20.19.2 Intersection of Two Sets
		20.19.3 Difference of Two Sets
		20.19.4 Symmetric Difference of Two Sets
		20.19.5 Complement of a Set
	20.20 LAWS Followed by Set Operations ∪ ∩ and Δ
	20.21 De-Morgan’s Principle
	20.22 Inclusive-Exclusive Principle
	20.23 Some Results on Cardinal Numbers
		20.23.1 Cartesian Product of Two Sets
		20.23.2 Number of Elements in Cartesian Product A × B
		20.23.3 Properties and Laws of Cartesian Product
	20.24 Relations
	20.25 Domain, Co-Domain and Range of Relation
	20.26 Universal Relation from Set A to Set B
	20.27 Number of Relations from Set A to Set B
	20.28 Relation on a Set
	20.29 Representation of Relation in Different Forms
	20.30 Classification of Relations
	20.31 Into Relation
		20.31.1 One-One-Onto Relation (Bijective Relation)
	20.32 Types of Relations
		20.32.1 Reflexive Relation
		20.32.2 Identity Relation
		20.32.3 Transitive Relation
		20.32.4 Anti-symmetric Relation
		20.32.5 Equivalence Relation
	20.33 Composition of Relations
	20.34 Inverse of a Relation
Chapter 21 : Functions
	21.1 Definition of Function
	21.2 Representation of a Function
	21.3 Some Standard Function
	21.4 Equal or Identical Functions
	21.5 Properties of Greatest Integer Function(Bracket Function)
		21.5.1 Properties of Least Integer Function
		21.5.2 Properties of Fractional Part Function
		21.5.3 Properties of Nearest Integer Function
	21.6 Classification of Functions
		21.6.1 One-one (Injective) Function
	21.7 Many-One Functions
		21.7.1 Onto (Surjective) Function
	21.8 Method of Testing for Injectivity
	21.9 Into (Non-Surjective) Function
	21.10 One-One Onto Function (Bijective Function)
	21.11 Testing of a Function for Surjective
	21.12 Number of Relations and Functions
	21.13 Composition of Non-Uniformly Defined Functions
	21.14 Properties of Composition of Function
		21.14.1 Definition of Inverse of a Function
	21.15 Condition for Invisibility of a Function
		21.15.1 Method to Find Inverse of a Given Function
	21.16 Properties of Inverse of a Function
	21.17 Even Function
		21.17.1 Properties of Even Functions
		21.17.2 Odd Function
		21.17.3 Properties of Odd Functions
	21.18 Algebra of Even-Odd Functions
	21.19 Even Extension of Function
	21.20 Odd Extension of Function
		21.20.1 Definition of Periodic Function
	21.21 Facts and Properties Regarding Periodicity
	21.22 Period of Composite Functions
	21.23 Periodicity of Modulus/Power of a Function
	21.24 Exception to LCM Rule
	21.25 Periodicity of Functions Expressed by Functional Equations
	21.26 Tips for Finding Domain and Range of a Function
Chapter 22 : Limits, Continuity and Differentiability
	Limit
		22.1 Limit of a Function
		22.2 Limit of Function F(X) At X = A
		22.3 Existence of Limit of a Function
		22.4 Non-Existence of Limit of a Function
		22.5 Algebra of Limits
		22.6 Indeterminate Forms
		22.7 Some Standard Limits
		22.8 Limits of Some Standard Composite Functions
		22.9 Some Useful Transformations
		22.10 Some Important Expansions
		22.11 Some Standard Approaches to Find Limit of a Function
			22.11.1 Sandwitch Theorem or Squeeze Play Theorem
	Continuity
		22.12 Continuity of F(X) At X = A
			22.12.1 Reasons of Discontinuity of f(x) at x = a
		22.13 Discontinuity of First Kind
			22.13.1 Discontinuity of Second Kind
		22.14 Pole Discontinuity
		22.15 Single Point Continuity
		22.16 One Sided Continuity
			22.16.1 Continuity of an Even and Odd Function
		22.17 Algebra of Continuity
		22.18 Continuity of a Function on a Set
			22.18.1 Domain of Continuity of Some Standard Function
			22.18.2 Continuity in an Open Interval
		22.19 Continuity of a Function on a Closed Interval
		22.20 Properties of Continuous Function
	Differentiability
		22.21 Differentiability at a Point
			22.21.1 Physical Significance
			22.21.2 Geometrical Significance
		22.22 Concept of Tangent and Its Association with Derivability
			22.22.1 Theorem Relating to Continuity and Differentiability
			22.22.2 Reasons of Non-Differentiability of a Function at x = a
		22.23 Algebra of Differentiability
			22.23.1 Domain of Differentiability
		22.24 Domain of Differentiability of Some Standard Functions
			22.24.1 Differentiability in Open and Closed Interval
		22.25 Miscellaneous Results on Differentiability
			22.25.1 Alternative Limit Form of Derivatives
		22.26 Differentiability of Parametric Functions
		22.27 Repeatedly Differentiable Functions
		22.28 Functional Equation
			22.28.1 Solution of a Functional Equation
Chapter 23 : Method of Differentiation
	23.1 Method of Differentiation
		23.1.1 Derivatives Using First Principle (Ab-initio) Method
		23.1.2 Method of Using First Principle
	23.2 Algebra of Differentiation
	23.3 Chain Rule
	23.4 Derivatives of Some Standard Functions
		23.4.1 Algebraic Functions
		23.4.2 Logarithmic and Exponential Functions
		23.4.3 Trigonometric Functions
		23.4.4 Inverse Circular Functions
	23.5 Differentiation of a Function with Respect
to Another Function
	23.6 Logarithmic and Exponential Differentiation
	23.7 Differentiation of Inverse Function
		23.7.1 Rules of Higher Order Derivative
	23.8 Implicit Differentiation
		23.8.1 Procedure to Find dy/dx for Implicit Function
		23.8.2 Shortcut Method to Find dy/dx for Implicit Functions
	23.9 Parametric Differentiation
	23.10 Determinant Forms of Differentiation
	23.11 Leibnitz’s Theorem for the Nth Derivative of the Product of Two Functions of X
	23.12 Successive Differentiation
	23.13 Some Standard Substitution
Chapter 24 : Application of Derivatives
	Rate of Change
		24.1 Instantaneous Rate of Change of Quantities
		24.2 Application of Rate of Change of Quantities
		24.3 Errors and Approximations
			24.3.1 Types of Errors
		24.4 Calculation of δY Corresponding to δX
	Tangent and Normal
		24.5 Tangents from an External Point
		24.6 Tangents/Normals to Second Degree
		24.7 Tangent at Origin
		24.8 Angles of Intersection of two Curves
			24.8.1 Algorithm to Find Angle of Intersection
		24.9 Orthogonal Curves
		24.10 Common Tangent
		24.11 Shortest Distance Between Two Non-Intersecting Curves
	Monotonicity
		24.12 Monotonicity of a Function on an Interval
		24.13 Condition for Monotonicity of Differentiable Functions on an Interval
		24.14 Monotonicity of Function on its Domain
		24.15 Domain of Monotonicity of a Function
		24.16 Critical Point
		24.17 Intervals of Monotonicity for Discontinuous Function
		24.18 Properties of Monotonic Function
		24.19 Application of Monotonicity
	Curvature of a Function
		24.20 Hyper Critical Point
		24.21 Points of Inflexion
		24.22 Method to Find the Points of Inflexion of the Curve Y = F(X)
		24.23 Type of Monotonic Function
	Rolle's and Mean Value Theorem
		24.24 Rolle’s Theorem
		24.25 Application of Rolle ’s Theorem
		24.26 Lagrange’s Mean Value Theorem
		24.27 Alternative form of LMVT
	Maxima and Minima
		24.28 Local Maxima
			24.28.1 Local Minima
		24.29 First Derivative Test (For Continous Functions)
		24.30 Point of Inflection and Saddle Point
		24.31 Global or Absolute Maxima and Minima
		24.32 Algebra of Global Extrema
		24.33 Even/Odd Functions
		24.34 Miscellaneous Method
		24.35 Second/Higher Order Derivative Test
		24.36 First Derivative Test for Parametric Functions
		24.37 Second Derivative Test for Parametric Function
		24.38 Darboux Theorem
		24.39 Fork Extremum Theorem
		24.40 Extrema of Discontinuous Functions
		24.41 Maximum and Minimum for Discrete Valued Functions
		24.42 Surface Area and Volume of Solids and Area, Perimeters of Plane Figures
		24.43 General Concept (Shortest Distance of a Pointfrom a Curve)
Chapter 25 : Indefinite Integration
	25.1 Introduction
		25.1.1 Anti-derivative of a Function
		25.1.2 Notation of Anti-derivatives or Indefinite Integral
		25.1.3 Algebra of Integration
	25.2 Integral of Some Standard Functions
	25.3 The Method of Substitution
		25.3.1 List of Some Standard Substitutions
		25.3.2 List of Some Standard Substitutions
		25.3.3 List of Some Standard Substitutions, for Integrand Function
		25.3.4 List of Some Standard Substitutions for Integrand Having
		25.3.5 Substitution after Taking xn Common
	25.4 Intregration of { sinm x cosn x dx ; m,n Є Z
		25.4.1 To Slove Integral of the Form
	25.5 Integration by Partial Fraction
	25.6 Integration of Rational and Irrational Expressions
	25.7 To Solve Integral of the Form
		25.7.1 Integral of Type
	25.8 Integral of Irrational Functions
		25.8.1 Euler's Substitution
	25.9 Integrating Inverses of Functions
	25.10 Integration of a Complex Function of a Real Variable
	25.11 Multiple Integration by Parts
Chapter 26 : Definite Integration and Area Under the Curve
	26.1 Area Function
	26.2 First Fundamental Theorem
	26.3 Second Fundamental Theorem
	26.4 Linearity of Definite Integral
	26.5 Properties of Definite Integral
		26.5.1 Generalization
		26.5.2 Condition of Substitution
	26.6 Convergent and Divergent Improper Integrals
	26.7 Applications
		26.7.1 Evaluation of Limit Under Integral Sign
		26.7.2 Leibnitz’s Rule for the Differentiation Under the Integral Sign
		26.7.3 Evaluate of Limit of Infinite Sum Using Integration
	26.8 Walli’s Formulae
		26.8.1 Walli’s Product
		26.8.2 Some Important Expansion
	26.9 Beta Function
	26.10 Gamma Function
		26.10.1 Properties of Gamma Function
		26.10.2 Relation Between Beta and Gamma Functions
	26.11 Weighted Mean Value Theorem
		26.11.1 Generalized Mean Value Theorem
	26.12 Determination of Function by Using Integration
	Area Under the Curve
		26.13 Area Bounded by Single Curve with X-Axis
			26.13.1 Area Bounded by Single Curve with y-axis
			26.13.2 Sign Conversion for Finding the Area Using Integration
			26.13.3 Area Bounded Between Two Curves
			26.13.4 Area Enclosed by Inverse Function
			26.13.5 Variable Area its Optimization and Determination of Parameters
			26.13.6 Determination of Curve When Area Function is Given
		26.14 Area Enclosed in Curved Loop
			26.14.1 Graphical Solution of the Intersection of Polar Curves
Chapter 27 : Differential Equation
	27.1 Introduction
	27.2 Differential Equation
		27.2.1 Types of Differential Equation
		27.2.2 Order and Degree of Differential Equation
	27.3 Linear Differential Equation
	27.4 Non-Linear Differential Equations
		27.4.1 Formation of Family of Curves
		27.4.2 Formation of Differential Equation
	27.5 Solution of Differential Equation
	27.6 Classification of Solution
		27.6.1 General Solution
	27.7 Variable Separable Form
		27.7.1 Equations Reducible to Variable Separable Form
		27.7.2 Homogeneous Differential Equation
	27.8 Solution of Homogeneous Differential Equation
	27.9 Equations Reducible to the Homogeneous Form
	27.10 Exact and Non-Exact Differential Equation
		27.10.1 Method of Solving an Exact Differential Equation
	27.11 Non-Exact Differential Equation
		27.11.1 Integrating Factor
		27.11.2 Leibnitz Linear Differential Equation
		27.11.3 First Order Linear Differential Equation
		27.11.4 Differential Equation of First Order and Higher Degree
	27.12 Higher Order Differential Equation
	27.13 Integral Equations and their Solving Method
		27.13.1 Orthogonal Trajectory of a Given Curve
	27.14 Application of Differential Equation
Chapter 28 : Vectors
	28.1 Physical Quantities
		28.1.1 Equality of Two Vectors
	28.2 Classification of Vectors
		28.2.1 Representation of a Free Vector in Component Form
		28.2.2 Direction Cosine and Direction Ratios of Vectors
	28.3 Addition of Vectors
	28.4 Subtraction of Vectors
		28.4.1 Properties of Vector Subtraction
	28.5 Collinear Vectors
		28.5.1 Conditions for Vectors to be Collinear
	28.6 Section Formula
		28.6.1 Collinearity of the Points
		28.6.2 Linear Combination of Vectors
		28.6.3 Linearly Dependent Vectors
		28.6.4 Linearly Independent Vectors
		28.6.5 Product of Two Vectors
		28.6.6 Scalar Product of Four Vectors
		28.6.7 Vector Product of Four Vectors
	28.7 Vector Equation and Method of Solving
Chapter 29 : Three-Dimensional Geometry
	29.1 Introduction
		29.1.1 Section Formula
		29.1.2 Corollary
		29.1.3 Centroid of a Triangle
		29.1.4 Centroid of a Tetrahedron
		29.1.5 Direction Ratios (DR’s)
		29.1.6 Relation Between the DC’s and DR’s
		29.1.7 The Angle Between Two Lines
		29.1.8 Projection of a Line Joining Two Points
		29.1.9 Vector Equation of a Curve
		29.1.10 Angle Between two Lines
		29.1.11 Condition of Parallelism
		29.1.12 Condition of Perpendicularity
		29.1.13 Condition of Coincidence
		29.1.14 Foot of Perpendicular Drawn From a Point P(x1, y1, z1)
		29.1.15 Distance of a Point P (x1,y1,z1) From the Line L
		29.1.16 Co-ordinates of Point of Intersection
	29.2 Skew lines
		29.2.1 Equation of Line of Shortest Distance
	29.3 Plane
	29.4 Area of Triangle
		29.4.1 Angle Between Two Planes (Angle Between the Normal Vector)
		29.4.2 Angle Between Line and Plane
		29.4.3 Distance Between Two Parallel Planes
		29.4.4 Distance of a Point from a Given Plane
		29.4.5 Foot of Perpendicular Drawn From a Point on Plane
		29.4.6 Equation of Bisectors of the Angle Between Two Planes
Chapter 30 : Probability
	30.1 Experiments
		30.1.1 Event
		30.1.2 Exhaustive Events
		30.1.3 Equally Likely Events
		30.1.4 Disjoint Events
		30.1.5 Independent and Dependent Events
		30.1.6 Mutually Exclusive and Exhaustive Events
		30.1.7 Conditional Probability
		30.1.8 Generalized Form
		30.1.9 Baye’s Theorem
	30.2 Geometrical Probability
Chapter 31 : Matrices and Determinants
	31.1 Matrix
	31.2 Sub Matrix
		31.2.1 Equal Matrices
	31.3 Multiplication of Matrix
		31.3.1 Properties of Multiplication of Matrices
		31.3.2 Transpose of a Matrix
		31.3.3 Symmetric Matrix
	31.4 Hermitian Matrix
		31.4.1 Properties of Hermitian Matrices
		31.4.2 Skew-Hermitian Matrix
		31.4.3 Orthogonal Matrix
		31.4.4 Idempotent Matrix
		31.4.5 Periodic Matrix
		31.4.6 Nilpotent Matrix
		31.4.7 Involutory Matrix
		31.4.8 Unitary Matrix
		31.4.9 Non-singular Matrix
	31.5 Adjoint of a Square Matrix
		31.5.1 Properties of Adjoint of Square Matrix A
		31.5.2 Inverse of Non-singular Square Matrix
	31.6 Matrix Polynomial
		31.6.1 Cayley Hamilton Theorem
		31.6.2 Elementry Transformation
		31.6.3 Elementary Matrix
		31.6.4 Equivalent Matrices
	31.7 Determinant Method (Cramer's Rule) for Solving Non-Homogenous Equations
		31.7.1 For Two Variables
		31.7.2 For Three Variables
	31.8 Solution of Non-Homogeneous Linear Equations by Elementary Row or Column Operations
		31.8.1 Solutions of Homogenous System of Equation
	31.9 Eliminant
		31.9.1 Linear Transformation
		31.9.2 Compound Transformation
		31.9.3 Application of Determinant
		31.9.4 Properties of Determinants
		31.9.5 Caution
	31.10 Special Determinant
		31.10.1 Symmetric Determinant
		31.10.2 Skew-Symmetric Determinant
		31.10.3 Cyclic Determinants
		31.10.4 Circulants
		31.10.5 Product of two Determinant
		31.10.6 Adjoint or Adjugate of Determinant
	31.11 Differentiation of Determinants
		31.11.1 Integration of a Determinant
Chapter 32 : Statistics
	32.1 Measures of Central Tendency
	32.2 Types of Distribution
		32.2.1 Arithmetic Mean
		32.2.2 Weighted Arithmetic Mean
	32.3 Combined Mean
		32.3.1 Properties of Arithmetic Mean
	32.4 Geometric Mean
	32.5 Harmonic Mean
	32.6 Order of A.M., G.M. and H.M.
	32.7 Median
	32.8 Mode
		32.8.1 Computation of Mode
	32.9 Measures of Dispersion
	32.10 Standard Deviation
	32.11 Variance
	32.12 Combined Standard Deviation