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NATIONAL DEFENCE ACADEMY AND NAVAL ACADEMY EXAMINATION Study Materials

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ALGEBRA

Contents

 

1. COMPLEX NUMBERS AND QUADRATIC EQUATIONS

1.1 Complex Numbers

1.1.1    Definition

1.1.2    Algebra of Complex number

1.1.2.1    Addition

1.1.2.2    Subtraction

1.1.2.3    Multiplication

1.1.2.4    Division

1.1.2.5    Equality in Complex Numbers

1.1.3    Modulus of a Complex number

1.1.4    Square root of a Complex number

1.1.5    Representation of a Complex number

1.1.5.1    Cartesian form (Geometric Representation)

                 1.1.5.1.1 Argument of a Complex number

1.1.5.2    Vectorial Representation

1.1.5.3    Trigonometric or Polar Representation

1.1.6    Geometrical representation of fundamental operations

1.1.6.1    Geometrical Representation of Addition

1.1.6.2    Geometrical Representation of Subtraction

1.1.6.3    Geometrical Representation of Multiplication of Complex Numbers

          1.1.6.3.1 Modulus and Argument Multiplication of two Complex Numbers

1.1.6.4    Geometrical Representation of Division of Complex Numbers

          1.1.6.4.1 Modulus and Argument Division of two Complex Numbers

1.1.7    Conjugate of a complex number

1.1.7.1    Geometrical Representation of Conjugate of Complex Number

1.1.7.2    Properties of Conjugate of Complex Number

1.1.8    Triangular inequality

1.1.9    Important results in context with rotation

1.1.9.1    Rotation Theorem

1.1.9.2    Some Results on Locus in Argand Plane

1.1.9.3    Dot and Cross product of complex number

1.1.10   Demoivre’s Theorem

1.1.11   Cube Root Of Unity

1.1.12   nth root of unity

1.1.13   Reflection Points for a straight line and Ptolemy’s theorem

1.1.13.1   Inverse points w.r.to a Circle

1.1.13.2   Ptolemy’s Theorem

1.1.14   Solved Examples

1.2 Quadratic Equation

1.2.1       Basic concepts

1.2.2       Relation between roots and coefficients

1.2.3       Nature of roots

1.2.4       Symmetric functions of roots

1.2.5       Formation of Quadratic Equations

1.2.6       Condition for two Quadratic Equations to have a Common root

1.2.7       Solved Examples

2.     PERMUTATIONS AND COMBINATION

2.1  Fundamental Principle of Counting

2.1.1    Rule of Product or Multiplication Principle

2.1.2    Rule of Sum or Addition Principle

2.2  Permutations

2.3  Circular Permutations

2.4  Combinations

2.5  Permutations vs. Combinations

2.6  Restricted Selection and Arrangement

2.7  Division and Distribution of Objects

2.8  Derangements and Multinomial Theorem

2.8.1    Derangements

2.8.2    Multinomial Theorem

2.9  Points to Remember

2.10             Solved Examples

3.   MATHEMATICAL INDUCTION AND BINOMIAL THEOREM

3.1  Mathematical Induction

3.1.1 The principle of Mathematical Induction

3.1.1.1    First principle of mathematical induction

3.1.1.2    Second principle of mathematical induction

3.1.2 Solved Examples

3.2 Binomial Theorem

3.2.1 Introduction

        3.2.1.1 Binomial Theorem

3.2.2 Properties of Binomial Expansion

3.2.3 Binomial Coefficients

3.2.4 Sum of Binomial Coefficients

3.2.5 Coefficient of a Particular Term

3.2.5.1    General Term

3.2.5.2    Middle Term

3.2.6 Greatest Binomial Coefficient

3.2.7 Properties of Binomial Coefficients

3.2.8 Some Important Results

3.2.9 Application of Binomial Expression

3.2.10   Solved Examples

 

 

4.   SEQUENCE AND SERIES

4.1        Basic Concepts

4.2        Arithmetic Progression

4.2.1                      Sum of n terms of an A.P.

4.2.2                      Properties of A.P.

4.2.3                      Arithmetic Mean

4.3        Geometric Progression

4.3.1 Sum of n terms of G.P.

4.3.2 Properties of G.P.

4.3.3 Geometric mean

4.4  Arithmetico-Geometric Progression

4.5  Harmonic Progression

4.5.1 Harmonic mean

4.6  Relation between A.M. ,G.M. and H.M.

4.7        Summation of Series

4.8  Method of Differences

4.9  Solved Examples  

 

5.     MATRICES AND DETERMINANTS

5.1 Matrices

5.1.1 Definition

5.1.2 Types of Matrices

5.1.2.1       Row Matrix

5.1.2.2       Column Matrix

5.1.2.3       Square Matrix

5.1.2.4       Traces of a Matrix

5.1.2.5       Diagonal Matrix

5.1.2.6       Scalar Matrix

5.1.2.7       Unit Matrix or Identity Matrix

5.1.2.8       Triangular Matrix

5.1.2.9       Null Matrix

5.1.2.10   Transpose of a Matrix

5.1.2.10.1     Properties of Transpose

5.1.2.11                  Conjugate of a Matrix

5.1.2.11.1     Properties of Conjugate

5.1.2.12                  Transpose conjugate of a Matrix



5.1.2.12.1     Properties of Transpose conjugate

5.1.3 Algebra of Matrices

5.1.3.1       Addition and Subtraction of Matrices

5.1.3.2       Scalar Multiplication

5.1.3.3       Multiplication of Matrices

5.1.3.3.1     Properties of Multiplication

5.1.4 Special Matrices

5.1.4.1       Symmetric and Skew Symmetric Matrices

5.1.4.2       Hermitian and Skew - Hermitian Matrices

5.1.4.3       Singular and Non-singular Matrices

5.1.4.4       Unitary Matrix

5.1.4.5       Orthogonal Matrix

5.1.4.6       Idempotent Matrix

5.1.4.7       Involuntary Matrix

5.1.4.8       Nilpotent Matrix

5.1.5 Adjoint of a Square Matrix

5.1.6 Inverse of a Matrix

5.1.6.1       Properties of Inverse of a Matrix

5.1.7 Elementary Operations on a Matrix

5.1.8 System of Simultaneous Linear Equations

5.1.8.1       Homogeneous and Non-Homogeneous System of Linear Equations

5.1.8.2       Solution of a Non-Homogeneous System of Linear Equations

5.1.8.3       Solution of Homogeneous System of Linear Equations

5.2 Determinants

5.2.1             Definitions

5.2.2             Properties of Determinants

5.2.3             Minors and Cofactors

5.2.4             Evaluation of a Determinant

5.2.4.1       Sarrus Rule

5.2.5 Operations on Determinants

5.2.5.1       Multiplication of two Determinants

5.2.5.2       Differentiation of a Determinant

5.2.5.3       Summation of Determinants

5.2.6 Special Determinants

5.2.6.1       Symmetric determinant

5.2.6.2       Skew symmetric determinant

5.2.6.3       Circulant determinant

5.2.7 Determinants: System of Linear Equations

5.2.7.1       Cramer’s Rule

5.2.7.2       Consistency of the System of the Equations

5.2.7.3       System of homogeneous linear equations

5.3   Solved Examples


TWO DIMENSIONAL COORDINATE GEOMETRY

 

Contents

 

1.         CARTESIAN PLANE AND LINES

1.1   Fundamental concept of 2D

1.1.1 Representation of points in a plane

1.1.2 Distance between two points

1.1.3 Section Formula  

1.1.4 Centroid , Incentre , Circum Centre and Orthocenter

1.1.5 Area of a triangle

1.1.6   Locus

1.1.7   Transformation of Axes

1.2 Straight Lines

1.2.1 Introduction

1.2.2 Various Forms of the Equations of the Straight Line

1.2.3 Point of Intersection of Two Lines

1.2.4 Condition of Concurrency 

1.2.5 Position of two points with respect to a given line

1.2.6 Length of the Perpendicular from a Point on a Line

1.2.7 Distance between parallel lines

1.2.8 Family of lines

1.2.9 Angle Bisectors

1.3 Pair of Straight Lines

1.3.1 Introduction

1.3.2 Angle between Pair of lines

1.3.3 Combined equation of the Angle bisectors of the Pair of lines

1.3.4 Homogenisation 

2.   CIRCLES

2.1 Circle –Definition

2.1.1 Equation of the Circle in various forms

2.1.2 General Equation of the Circle

2.1.3 Intercepts made by the circle on axes

2.1.4 Parametric Equation of a circle

2.1.5 Position of a point with respect to a circle

2.1.6 Intersection of a line and a Circle

 

2.2 Contact of Two circles

2.2.1   Angle of Intersection of Two circles

2.2.2   Orthogonal Intersection of two circles

2.3 Chord of a Circle

2.3.1   Common Chord of Two Circles

2.3.2   Diameter of a Circle

2.4 Tangent and Normal

2.4.1 Tangent to the Circle

2.4.2 Director Circle

2.4.3 Normal to the Circle

2.4.4 Common Tangents to Two circles

           2.4.4.1    Direct common tangents

           2.4.4.2    Transverse Common tangents

2.4.5 Chord of Contact

2.5 Family of Circles

2.6 Radical Axis

2.6.1   Properties of the Radical Axis

2.7 Co-axial System of Circles

2.7.1   Limiting Points of a Co-axial system

2.8 Solved Examples

3.   CONICS               

3.1 Parabola

3.1.1   Definition

3.1.2   Standard Equation of Parabola

                  3.1.2.1    Important Terms

                       3.1.2.2    Standard Forms of Parabola

                       3.1.2.3    Position of a Point Relative to a Parabola

                       3.1.2.4    Intersection of a line and Parabola

              3.1.3    Parametric Equation of a Parabola

              3.1.4    Chords of Parabola

                       3.1.4.1    Equation of Chord

                       3.1.4.2    Condition for the chord to be focal chord

                       3.1.4.3    Diameter of Parabola

            3.1.5   Tangent and Normal

                      3.1.5.1     Equation of Tangent in Different forms

                      3.1.5.2     Equation of Pair of Tangents

                      3.1.5.3     Chord of Contact

                      3.1.5.4     Equation of Normal in Different forms

                      3.1.5.5     Co-normal Points

            3.1.6   Pole and Polar

            3.1.7   Solved Examples

     3.2 Ellipse

        3.2.1   Definition

        3.2.2   Standard Equation of Ellipse

                           3.2.2.1   Important Terms

                 3.2.2.2   Equation of ellipse in other form

                 3.2.2.3   Auxiliary circle

                      3.2.2.4   Position of a Point Relative to an Ellipse

                      3.2.2.5   Intersection of a line and an Ellipse

       3.2.3   Parametric Equation of the Ellipse

            3.2.4   Chords of Ellipse

                      3.2.4.1   Equation of Chord

                      3.2.4.2   Diameter of an Ellipse

                      3.2.4.3   Conjugate Diameters

            3.2.5   Tangent and Normal

                      3.2.5.1   Equation of Tangent in Different forms

                      3.2.5.2   Equation of Pair of Tangents

                      3.2.5.3   Chord of Contact

                      3.2.5.4   Director Circle

                      3.2.5.5   Equation of Normal in Different forms

             3.2.6   Pole and Polar

             3.2.7   Solved Examples

3.3 Hyperbola

3.3.1   Definition

3.3.2   Standard Equation of hyperbola

                 3.3.2.1 Important Terms

                      3.3.2.2 Position of a Point Relative to a Hyperbola

                      3.3.2.3 Intersection of a line and Hyperbola

            3.3.3   Parametric Equation of Hyperbola

            3.3.4   Conjugate Hyperbola

            3.3.5   Chords of Hyperbola

                      3.3.5.1 Equation of Chord

                      3.3.5.2 Diameter of Hyperbola

                      3.3.5.3 Conjugate Diameters

3.3.6   Tangent and Normal

                      3.3.6.1 Equation of Tangent in Different forms

                      3.3.6.2 Equation of Pair of Tangents

                      3.3.6.3 Chord of Contact

                      3.3.6.4 Director Circle

                      3.3.6.5 Equation of Normal in Different forms

3.3.7    Pole and Polar

3.3.8    Asymptotes

3.3.9    Rectangular Hyperbola

3.3.10   Solved Examples


VECTORS

&

THREE DIMENSIONAL COORDINATE GEOMETRY

Contents

1. VECTOR ALGEBRA

     1.1 Introduction 

1.1.1 Definitions

1.1.2 Type of vectors

     1.2 Addition and subtraction of vectors

     1.3 Important properties of Vectors

     1.4 Collinear and Co-planar vectors

     1.5 Section formula

     1.6 Orthogonal System of vectors

           1.6.1 Direction cosines and direction ratios

     1.7 Multiplication of vectors

1.7.1 Scalar (or dot) product

1.7.2 Vector (or cross) product

1.7.3 Scalar triple product

1.7.4 Vector triple product

     1.8 Reciprocal System of Vectors

     1.9 Applications and Geometrical results

2. THREE DIMENSIONAL GEOMETRY

2.1 Introduction

2.2 Co-ordinates of a Point

      2.2.1 Distance formula

      2.2.2 Section formula

2.3 Plane

      2.3.1 Equation of a plane

      2.3.2 System of planes

      2.3.3 Angle between two planes

      2.3.4 Conditions for two planes to be parallel or perpendicular

      2.3.5 Position of a point with respect to a plane

      2.3.6 Distance of a point from a plane

      2.3.7 Bisectors of angles between two planes

      2.3.8 Tetrahedron

2.4 Straight Line 

      2.4.1 Equation of a straight line

      2.4.2 Equation of a line through the intersection of given lines

       2.4.3 Angle between two lines

       2.4.4 Projection of a line segment

       2.4.5 Distance of a point from a line

       2.4.6 Shortest distance between two lines

       2.4.7 A Plane and a straight line

  

DIFFERENTIAL CALCULUS

 

Contents

 

1. SETS, RELATIONS AND FUNCTIONS

1.1 Set theory

1.1.1       Set and its Representation

1.1.2       Subset of a Set

1.1.3       Kinds of Set

1.1.3.1       Empty and Singleton Sets

1.1.3.2       Finite and Infinite Sets

1.1.3.3       Equivalent and Equal Sets

1.1.4       Universal Set

1.1.5       Power Set

1.1.6       Venn Diagrams

1.1.7       Operation on Sets

1.1.7.1          Union of Sets

1.1.7.2          Intersection of Sets

1.1.7.3          Disjoint Sets

1.1.7.4          Difference of Sets

             1.1.7.4.1 Symmetric Difference of Sets

1.1.7.5          Complement of a Set

1.1.8       Laws of Algebra of Sets

1.2 Ordered Pairs and Cartesian product

1.2.1       Ordered Pairs

1.2.2       Cartesian Product of sets

1.3 Relations

1.3.1       Introduction

1.3.2       Inverse of a Relation

1.3.3       Types of Relations

1.3.3.1       Identity Relation

1.3.3.2       Universal Relation

1.3.3.3       Void Relation

1.3.3.4       Reflexive Relation

1.3.3.5       Symmetric Relation

1.3.3.6       Antisymmetric Relation

1.3.3.7       Transitive Relation

1.3.3.8       Equivalence Relations

1.4 Functions

1.4.1       Introduction to Functions

1.4.2       Equal Function

1.4.3       Kinds of Functions

1.4.3.1       One-One Function(Injection)

1.4.3.2       Onto Function(Surjection)

1.4.3.3       Bijection (One-One Onto Function)

1.4.4       Real Functions

1.4.4.1 Some Standard Real Functions

1.4.4.1.1         Constant and Identity Functions

1.4.4.1.2         Modulus Function

1.4.4.1.3         Greatest and Smallest Integer Functions

1.4.4.1.4         Even and Odd Functions

1.4.4.1.5         Explicit and Implicit Functions

1.4.4.1.6         Periodic Functions

1.4.4.2 Summary of function and their Graphs

1.4.4.3 Algebra of Real functions

1.4.5       Composite Functions

1.4.6       Inverse Functions

1.4.6.1       Method to Find Inverse of a Function

1.4.6.2       Some Standard Functions along with their Inverse  Functions

1.4.7       Solved Examples

1.5 Binary Operation

 

2. LIMITS, CONTINUITY AND DIFFERENTIABILITY

2.1    Limits

2.1.1       Concept of limits

2.1.2       Definition of limit

2.1.2.1    Left hand limit

2.1.2.2    Right hand limit

2.1.3       Algebra of limits

2.1.4       Evaluation of limits

2.1.4.1          Algebraic limits

2.1.4.2          Trigonometric limits

2.1.4.3          Exponential and Logarithmic limits

2.1.5       Indeterminate forms

2.1.5.1          L’ Hospital’s rule

2.1.6       Solved Examples

2.2    Continuity and Differentiability

2.2.1       Continuity

2.2.1.1       Continuity at a point

2.2.1.2       Continuity on an interval

2.2.1.3       Geometrical meaning of continuity

2.2.1.4       Discontinuity of a function

2.2.1.5       Types of discontinuity

2.2.1.6        Important results on continuous function

2.2.2       Differentiability

2.2.2.1          Differentiability of a function

2.2.2.2          Differentiability of a function at a point

2.2.2.3          Differentiability of a function on an interval

2.2.2.4          Some Important results on differentiability

2.2.2.5          Relation between Continuity and differentiability

2.2.3       Solved Examples

3. Differentiation

3.1     Derivative or differential coefficient of a function

3.2     Differentiation from first principle

3.3     Standard Derivatives

3.4     Fundamental Rules for Differentiation

3.4.1             Product Rule of Differentiation

3.4.2             Quotient Rule of Differentiation

3.4.3             Chain Rule of Differentiation

3.5     Some more methods of Differentiation

3.5.1       Derivative of Parametric functions

3.5.2       Derivative of Implicit functions

3.5.3       Logarithmic Differentiation

3.5.4       Differentiation by Substitution

3.5.5       Differentiation of Infinite series

3.5.6       Differentiation of a function with respect to another function

3.5.7       Differentiation of Determinants

3.6     Higher Order Derivatives

3.7     Solved Examples

4. PRODUCT RULE (DIFFERENTIATION)

QUOTIENT RULE (DIFFERENTIATION)

CHAIN RULE (DIFFERENTIATION)

SOME MORE METHODS OF DIFFERENTIATION

SUCESSIVE DIFFERENTIATION

PARTIAL DIFFERENTIATION

Application of Derivatives

4.1     Rolle’s and Lagrange’s Mean Value Theorems

4.2     Rate of Change

4.3     Errors and Approximations

4.4     Tangents and Normals

4.4.1       Equation of Tangent

4.4.2       Equation of Normal

4.4.3       Angle of Intersection of two curves

4.4.4       Important terms

4.5     Increasing and Decreasing Functions

4.6     Maxima and Minima

4.6.1       Maxima and Minima at end point

4.6.2       Extrema of continuous functions

4.6.3       Determination of points of Local Maxima and Local Minima

    4.6.3.1 First Derivative Test

    4.6.3.2 Second Derivative Test

4.6.4       Global maximum/minimum points

 

 


 

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