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NDA 2019
Study Materials

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1.1 Complex Numbers

1.1.1    Definition

1.1.2    Algebra of Complex number    Addition    Subtraction    Multiplication    Division    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    Cartesian form (Geometric Representation)

        Argument of a Complex number    Vectorial Representation    Trigonometric or Polar Representation

1.1.6    Geometrical representation of fundamental operations    Geometrical Representation of Addition    Geometrical Representation of Subtraction    Geometrical Representation of Multiplication of Complex Numbers

 Modulus and Argument Multiplication of two Complex Numbers    Geometrical Representation of Division of Complex Numbers

 Modulus and Argument Division of two Complex Numbers

1.1.7    Conjugate of a complex number    Geometrical Representation of Conjugate of Complex Number    Properties of Conjugate of Complex Number

1.1.8    Triangular inequality

1.1.9    Important results in context with rotation    Rotation Theorem    Some Results on Locus in Argand Plane    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   Inverse points w.r.to a Circle   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


2.1  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


3.1  Mathematical Induction

3.1.1 The principle of Mathematical Induction    First principle of mathematical induction    Second principle of mathematical induction

3.1.2 Solved Examples

3.2 Binomial Theorem

3.2.1 Introduction 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    General Term    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




4.1        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  



5.1 Matrices

5.1.1 Definition

5.1.2 Types of Matrices       Row Matrix       Column Matrix       Square Matrix       Traces of a Matrix       Diagonal Matrix       Scalar Matrix       Unit Matrix or Identity Matrix       Triangular Matrix       Null Matrix   Transpose of a Matrix     Properties of Transpose                  Conjugate of a Matrix     Properties of Conjugate                  Transpose conjugate of a Matrix     Properties of Transpose conjugate

5.1.3 Algebra of Matrices       Addition and Subtraction of Matrices       Scalar Multiplication       Multiplication of Matrices     Properties of Multiplication

5.1.4 Special Matrices       Symmetric and Skew Symmetric Matrices       Hermitian and Skew - Hermitian Matrices       Singular and Non-singular Matrices       Unitary Matrix       Orthogonal Matrix       Idempotent Matrix       Involuntary Matrix       Nilpotent Matrix

5.1.5 Adjoint of a Square Matrix

5.1.6 Inverse of a Matrix       Properties of Inverse of a Matrix

5.1.7 Elementary Operations on a Matrix

5.1.8 System of Simultaneous Linear Equations       Homogeneous and Non-Homogeneous System of Linear Equations       Solution of a Non-Homogeneous System of Linear Equations       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       Sarrus Rule

5.2.5 Operations on Determinants       Multiplication of two Determinants       Differentiation of a Determinant       Summation of Determinants

5.2.6 Special Determinants       Symmetric determinant       Skew symmetric determinant       Circulant determinant

5.2.7 Determinants: System of Linear Equations       Cramer’s Rule       Consistency of the System of the Equations       System of homogeneous linear equations

5.3   Solved Examples






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 


2.1 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

     Direct common tangents

     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

            Important Terms

                 Standard Forms of Parabola

                 Position of a Point Relative to a Parabola

                 Intersection of a line and Parabola

              3.1.3    Parametric Equation of a Parabola

              3.1.4    Chords of Parabola

                 Equation of Chord

                 Condition for the chord to be focal chord

                 Diameter of Parabola

            3.1.5   Tangent and Normal

                 Equation of Tangent in Different forms

                 Equation of Pair of Tangents

                 Chord of Contact

                 Equation of Normal in Different forms

                 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

                    Important Terms

          Equation of ellipse in other form

          Auxiliary circle

               Position of a Point Relative to an Ellipse

               Intersection of a line and an Ellipse

       3.2.3   Parametric Equation of the Ellipse

            3.2.4   Chords of Ellipse

               Equation of Chord

               Diameter of an Ellipse

               Conjugate Diameters

            3.2.5   Tangent and Normal

               Equation of Tangent in Different forms

               Equation of Pair of Tangents

               Chord of Contact

               Director Circle

               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

        Important Terms

             Position of a Point Relative to a Hyperbola

             Intersection of a line and Hyperbola

            3.3.3   Parametric Equation of Hyperbola

            3.3.4   Conjugate Hyperbola

            3.3.5   Chords of Hyperbola

             Equation of Chord

             Diameter of Hyperbola

             Conjugate Diameters

3.3.6   Tangent and Normal

             Equation of Tangent in Different forms

             Equation of Pair of Tangents

             Chord of Contact

             Director Circle

             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






     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


2.1 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







1.1 Set theory

1.1.1       Set and its Representation

1.1.2       Subset of a Set

1.1.3       Kinds of Set       Empty and Singleton Sets       Finite and Infinite Sets       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          Union of Sets          Intersection of Sets          Disjoint Sets          Difference of Sets

    Symmetric Difference of Sets          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       Identity Relation       Universal Relation       Void Relation       Reflexive Relation       Symmetric Relation       Antisymmetric Relation       Transitive Relation       Equivalence Relations

1.4 Functions

1.4.1       Introduction to Functions

1.4.2       Equal Function

1.4.3       Kinds of Functions       One-One Function(Injection)       Onto Function(Surjection)       Bijection (One-One Onto Function)

1.4.4       Real Functions Some Standard Real Functions         Constant and Identity Functions         Modulus Function         Greatest and Smallest Integer Functions         Even and Odd Functions         Explicit and Implicit Functions         Periodic Functions Summary of function and their Graphs Algebra of Real functions

1.4.5       Composite Functions

1.4.6       Inverse Functions       Method to Find Inverse of a Function       Some Standard Functions along with their Inverse  Functions

1.4.7       Solved Examples

1.5 Binary Operation



2.1    Limits

2.1.1       Concept of limits

2.1.2       Definition of limit    Left hand limit    Right hand limit

2.1.3       Algebra of limits

2.1.4       Evaluation of limits          Algebraic limits          Trigonometric limits          Exponential and Logarithmic limits

2.1.5       Indeterminate forms          L’ Hospital’s rule

2.1.6       Solved Examples

2.2    Continuity and Differentiability

2.2.1       Continuity       Continuity at a point       Continuity on an interval       Geometrical meaning of continuity       Discontinuity of a function       Types of discontinuity        Important results on continuous function

2.2.2       Differentiability          Differentiability of a function          Differentiability of a function at a point          Differentiability of a function on an interval          Some Important results on differentiability          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







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 First Derivative Test Second Derivative Test

4.6.4       Global maximum/minimum points