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ALGEBRA

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

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.

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

Some Basic Concepts of chemistry

&

          CHEMISTRY IN EVERYDAY LIFE 

Contents

 

1.     Some Basic Concepts of Chemistry

1.1      Introduction

1.2      Measurement of Physical Quantities

1.2.1 Precision and Accuracy

1.2.2 Significant Figures

1.2.3 Dimensional Analysis

1.3    Matter

1.3.1 Physical classification

1.3.1.1 Solid

1.3.1.2 Liquid

1.3.1.3 Gas

1.3.2 Chemical Classification

1.3.2.1 Element

1.3.2.2 Compound

1.3.2.3 Mixture

1.4    Laws of Chemical Combination

1.4.1 Law of Conservation of mass

1.4.2 Law of constant composition/definite proportions

1.4.3 Law of Multiple Proportions

1.4.4 Law of Reciprocal Proportions

1.4.5 Gay Lussac’s Law of Gaseous Volumes

1.4.6 Avogadro’s law

1.5    Dalton’s Atomic Theory

1.5.1 Modified Dalton’s Atomic Theory

1.6    Atomic and Molecular Masses

1.6.1 Atomic Mass Unit

1.6.2 Atomic Mass

1.6.3 Gram Atomic Mass

1.6.4 Molecular Mass

1.6.5 Gram Molecular Mass

1.7    Mole concept

1.8    Chemical Formula

1.8.1 Empirical formula

1.8.2 Molecular formula

1.9  Chemical equation and Stoichiometry

1.9.1 Balancing of Chemical Equation

1.9.2 Stoichiometry

1.9.2.1 Limiting reagent

1.10 Mole concept in Solutions

1.10.1 Mass percent or weight percent (w/w %)

1.10.2 Mole fraction

1.10.3 Molarity

1.10.4 Molality

1.10.5 Normality

2.     Chemistry in Everyday Life

2.1    Introduction

2.2    Chemistry in Dyes

                2.2.1 Colour and Constitution

                 2.2.1.1 Chromophores

   2.2.1.2 Auxochromes

2.2.2 Classification of Dyes

2.2.2.1 On the basis of chemical constitution

2.2.2.2 On the basis of application

     2.3    Chemistry in Medicines

                2.3.1 Drugs or Medicines

                2.3.2 Classification of Drugs

                       2.3.2.1 On the basis of Pharmacological Effect

                       2.3.2.2 On the basis of Drug Action

                       2.3.2.3 On the basis of Chemical Structure

                       2.3.2.4 On the basis of molecular targets

2.3.3 Drug-Target interaction

          2.3.3.1 Enzymes as Drug Targets

         2.3.3.2 Receptors as Drug Targets

2.3.4 Types of Drugs on the basis of drug action

2.3.4.1 Antacids

            2.3.4.2 Antihistamines

           2.3.4.3 Tranquillizers

           2.3.4.4 Analgesics

           2.3.4.5 Antipyretics

           2.3.4.6 Antimicrobials

           2.3.4.7 Antibiotics

2.3.4.8 Sulpha Durgs

2.3.4.9 Antimalarials

2.3.4.10 Antiseptics and Disinfectants

2.3.4.11 Antifertility drugs

2.4    Chemistry in Food

2.4.1 Artificial Sweetening Agents

2.4.1.1 Saccharin (o- sulphobenzimide)

2.4.1.2 Aspartame

          2.4.1.3 Alitame

2.4.1.4 Sucralose

2.4.1.5 Cyclamate

2.4.1.6 L-Glucose

               2.4.2. Preservatives.

                      2.4.2.1 Table Salt

                      2.4.2.2 Sugar

                      2.4.2.3 Vegetable Oils

                      2.4.2.4 Sodium benzoate

                      2.4.2.5 Sodium metabisulphite

                      2.4.2.6 Sorbic acid and its salts

                      2.4.2.7 Epoxides

                      2.4.2.8 p-Hydroxybenzoate esters

2.5    Chemistry of Cleansing Agents 

2.5.1 Soaps

2.5.1.1 Manufacture of Soaps

          2.5.1.2 Types of Soaps

          2.5.1.3 Advantages and Disadvantages of using soap as cleansing agents

2.5.2 Synthetic Detergents or Soapless Soaps

2.5.2.1 Classification, synthesis and uses of detergents

   2.5.2.2 Advantages of synthetic detergents over soaps

   2.5.2.3 Disadvantages of Detergents

2.6    Chemicals in Rocket Fuels

2.6.1 Introduction to Space Chemistry

2.6.2 Propellants of Rocket and Guided Missiles 

States of Matter

Contents

1. Introduction

2. Gaseous State

    2.1 Measurable properties of gases

    2.2 Gas laws

         2.2.1 Boyle’s law

          2.2.2 Charles’ law

2.2.3 Avogadro’s law

         2.2.4 Combined Ideal Gas Law 

         2.2.5 Dalton’s law of partial pressure

         2.2.6 Graham’s law of diffusion

   2.3 Kinetic Molecular theory of gases

         2.3.1 Maxwell – Boltzmann distribution of molecular speeds

          2.3.2 Kinetic Gas Equation

          2.3.3 Average Kinetic Energy and Absolute Temperature

          2.3.4 Collision Properties and Velocities of Molecules

   2.4 Behaviour of Real gases

2.4.1 Vander Waals equation

2.4.2 Liquefaction of gases

2.4.3 Critical Constants

         2.4.4 Boyle’s Temperature and Inversion Temperature

3. Liquid State

    3.1 Properties of Liquids

3.1.1 Vapour Pressure

3.1.2 Viscosity

3.1.3 Surface tension

4. Solid State

    4.1 Properties of solids

    4.2 Types of solids

          4.2.1 Amorphous solids

            4.2.2 Crystalline solids

4.3 Classification of Crystalline solids

     4.3.1 Molecular crystals

     4.3.2 Ionic crystals

     4.3.3 Covalent or Network crystals

     4.3.4 Metallic crystals        

4.4 X-Ray Studies of Crystalline solids

4.5 Crystal Lattices and Unit Cell

     4.5.1 Types of Unit cell

     4.5.2 Packing of Constituent Particles in Crystals

     4.5.3 Packing efficiency

4.6 Imperfection in Crystalline solids

     4.6.1 Stoichiometric defects

     4.6.2 Non - Stoichiometric defects

     4.6.3 Impurity defects

    4.7 Properties of Crystalline solids

          4.7.1 Electrical Properties of Solids

          4.7.2 Magnetic Properties of Solids

          4.7.3 Dielectric Properties of Solids

SOLUTIONS

Contents

1.     Introduction

2.     Methods of expressing the concentration of a solution

            2.1  Percentage

                          2.1.1 Mass Percentage or Percent by Mass

                          2.1.2 Volume Percentage or Percent by Volume

2.2 Strength (Concentration in grams per litre)

     2.3 Parts Per Million (ppm)

     2.4 Parts Per Billion (ppb)

     2.5 Molarity

              2.5.1 Molarity of dilution

            2.5.2 Molarity of mixing

     2.6 Normality

          2.6.1 Normality of dilution

          2.6.2 Normality of mixing

     2.7 Molality

     2.8 Formality

     2.9 Mole Fraction

     2.10 Mole %

     2.11 Mass Fraction

3.  Types of Solutions

3.1 Solutions of Gases in Liquids

           3.1.1 Henry’s Law of Vapour Pressure

                     3.1.1.1 Applications of Henry’s Law

               3.1.1.2 Limitations of Henry’s Law

     3.2 Solutions of Liquids in Liquids

            3.2.1 Solubility Principles for Liquid-Liquid Solutions

            3.2.2 Vapour Pressure of Liquid Solutions

            3.2.3 Raoult’s Law of Vapour Pressure for Volatile Solutes

           3.2.4 Ideal and Non-ideal Solutions

                   3.2.4.1 Ideal Solutions

                    3.2.4.2 Non-ideal Solutions

                    3.2.4.3 Azeotropic Mixtures or Constant Boiling Mixtures

    3.3   Solutions of Solids In Liquids

           3.3.1 Raoult’s Law of Vapour Pressure for Non-volatile Solutes         

           3.3.2 Colligative Properties of Dilute Solutions

              3.3.2.1 Relative Lowering in the Vapour Pressure

              3.3.2.2 Elevation of Boiling Point

              3.3.2.3 Depression of Freezing Point

                     3.3.2.4 Osmotic Pressure

    3.4   Abnormal Colligative Properties and Abnormal Molecular Masses

           3.4.1  van’t Hoff factor (i) in case of dissociation

            3.4.2  van’t Hoff factor (i) in case of association 

                                                                                        ATOMIC STRUCTURE

&

CHEMICAL BONDING

Contents

1. Atomic Structure  

1.1 Introduction 

1.2 Discovery of Fundamental Particles

1.2.1 Discovery of Electron

1.2.2 Discovery of Proton

1.2.3 Discovery of Neutron

1.2.3.1 Moseley Experiment–Atomic Number

1.3 Atomic Models

1.3.1 Thomson’s Model

1.3.2 Rutherford Model

1.3.3 Bohr’s Model for Hydrogen Atom

1.3.4 Sommerfeld Extension to Bohr’s Model

       1.3.5 Quantum Mechanical Model of Atom

1.4 Dual Nature of Radiation and Matter

1.4.1 Wave Nature of Electromagnetic Radiation

1.4.1.1 Electromagnetic Wave Theory

1.4.1.2 Atomic Spectra

1.4.2 Particle Nature of Electromagnetic Radiation

1.4.2.1 Black Body Radiation and Photoelectric Effect

1.4.2.2 Planck’s Quantum Theory

1.4.3 Dual Nature of Matter

1.4.3.1 The De Broglie Relationship

1.4.3.2 Heisenberg’s Uncertainty Principle

1.5 Quantum Numbers

1.5.1 Principal Quantum Number

1.5.2 Azimuthal Quantum Number

1.5.3 Magnetic Quantum Number

1.5.4 Spin Quantum Number

1.6 Atomic Orbitals

1.6.1 Shapes of Atomic Orbitals

1.6.2 Energies of Atomic Orbitals

1.6.3 Filling of Orbitals in Atoms

1.7 Electronic Configuration of Atoms

1.8 Atomic Terms

1.8.1 Atomic Number

1.8.2 Nucleons

1.8.3 Mass Number

1.8.4 Atomic Mass Unit

1.8.5 Atomic Species

2. CHEMICAL BONDING

2.1 Introduction

2.2 Electronic Theory Of Valency

2.3 Ionic bond

      2.3.1 Conditions for the formation of ionic bond

      2.3.2 Factors Governing the Formation of lonic Bonds

      2.3.3 General characteristics of ionic compounds

2.4 Covalent Bond

      2.4.1 Formal Charge (F.C.) on an atom in a molecule/ion

       2.4.2 General characteristics of covalent compounds

       2.4.3 Types of Covalent Bonds-Sigma and Pi Bonds  

              2.4.3.1 Sigma (s) bond

              2.4.3.2 Pi (p) Bond

       2.4.4 Some Important Bond Characteristics

2.5 Polar Covalent Bonds – Electronegatitivy

2.6 Dipole Moments

      2.6.1 Dipole moment and percentage ionic character

      2.6.2 Transition from ionic to covalent bond – Fajans’ rule

2.6.3 Factors influencing ion – deformation or increasing covalent   character

2.7 Lewis Structures Of Molecules

      2.7.1  Limitations of Lewis theory of drawing structure

2.8 Co-Ordinate Covalent Bond or Dative Bond

      2.8.1 General characteristics of coordinate

Covalent   compounds

      2.8.1.1 Resonance

2.9  Valence shell electron pair repulsion (VSEPR) theory

       2.9.1 Molecules in which the central atom has one or more lone  

                   pairs

       2.9.2 Prediciting geometry of species using vsepr theory

       2.9.3 Limitations of vsepr theory and advent of valence bond

2.10 Valence Bond Theory

2.11  Hybridization of Atomic Orbitals and the Shape of Molecules

         2.11.1 Summary of Hybridization

         2.11.2 Electron-Pair Geometry and Molecular Geometry

2.12 Molecular Orbital Theory

        2.12.1 Comparison of Valence Bond and Molecular orbital theories

 2.13 Hydrogen bonding

         2.13.1 Types of Hydrogen Bonding

         2.13.2 Properties of Hydrogen Bond

         2.13.3 Significance of hydrogen bonding in biological systems

GENERAL PHYSICS 

&

MECHANICS

Contents

1.     General physics (Physics and Measurement)

1.1  Physics, technology and society

1.1.1    What is physics?

1.1.2    Physics, technology and society

1.1.3    Link between technology and physics

1.2    Unit and Dimensions

1.2.1    Unit

1.2.1.1       System of units

1.2.1.2       Derived units

1.2.2    Dimensions

1.2.2.1 Dimensional equation

1.2.2.2 Principle of homogeneity

1.2.2.3 Uses of dimensional analysis

1.2.2.4 Limitations of the theory of dimensions

1.3    Significant figures and errors analysis

1.3.1    Significant figures

1.3.1.1 Rules for calculating significant figures

1.3.1.2 Significant figures in algebraic operations

1.3.2    Errors

1.3.2.1 Percentage error

1.3.2.2 Propagation of errors

1.4    Least count, accuracy and precision of measuring instruments

1.4.1    Least count

1.4.2    Accuracy of measurement

1.4.3    Precision

2.  Kinematics (Motion in one, two and three dimensions)

2.1    Motion in One Dimension

2.1.1    A body

2.1.2    Particle

2.1.3    Motion in one dimension

2.1.3.1 Motion

2.1.3.2 Frame of reference

2.1.3.3 Position of an object

2.1.3.4 Position vector

2.1.3.5 Displacement

2.1.3.6 Distance

2.1.3.7 Difference between distance and displacement

2.1.3.8 Velocity

2.1.3.9 Acceleration

2.1.4    Uniform and non uniform motion

2.1.4.1 Graphical representation of the uniform motion

2.1.4.2       Velocity vector in non uniform motion

2.1.4.3       Instantaneous velocity

2.1.4.4       Acceleration vector in non uniform motion

2.1.5    Relative velocity

2.1.6    Equations of motion

2.1.6.1 Distance travelled in nth second

2.1.7    Graphs (straight line motion)

2.1.7.1 Position time graph

2.1.7.2 Velocity time graph

2.1.7.3 Acceleration time graph

2.1.8    Analysis of uniformly accelerated motion

2.1.9    Motion under gravity

2.2    Scalars and vectors

2.2.1    Scalars

2.2.2    Vectors

2.2.2.1 Representation of a vector

2.2.2.2 Unit vector

2.2.2.3 Parallel vectors

2.2.2.4 Equal vectors

2.2.2.5 Negative vector

2.2.2.6 Null vector

2.2.2.7 Invariance of the vector

2.2.3    Addition and subtraction of vectors

2.2.3.1 Geometrical method

2.2.3.2 Resolution of a vector

2.2.3.3 Law of parallelogram of vectors

2.2.3.4 Vector subtraction

2.2.3.5 Properties of vector addition

2.2.4    Multiplication of vectors

2.2.4.1 Multiplication of a vector by a scalar

2.2.4.2 Multiplication of a vector by vector

2.2.4.3 Dot product or scalar product

2.2.4.4 Cross product or vector product

2.3    Motion in Two Dimensions

2.3.1    Projectile

2.3.2    Motion of projectile

2.3.2.1 Horizontal projection

2.3.2.2 Projectile motion on an inclined plane

2.3.2.3 Motion down the plane

2.4    Uniform Circular Motion

2.4.1    Circular motion

2.4.2    Uniform circular motion

2.5    Relative Motion

2.5.1    Relative velocity

2.5.2    Physical significance of relative velocity

2.5.3    Relative motion between rain and man

2.5.4    Relative motion of a swimmer in flowing water

2.5.5    Crossing of the river in minimum time

2.5.6    Velocity of separation/Approach or relative angular velocity

3.     Laws of motion

3.1    Newton’s first law of motion

3.1.1    Inertia

3.1.2    Momentum

3.2    Newton’s second law of motion

3.2.1    Impulse

3.3    Newton’s third law of motion

3.3.1    Illustrations of Newton’s third law of motion

3.4    Types of forces

3.4.1    Force

3.4.2    Classification of forces

3.5    Apparent weight of a man in a lift/Elevator

3.6    Constraint relations

3.7    Law of conservation of linear momentum

3.7.1    Applications of law of conservation of linear momentum

3.8    Friction

3.8.1    Sliding friction

3.8.2    Cause of sliding friction

3.8.3    Types of friction

3.8.4    Rolling friction

3.8.5    Cause and direction of rolling friction

3.8.6    Laws of friction

3.8.7    Coefficient of friction

3.8.8    Angle of friction

3.8.9    Angle of response

3.9    Dynamics of circular motion

3.9.1    Definition of circular motion

3.9.2    Kinematics of circular motion

3.9.2.1 Angular variables

3.9.3    Centripetal force

3.9.4    Centrifugal force

3.9.5    Applications of centripetal and centrifugal forces

3.9.6    Banking of roads

3.9.6.1       Purpose of banking

3.9.6.2       What is skidding?

3.9.6.3       How to avoid skidding?

3.9.6.4       Overturning

3.10  Solved examples

4.     Work, Energy and Power

4.1    Work

4.1.1 Work Done by a Constant Force

4.1.2 Work depends on the frame of reference

4.1.3    Work Done by a Variable Force

4.2    Conservative and Non-Conservative Forces

4.3    Mechanical Energy

4.3.1 Kinetic Energy

4.3.2 Potential Energy

4.3.2.1 Gravitational Potential Energy

4.3.2.2 Elastic Potential Energy and Kinetic Energy

4.3.2.3 Work Energy Theorem

4.3.2.4       Conservation of Mechanical Energy

4.4 Collisions

4.4.1 Types of Collision

4.4.1.1 Elastic collision

4.4.1.2 Inelastic collision

4.4.2 Coefficient of restitution

4.4.3  Velocities of colliding bodies after collision

4.4.4 Elastic collision in one dimension

4.4.5  Perfectly inelastic collision in one dimension

4.4.6    Elastic collision in two dimensions

4.5 Power

4.6 Solved Examples

5.     Rotational Motion

5.1 Centre of mass

5.1.1 Centre of Mass for a Continuous Distribution

5.1.2 Motion of Centre of Mass

5.1.3 Velocity of the centre of mass of a system of particles

5.2 Moment of a force or torque

5.2.1 Expression for torque in Cartesian coordinates

5.3 Angular Momentum of a Particle

5.3.1 Principle of conservation of angular momentum

5.3.2 Some examples of conservation of angular momentum

5.4 Moment of inertia

5.4.1 Moment of inertia of some bodies of regular shape

5.4.2 Radius of gyration

5.4.3 Torque and moment of inertia

5.4.4 Angular momentum and moment of inertia

5.5 Theorem of parallel axes

5.6 Theorem of perpendicular axes

5.7 Application of theorem of parallel axes

5.7.1 Moment of inertia of a thin circular ring

5.7.2 Moment of inertia of a uniform circular ring

5.7.3 Moment of inertia of a uniform circular disc

5.8 Rigid body dynamics

5.8.1 Laws of rotational motion

5.8.2 Equations of rotational motion

5.9    Solved examples

6.     Gravitation

6.1 Newton’s Law of Gravitation

6.2 Acceleration due to Gravity (g)

6.2.1 Variation of Acceleration due to Gravity

6.3  Kepler’s Laws of Planetary motion

6.3.1 Kepler’s Second Law

6.4 Gravitational Field and Intensity

6.5 Gravitational Potential Energy

6.6 Gravitational Potential

6.6.1 Gravitational potential due to a shell

6.6.2 Gravitational potential (V) due to a uniform solid sphere

6.6.3 Binding Energy

6.7 Escape velocity

6.8 Orbital velocity

6.8.1 Expression for orbital velocity

6.9. Time period of satellite

6.10 Geostationary Satellite

6.11 Solved examples

ELECTRICITY AND MAGNETISM 

Contents 

1. Electrostatics

1.1 Introduction

1.2 Electric Charge

1.2.1 Two kinds of electric charges

1.2.2 Additive nature of charge

1.2.3 Quantization of charge

1.2.4 Conservation of charge

1.2.5 Properties of electric charge

1.3 Coulomb’s Law

1.3.1 Relative permittivity (dielectric constant)

1.3.2 Principle of superposition

1.4 Electric Field

1.4.1 Electric field intensity

1.4.2 Electric field due to a point charge   

1.4.3 Representation electric field

1.4.4 Field lines in case of system of two charges

1.4.5 Electric field due to a discrete distribution of charges

1.4.6 Electrical Lines of forces

1.4.6.1 Properties of electric lines force

1.5 Electric dipole

1.5.1 Electric dipole moment

1.5.2 Electric field on axial line of an electric dipole

1.5.3 Electric field on equatorial line of an electric dipole

1.5.4 Electric field at any point due to an electric dipole

     1.5.5 Torque on a dipole in a uniform electric field

1.6 Electric flux

      1.6.1 Gauss’ theorem

      1.6.2 Gaussian surface

      1.6.3 To deduce Coulomb’s law from gauss’ theorem

      1.6.4 Application of gauss law

      1.6.5 Electric field due to a uniformly charged spherical shell

1.7 Electric Potential

1.7.1 Principle of superposition

1.7.2 Equipotential surface

1.7.3 Electric Potential Energy

1.7.4 Relationship between Electric potential energy (U) and electric force

1.7.5 Earthing

1.8 Conductors and Insulators

1.9 Dielectrics

1.9.1 Polar dielectrics

1.9.2 Non-polar dielectric

1.10 Polarization

1.10.1 Polarization of a dielectric slab

1.10.2 Dielectric constant

1.10.3 Polarization density

1.10.4 Electric susceptibility

1.11. Capacitance

1.11.1 Parallel plate capacitor

1.11.2 Grouping of capacitors

1.11.3 Capacitance of parallel plate capacitor with a dielectric slab between the plates

1.11.4 Energy stored in a capacitor

1.11.5 Stored energy in terms of the Electric field

1.12. Solved Examples

2. Current Electricity

2.1 Electric Current

2.1.1 Current density

2.1.2 Drift Velocity

2.1.3 Current and drift velocity

2.1.4 Ohm’s law

2.1.5 Electrical resistance

2.1.6 Electrical resistivity

2.1.7 Resistivity of various materials

2.1.8 Temperature dependence of resistivity and resistance

2.1.9 V-I characteristics of ohmic and non ohmic conductors

2.1.10 Electric energy

2.1.11 Electric power

2.1.12 Colour code for carbon resistors

2.1.13 Grouping of Resistors

2.2 Electric cell

2.2.1 Internal resistance of a cell

2.2.2 Factors affecting Internal Resistance of a cell

2.2.3 Internal Resistance of a cell in terms of E, V and R

2.2.4 Electromotive force (emf) and potential difference of a cell

2.2.5 Grouping of cells

2.3 Kirchhoff’s laws

2.4 Wheatstone bridge

2.5 Meter Bridge

2.5.1 Applications of Meter Bridge

2.6 Potentiometer

2.6.1 A