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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

5.2.4.1Sarrus 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

2.2.2Orthogonal 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

3.3Hyperbola

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

1.9Applications and Geometrical results

2. THREE DIMENSIONAL GEOMETRY

2.1Introduction

2.2Co-ordinates of a Point

2.2.1 Distance formula

2.2.2 Section formula

2.3Plane

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

2.4Straight 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

2.1Limits

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

2.2Continuity and Differentiability

2.2.1Continuity

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

2.2.2Differentiability

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

3.4Fundamental Rules for Differentiation

3.4.1Product Rule of Differentiation

3.4.2Quotient Rule of Differentiation

3.4.3Chain Rule of Differentiation

3.5 Some more methods of Differentiation

3.5.1 Derivative of Parametric functions

3.5.2 Derivative ofImplicit 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 anotherfunction

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

4.2Rate of Change

4.3Errors and Approximations

4.4Tangents 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

4.5Increasing and Decreasing Functions

4.6Maxima 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

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 Applications of a potentiometer

2.7 Solved Examples

3. Magnetic Effect of Current and Magnetism

3.1 Magnetic effect of current

3.1.1 Introduction

3.1.1.1 Oersted’s Experiment

3.1.1.2 Rules to determine the direction of magnetic field

3.1.2 Magnetic Force

3.1.2.1 Magnetic Field, Lorentz Force

3.1.3 The Biot - Savart Law

3.1.3.1 Magnetic Field due to a Straight Wire carrying current

3.1.3.2 Magnetic Field Lines

3.1.3.3 Magnetic Field due to a Circular Loop carrying current:

3.1.3.4 Magnetic Field due to a Solenoid

3.1.4 Ampere’s Circuital Law

3.1.4.1 Applications of Ampere’s circuital law

3.1.5 Force on a charge

3.1.5.1 Force on a charge in electric field

3.1.5.2 Motion of a charge inside electric field

3.1.5.3 Force on a charge moving inside a magnetic field

3.1.5.4 Motion of charge inside magnetic field

3.1.5.5 Cyclotron

3.1.5.6 Force on a current carrying conductor placed inside a magnetic field

3.1.5.7 Force between two infinitely long parallel current carrying conductors

3.1.5.8 Torque on a current carrying loop placed inside a magnetic field

3.1.6 Moving coil galvanometer

3.1.6.1 Sensitivity of a galvanometer

3.1.6.2 Conversion of galvanometer to Ammeter

3.1.6.3 Conversion of galvanometer to Voltmeter

3.1.7 Solved Examples

3.2 Magnetism

3.2.1 Magnetic dipole

3.2.1.1 Magnetic dipole moment

3.2.1.2 Current loop and magnetic dipole

3.2.1.3 Bar Magnet as an Equivalent Solenoid

3.2.1.4 Magnetic field due to a bar magnet

3.2.1.5 Torque on a magnetic dipole in a magnetic field

3.2.1.6 Potential energy stored in a magnetic dipole on rotating inside a magnetic field

3.2.2 Magnetic Field Lines

3.2.2.1 Properties of Magnetic Lines of Force

3.2.3 Earth’s magnetic field and magnetic elements

3.2.4 Tangent law

3.2.4.1 Tangent galvanometer

3.2.4.2 Vibration magnetometer

3.2.5 Classification of Magnetic materials

3.2.5.1 Intensity of magnetization

3.2.5.2 Magnetic induction

3.2.5.3 Magnetic susceptibility

3.2.5.4 Magnetic permeability

3.2.5.5 Curie temperature

3.2.5.6 Curie’s law

3.2.5.7 Hysteresis

3.2.5.8 Permanent Magnets

3.2.5.9 Electromagnets

3.2.5.10 Some Useful Facts

3.2.6 Solved Examples

4. Electromagnetic Induction and Alternating Currents

4.1 Electromagnetic induction

4.1.1 Introduction

4.1.2 Magnetic flux

4.1.3 Faraday’s laws of electromagnetic induction

4.1.4 Induced EMF and Current

4.1.5 Lenz’s law

4.1.6 Lenz’s Law and Conservation of Energy

4.1.7 Motional emf

4.1.8 Eddy currents

4.1.9 Self Induction

4.1.10 Coefficient of self induction

4.1.11 Grouping of inductors

4.1.12 Energy stored in an inductor

4.1.13 Self inductance of a long solenoid

4.1.14 Energy stored in a solenoid

4.1.15 Mutual induction

4.1.16 Coefficient of mutual induction

4.1.17 Mutual inductance of two long solenoids

4.2 Alternating Currents

4.2.1 Alternating Current and Voltage

4.2.2 AC Circuit Elements

4.2.2.1 Pure Resistive Circuit

4.2.2.2 Pure inductive Circuit

4.2.2.3 Pure Capacitive Circuit

4.2.3 Reactance and Impedance

4.2.4 Power in an AC Circuit

4.2.5 Series AC Circuits

4.2.5.1 Series LR Circuit

4.2.5.2 Series RC Circuit

4.2.5.3 Series LC Circuit

4.2.5.4 Series LCR Circuit

4.2.6 Wattless Current

4.2.7 Electric Generator

4.2.7.1 AC Generator

4.2.8 Transformer

4.3 Solved Examples

5. Electromagnetic Waves

5.1 Conduction current

5.2 Displacement current

5.3 Modified Ampere circuital law

5.4 Maxwell’s equations

5.5 Electromagnetic waves

5.5.1 Characteristics of electromagnetic waves

5.5.2 Sources of electromagnetic radiation

5.5.3 Detectors of electromagnetic radiation

5.5.4 Properties of electromagnetic waves

5.6 Electromagnetic spectrum

5.6.1 Applications of Electromagnetic Spectrum

5.7 Solved Examples

PROPERTIES OF MATTER

Contents

1. Properties of Solids

1.1 Elasticity & plasticity

1.1.1 Factors affecting elasticity

1.2 Stress & strain

1.2.1 Stress

1.2.1.1 Longitudinal stress

1.2.1.2 Bulk stress

1.2.1.3 Shearing stress or Tangential stress

1.2.2 Strain

1.2.2.1 Longitudinal strain

1.2.2.2 Volume strain

1.2.2.3 Shearing strain

1.2.3 Stress-strain relationship in a wire

1.2.3.1 Breaking stress

1.2.3.2 Elastic Hysteresis

1.2.3.3 Elastic after effect

1.2.3.4 Elastic fatigue

1.3 Hooke’s law

1.4 Modulus of rigidity

1.4.1 Young’s modulus

1.4.2 Bulk modulus

1.4.3 Rigidity modulus

1.4.4 Factors on which Y, B, G depend

1.4.5 Relations among Y, B, G andσ

1.5 Thermal stress

1.6 Poisson’s ratio

1.7 Force constant

1.8 Solved Examples

2. Properties of liquids

2.1 Fluid Statics

2.1.1 Fluid

2.1.2 Fluid Pressure

2.1.3 Atmospheric Pressure

2.1.4 Variation of pressure with depth

2.2 Pascal’s law

2.2.1 Applications of Pascal’s law

2.3 Archimedes’s principle

2.3.1 Laws of flotation

2.4 Viscosity

2.4.1 Coefficient of viscosity

2.4.2 Similarity between viscosity and solid friction

2.4.3 Poiseuille’s formula

2.5 Stoke’s law

2.5.1 Importance of Stoke’s law

2.5.2 Terminal velocity

2.5.3 Variation of viscosity

2.5.4 Practical uses of the knowledge of viscosity

2.6 Streamline flow

2.7 Laminar flow

2.8 Turbulent flow

2.9 Critical velocity

2.10 Reynold number

2.11 Equation of continuity

2.12 Energies of a fluid

2.13 Bernoulli’s theorem

2.13.1 Limitations of Bernoulli’s theorem

2.13.2 Applications Bernoulli’s theorem

2.13.3 Venturimeter

2.13.4 Torricelli’s theorem

2.14 Surface Tension

2.14.1 Adhesive force

2.14.2 Cohesive force

2.14.3 Molecular range

2.14.4 Sphere of influence

2.14.5 Surface film

2.14.6 Surface tension

2.14.7 Surface energy

2.14.8 Work done in blowing a liquid drop or soap bubble

2.14.9 Formation of a bigger drop by a number of smaller drops

2.14.10 Pressure difference across curved surfaces of radii of curvature R₁and R₂.

2.14.11 Angle of contact

2.14.12 Capillary action or capillarity

2.14.13 Dependence of surface tension

2.14.14 Radius of the new bubble formed when two bubbles coalesce

2.14.15 Radius of interface when two soap bubbles of different radii are in contact

2.15 Solved examples

3. Thermal properties of matter

3.1 Heat

3.2 Internal energy

3.3 Specific Heat

3.4 Thermal Equilibrium

3.5 Zeroth Law of Thermodynamics and Temperature

3.5.1 Temperature

3.5.2 Measurement of Temperature

3.5.3 Triple Point

3.5.4 What is a thermometer?

3.5.5 Temperature scales

3.6 Thermal Expansion

3.6.1 Linear expansion

3.6.2 Area expansion

3.6.3 Volume expansion

3.6.4 Relation betweenα,βandγ

3.7 Calorimetry

3.8 Change of state

3.9 Latent heat

4. Kinetic Theory of Gases

4.1 Molecular theory of matter

4.2 Ideal gas or perfect gas

4.3 Avogadro’s hypothesis

4.3.1 Numerical value of R

4.4 Assumptions of kinetic theory of gases

4.5 Concept of pressure exerted by a gas

4.5.1 Expression for pressure due to an ideal gas

4.5.2 Relation between pressure and kinetic energy of the gas

4.6 Average kinetic energy per molecule of the gas

4.6.1 Kinetic interpretation of temperature

4.7 Most probable speed

4.8 Mean speed or average speed

4.9 Root mean square speed

4.10 Degrees of freedom

4.10.1 Degrees of freedom of mono-atomic gases

4.10.2 Degrees of freedom of di-atomic gases

4.10.3 Degrees of freedom of tri-atomic gases

4.11 Law of equipartition of energy

4.12 Specific heat capacity

4.12.1 Specific heat capacity of mono-atomic gases

4.12.2 Specific heat capacity of di-atomic gases

4.12.3 Specific heat capacity of Tri-atomic gases

4.12.4 Specific heat capacity of Poly-atomic gases

4.12.5 Determination of γ from the degrees of freedom

4.13 Concept of mean free path

4.13.1 Brownian motion

4.14 Solved examples

OPTICS

Contents

1. Ray Optics

1.1 Introduction

1.2 Reflection of light

1.2.1 Important terms related to reflection

1.2.2 Laws of reflection

1.2.3 Image formation

1.2.4 Characteristics of the image formed by a plane mirror

1.2.5 Spherical mirrors

1.2.6 Some important definition

1.2.7 Relation between F and R

1.2.8 Rules for Image formation

1.2.9 Image formation by concave mirror

1.2.10 Sign Convention

1.2.11 Mirror Formula

1.2.12 Linear Magnification

1.3 Refraction

1.3.1 Laws of Refraction

1.3.2 Refractive index

1.3.3 Lateral Deviation (through a glass slabs)

1.3.4 Apparent depth

1.3.5 Critical angle & total internal reflection

1.3.5.1 Applications of Total internal reflection

1.3.6 Refraction at a Single Spherical Surface

1.3.7 Lens

1.3.7.1 Types of lenses

1.3.7.2 Optical Centre

1.3.7.3 Principal Axis

1.3.7.4 Focus (F)

1.3.7.5 Rules for Image Formation

1.3.7.6 Lens Makers Formula

1.3.7.7 Lens Formula

1.3.7.8 Power of Lens

1.3.7.9 Combination of Lenses

1.4 Dispersion of light

1.4.1 Refraction through Prism

1.4.2 Minimum Deviation

1.4.3 Dispersion of Light through Prism

1.4.4 Angular Dispersion

1.4.5 Dispersive Power (W)

1.4.6 Dispersion without Deviation

1.4.7 Deviation without Dispersion (Achromatic Combination of Prism)

1.5 Spectrometer

1.6 Absorption and emission spectra

1.7 Scattering of light

1.7.1 Illustrations of Rayleigh’s scattering of light

1.8 Formation of rainbow

1.9 Solved examples

2. Optical instruments

2.1 Human eye

2.1.1 The Important Parts of the Eye and their Functions

2.1.2 Accommodation of the Eye

2.1.3 Power of Accommodation

2.1.4 Near point or Least Distance of Distinct Vision

2.1.5 Far Point

2.1.6 Range of Vision

2.1.7 Defects of Vision and theirCorrection

2.2 Simple Microscope

2.3 Compound Microscope

2.4 Telescope

2.4.1 Astronomical Telescope (Refracting Type)

2.4.2 Reflecting Type Telescope (Cassegrain Telescope)

2.4.3 Some Important Features of a Telescope

2.5 Solved Examples

3. Wave Optics

3.1 Wave Front

3.2 Huygens’s Principle

3.3 Laws of Reflection on Wave Theory

3.4 Refraction On The Basis Of Wave Theory

3.5 Principle of Superposition

3.5.1 Coherence

3.5.2 Interference

3.6 Young’s double slit Experiment

3.7 Diffraction of Light

3.7.1 Diffraction at a Single Slit (Fraunhofer Diffraction)

3.8 Rayleigh’s Criterion of Limiting Resolution

3.9 Resolving Power of a Microscope

3.10 Resolving Power of a Telescope

3.11 Polarisation of Light

3.11.1 To Detect Plane Polarised Light

3.11.2 Polarisation by Reflection

3.11.3 Brewster’s Law

3.11.4 Polarisation by Scattering

3.11.5 Law of Malus

3.11.6 Polaroids

3.12 Solved Examples

HEAT AND THERMODYNAMICS

&

OSCILLATIONS AND WAVES

Contents

1. Heat and Thermodynamics

1.1 Introduction

1.2 Thermal equilibrium

1.3 Zeroth law of thermodynamic

1.3.1 Three different Scales of Temperature

1.3.2 Conversion of temperature from one scale to another

1.3.3 Constant volume Gas Thermometer

1.3.4 Constant Pressure Thermometer

1.3.5 Platinum resistance thermometer

1.4 Thermodynamic state variables

1.5 Thermodynamic equation of state

1.6 Thermodynamic processes

1.7 Indicator diagram or P-V diagram

1.8 Pressure temperature phase diagram

1.9 Heat, Internal energy and Work

1.10 First law of Thermodynamics

1.10.1 Important points regarding first law of thermodynamics

1.10.2 Applications of the First law of thermodynamics

1.11 Specific heats of gases

1.11.1 Relation between Specific heats of the gas

1.12 Isothermal Process

1.13 Adiabatic process

1.14 Graphs of expansion process

1.15 Graphs of compression process

1.16 Second law of Thermodynamics

1.17 Reversible and irreversible process

1.18 Heat engine

1.18.1 Thermal efficiency of a heat engine

1.18.2 Types of Heat engines

1.18.3 Principle of a refrigerator (or heat pump)

1.19 Carnot Cycle

1.19.1 Efficiency of Carnot engine

1.19.2 Carnot theorem

1.20 Entropy

1.21 Solved examples

1.22 Heat Transfer

1.22.1 Introduction

1.22.2 Thermal conductivity

1.22.3 Absorptive Power

1.22.4 Emissive Power E

1.22.4.1 Emissitivity

1.22.5 Black Body

1.22.6 Kirchoff’s Law

1.22.7 Stefan’s Law of Radiation

1.22.8 Newton’s Law of Cooling

1.22.9 Wien’s law

1.22.10 Solar Constant

1.22.10.1 Surface temperature of sun

1.22.11 Solved examples

2. Oscillations and waves

2.1 Oscillations

2.1.1 Periodic motion

2.1.1.1 Oscillatory motion

2.1.1.2 Harmonic functions

2.1.1.3 Simple Harmonic oscillations

2.1.1.3.1 Mathematical representation of simple harmonic oscillations

2.1.1.3.2 Graphical representation of simple harmonic oscillations

2.1.1.4 Non-Harmonic oscillations

2.1.1.4.1 Graphical representation of non-harmonic oscillations

2.1.1.5 Terms related to periodic motion

2.1.1.6 Periodic function

2.1.2 Simple Harmonic motion

2.1.2.1 Geometrical interpretation of S.H.M.

2.1.2.2 Characteristics of Simple Harmonic Motion

2.1.3 Total energy in S.H.M.

2.1.3.1 Potential Energy

2.1.3.2 Kinetic Energy

2.1.3.3 Total Energy

2.1.3.4 Graphical representation of total energy of SHM

2.1.4 Oscillations of a loaded spring

2.1.4.1 Vibrations in the horizontal direction

2.1.4.2 Vibrations of a vertical spring

2.1.4.3 Oscillations of loaded spring combination

2.1.5 Simple pendulum

2.1.6 Free oscillations

2.1.7 Forced oscillations

2.1.8 Undamped simple harmonic oscillations

2.1.9 Damped simple harmonic oscillations

2.1.10 Resonant oscillations

2.1.11 Solved Examples

2.2 Waves

2.2.1 Introduction

2.2.1.1 Characteristics of a wave

2.2.1.2 Properties of waves

2.2.1.3 Wave Motion

2.2.1.4 Types of Wave

2.2.1.5 Types of mechanical Wave Motion

2.2.1.6 Some important points related with the wave motion

2.2.1.7 Characteristics of material medium for the propagation of transverse wave

2.2.1.8 Characteristics of transverse waves

2.2.1.9 Characteristics of longitudinal waves

2.2.1.10 Characteristics of wave motion

2.2.2 Equation of a plane progressive simple harmonic Wave

2.2.2.1 Phase and phase difference

2.2.2.2 Relation between particle velocity and wave velocity

2.2.2.3 Particle acceleration

2.2.3 Sound waves

2.2.3.1 Speed of a travelling wave

2.2.3.2 Speed of transverse waves on a stretched string

2.2.3.3 Speed of Longitudinal waves

2.2.3.4 Newton’s Formula for velocity of sound in Gases

2.2.3.5 Error in Newton’s Formula

2.2.3.6 Laplace’s Correction

2.2.3.7 Factors affecting velocity of sound

2.2.4 Reflection of waves

2.2.4.1 Reflection from a hard boundary

2.2.4.2 Reflection from a Soft boundary

2.2.4.3 Reflection of Circular Waves

2.2.4.4 Point source of sound reflecting from a plane surface

2.2.4.5 Reflection from Concave Surface

2.2.4.6 Reflection of waves at a closed end (denser medium)

2.2.4.7 Reflection of waves at an open end (rarer medium)

2.2.4.8 Important practical applications of reflection of sound waves

2.2.4.9 The principle of superposition of waves

2.2.5 Standing waves or Stationary waves

2.2.5.1 Characteristics of a standing waves or stationary waves

2.2.5.2 Standing waves on a string

2.2.5.3 Laws of vibrations of stretched strings

2.2.5.4 Standing waves in pipes

2.2.5.4.1 Standing waves in closed organ pipes

2.2.5.4.2 Standing waves in open organ pipes

2.2.6 Beats

2.2.6.1 Formation of Beats

2.2.6.2 Demonstration of Beats

2.2.6.3 Applications of the phenomenon of beats

2.2.7 Doppler Effect in sound

2.2.7.1 Doppler shift

2.2.7.2 Application of Doppler Effect

2.2.8 Solved Examples

QUANTITATIVE APTITUDE

Contents

1. Basic Tips For Calculation

2. Arithmetic

2.1 Number System

2.1.1 Introduction

2.1.2 Classification of Numbers

2.1.3 Classification of Natural Numbers

2.1.4 Properties of Rational and Irrational Numbers

2.1.5 Some important results on Numbers

2.1.6 Tests of Divisibility

2.1.7 Modulus of a Real number

2.2 HCF and LCM

2.2.1 Factors and multiples

2.2.2 Common factors

2.2.3 Highest Common factor (HCF)

2.2.3.1 Prime factorization method

2.2.3.2 Division method

2.2.3.3 HCF of Decimals

2.2.3.4 HCF of vulgar fractions

2.2.4 Least Common multiple (LCM)

2.2.4.1 Prime factorization method

2.2.4.2 Common Division method

2.2.4.3 LCM of Decimals

2.2.4.4 LCM of Fractions

2.2.5 Relation between HCF and LCM

2.2.6 Comparison of fractions using LCM

2.3 Simplification

2.3.1 VBODMAS Rule

2.3.2 Basic arithmetic operations and symbols

2.3.3 Short cut methods for basic operations

2.3.3.1 Short cuts in Addition

2.3.3.2 Short cuts in Subtraction

2.3.3.3 Short-cuts in Multiplication

2.3.3.4 Short cuts in Division

2.3.4 Some algebraic formulae for Simplification

2.4 Approximation

2.4.1 Addition and subtraction

2.4.2 Multiplication

2.4.3 Division

2.5 Ratio and Proportion

2.5.1 Definition and basic concepts

2.5.2 Ratio between more than two quantities

2.5.3 Compound ratio

2.5.4 Comparison of ratios

2.5.5 Proportion

2.5.6 Types of Proportion

2.5.7 Some specific cases

2.6 Average

2.6.1 Definition

2.6.2 Weighted average

2.6.3 Specific cases of average

2.6.3.1 Average speed

2.6.3.2 Average of Series of Numbers

2.6.3.3 Miscellaneous cases

2.7 Additional Topics in Arithmetic

2.7.1 Problems on Trains

2.7.1.1 Basic concepts

2.7.1.2 Specific cases

2.7.1.2.1 Train moving past a stationary point object

2.7.1.2.2 Train moving past a stationary lengthy object

2.7.1.2.3 Train moving past a moving point object

2.7.1.2.4 Train moving past a moving lengthy object

2.7.2 Unitary Method

2.7.2.1 Time and Distance

2.7.2.1.1 Basic definitions and concepts

2.7.2.1.2 Formulae for specific cases

2.7.2.2 Time and Work

2.7.2.2.1 Basic concepts

2.7.2.2.2 Formulae for specific cases

2.7.2.3 Pipes and Cisterns

2.7.2.3.1 Basic concepts

2.7.2.3.2 Basic principles

2.7.2.3.3 Specific cases and formulae

2.7.2.4 Boat and Streams

2.7.2.4.1 Basic concepts

2.7.2.4.2 Specific cases

3. Commercial Maths

3.1 Interest

3.1.1 Simple Interest

3.1.1.1 Basic definitions and concepts

3.1.1.2 Some specific cases

3.1.2 Compound Interest

3.1.2.1 Basic definitions and concepts

3.1.2.2 Compounding for different time periods and rates

3.1.2.3 Some specific cases

3.1.2.4 Cases relating Simple and Compound interest

3.2 Percentage

3.2.1 Definition

3.2.2 Conversion between Percent, fraction and decimal

3.2.3 Specific cases of Percentage

3.2.3.1 Percent relationship between quantities

3.2.3.2 Examinations and marks

3.2.3.3 Geometry

3.2.3.4 Depreciation

3.2.3.5 Population growth

3.2.3.6 Price and consumption

3.3 Profit and Loss

3.3.1 Definition

3.3.2 Rule of fraction

3.3.3 Specific cases of Profit and Loss

3.3.3.1 Relation between Cost price and selling price

3.3.3.2 Using false weights

3.3.3.3 Discounts

3.3.3.4 Multiple transactions

3.4 Partnership

3.4.1 Basic concepts

3.4.2 Basic principle of profit sharing

3.4.2.1 Simple partnership

3.4.2.3 Compound partnership

3.4.3 Specific cases

4. Series

4.1 Definition

4.2 Types of Series

4.2.1 Arithmetic Series

4.2.2 Geometric Series

4.2.3 Series of squares and cubes

4.2.4 Mixed series

4.2.5 Two tier Arithmetic series

4.2.6 Three tier Arithmetic series

4.2.7 Arithmetico-Geometric series

4.2.8 Geometrico-Arithmetic series

4.2.9 Twin series

4.3 Suggested steps for solving series problems

4.4 Different types of problems on Series

4.4.1 Finding the next term

4.4.2 Finding wrong/missing number

4.4.3 Two line number series

4.5 Some unique series

GENERAL MENTAL ABILITY

Contents

1. VERBAL TESTS

1.1 Reasoning Tests

1.1.1 Direction and Distance Test

1.1.2 Classification (Odd Man Out) Test

1.1.3 Coding-Decoding Tests

1.1.4 Series Test

1.1.5 Relationships Test

1.1.5.1 Blood Relationships Test

1.1.5.2 Coded Relationships Tests

1.1.6 Alphabet Test

1.1.7 Number, Ranking and Time Sequence Test

1.1.8 Mathematical Operations

1.1.9 Logical Word Sequence Test

1.1.10 Cubes and Cubical Dice Tests

1.1.11 Data Arrangement Test

1.1.12 Data Sufficiency Test

1.2 Logic Tests

1.2.1 Syllogism

1.2.2 Statements Conclusions

1.2.2.1 Coded Inequality

2. NON VERBAL TESTS

2.1Problems Based On Symmetry

2.1.1 Problems On Mirror Images

2.1.2 Problems On Water Images

2.1.3 Completion Of An Incomplete Pattern

2.1.4 Problems On Paper Folding

2.1.5 Problems On Paper Cutting

2.2 Problem Based On Visual Ability

2.3Classification

2.4Series

2.5Analogy

ENGLISH

Contents

1. Parts of Speech

1.1 Noun

1.1.1 Kinds of Noun

1.1.2 Gender

1.1.3 Number

1.1.4 Case

1.1.5 Rules

1.2 Adjective

1.2.1 Kinds of Adjectives

1.2.2 Degrees of Adjectives

1.2.3 Rules

1.3 Pronoun

1.3.1 Kinds of Pronoun

1.3.2 Rules

1.4 Verb

1.4.1 Different Kinds of Verbs

1.4.2 Subject Verb Agreement

1.5 Adverb

1.5.1 The Kinds of Adverbs

1.5.2 Comparison of Adverbs

1.5.3 Some Rules Related to the Use of Adverbs

1.6 Preposition

1.6.1 Rules

1.6.2 Short List of One-Word Prepositions

1.6.3 General Rules Regarding Certain Prepositions

1.7 Conjunction

1.7.1 Classification of Conjunctions

1.7.2 Some Important Rules

1.8 Interjection

2. Articles

2.1 Kinds of Articles

2.1.1 The Indefinite Article ‘a/an’

2.1.2 The Definite Article ‘the’

3. Tenses

3.1 Classification of Tenses

3.1.1 Present Tense

3.1.2 Past Tense

3.1.3 Future Tense

3.2 Summary of Rules with Examples in Tabular Form

4. Punctuations and Capitals

4.1 List of Important Punctuation Marks

4.2 Rules Regarding Usages

4.3 Use of Capital Letters

5. Idioms and Phrases

5.1 Idioms

5.2 Phrases

5.3 Clauses

6. Sentences

6.1 Kinds of Sentences

6.2Subject and Predicate

7. Active and Passive Voice

7.1 Active Voice

7.2 Passive Voice

7.3 Rules for Changing the Voice

7.4 Transformation of Sentences-Voice Change

7.5 Summary of Rules with Examples in Tabular Form

8. Direct and Indirect Speech

8.1 Direct Speech

8.2 Indirect Speech

8.3 Rules for Changing Direct into Indirect Speech

9. English Vocabulary

9.1 Antonyms

9.1.1 Strategies for Antonyms

9.2 Synonyms

9.2.1 Strategies for Synonyms

10. Fill in the Blanks

10.1 Types of Fill in the Blanks

10.1.1 Logical Fill in the Blanks

10.1.2 Paired Logical Fill in the Blanks

10.1.3. Grammar Based Fill in the Blanks

10.2 Strategies for Fill in the Blanks

11. Cloze Test

11.1 Strategies for Cloze Test

12. English Usage Errors

12.1 Types of Errors

12.1.1 Error in Subject Verb Agreement

12.1.2 Error in the Tense or Form of a Verb

12.1.3 Error in Subjunctive Mood

12.1.4 Error in Comparison

12.1.5 Error in the use of Adjective for Adverb or Vice-Versa

12.1.6 Error of Parallel Construction

12.1.7 Error in Diction or Idiom

12.2 Types of Common Errors

12.2.1 Common Errors in the Use of Articles

12.2.2 Common Errors in the Use of Nouns

12.2.3 Common Errors in the Use of Pronouns

12.2.4 Common Errors in the Use of Adjectives

12.2.5 Common Errors in the Use of Verbs

12.2.6 Common Errors in the Use of Adverbs

12.2.7 Common Errors in the Use of Prepositions

12.2.8 Common Errors in the Use of Conjunction

12.2.9 Common Errors in the Use of Tenses

12.2.10 Common Errors in the Use of Modals

13. Sentence Improvement

13.1 Strategies for Sentence Improvement

14. Rearrangement of Sentences in Paragraph

14.1 Strategies for Rearrangement of Sentences in Paragraph

15. English Comprehension

15.1 Types of Comprehension Based Questions

15.2 Strategies for Comprehension Based Questions

GENERAL KNOWLEDGE

Contents

1. General Knowledge (World)

1.1 Books and Authors

1.2 World’s Great Structures (Building)

1.3 Calendars

1.4 Countries

1.4.1 Countries and their Capital

1.4.2 Countries and Their Emblems

1.4.3 Countries and Their Parliaments

1.4.4 Lines and Boundaries

1.4.5 Minerals and Producing Countries

1.4.6 Major Industrial Towns

1.4.7 Major Riverside Cities

1.4.8 Tribes and Races

1.5 Currencies

1.6 Music and Dance

1.7 Disasters

1.8 Famous Places of World

1.9 Famous Personalities

1.10 First, Largest, Longest of World

1.10.1 ‘First’ in the World

1.10.2 Important Geographical Facts

1.11 Inventions and Discoveries

1.12 Languages

1.13 Religions

1.14 Wars

1.15 Weathers and Climate

1.16 World organization

1.16.1 United Nations Organization (UNO)

1.16.2 The Commonwealth

1.16.3 The Non–Aligned Movement (NAM)

1.16.4 South Asian Association for Regional Cooperation (SAARC)

1.16.5 European Union (EU)

1.16.6 Organisation of Petroleum Exporting Countries (OPEC)

1.16.7 North Atlantic Treaty Organisation (NATO)

1.16.8 G-7 (Now G-8)

1.16.9 Association of South-East Asian Nation (ASEAN)

1.16.10 World Trade Prganisation (WTO)

1.17 Statistical Data of World

1.17.1 Census of world

1.18 Prize and Awards

2. General Knowledge (India)

2.1 Books and Authors(I)

2.1.1 Indian Authors (English)

2.2.2 Ancient Classical Authors

2.2 Dances(I)

2.2.1 Classical Dance of India

2.2.2 Folk Dance of India

2.3 Famous Places

2.4 Famous Personalities

2.5 First, Largest, Longest(I)

2.5.1 Record Makers (India)

2.5.2 India’s Superlatives Structures

Music(I)

2.6.1 Hindustani classical music

2.6.2 Carnatic music

2.7 Religions of India

2.8 Sanctuaries(I)

2.9 National Symbols (I)

2.10 Statistical Data of India

2.10.1 Census of India

2.11 Prize and Awards(I)

2.11.1 National Awards

2.11.2 Other Awards