Notes for JEE Main & Advanced Mathematics

Recent Notes

Matrices of Rotation of Axes
Geometrical Transformations
Cayley-Hamilton Theorem
Consistency of a System of Linear Equation \[\mathbf{AX=B,}\] where \[\mathbf{A}\] is a square matrix
Homogeneous and Non-homogeneous Systems of Linear Equations
Echelon Form of a Matrix
Rank of Matrix
Inverse of a Matrix
Adjoint of a Square Matrix
Special Types of Matrices
Transpose of a Matrix
Positive Integral Powers of a Matrix
Multiplication of Matrices
Scalar Multiplication of Matrices
Addition and Subtraction of Matrices
Trace of a Matrix
Types of Matrices
Equality of Matrices
Order of a Matrix
Definition
Some Special Determinants
Application of Determinants in Solving a System of Linear Equations
Differentiation and Integration of Determinants
Product of Two Determinants
Minors and Cofactors
Properties of Determinants
Definition
Lagrange's Mean Value Theorem
Rolle's Theorem
Greatest and Least Values of a Function Defined on an Interval \[[a,\,\,b]\]
Properties of Maxima and Minima
Higher Order Derivative Test
Sufficient Criteria for Extreme Values (1st Derivative Test)
Necessary Condition for Extreme Values
Definition
Definition
Length of Perpendicular from Origin to the Tangent
Length of Intercept Made on Axes by The Tangent
Length of Tangent, Normal, Subtangent, Subnormal
Angle of Intersection of Two Curves
Equation of the Tangent and Normal
Slope of the Tangent and Normal
Velocity and Acceleration in Rectilinear Motion
Derivative as the Rate of Change
Deduction of Euler?s Theorem
Euler's Theorem on Homogeneous Functions
Higher Partial Derivatives
Definition
Leibnitz?s Theorem
Differentiation of Integral Function
\[{{n}^{th}}\] Derivative Using Partial Fractions
Successive Differentiation or Higher Order Derivatives
Differentiation of a Function with Respect to Another Function
Methods of Differentiation
Theorems for Differentiation
Some Standard Differentiation
Introduction
Relation Between A.P., G.P. and H.P.
Properties of Arithmetic, Geometric, Harmonic Means Between Two Given Numbers
Special Series
Method of Difference
Method for Finding Sum
Sum of A.G.P.
\[{{n}^{th}}\]term of A.G.P.
Definition
Properties of H.P.
Harmonic Mean
General Term of an H.P.
Definition
Properties of G.P.
Geometric Mean
Sum of infinite terms of a G.P.
Sum of first 'n' terms of a G.P.
Selection of Terms in a G.P.
General term of a G.P.
Definition
Properties of A.P.
Arithmetic Mean
Sum of n terms of an A.P.
Selection of Terms in an A.P.
General Term of an A.P.
Definition
Some Important Results
Some Properties Related to Circle
Some Terminology Related to Height and Distance
Solutions of Triangles
Regular Polygon
Cyclic Quadrilateral
Ex-central Triangle
Pedal Triangle
Circle Connected With Triangle
Half Angle Formulae
Area of Triangle
Napier's Analogy (Law of Tangents)
Theorem of The Medians: (Apollonius Theorem)
Projection Formulae
The Law of Cosines or Cosine Rule
Relation Between Sides and Angles
Periodic Functions
Important Points to be Taken in Case of While Solving Trigonometrical Equations
Method for Finding Principal Value
General Solution of The Form \[\mathbf{acos}\,\mathbf{\theta +bsin}\,\mathbf{\theta =c}\]
Trigonometrical Equations With Their General Solution
Trigonometrical Equations and Inequations Definition
Differentiability of a Function at a Point
Discontinuous Function
Continuity From Left and Right
Continuity of a Function at a Point
Introduction
Methods of Evaluation of Limits
Fundamental Theorems on Limits
Limit of a Function
Inverse Function
Composite Function
Periodic Function
Even and Odd Function
Conditional Trigonometrical Identities
Kinds of function
Algebra of Functions
Domain, Co-domain and Range of Function
Definition of Function
Intervals
Maximum and Minimum Value of a \[\mathbf{cos}\,\,\mathbf{\theta }\,\,\mathbf{+}\,\mathbf{b}\,\,\mathbf{sin}\,\,\mathbf{\theta }\]
Some Important Definitions
Trigonometric Ratio of Sub-multiple of an Angle
Trigonometric Ratio of Multiple of an Angle
Formulae to Rransform The Product Into Sum or Difference
Trigonometrical Ratios In Terms of Each Other
Trigonometrical Ratios for Some Special Angles
Trigonometrical Ratios for Various Angles
Trigonometrical Ratios of Allied Angles
Trigonometrical Ratios or Functions
Domain and Range of a Trigonometrical Function
Angle of Intersection of Two Spheres
Intersection of Straight Line and a Sphere
Condition of Tangency of a Plane to a Sphere
Relation Between an arc and an Angle
Relation Between Three Systems of Measurement of an Angle
System of Measurement of Angles   
Section of a Sphere by a Plane
Equation in Sphere in Various Forms
General Equation of Sphere
Projection of a Line on a Plane
Angle Between Line and Plane
Intersection Point of a Line and Plane
Introduction
Equation of Plane Through a Given Line
Coplanar Lines
Image of a Point in a Plane
Equation of Planes Bisecting Angle Between Two Given Planes
Locus
Angle Between Two Planes
Foot of Perpendicular from a Point \[\mathbf{A(\alpha ,}\,\,\mathbf{\beta ,}\,\,\mathbf{\gamma )}\]to a given plane \[\mathbf{ax+by+cz+d=0}\].
Transformation of Axes
Equation of Plane Passing Through the Given Point
Area of Some Geometrical Figures
Definition of Plane and Its Equations
Shortest Distance Between two Straight Lines
Foot of Perpendicular from a Point to The Line
Intersection of Two Lines
Changing Unsymmetrical form to Symmetrical Form
Some Points of a Triangle
Straight Line in Space
Angle Between Two Lines
Projection
Direction Cosines and Direction Ratios
Triangle and Tetrahedron
Section Formulae
Section Formula
Distance Formula
Properties of Some Geometrical Figures
Co-ordinates of a Point in Space
Distance Formula
Polar Co-ordinates
Cartesian Co-ordinates of a Point
Introduction
Binomial Distribution
Total Probability and Baye's Rule
Conditional Probability
Addition Theorems on Probability
Odds in Favour and Odds Against an Event
Problems Based on Combination and Permutation
Classical Definition of Probability
Definitions of Various Terms
Distance Between The Pair of Parallel Straight Lines     
Removal of the term \[\mathbf{xy}\] from \[\mathbf{f(X,}\,\mathbf{Y)=a}{{\mathbf{x}}^{\mathbf{2}}}\mathbf{+2hxy+b}{{\mathbf{y}}^{\mathbf{2}}}\] Without Changing the Origin
Removal of First Degree Terms
Equation of the Lines Joining the Origin to the Points of Intersection of a Given Line and a Given Curve
Bisectors of the Angles Between the Lines
Angle Between the Pair of Lines
Point of Intersection of Lines Represented by \[\mathbf{a}{{\mathbf{x}}^{\mathbf{2}}}\mathbf{+ 2hxy+b}{{\mathbf{y}}^{\mathbf{2}}}\mathbf{+ 2gx+ 2fy+c= 0}\]
Equation of Pair of Straight Lines
Logic Gates
Switching Circuits
Boolean Functions
Introduction
Duality
Algebra of Statements
Tautologies and Contradictions
Negation of Compound Statements
Logical Equivalence
Basic Logical Connectives or Logical Operators
Truth Tables
Types of Statements
Use of Venn Diagrams in Checking Truth and Falsity of Statements
Statements or Propositions
Introduction
Advantages and Limitations of L.P.P.
To Find The Vertices of Simple Feasible Region Without Drawing a Graph
Graphical Solution of Two Variable Linear Programming Problem
Mathematical Formulation of a Linear Programming Problem
Terms of Linear Programming
Linear Inequations
Inverse Trigonometric Ratios of Multiple Angles
Formulae for Sum, Difference of Inverse Trigonometric Function
Properties of inverse trigonometric functions
Domain and Range of Inverse Trigonometric Functions
Graphs of Inverse Trigonometric Functions
Integration of Hyperbolic Functions
Integration of Trigonometric Functions
Integration of Rational Functions by Using Partial Fractions
Integrals of the form \[\int{\frac{a sinx+b cosx}{c sinx+d cosx} }\]and \[\int{\frac{a\,sinx+bcos\,x+q}{c\,sinx+d\,cosx+r}}\]
Integrals of the form \[\int_{{}}^{{}}{\frac{dx}{a+b\mathbf{co}{{\mathbf{s}}^{\mathbf{2}}}x},\,\int_{{}}^{{}}{\frac{dx}{a+b\,\mathbf{si}{{\mathbf{n}}^{\mathbf{2}}}x},}}\]\[\int_{{}}^{{}}{\frac{dx}{a\,\mathbf{si}{{\mathbf{n}}^{\mathbf{2}}}x+b\,\mathbf{co}{{\mathbf{s}}^{\mathbf{2}}}x}\mathbf{,}}\]\[\int_{{}}^{{}}{\frac{dx}{{{\mathbf{(}a\,\sin x+b\,\mathbf{cos}\,x\mathbf{)}}^{\mathbf{2}}}}\mathbf{,}}\int_{{}}^{{}}{\frac{dx}{a\,+b\mathbf{si}{{\mathbf{n}}^{\mathbf{2}}}x+c\,\mathbf{co}{{\mathbf{s}}^{\mathbf{2}}}x}}\]
Integrals of the Form \[\int{\frac{dx}{a+bcosx+csinx}}\,,\,\,\int{\frac{dx}{asinx+bcosx}}\]
Integrals of the Form \[\int{\frac{dx}{a+b\,cos\,x}}\] and \[\int{\frac{dx}{a+b\,sin\,x}}\]
Evaluation of the Various Forms of Integrals by Use of Standard Results
Position of Centre of Gravity in Some Special Cases
Integration by Parts
Centre of Gravity of a Compound Body and Remainder
Centre of Gravity
Limiting Equilibrium on an Inclined Plane
Coefficient of Friction
Friction
Equilibrium of Coplanar Forces
Triangle Theorem of Couples
Integration By Substitution
Couples
Moment
Parallel Forces
Lami's Theorem
Polygon Law of Forces
Triangle Law of Forces
Parallelogram Law of Forces
Fundamental Integration Formulae
Introduction
Definition
Image of a Point in Different Cases
Reflection on The Surface
Concurrent Lines
Position of Two Points with Respect to a Line
Position of a Point with Respect to a Line
Length of Perpendicular
Equations of the Bisectors of The Angles Between two Straight Lines
Equation of Straight Line Through a Given Point Making a Given Angle With a Given Line
Angle Between two Non-parallel Lines
General Equation of Lines Through the Intersection of Two Given Lines
Point of Intersection of Two Lines
General Equation of a Straight Line and its Transformation in Standard Forms
Equation of Parallel and Perpendicular Lines to a Given Line
Equations of Straight Line in Different Forms
Slope (Gradient) of a Line
Definition
Miscellaneous Differential Equation
Application of Differential Equation
Linear Differential Equation
Solution by Inspection
Exact Differential Equation
Homogeneous Differential Equation
Variable Separable Type Differential Equation
Formation of Differential Equation
Definition
Symmetrical Area
Application of Vectors in 3-dimensional Geometry
Rotation of a Vector About an Axis
Vector Product of Four Vectors
Area of Bounded Regions
Scalar Product of Four Vectors
Vector Triple Product
Integration of Piecewise Continuous Functions
Scalar Triple Product
Volumes and Surfaces of Solids of Revolution
Some Important Results of Definite Integral
Area Between Two Curves
Vector or Cross Product
Scalar or Dot Product
Linear Independence and Dependence of Vectors
Linear Combination of Vectors
Position Vector
Leibnitz's Rule
Properties of Vectors
Walli's Formula
Types of Vector
Reduction Formulae for Definite Integration
Gamma Function
Summation of Series by Integration
Representation of Vectors
Introduction
Properties of Definite Integral
Evaluation of Definite Integral by Substitution
Definition
Rectangular or Equilateral Hyperbola
Asymptotes of a Hyperbola
Subtangent and Subnormal of the Hyperbola
Diameter of The Hyperbola
Pole and Polar
Equation of The Chord Joining Two Points on The Hyperbola
Equation of the Chord of the Hyperbola Whose mid point \[\mathbf{(}{{\mathbf{x}}_{\mathbf{1}}}\mathbf{,}\,\,{{\mathbf{y}}_{\mathbf{1}}}\mathbf{)}\] is given
Equation of Chord of Contact of Tangents Drawn from a Point to a Hyperbola
Equations of Normal in Different Forms
Equation of Pair of Tangents
Equations of Tangent in Different Forms
Intersection of a Line and a Hyperbola
Position of a Point With Respect to a Hyperbola
Parametric Equations of Hyperbola
Auxiliary Circle of Hyperbola
Special form of Hyperbola
Conjugate Hyperbola
Standard Equation of the Hyperbola
Definition
Subtangent and Subnormal
Diameter of The Ellipse
Pole and Polar
Equation of The Chord Joining Two Points on an Ellipse
Equation of Chord With Mid Point \[\mathbf{(}{{\mathbf{x}}_{\mathbf{1}}}\mathbf{,}{{\mathbf{y}}_{\mathbf{1}}}\mathbf{)}\]
Chord of Contact
Auxiliary Circle
Equations of Normal in Different Forms
Equation of Pair of Tangents \[\mathbf{S}{{\mathbf{S}}_{\mathbf{1}}}\mathbf{=}{{\mathbf{T}}^{\mathbf{2}}}\]
Equations of Tangent in Different Forms
Intersection of a Line and an Ellipse
Position of a Point with Respect to an Ellipse
Special Forms of an Ellipse
Parametric Form of The Ellipse
Standard Equation of the Ellipse
Definition
Pole and Polar
Length of Tangent, Subtangent, Normal and Subnormal to \[{{\mathbf{y}}^{\mathbf{2}}}\mathbf{= 4ax}\,\,\mathbf{at }\,\mathbf{(a}{{\mathbf{t}}^{\mathbf{2}}}\mathbf{,2at)}\]
Length of Tangent, Subtangent, Normal, Subnormal
Diameter of a Parabola
Equation of the Chord Joining any Two Points on the Parabola 
Equation of the Chord of the Parabola Which is Bisected at a Given Point  
Equation of The Chord of Contact of Tangents to a Parabola 
Co-normal Points
Relation Between \[\mathbf{'}{{\mathbf{t}}_{\mathbf{1}}}\mathbf{'}\] and \[\mathbf{'}{{\mathbf{t}}_{\mathbf{2}}}\mathbf{'}\] if Normal at \[\mathbf{'}{{\mathbf{t}}_{\mathbf{1}}}\mathbf{'}\] Meets the Parabola Again at \[\mathbf{'}{{\mathbf{t}}_{\mathbf{2}}}\mathbf{'}\]
Point of Intersection of Normals at Any Two Points on The Parabola
Equations of Normal in Different Forms
Equation of Pair of Tangents From a Point to a Parabola
Point of Intersection of Tangents at Any Two Points on The Parabola
Equations of Tangent in Different Forms
Position of a Point and a Line With Respect to a Parabola
Parametric Equations of a Parabola
Special form of Parabola \[{{\mathbf{(yk)}}^{\mathbf{2~}}}\mathbf{= 4a}\,\,\mathbf{(xh)= a}\]
Standard Equation of The Parabola
Definition
Recognisation of Conics
General Equation of a Conic Section When Its Focus, Directrix and Eccentricity are Given
Definitions of Various Important Terms
Definition
Image of The Circle by The Line Mirror
Limiting Points
Co-axial System of Circles
Radical Centre
Radical Axis
Family of Circles
Angle of Intersection of Two Circles
Common Tangents to Two Circles
Pole and Polar
Diameter of a Circle
Common Chord of Two Circles
Chord of Contact of Tangents
Normal to a Circle at a Given Point
Power of Point with Respect to a Circle
Director Circle
Pair of Tangents
Length of Tangent
Tangent to a Circle at a Given Point
Intersection of a Line and a Circle
Position of a Point With Respect to a Circle
Intercepts on the Axes
Equation of a Circle in Some Special Cases
Standard Forms of Equation of a Circle
Definition
Divisibility Problems
Second Principle of Mathematical Induction
First Principle of Mathematical Induction
Some Important Points
Three / Four Consecutive Terms or Coefficients
Binomial Theorem for any Index
Multinomial Theorem (for Positive Integral Index)
An Important Theorem
Use of Differentiation and Integration in Binomial Theorem
Properties of Binomial Coefficients
Greatest Term and Greatest Coefficient
To Determine a Particular Term in the Expansion
Middle Term
Number of Terms In The Expansion of \[{{(a+b+c)}^{n}}\] and \[{{(a+b+c+d)}^{n}}\]
Independent Term or Constant Term
General Term
Some Important Expansions
Binomial Theorem for Positive Integral Index
Binomial Expression
Number of Divisors
Multinomial Theorem
Some Important Results For Geometrical Problems
Derangement
Division Into Groups
Conditional Combinations
Number of Combinations With Repetition and All Possible Selections
Number of Combinations Without Repetition
Definition
Circular Permutations
Conditional Permutations
Number of permutations with repetition
Number of Permutations Without Repetition
Definition of Permutation
Introduction
Some Important Result
Equations Which Can Be Reduced to Linear, Quadratic and Biquadratic Equations
Rational Algebraic Inequations
Descarte's Rule of Signs
Position of Roots
Wavy Curve Method
The Quadratic Expression
Properties of Quadratic Equation
Condition for Common Roots
Higher Degree Equations
Relations Between Roots and Coefficients
Nature of Roots
Solution of Quadratic Equation
Types of Quadratic Equation
Polynomial
Composition of Relations
Equivalence Classes of An Equivalence Relation
Types of Relations
Inverse Relation
Definition
Cartesian Product of Sets
Laws of Algebra of Sets
Some important results on number of elements in sets
Operations on sets
Venn-Euler Diagrams
Types of Sets
Introduction

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