Relations & Functions-Types of Functions: Classification, One-One, Onto, Examples-Class 12 Math Chapter 1 Notes Download free pdf
Functions are defined as the relations which give a particular output for a particular input value. A function has a ...
Read moreComposite functions-Relations & functions Class 12 Math Chapter 1 Notes Study Material Download free pdf
Composite Functions Let f : A->B and g : B->C be two functions. Then the composition of f and g, ...
Read moreInvertible Functions-Graph, Solved Examples & FAQs, Relations & functions Class 12 Math Chapter1 Notes Study Material Download free pdf
As the name suggests Invertible means “inverse“, and Invertible function means the inverse of the function. Invertible functions, in the ...
Read moreComposition of Functions: Definition, Domain, Range, Examples, Relations & functions Class 12 Math Chapter 1 Notes Study Material Download free pdf
Composition of Function is the process or operation which combines two or more functions together into a single function. We ...
Read moreInverse Functions, Graphs & Solved Examples, Relations & functions Class 12 Math Chapter 1 Notes Study Material Download free pdf
Inverse Functions are an important concept in mathematics. An inverse function basically reverses the effect of the original function. If you ...
Read moreInverse Trigonometric Functions-Formulas, Types, Graphs & Solved Examples, Class 12 Math Chapter 2 Notes Study Material Download free pdf
Inverse trigonometric functions are the inverse functions of basic trigonometric functions. In mathematics, inverse trigonometric functions are also known as ...
Read moreGraphs of Inverse Trigonometric Functions-Formulas, Solved Examples, Class 12 Math Chapter 2 Notes Study Material Download free pdf
Inverse trigonometric functions are the inverse functions of the trigonometric ratios i.e. sin, cos, tan, cot, sec, cosec. These functions ...
Read moreProperties of Inverse Trigonometric Functions-Formulas, Solved Examples, Class 12 Math Chapter 2 Notes Study Material Download free pdf
Inverse trigonometric functions, also known as arc functions, are used to find the angle that corresponds to a given trigonometric ...
Read moreInverse Trigonometric Identities-Domain, Range, Formulas, Properties, Solved Examples, Class 12 Math Chapter 2 Notes Study Material Download free pdf
Inverse Trigonometric Identities: In mathematics, inverse trigonometric functions are also known as arcus functions or anti-trigonometric functions. The inverse trigonometric functions are the inverse ...
Read moreRelations & Functions Exercise 2.1 NCERT Solutions for Class 11 Maths Chapter 2 Free PDF Download
NCERT Solutions for Class 11 – Chapter 2 Relations and Functions – Exercise 2.1 This section provides detailed solutions for ...
Read moreRelations & Functions Exercise 2.2 NCERT Solutions for Class 11 Maths Chapter 2 Free PDF Download
NCERT Solutions for Class 11 – Mathematics – Chapter 2 Relations and Functions – Exercise 2.2 This section provides detailed ...
Read moreRelations & Functions Exercise 2.3 NCERT Solutions for Class 11 Maths Chapter 2 Free PDF Download
NCERT Solutions for Class 11 – Mathematics – Chapter 2 Relations and Functions – Exercise 2.3 Exercise 2.3 deals with ...
Read moreRelations & Functions Miscellaneous Exercise NCERT Solutions for Class 11 Maths Chapter 2 Free PDF Download
In this article, we will be going to solve the entire Miscellaneous Exercise of Chapter 2 of the NCERT textbook. ...
Read moreRelations & Functions Exercise 1.1 NCERT Solutions Class 12 Math Chapter 1 free PDF Download
Relation and Function are two ways of establishing links between two sets in mathematics. Relation and Function in maths are ...
Read moreRelations & Functions Exercise 1.2 NCERT Solutions Class 12 Math Chapter 1 free PDF Download
In Class 12 Mathematics, Chapter 1, “Relations and Functions” students delve into the foundational concepts of relations and functions. Exercise ...
Read moreRelations & Functions Exercise 1.3 NCERT Solutions Class 12 Math Chapter 1 free PDF Download
Exercise 1.3 delves into the advanced concepts of function composition and invertible functions, building upon the foundational knowledge of relations ...
Read moreRelations & Functions Miscellaneous Exercise NCERT Solutions Class 12 Math Chapter 1 free PDF Download
Class 12 NCERT Solutions- Mathematics Chapter 1 Relations And Functions -Miscellaneous Exercise on Chapter 1 Question 1. Let f : R ...
Read moreMatrices Exercise 3.1 NCERT Solutions Class 12 Math Chapter 3 free PDF Download
Matrices are one of the important chapters of NCERT Solutions for Class 12 Maths. It is also a powerful and essential ...
Read moreMatrices Exercise 3.2 NCERT Solutions Class 12 Math Chapter 3 free PDF Download
Matrices are a fundamental concept in linear algebra, used extensively in mathematics, physics, engineering, computer science, and various other fields. ...
Read moreMatrices Exercise 3.3 NCERT Solutions Class 12 Math Chapter 3 free PDF Download
Chapter 3 of the Class 12 NCERT Mathematics textbook, titled “Matrices,” delves into the fundamental concepts of matrices, including their ...
Read moreMatrices Miscellaneous Exercise NCERT Solutions Class 12 Math Chapter 3 free PDF Download
Chapter 3 of the Class 12 NCERT Mathematics textbook, titled “Matrices,” covers fundamental concepts related to matrices, including operations such ...
Read moreMatrices Definition, Properties, Types, Formulas, Solved Examples
A rectangular array of m × n numbers (real or complex) in the form of m horizontal lines (called rows) ...
Read moreMatrices Types, Properties:Row, Column, Zero or Null, Singleton, Horizontal, Vertical, Square, Diagonal, Scalar, Unit or Identity Matrix, Equal Matrices, Triangular, Singular & Non-Singular Matrix, Symmetric & Skew Symmetric Matrices, Hermitian & Skew-Hermitian Matrices, Idempotent, Nilpotent, Periodic, Involutory Matrix
There are different types of matrices, and they are basically categorised on the basis of the value of their elements, ...
Read moreAlgebra of Matrices:Addition, Subtraction, Scalar Multiplication, Matrices Multiplication, Rules & Solved Examples
Algebra of Matrices is the branch of mathematics, which deals with the vector spaces between different dimensions. The innovation of ...
Read moreTranspose of a Matrix-Addition & Multiplication Property of Transpose, Solved Examples
Transpose of a matrix is one of the most commonly used methods in matrix transformation. For a given matrix, the ...
Read moreSymmetric & Skew Symmetric Matrix-Properties, Solved Examples
A symmetric matrix and skew-symmetric matrix both are square matrices. But the difference between them is, the symmetric matrix is ...
Read moreElementary Operation of Matrix with Solved Examples-Class 12 Math Matrices Notes
A matrix is an array of numbers arranged in the form of rows and columns. The number of rows and ...
Read moreInvertible Matrices | Invertible Matrix Theorems, Proofs, Applications & Properties, Solved Examples, Class 12 Matrix Notes
A matrix is an array of numbers arranged in the form of rows and columns. The number of rows and ...
Read moreDeterminants-Properties-Differentiation and Integration of Determinants-Class 12 Math Notes Study Material free pdf download
In linear algebra, a determinant is a special number that can be determined from a square matrix. The determinant of ...
Read moreDeterminant To Find Area Of A Triangle – Solved Examples-Class 12 Math Determinants Notes Study Material free pdf download
We know what a determinant is, let us know how to use Determinant to find Area of a Triangle Suppose ...
Read moreMinors and Cofactors in Matrix with Solved Examples Problems-Class 12 Math Determinants Notes Study Material free pdf download
Minors and cofactors are two of the most important concepts in matrices, as they are crucial in finding the adjoint ...
Read moreAdjoint and Inverse of a Matrix With Their Relation, Properties, Solved Examples Problems, Class 12 Math Determinants Notes Study Material free pdf download
The adjoint of a matrix, also called the adjugate of a matrix, is defined as the transpose of the cofactor ...
Read moreContinuity and Differentiability of a Function-Solved Examples Problems, Class 12 Math Notes Study Material free pdf download
Continuity and Differentiability is one of the most important topics which help students to understand the concepts like, continuity at a ...
Read moreExponential Functions, Logarithmic Functions-Definition, Formula, Properties, Rules, Graphs, Derivatives, Exponential Series & Solved Examples, Class 12 Math Notes Study Material Download Free PDF
An exponential function is a mathematical function, which is used in many real-world situations. It is mainly used to find ...
Read moreLogarithmic Differentiation – Formula, Solutions & Solved Examples Problems, Class 12 Math Notes Study Material Download Free PDF
Logarithmic differentiation is a method to find the derivatives of some complicated functions, using logarithms. There are cases in which ...
Read moreDerivatives Of A Function In Parametric Form-Solved Examples, Class 12 Math Notes Study Material Download Free PDF
Derivatives of a function in parametric form: There are instances when rather than defining a function explicitly or implicitly we ...
Read moreSecond Order Derivative | Explanation with Examples, Class 12 Math Notes Study Material Download Free PDF
For understanding the second-order derivative, let us step back a bit and understand what a first derivative is. The first ...
Read moreMean Value Theorem Formula Equation | Mean Value Theorem For Integrals, Class 12 Math Notes Study Material Download Free PDF
In mathematics, the mean value theorem states, roughly, that given a planar arc between two endpoints, there is at least ...
Read moreIncreasing & Decreasing Functions Monotonicity with Examples for Functions, Class 12 Math Notes Study Material Download Free PDF
Before explaining the increasing and decreasing function along with monotonicity, let us understand what functions are. A function is basically a ...
Read moreEquation of Tangent And Normal to a Curve with Examples, Class 12 Math Notes Study Material Download Free PDF
The applications of derivatives are: In this article, we will learn to use differentiation to find the equation of the tangent ...
Read moreDifferential Calculus & Approximation | Tangent Line Approximation, Class 12 Math Notes Study Material Download Free PDF
In mathematics, differential calculus is a branch of mathematics that deals with the process of finding the derivative of a ...
Read moreIntegral Calculus – Definition, Formulas, Methods, Applications, Examples, Class 12 Math Notes Study Material Download Free PDF
Integral Calculus is the branch of calculus where we study integrals and their properties. Integration is an essential concept which ...
Read moreIntegration in Maths – Definition, Formulas and Types, Class 12 Math Notes Study Material Download Free PDF
Integration is the calculation of an integral. Integrals in maths are used to find many useful quantities such as areas, ...
Read moreIntegrals | Definition, Meaning, Formula, Application and Examples, Class 12 Math Notes Study Material Download Free PDF
An integral in mathematics is a continuous analog of a sum that is used to determine areas, volumes, and their generalizations. ...
Read moreIntegration by Substitution Method – Formula, Examples & Questions, Class 12 Math Notes Study Material Download Free PDF
Integration by substitution or u-substitution is a highly used method of finding the integration of a complex function by reducing it to a ...
Read moreIntegration by Partial Fractions – Method, Examples & Practice Problems, Class 12 Math Notes Study Material Download Free PDF
Integration by Partial Fractions is one of the methods of integration, which is used to find the integral of the ...
Read moreIntegration by Parts | Formula, Derivation and Examples, Class 12 Math Notes Study Material Download Free PDF
Integration by Parts or Partial Integration, is a technique used in calculus to evaluate the integral of a product of two functions. The formula ...
Read moreIntegration by Substitution Formula | Sample Problems | Class 12 Math Notes Study Material Download Free PDF
The process of finding the anti-derivative of a function is the inverse process of differentiation i.e. finding integral is the ...
Read moreIntegral of Particular Functions (With Proof and Example) | Class 12 Math Notes Study Material Download Free PDF
Integral is the method to sum the functions on a large scale. Hence, here we have discussed integrals of some ...
Read morePartial Fraction in Integration – Definition, Formula, Decomposition, Examples | Class 12 Math Notes Study Material Download Free PDF
Partial fractions are the fractions used for the decomposition of a rational expression. When an algebraic expression is split into ...
Read moreIntegration by Parts – Formula, ILATE Rule & Solved Examples | Class 12 Math Notes Study Material Download Free PDF
Integration by parts is a special technique of integration of two functions when they are multiplied. This method is also ...
Read moreDefinite Integral – Definition, Formulas, Properties and Solved Examples | Class 12 Math Notes Study Material Download Free PDF
An integral assigns numbers to functions in mathematics to define displacement, area, volume, and other notions that arise by connecting ...
Read moreArea Under The Curve Using Integration | Class 12 Math Notes Study Material Download Free PDF
In Calculus, we have seen definite integral as a limit of a sum and we know how to evaluate it ...
Read moreArea under Curves – Tricks for Solving Area under Curves | Class 12 Math Notes Study Material Download Free PDF
Important Concepts on Area under the Curves If the area under one curve and another curve can be found, then ...
Read moreApplication of Integrals MCQs (With Answers) | Class 12 Math Notes Study Material Download Free PDF
MCQs on Class 12 Maths Chapter Application of Integrals Check out the Class 12 Maths Chapter 8 Application of Integrals ...
Read moreIndefinite Integrals MCQs With Answers | Class 12 Math Notes Study Material Download Free PDF
MCQs on Class 12 Maths Chapter 7 Integrals Check out the MCQs for Class 12 Maths Chapter 7 Integrals. Each ...
Read moreImportant Integration Questions With Answers | Class 12 Math Notes Study Material Download Free PDF
Integration questions with answers are available here for students of Class 11 and Class 12. Integration is an important topic ...
Read moreArea Between Two Curves Using Integration | Class 12 Math Notes Study Material Download Free PDF
As we all know, integration means calculating the area by dividing the region into many elementary strips and then adding ...
Read moreDifferential Equations | Types, Order, Degree, Solved Examples | Class 12 Math Notes Study Material Download Free PDF
Differential Equation Definition A differential equation is an equation which contains one or more terms and the derivatives of one ...
Read moreTypes of Differential Equations | Class 12 Math Notes Study Material Download Free PDF
An equation with one or more terms that involve derivatives of the dependent variable with respect to an independent variable ...
Read moreSolution Of A Differential Equation -General & Particular | Class 12 Math Notes Study Material Download Free PDF
Solving Differential Equations The solution of a differential equation – General and particular will use integration in some steps to ...
Read moreSeparable Differential Equation-Solved Examples | Class 12 Math Notes Study Material Download Free PDF
A differential equation is an equation of the form \(\begin{array}{l}\frac{\mathrm{d} y}{\mathrm{d} x} = g(x),\;\;\;\; where\;\; y = f(x)\end{array} \). General ...
Read moreVectors in Maths-Unit Vector, Zero or null Vector, Magnitude of Vector, Operations on Vectors, Dot product of Vectors, Scalar Triple Product
Vectors, in Maths, are objects which have both, magnitude and direction. Magnitude defines the size of the vector. It is represented ...
Read moreTypes of Vectors | Zero Vector, Unit Vector, Position Vector, Co-initial Vector, Like & Unlike Vectors, Co-planar Vectors, Collinear Vectors, Equal Vectors, Displacement Vector, Negative of a Vector
What is a vector? Vector is a physical quantity that has both direction and magnitude. In other words, the vectors ...
Read moreLaws of Vector Addition- Parallelogram, Triangle Law, Vector Addition | Class 12 Math Notes Study Material Download Free PDF
Laws of Vector Addition A vector is a physical quantity which is represented both in direction and magnitude. In the ...
Read moreMultiplication Of Vectors by a Scalar Quantity with examples | Class 12 Math Notes Study Material Download Free PDF
A vector relates two given points. It is a mathematical quantity having both the Magnitude and the direction. Multiplication of ...
Read moreScalar or Dot Product Of Two Vectors | Projection of Vector | Definition, Properties, Formulas and Examples | Class 12 Math Notes Study Material Download Free PDF
Vector is a quantity that has both magnitude and direction. Some mathematical operations can be performed on vectors such as ...
Read moreDirection Cosines & Direction Ratios – Definitions & Examples | Class 12 Math Notes Study Material Download Free PDF
Direction Cosines When a directed line OP passing through the origin makes \(\begin{array}{l}\alpha \end{array} \) , \(\begin{array}{l}\beta\end{array} \) and \(\begin{array}{l} ...
Read moreEquation of Line in Three Dimensions | Cartesian & vector Equation | Class 12 Math Notes Study Material Download Free PDF
Equation of a line is defined as y= mx+c, where c is the y-intercept and m is the slope. Vectors ...
Read moreAngles Between two Lines in 3D Space: Solved Examples | Class 12 Math Notes Study Material Download Free PDF
A line in mathematics and geometry is a fundamental concept representing a straight, one-dimensional figure that extends infinitely in both ...
Read moreShortest Distance Between Two Lines in 3D Space, Solved Examples | Class 12 Math Notes Study Material Download Free PDF
The shortest distance between two lines in three-dimensional space is the length of the perpendicular segment drawn from a point ...
Read moreProbability-Definition, Formula, Types, Problems & Solutions | Class 12 Math Notes Study Material Download Free PDF
Probability means possibility. It is a branch of mathematics that deals with the occurrence of a random event. The value is ...
Read moreConditional Probability, Solved Examples | Class 12 Math Notes Study Material Download Free PDF
Imagine a student who takes leave from school twice a week excluding Sunday. If it is known that he will ...
Read moreMultiplication Rule of Probability | Proof & Solved Examples | Class 12 Math Notes Study Material Download Free PDF
The multiplication rule of probability explains the condition between two events. For two events A and B associated with a ...
Read moreIndependent & Mutually Exclusive Events in Probability-Definition, Venn Diagram & Solved Examples | Class 12 Math Notes Study Material Download Free PDF
Independent events are those events whose occurrence is not dependent on any other event. For example, if we flip a ...
Read moreBayes Theorem | Definition, Proof, Formula, Derivation & Solved Examples | Class 12 Math Notes Study Material Download Free PDF
Bayes’ theorem describes the probability of occurrence of an event related to any condition. It is also considered for the ...
Read moreProbability Distribution | Definition, Types, Formula, Examples | Class 12 Math Notes Study Material Download Free PDF
In Statistics, the probability distribution gives the possibility of each outcome of a random experiment or event. It provides the ...
Read moreProbability Bernoulli Trial & Binomial Distribution of Random Variables | Class 12 Math Notes Study Material Download Free PDF
Bernoulli Trials and Binomial Distribution are explained here in a brief manner. Bernoulli’s trial is also said to be a ...
Read moreLinear Programming-Definition, Methods & Examples | Class 12 Math Notes Study Material Download Free PDF
Linear programming is a method of optimising operations with some constraints. The main objective of linear programming is to maximize ...
Read moreLinear Programming Problem LPP-Simplex & Graphical Method | Class 12 Math Notes Study Material Download Free PDF
Linear Programming Problems (LPP): Linear programming or linear optimization is a process which takes into consideration certain linear relationships to ...
Read moreTypes of Linear Programming Problems | Class 12 Math Notes Study Material Download Free PDF
Linear programming or linear optimization is a process that takes into consideration certain linear relationships to obtain the best possible ...
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