JEE Maths TutorialWe will cover below topics.
- Sets, Relations and Functions, functions as mappings, domain, codomain, range of functions, invertible functions etc
- Algebra - Quadratic equations, Arithmetic and geometric progressions, Logarithms and their properties, permutations and combinations, binomial theorem
- Probability and Statistics - Bayes Theorem
- Trigonometry - Trigonometry Function and Inverse trigonometric functions
- Analytical Geometry - 2D and 3D, Equation of a straight line, parabola, ellipse
- Differential Calculus - Limit of a function, continuity of function, derivative etc
- Integral Calculus
Sets and Relations
Types of SetsEmpty, finite, infinite
Algebra of Setsintersection, complement, difference and symmetric difference of sets and their algebraic properties, De-Morgan's laws on union, intersection, difference (for finite number of sets) and practical problems based on them.
Function mappingDomain and codomain
Range of functions
Even and odd functions
Into, onto and one-to-one functions
Special functionspolynomial, trigonometric, exponential, logarithmic, power, absolute value, greatest integer
Function Operationssum, difference, product and composition of functions
Algebra of complex numbersaddition, multiplication, conjugation, polar representation, properties of modulus and principal argument, triangle inequality, cube roots of unity, geometric interpretations. Statement of fundamental theorem of algebra, Quadratic equations with real coefficients, relations between roots and coefficients, formation of quadratic equations with given roots, symmetric functions of roots.
Sequences and SeriesArithmetic and geometric progressions, arithmetic and geometric means, sums of finite arithmetic and geometric progressions, infinite geometric series, sum of the first n natural numbers, sums of squares and cubes of the first n natural numbers.
LogarithmsLogarithms and their properties,
permutations and combinations
Mathematical Induction and Binomial TheoremBinomial theorem for a positive integral index, properties of binomial coefficients.
Matrix operationsMatrices as a rectangular array of real numbers, equality of matrices, addition, multiplication by a scalar and product of matrices, transpose of a matrix, elementary row and column transformations, determinant of a square matrix of order up to three, adjoint of a matrix, inverse of a square matrix of order up to three
Properties of Matrix operationsdiagonal, symmetric and skew-symmetric matrices and their properties, solutions of simultaneous linear equations in two or three variables.
Probability and Statistics
ProbabilityRandom experiment, sample space, different types of events (impossible, simple, compound), addition and multiplication rules of probability, conditional probability, independence of events, total probability, Bayes Theorem, computation of probability of events using permutations and combinations.
StatisticsMeasure of central tendency and dispersion, mean, median, mode, mean deviation, standard deviation and variance of grouped and ungrouped data, analysis of the frequency distribution with same mean but different variance, random variable, mean and variance of the random variable.
Trigonometric Functions and IdentitiesTrigonometric functions, their periodicity and graphs, addition and subtraction formulae, formulae involving multiple and sub-multiple angles,
Trigonometric EquationsGeneral solution of trigonometric equations. Inverse trigonometric functions (principal value only) and their elementary properties
Two dimensionsCartesian coordinates, distance between two points, section formulae, shift of origin. Equation of a straight line in various forms, angle between two lines, distance of a point from a line; Lines through the point of intersection of two given lines, equation of the bisector of the angle between two lines, concurrency of lines; Centroid, orthocentre, incentre and circumcentre of a triangle. Equation of a circle in various forms, equations of tangent, normal and chord. Parametric equations of a circle, intersection of a circle with a straight line or a circle, equation of a circle through the points of intersection of two circles and those of a circle and a straight line. Equations of a parabola, ellipse and hyperbola in standard form, their foci, directrices and eccentricity, parametric equations, equations of tangent and normal. Locus problems.
Three dimensionsDistance between two points, direction cosines and direction ratios, equation of a straight line in space, skew lines, shortest distance between two lines, equation of a plane, distance of a point from a plane, angle between two lines, angle between two planes, angle between a line and the plane, coplanar lines.
Calculus is a branch of mathematics that focuses on the study of rates of change and accumulation of quantities. It is divided into two main branches: differential calculus and integral calculus.
Calculus provides powerful tools for understanding and modeling the behavior of functions, solving real-world problems, and making predictions. It is a fundamental subject in mathematics with wide-ranging applications.
Differential calculus is concerned with understanding how a function changes as its input (independent variable) changes. Key concepts in differential calculus include:
- Derivative: The derivative of a function f(x), denoted asf′(x) or dxdy, measures the rate of change of the function with respect to the independent variable x. It represents the slope of the tangent line to the graph of the function at a given point.
- Rules of Differentiation: There are rules and formulas for finding derivatives of various types of functions, including power functions, exponential functions, trigonometric functions, and more.
- Applications: Differential calculus is used in physics, engineering, economics, and many other fields to analyze and solve problems involving change, motion, optimization, and more.
In mathematics, a continuous function is a fundamental concept that describes the behavior of functions in a smooth and connected manner. A continuous function is characterized by several key properties:
A continuous function is a function f(x) such that its graph is smooth and unbroken. This means that as you trace the graph, there are no abrupt changes or jumps. You can draw the graph of a continuous function without lifting your pencil from the paper.
Continuous functions have no gaps, holes, or breaks in their graphs. In other words, there are no points in the domain of the function where the graph suddenly "jumps" or exhibits a discontinuity. Discontinuities can take various forms, including jump discontinuities, removable discontinuities (holes), and infinite discontinuities (vertical asymptotes).
One of the important properties of continuous functions is theIntermediate Value Theorem. It states that if you have a continuous functionf(x) defined on a closed interval [a,b] , then for any value y between f(a)and f(b) , there exists at least one x in the interval[a,b] such that f(x)=y. This theorem highlights the connectedness of continuous functions.
The concept of limits is closely associated with continuity. A function f(x) is considered continuous at a point x=c if and only if the limit of the function as xapproaches c exists and is equal to the function's value atc:limx→cf(x)=f(c).
Continuous functions appear in various branches of mathematics and science. Here are some examples:
- Linear functions: f(x)=mx+bwhere m and b are constants.
- Polynomial functions:f(x)=anxn+an−1xn−1+…+a1x+a0where ai are constants and n is a non-negative integer.
- Trigonometric functions: f(x)=sin(x),f(x)=cos(x), f(x)=tan(x) etc.
- Exponential functions: f(x)=ex.
Continuous functions play a crucial role in calculus, analysis, and many areas of mathematics and science. Their smooth and connected nature makes them valuable for modeling real-world phenomena.
Integral calculus is concerned with finding the accumulation of quantities over an interval or area. Key concepts in integral calculus include:
- Integral: The integral of a functionf(x), denoted as∫f(x)dx, represents the area under the curve of the function over a specified interval. It is the reverse process of differentiation.
- Definite and Indefinite Integrals: Integrals can be classified as definite integrals, which find the accumulated quantity over a specific interval, and indefinite integrals, which find a general antiderivative of a function.
- Applications: Integral calculus is used in calculating areas, volumes, work, and various physical and mathematical problems that involve accumulation.