Course

# JEE Maths Tutorial

We 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
• Matrices
• 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
• Vectors

# Sets and Relations

## Types of Sets

Empty, finite, infinite

## Algebra of Sets

intersection, 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.

# Functions

## Function mapping

Domain and codomain

## Special functions

polynomial, trigonometric, exponential, logarithmic, power, absolute value, greatest integer

## Function Operations

sum, difference, product and composition of functions

# Algebra

## Algebra of complex numbers

addition, 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 Series

Arithmetic 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.

## Logarithms

Logarithms and their properties,

## Mathematical Induction and Binomial Theorem

Binomial theorem for a positive integral index, properties of binomial coefficients.

# Matrices

## Matrix operations

Matrices 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 operations

diagonal, symmetric and skew-symmetric matrices and their properties, solutions of simultaneous linear equations in two or three variables.

# Probability and Statistics

## Probability

Random 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.

## Statistics

Measure 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.

# Trigonometry

## Trigonometric Functions and Identities

Trigonometric functions, their periodicity and graphs, addition and subtraction formulae, formulae involving multiple and sub-multiple angles,

## Trigonometric Equations

General solution of trigonometric equations. Inverse trigonometric functions (principal value only) and their elementary properties

# Analytical Geometry

## Two dimensions

Cartesian 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 dimensions

Distance 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

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

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 as$f'(x)$ or $\frac{{dy}}{{dx}}$, 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.
Limit of a function at a real number, continuity of a function, limit and continuity of the sum, difference, product and quotient of two functions, L'Hospital rule of evaluation of limits of functions. Continuity of composite functions, intermediate value property of continuous functions. Derivative of a function, derivative of the sum, difference, product and quotient of two functions, chain rule, derivatives of polynomial, rational, trigonometric, inverse trigonometric, exponential and logarithmic functions. Tangents and normals, increasing and decreasing functions, derivatives of order two, maximum and minimum values of a function, Rolle's theorem and Lagrange's mean value theorem, geometric interpretation of the two theorems, derivatives up

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 function$f(x)$ 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 $x$approaches $c$ exists and is equal to the function's value at$c : \lim_{{x \to c}} f(x) = f(c)$.

Continuous functions appear in various branches of mathematics and science. Here are some examples:

• Linear functions: $f(x) = mx + b$where $m$ and $b$ are constants.
• Polynomial functions:$f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$where $a_i$ 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: $f(x) = e^x$.

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

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 function$f(x)$, denoted as$\int f(x) \, dx$, 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.
Integration as the inverse process of differentiation, indefinite integrals of standard functions, definite integrals as the limit of sums, definite integral and their properties, fundamental theorem of integral calculus. Integration by parts, integration by the methods of substitution and partial fractions, application of definite integrals to the determination of areas bounded by simple curves. Formation of ordinary differential equations, solution of homogeneous differential equations of first order and first degree, separation of variables method, linear first order differential equations.

# Vectors and 3D Geometry

## Vectors

Addition of vectors, scalar multiplication, dot and cross products, scalar and vector triple products, and their geometrical interpretations