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Look around carefully you will notice that there is not a single thing or activity in our everyday lives that does not involve maths. Maths is a subject that requires more smart work, unlike other subjects. It can be a nightmare for some while some can absolutely fall in love with the same. The vastness of mathematics has been ever-evolving in its concepts, and theories have been challenged by famous Indian Mathematicians. 

Let's explore the contributions of some of the famous Indian mathematicians who are recognized worldwide.

Famous Indian Mathematicians

Brahmagupta

Brahmagupta is popularly recognized for his high contribution to the area of geometry and number system. His major offering was the recognization of the number zero and this was trailblazing for the ancient world of mathematics. It is due to his contribution of zero that resulted in the present-day number system in mathematics. The major discovery of his time was Brahmagupta’s theorem for cyclic quadrilaterals.

Aryabhatta

Aryabhatta was a man of vast knowledge and an excellent teacher in ancient India. His ocean full of knowledge has covered areas of mathematics, astronomy, and science. In his knowledgeable treaty named “Aryabhatiya,” he has covered several trigonometry and algebra. He unearthed the concept of the number system and gave an estimation of the value of pi. 

Bhaskara

Bhaskara was an esteemed astronomer and a brilliant Indian mathematician who made significant progress in both fields in ancient times. He carved into the number system and formulated easy formulas concerning the squaring and multiplication of numbers. He laid the strong foundation of mathematics in India through his supreme contribution.

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Srinivasa Ramanujan

Srinivasa Ramanujan is the greatest mathematician who has played with elliptic functions and the analytical theory of numbers. He was selected as the first Indian Fellow of Trinity College, Cambridge University.

P.C. Mahalanobis

Also known as the father of Indian statistics, P.C Mahalanobis has made a major contribution to the field of statistics. His contribution in this field includes the concept of the pilot and large-scale surveys that has had his name listed among the famous Indian mathematicians. P.C Mahalanobis is the founder of the Indian Institute of Statistical Science. Provided his contributions 29th January is celebrated as National Statistics Day in his honor. 

C.R Rao

Calyampudi Radhakrishna Rao is one of the greatest mathematicians and has contributed immensely to the field of estimation theory, biometry, statistical inference, and functional equations. He held the view of putting India on the global map in the field of statistics. His aim was to help brainy students in pursuing their careers in the field of statistics and still, his contributions continue to push them to explore the unexplored. 

Ashutosh Mukherjee

Ashutosh Mukherjee is among the famous Indian mathematicians due to his astounding contribution to the field of mathematics. His major contribution lies in the discovery of famous talents like other individuals like C.V. Raman and S. Radhakrishan. He drafted many scholarly papers on mathematics and physics. He was an important member of a number of mathematical societies across the countries and provided immense research in mathematics. 

Raj Chandra Bose

His passion for the field of mathematics persuaded him to fight relentlessly for what he wanted to pursue. In the face of numerous issues including the financial aspect, he covered his bachelor’s and master’s degrees. Due to his brilliant reach in the subject, he was invited by Mahalanobis to ISI to undertake research in the field of statistics and take a part-time post. The contributions done by Raj Chandra Bose still continue to be valued by the entire world.

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K.R. Parthsarthy

K.R. Parthsarthy broadened the horizons in the field of mathematics and physics and is a renowned statistician, probabilist, and mathematician. He explored quantum stochastic calculus and was then referred to as the pioneer of the same. He dedicated years of his life to this field and made it an aim to devote his life to conducting research in the field of mathematics. As it was rightly said K.R. Parthsarthy was indeed a ‘different kind of mind’.

Phoolan Prasad

Phoolan Prasad has received multiple awards and is recognized in and outside the country for his major contributions to the field of differential equations, and fluid mechanics. He has authored several books that include all his research which is not only distinctive but also has been written from a futuristic blend of mathematics and Physics. His brilliance went on to evaluate the properties of equations of several physical phenomena and put them into use to develop new principles of physics.

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Concluding lines

And here our list of Famous Indian Mathematicians comes to an end. It was these heroes who brought a major revolution in the subject and made distinct discoveries. Their broader horizons made some concepts easy to understand and turned around some concepts to another level. And it is due to these major contributions and discoveries that these are recognized and honored till date.

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To provide you with a sound understanding of the concepts we are bringing you the Class 12 maths NCERT solutions. A good understanding of the solutions is important both for the boards and for the JEE Mains and Advanced as it will equip you with the information related to class 12 Maths. The class 12th Maths includes 13 important chapters that you must be well-versed with. So, to make your board exams easy we have provided solutions that are not just concise but also provide alternative ways to solve some questions. In this article, we will reflect on some of the ways to get solutions to your problems of class 12th boards and JEE exam. 

The CBSE schools follow the curriculum provided by the NCERTs and hence to aid you with the class 12 maths NCERT solutions this is the best place to be. 

Why is familiarizing with the Class 12 maths NCERT solutions crucial? 

Your score on the boards or in the JEE exam is a deciding factor in the college you get admission to. Be it the 12th board or the JEE exam, clearing both of these is not an easy task. Thus, you have to bend over backward to get a good percentage in both. And Math being one of the most hard going subjects requires a little extra effort.

Therefore, familiarizing yourself with the Class 12 maths NCERT solutions will help you beat the exam. 

Chapter-wise Class 12 maths NCERT solutions

Relations and functions

This chapter deals with concepts of relations, functions, domains, and codomains introduced in Class 11. Students are taught about several real-valued functions and their graphs. This chapter includes 4 exercises that focus on strengthening the concepts explained in the chapter. 

Inverse Trigonometric Functions

This chapter involves a constricted array of trigonometric functions that empowers them to be one on one and thus makes the inverse of the functions clear. This chapter digs into inverse trigonometry through its two exercises that acknowledge the students about the behavior of the functions, their graphical representations, and the elementary properties. This later provides great help in integers later.

Matrix

This chapter fills you in on the most important aspect of mathematics, i.e, Matrices. It gives an insight into matrices and Matrix algebra. 

Determinants

After a good intro with the matrixes in chapter 3, chapter 4 proceeds with another important Class 12 maths NCERT solutions which are Determinants. This chapter digs into the determinants of orders up to 3, and their cofactors and inverses. It further involves the applications of determinants, like calculation of the area of triangles, adjoint of matrices, and consistency of a set of linear equations.

Continuity and Differentiability

This chapter proceeds after the  Differentiation of Functions taught in Class 11. This chapter carries on with the important concepts of  Continuity, and Differentiability and puts in place the relation between them. 

Along with this, the chapter involves the continuity and differentiability of Inverse Trigonometric Functions. Students are also introduced to exponential and logarithmic functions through this chapter. 

Application of Derivatives

Continuity and Differentiability give way to the beginning of calculus. This chapter approaches the differentiability of functions and introduction to derivatives. This chapter dwells into the derivatives of specific functions, and their applications, namely the tangents and normal to curves, and the rate of change in some quantities. It also focuses on the increasing and decreasing of the function. 

Integrals

This is one of the pivotal chapters of class 12th Mathematics and questions related to it require more time to be solved. This chapter covers the integration of functions, and their applications in terms of the calculation of areas and introduces integral calculus. This chapter provides a total of 11 exercises making it part of the chapter that requires ample practice. 

Application of Integrals

This chapter deals with the applications of integrals in calculating areas under simple curves, lines, parabolas, and ellipses. This chapter provides 2 exercises that are enough for the practice of the area of calculus. 

Differential Equations

This chapter instigates the concept of Differential equations, the general and particular solutions, and the order and degree of an equation. It will also provide sound knowledge to the students about the applications of differential equations in the six exercises. 

Vector Equations

Concepts like vector quantities, that is, the quantities with both magnitude and direction are covered in this chapter. It focuses on vector algebra that encapsulates rules of addition of vector quantities and other important properties. Vector Algebra plays a crucial role in Physics. 

Three-Dimensional Geometry

This chapter emphasizes the utilization of vectors to calculate the distance between lines, a point, and a line, and a line and a plane, along with the angles between them. It also familiarizes with the concepts of direction cosines.

Linear Programming

This chapter deals with the concepts of linear programming problems, and their solutions with the help of graphical methods only.

Probability

This chapter explores the concept of Conditional Probability and its use in our daily lives. It covers the multiplication rule and the independence of events.

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An equation that contains one or more functions with their derivatives is known as a differential equations. The derivatives indicate the rate at which the functions change at a point.  It is predominantly used in fields like Physics, Engineering, and Biology. The key purpose of the differential equation is the examination of the solution that sits well with the equation and properties of the solution.

The widely feared differential equation is not as difficult. With the usage of explicit formulas given in this article, you can easily solve this type of equation. Let’s discuss the definition, types, and methods to solve the differential equation, the order, and degree of the differential equation, ordinary differential equations with real-world examples, and a solved problem.

Also check: Complete mathematics course

What is a differential equation?

An equation that contains one or more terms and the derivatives of a single variable in regard to the other variable.

dy/dx = f(x)

In the given equation ‘X’ is an independent variable and ‘Y’ is a dependent variable.

For example, dy/dx = 5x

A differential equation comprises derivatives that are either partial or ordinary derivatives. The derivatives represent a change in the rate, while a differential equation illustrates the relationship between the quantity that is constantly changing in regard to the change in the other quantity. You can use a variety of differential equation formulas to find solutions to the derivatives.

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Differential equation: The order

The highest order of the derivative present in the equation is the order of a differential equation. Given are some of the examples of different orders of the differential equations:

• dy/dx = 3x + 2 , the equation is in the order 1

• (d2y/dx2)+ 2 (dy/dx)+y = 0. The order is 2

• (dy/dt)+y = kt. The order is 1

First order of the differential equation

As shown in example 1, the first order of the differential equation has a degree equal to 1. The linear equations included in the form of derivatives in the equation are in the first order. It comprises a single derivative such as dy/dx, where X and Y are the two variables, which is shown as:

dy/dx = f(x, y) = y’

Second order of the differential equation

The equation that comprises a second-order derivative is referred to as a second-order differential equation.

The second-order differential equation is written as:

d/dx(dy/dx) = d2y/dx2 = f”(x) = y”

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The Degree of differential equation

The power of the highest-order derivative wherein the original equation is characterized in the form of a Polynomial equation in the derivatives like y’,y”, y”’, and so on, is directed as the degree of the differential equation.

Example:

Assume (d2y/dx2)+ 2 (dy/dx)+y = 0 to be a differential equation, therefore the degree of the equation presented here is 1.

Given are some more examples of the same:

• dy/dx + 1 = 0, degree is 1

• (y”’)3 + 3y” + 6y’ – 12 = 0, the degree is 3.

• (dy/dx) + cos(dy/dx) = 0; the given is not a polynomial equation in y’ and the degree of such a differential equation cannot be explained.

Note: The order and degree are always positive integers of a differential equation.

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How many types of differential equations are there?

Differential equation can be classified into the following categories:

• Ordinary Differential Equation

• Partial Differential Equation

• Linear Differential Equation

• Nonlinear differential Equation

• Homogeneous Differential Equation

• Nonhomogeneous Differential Equation

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Differential Equation Solutions

The solution of the differential equation is the function that satisfies the given differential equation. Whereas, a solution that contains several arbitrary constants as the order of a differential equation is referred to as a general solution.

A solution that is unchained from arbitrary constants is called a particular solution.

There are two ways through which you can find a solution to a differential solution:

1. Separation of variables

2. Integrating factor

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Separation of variables

This approach is used when the differential equation can be written as dy/dx = f(y)g(x) where the function of y is f and the function of x is g. Considering this as an initial condition, note this problem again as 1/f(y)dy= g(x)dx and then integrate on both sides.

Integrating factor

This technique is put into use when the differential equation is in the form of dy/dx + p(x)y = q(x) where p and q both are the functions of x only.

The first order of the differential equation is shown as y’+ P(x)y = Q(x). where x has p and q as its functions and the first derivative of y.

An equation that contains derivatives of an unknown factor that can be either a partial or ordinary derivative is known as a high-order differential derivative. It can be presented in any order.

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Applications of Differential equation

Differential equations have their uses in several fields such as applied mathematics, science, and engineering. Apart from its uses in the technical field, it is also used to deal with real-world problems.

Let's familiarize ourselves with the uses of differential equations in real life:

1. It is used in describing various exponential growth and decay.

2. The differential equation is helpful in calculating the return on investment over a time period.

3. It is of major use in medical science in estimating the growth of cancer and the rolling of disease in the body.

4. It is also used in describing the movement of electricity.

5. Economists use it in discovering the optimum investment strategist.

6. Differential equations are helpful in describing the motion of waves or pendulums.

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The other areas of its usage are heat conduction analysis, and understanding the motion of waves. The ordinary differential equation can be used in the field of engineering for finding the relationship between various parts of the bridge.

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When planning to pursue your future in engineering the first question that pops up in your mind is what is the difference between JEE and IIT? These two terms are used along and are thought to be the same, but contrary to this these two terms might be interconnected but hold different meanings. 

Keep reading the given write-up to get thorough knowledge of the difference between JEE and IIT. 

What is the difference between JEE and IIT?

When talking of JEE, it stands for Joint Entrance Exam, it is an all-India level entrance exam that you are required to clear if you want to pursue engineering from the top engineering institutes of the country such as IITs, NITs, IIITs, and various other CFTIs. 

The JEE is a two-part exam, i.e, it is conducted in two parts. JEE Mains and JEE Advanced. JEE mains is the first-level paper on clearing which with a good score you become eligible for NITs, IIITs, and CFTIs. The top scorers of the JEE mains are eligible for the JEE Advanced cracking which with a high score you become eligible to apply for a seat in the IITs.

Now coming to IIT, it stands for Indian Institute of Technology which is the name of a highly desirable engineering college across the country with its branches situated across various states.

In other words, the major difference between JEE and IIT is that JEE is the exam that you take to get admission to IIT colleges.

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Does the state JEE exam differ from the main JEE exam?

You need to be very clear about the exam you will be appearing for, making an elementary mistake like this one can be a huge blunder for you. Confusion may occur because there are other entrance exams like the state engineering exams of West Bengal, and the West Bengal Joint Entrance Exam that has a postfix with the JEE. 

The JEE exam is conducted for courses like BE/ BTech, B. Arch, and B. Planning.

Also read: Best way to prepare for IIT JEE

More to the difference between IIT JEE and JEE

Still, confused about the exact difference between IIT-JEE and state JEE? So, let's burst this bubble of confusion. IIT JEE refers to the exam that you take to get admission to the IITs. Make sure that you do not get this mixed up with the joint entrance exams of other states. The exams of other states come abbreviated as JEE too but with the prefixed name of the state. For example West Bengal JEE. 

JEE is one of the most back-breaking exams conducted in the country. Around 9 lakhs students appear every year for the same but only around 12,000 get qualified for the top IITs. you need to work finger to the bone in order to get admission to the top Engineering colleges. 

This is the key difference between JEE and IIT.

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How are JEE Mains and JEE Advanced different?

Now that you are well familiar with the difference between IIT and JEE, now is the time you familiarize yourself with the key difference between JEE Mains and JEE Advanced. 

If IIT is your dream college then you need to appear for the JEE exam which is conducted by the National Testing Agency (NTA). You can apply for the JEE Advanced only if you clear the JEE Mains with a good score.  

The structure of the question paper of the JEE Mains and JEE Advanced is not considerable but the difficulty level varies.

What happens after the JEE exam? 

The JEE main exam is the first exam in the procedure of getting admission to the top IITs. After the announcement of the result, you will be allotted an All-India Rank. Given the rank, you will be eligible for the following: 

1. You will be eligible for any of the 32 NITs. 

2. You can apply for any of the 18 IIITs

3. Eligible for admission in any of the 19 CFTIs. 

4. You can apply to any of the participating engineering colleges. 

5. With a good score in the JEE Mains, you will be eligible for JEE Advanced.

Also Read: How to prepare for JEE without coaching

Eligibility criteria for NITs, IIITs and CFTIs

The Central seat allocation board is in charge of allocating seats in NITs, IIITs, and CFTIs for B.E./B.Tech/B.Arch/B.Planning courses.

You need to meet the given requirements in order to be eligible for the NITs, IIITs, and CFTIs: 

1. A qualifying All India Rank in the JEE Main 

2. 75% marks in the 12th board's examination

3. The students must rank among the top 20 percentile in class 12th boards 

4. The SC/ST students need to score 65% in the 12th board's examination 

Eligibility criteria of JEE Advanced

• Candidates belonging to the general category must qualify for the 12th board or equivalent qualifying examination with a minimum of 75% marks and the students belonging to the SC, ST, and PwD categories must score 65% marks in the same. 

• The candidate must be amongst the top 20 percentile of their respective Board examinations to be eligible for the JEE Advanced exam. 

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Eligibility criteria for the IITs

To be eligible for admission to the IITs you must fulfill the specific criteria. The eligibility criteria for admission to the IITs have been mentioned below: 

1. Age limit

2. Number of attempts

3. Year of passing the 12th board or equivalent examination

4. The other major criteria for admission to the IITs is that students need to qualify in  JEE mains and JEE Advanced both. 

The path to IITs demands serious hard work and consistency. Now that you are clear with the difference between JEE and IIT so, grab your spirit together and begin your preparation with a positive outlook. 

FAQs

What is the best way to prepare for entrance exams like JEE?

Best ways to prepare for IIT JEE Exam

• Get acquainted with the JEE 2023 syllabus

• Familiarize yourself with the JEE exam pattern

• Form a JEE 2023 preparation study plan

What are the easiest and hardest subjects to score in for the IIT-JEE Advanced exam?

The hardest subject of the JEE Advanced is Physics which is followed by Maths. Given are the important chapters from both subjects.

Important chapters Physics:

• Mechanics

• Modern Physics

• Electricity and Magnetism

• Optics

Important chapters Math:

• Matrices and Determinants

• Probability and Permutations & Combinations

• Coordinate Geometry

• Definite Integrals

• Vectors and 3D Geometry

What are the benefits of taking the JEE Main exam for an IIT aspirant?

Given are the top benefits of clearing the JEE exam:

1. By clearing the JEE exam with a good score you will get admission to the top IITs in the country

2. Preparing for this exam will provide you with enough wisdom to clear other engineering exams.

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