 This is not recording, now it's recording. Good. Hello. Sorry, sorry. Okay, so yes, Raghav. You. Anand. Limitless. Aadaksh. Aadaksh. Ideal. Stutthi. Stutthi means worshipping. You don't know how you must have known that. Aarind. Aarind means Aarind Grace. Aarind. No, the rest, North Indian Grace. They came. There is an Aarind in division theory. But anyways, I don't know whether it's true or not. What's your name? Vedhansh. Vedhansh. Medhaansh. So, Anshok Medhaav. Medhaavi. Talent. Part of talent. Become full talent. Okay. This is a disaster. So, Vaishnavi. Vedhansh. Aksh. Aksh means unheard is Aksh. Chak means heard. Aksh means which? You can't be? Third. Third. Okay, I forgot your name. Horse. Oh. I keep keep on eating. And keep on eating. Then you stay away. Okay. Physics was my part-time job. Last time. Who are these people? Ten. So, 10 grade mathematics will be a little interesting. And as I told you, mostly mathematics classes will be clear on. Okay. So, we are starting today. And we will be finishing your course by the month of June. So, 10 grade mathematics will be covered by June end. And then we will take you to 11 grade another. December. So, we will be good with 11 grade mathematics. Then we will come back again with 10 grade mathematics. And this will happen for all the courses. So, that by the end of March, you should be also acing. Sanctuary and other spectacles. Subjects as well as when you, the idea is when you come to 11 grades, you should get the ground running. So, there will be a dedicated patch for, this is a plan for all of you. That if whatever we have planned goes well, then the movement you join in 11 grade, your 11 grade service will be covered by June end itself. You will be done with your 10 grade courses. So, that the last and final year will be spring. You will have to run. And the expectation is those who will be writing J, it should be within top of all the line. And those who will be writing J, should also get your respect. You know, good qualities. And that's the only strategy you do it. In 9 grade, it was a little easy. Yes, we went a little easy. And we never looked care whether someone is in assignment and things are happening on time or not. But this year it will be a little more serious. And I would require all the movement as you have started showing about those glimpses toward the end of the last year. There will be prints and all that. That's another part of it. But yes, that grade will be, first of all, you have to take out from this, this notion from your head that there will be something which is a milestone year, but it's not. No more. So, don't give high power on 10th grade. Yes, we will be happy about that. Waste, so hence, from day one only to a little serious. So, don't worry about who takes part. We'll take care of everything. So, I'm anyways there. I'm connected to you individually as well. So, those who are, now please come. I don't know who. I think I'm not here because I will keep you. So, yeah. So, we understood the idea. So, we will be going pretty fast. Rapid fire. Why? So, I believe 10th grade, since it is now currently where we have to take care of so many people who are under privilege as well. So, hence, we have deliberately kept 10th grade motions couple of notches less than 9th grade. But in that course, what happens is you get misdirected and you think that this itself is so much. So, let's not keep that at least you should not be having this kind of mindset at all. We will be running fast. I will be in mathematics or physics for that matter. Whoever takes it. I will be helping you. So, you can connect with me any time during the day, you know the pattern, you know the text. Last time, logging was voluntary this year. It is compulsory. So, if you don't talk, you are out. Out means out. Don't come to the class. I will be very firm and very strict. I will teach only one. But listen, I will teach only one. But that one should be the best. So, if you have any problems, I know you've got it out, I don't really care. But this batch will be for all those who are going to perfect themselves for one year with me. And rest assured, you will be much ahead of time. For a 90-year cooperation. If you are lacking in sincerity, you will be out. No, nothing like last year. You will be sovereignly told not to come to the class. And yes, if you are struggling, you know, I have always been with you. Can you be your best friend? Because last time, mathematics first. Anyone knows what mathematics is? Subject. Subject which is putting us under tremendous misery. Just get rid of the subject as quickly as possible. So, how many of you have hatred for mathematics? Be open. Hatred for mathematics. Hatred. No, there's no part of it. I hate you because I hate you. I don't partly hate you. So, yes, be frank, no worries. But I wouldn't... Yeah, be open as up. So, I don't like mathematics. I didn't use the word don't like. I used the word hate. But yes, I wouldn't... I would rather be first of all not to be biased against any subject. It might be that the way of acquiring knowledge is, you know, or attitude of... Never mind, I'll try to turn it around. Okay, whatever, small knowledge I would be having. So, but then the question is, what is mathematics? So, are the NCIT problems, problems, limitless? What is mathematics? Theories of mathematics. Study of mathematics. Agree? What's your name? Midanj. Yeah, Midanj, you only need that study of mathematics. Study of numbers is mathematics. Everything is numbers. Not limited to numbers. They'll be saying, not limited to numbers. Then what else? Mathematics. Mathematics is mathematics. Oh, okay. Some dalmo, the peanut. You have mixture, right? Very tasty mixture. Okay, so it's a mixture of coefficients and variables. Yes, geometry. Grabs, what else? Trinometry. Okay, who started studying mathematics? See, first of all... You mean who discovered? Now, Shruti is asking who discovered. Now, tell me whether mathematics was invented or discovered. So, mathematics is discovered or invented? How many were discovered? Why do you keep looking at each other? I mean, your thoughts are your thoughts, right? 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. And rest of all, invented. How many were invented? Oh, you are up that many times, huh? Don't tackle in the middle. Invented. Yeah. Invented. Invented. Invented. Invented. Invented. 4, 5. More discovery. Why discovery is not invented? Numbers. Who? This one. Is it invented or discovered? Invented. Discover. This letter. This one. Invented or discovered. But discovery. Invented or discovered. Invented or discovered. So, 0, okay. See. I understand. So, hence, I won't die now. Listen. So, I'll take you through the history of mathematics first. Before we delve into what and why there is a game. So, sooner you will be studying in your similar time as a doctor. There is a guy called Theliz. Are you understanding? Theliz. Theliz. Theliz, not Theliz. Theliz is the wrong pronunciation. It's Theliz. Theliz theorem. Basic proportionality theorem. I don't know if you are aware of that. So, he was one of the most important. I don't see notebooks on a desk. So, no notebook. So, you'll have to maintain a notebook. Can I sit here? I charged 1000 rupees for this. You didn't even sit. Sad. Okay. Come on. Notebook. No notebook. You are thrown out. Okay. Write the date. Don't you guys are? What's the date? 12 March 2009. Yeah. Write the date. It's your mathematics. Then we will tell you the equations of what and why. So, yes. So, I was talking about this. While we were discussing physics, also we had this discussion. Oh. Which all civilizations existed? Which all civilizations do you know of? Indescribable. Indescribable. Indescribable. Who loves history here? History? Love. Yes, love. Yes. I like history. You like history. You don't love history. History? No, I love the history. No, NCRD is not history. Don't compare that. Okay, listen. Anyway. Now, listen, listen. Now I tell you the story. Listen to the story. So, there was a king. There was a queen. Not a story. So, there were four civilizations. Not four, exactly. Five, actually. Who picked up and where? Indescribable. Okay. This is the story. Indescribable. Thank you very much. Indescribable. Indescribable. Yes. 1,600 BC. 1,600 BC was Indescribable. Not connected. Indescribable. Indescribable. Around 5,000 BC. Indescribable. Okay. So, from now, around 7,000 years back, there was a civilization that had not been established beyond now. But whatever evidence that we have got, it appears that Indescribable age existed around 5,000 BC. Okay. Now, any other civilization? Mesopotamia. Mesopotamia. Which age? Today. Iraq. Iraq. Yeah. That was called Mesopotamia. Right? What now? Egypt. Egypt. Where is it? Egypt. Southern Africa. And Southern Europe also. What? Mediterranean. Mediterranean. Mediterranean. Yes. So, it was not an Mediterranean sea. So, that was Egypt. What now? East China. China. Okay. Yellow river. Young sea. Young. Have you heard of? Yeah. So, there was that civilization. And? And? Now, Greek. Greek. How many in Greeks? If you see, what do you know about Greek civilization? So, they are the perfect city planning. Yeah. Yeah. Perfect city planning. That means drainage. Roadways. And. Market environments. Yes. All greeneries. Public parking. And all that. And bricklaying structure. All these were characteristic features. Royalty. Right? Egyptian. Pyramids. Pyramids. And, anyone has visited Egypt? Egypt by chance? Yeah. No. You? You have seen pyramids? Tell me some characteristics of pyramids. First thing which gives you awe. What is its size? Magnitude. Magnitude. Magnitude. My God. Expands of it. And this was before any crane, helicopter or anything like this. Right? One stone block of tons. Have to be taken to, let's say, 50 feet high point. How was that possible? Without having knowledge of inclined planes, geometry, angles and all that. Do you think it was possible? Very difficult, right? Can you lift up all of you together? Can you lift up stone of what type? No. What do you need? The questions were busy in, let's say, applying the knowledge. There was one civilization which was busy in building the... Building the body of knowledge and that was... So Thales existed around 7th century BC. Right? And how it happened was people didn't have internet that time. No Instagram. Don't be on Instagram. I'm stalking you. So there was no internet. There's no internet. Listen, listen. All of you. So what was happening? People didn't have much job to do. So what did they do? They were raising stars. Yes, actually. And that's how astronomy came into being. And astronomy is mother of many, many sciences. And till today, lots of sciences are coming out of astronomy. And today was a scientific across astronomy. Geometry. Direct follow up of astronomy. Why people were interested in raising the stars? The movement of the stars. And how are they behaving? Because that time religion had a lot of say in human civilization. And people used to think that all these bright stars up there in the sky, the moon, the meteorites and all that are messengers of God. And they are trying to send a message which we humans must decipher to take ourselves to the next level. Okay? There was lightning and people used to get, you know, frightened. That maybe God is angry. Hence lightning strikes somewhere. Some people get killed. You understand? And they take it as, okay, we must have done something. And hence God is taking over. He's punishing us. Correct? So that's how it all started. Then came, you know, I was talking about these green people. Who started putting brains behind whatever was happening. And you remember, I gave you the story of premium and quadruped last year. Anyone remembers, premium? And quadruped media tells the story of premium. Okay, there were three subjects, four subjects in green schools. Do you remember? Okay. There, premium, the word, premium, premium comes from the root word, premium. Dreaming means very common. Very, you know, less important in current parallels. You see, it's less important. That time, premium was a set of three subjects, which every green student has to go through by default. Right? For example, 10th grade is considered to be by default. Everyone has to get a matrimonial certificate. Same way as in green schooling, premium. What are subjects? Logic. Then grammar. Then metaphysics, which is called religion. Somewhat related to religion. Religion, right? So these are the three subjects people used to study. Right? So logic was the key word. So people started asking questions. Why are there stars? Why is this? What is this? You know, so flat surface where all of us are inhabiting. So people started asking questions. What, why, where, where, like that. And they started exploring where have we come from. This led to the, you know, the work of subject of philosophy. Aristotle, Socrates, Plato, all these people, they have started thinking where have we come from. And today, whatever physics you study, let's say, gets back to Big Bang. By elaborating on this question, where have we come from, where have we come from, where have we come from. Now today we know that we have come from 30.7 billion years ago. There was a big explosion somewhere in space. And that's what we know so far. Yeah, we don't know. Before that one. Right? This was the question. And it took around more than 2,000 years, 2,500 years for us to come to this conclusion. Okay? So, came Thales, then came a guy called Pythagoras. Who was it? You know Pythagoras, what for? He is the one who discovered it. He invented it? No. He was the guy who proved it. Aristotle was no El Mesopotamian in 15th century BC. Pythagoras existed in 5th century BC. All the years before Pythagoras was on this earth, people were aware that in the right angle triangle, the sum of the squares of two sides will be equal to the square of Pythagoras. Pythagoras and his sect, he started as a sect, Pythagoreans there. They used to call themselves Pythagoreans. And it's a very closed group. The entry was not easy. These, this bubble group, what? Pythagoras? Yeah. And now you have, you have seen Pythagoras doing this guy and all that. Animations? Yeah? You know this, what I'm talking about. Anyway, so that was Pythagoras. Then came a very famous guy. You know him? He's a man of manners. I did. You did? When was he? 1st century BC. What was happening in India then? We were trying to defend Alexander. Mauryan Empire. Mauryan Empire. Chandramukta Maurya was the king in that time. And then Alexander had been made in India. And we were trying to defend our own land and all that was happening in Indian context. Meanwhile in Greece, this guy was busy writing the book, which has become a problem for you guys The name of the book was? The Elements. Are you aware of this name? The Euclid Study Mathematics 98. Yeah sir. Euclid, Euclid's geometry was there. Yeah. So he wrote a book called? Yeah. And he so studied all the books in which he has covered number theory, he has covered geometry and what not. And he was the, and you will be started and you know, astronomers will know that Euclid's Elements is the second most popular book after the Bible. Those many books, puppies have absorbed both of them in the world. Very close. The very popular book which he wrote was a, and this is the book which lots of star words have studied and changed the world. You can be one of them. Fermat, FDRMT. People call it Fermat. It's Fermat. So lots of star words, you know. Then all these, you know, all these Greek philosophers, philosophers, you know, Archimedes. So Eurodontists, then this guy called Euclid himself, then Archimedes. All these people, and at that time it was not, it was just logic. People were doing logic and they were coming up with several theories and all. Then came Christ and things changed forever. Why? Because from 1st century BC, sorry, 1st century around that time, till 15th century AD, a span of 1500 plus years, 1600 years, it's called dark age. Why? No development in mathematics. And yes, here and there, for example, in 10th century, yes, Indians were developing their own set of, this thing, so around 7th century AD, zero count came up. So till 6th, 7th century AD, no one knew of zero. No value of zero at all in any of these mathematics. You know, of literature. So there was no zero, no talk about it. So 6th century, you know, right? And how zero came into being, nothingness was now represented by a symbol. So coming back to that 1 and 2 and 3, these are all philosophies. These are all invented stuff. Oneness is a philosophy, unit is a philosophy. You represent it by a symbol. Isn't it? It's too tight. It's a philosophy. There are two clear but distinct things, or maybe ideas. All these are philosophies. So hence, to represent this philosophy, you came up with a symbol. And now to make it two. In Roman system it was I-N. I-N. So this was the story and then what happened? So till that time, people were busy or before Christ, all these Greek history times, people did only two things. One was geometry. Another was number theory. Today what you study as real numbers and number system was part of that number theory. You could also do a lot of work on number theory. Prime numbers, composite numbers, perfect numbers, this, that, number theory and geometry. So all you, what you learn, Pythagoras is theorem and similar triangle, circle, tangent, sequence, everything was developed before Christ himself. Okay? 10th century AD there was a guy in Arab world. And Arab world was also very much rich in mathematics. They came up with a very unique or very important topic in mathematics. That's why. That's why. Algebra, right? The word algebra itself, it says A-L. So where do you see A-L part of it? Al, al, al, al, al, al. Everywhere al is in Middle East. So algebra came from there. And the root word is algebra. Algebra. Where you have unknown. So I don't know. So there was a guy for Rismi. What's the name? A-L for K-H-W-A. Unknown. And then he took Indian numeric system including zero to Arab world. Using our 1, 2, 3, 4, 5, 6, 2, 0, this number system. Then there was another guy in 11th century AD called Fibonacci. He was an Italian. He came to Arab to understand mathematics. He figured out, oh my god, there is a very brilliant value system which is used in Arab world. Now it is called because we have proved that the source belongs to our own system. Then he took this system to western Europe. But no one paid attention to it. Till 16th century when there was a famous event which took place in Europe. What is that called? Renaissance. When everything started, everyone started revisiting old concepts. And all those Euclidean geometry all that Pythagoras used to have and everything started. The dark age is over. The church controlled our civilization. This kind of over people started revolting. Luther King, Kelvin, all these people came up. They started denying whatever church was trying to impose onto them. Everything started and hence came a guy called Galileo and he changed the world forever. Then came a very very famous guy Mr. Isaac Newton and he devastated all your life. Destroyer. But yes, we are enjoying the benefits of whatever theory he gave. And after that there was another very famous mathematician Boiler who came up in 17th century he gave the concept of something called imaginary numbers. So how can math be discovered? So there is dichotomy. So there are places where you think that it is invented and there are places where you think it is discovered. So people, mathematicians have two schools of thought. How else do you say it is? There you go then. You are just in nature. You are just empowering it. Right? Sometime back I had posted this thing on my Instagram. How many of you have seen that? There are four pictures in one. Anyone noticed it? You guys are very good guys and you are from social media. Don't follow me. That's a sin. Follow. Stay out of my Instagram. In case your mom and dad have come to see that. So basically what is common between... Have you seen lightning strikes? Have you seen a river tank? Yes. That's so so overwhelming for us. There is so much of similar patterns in nature. There were fractals when you go up in high mathematics. You study. See, mathematics don't find mathematics in books. Books are just a small representation of mathematics. Find mathematics around you. Don't find mathematics in books. It's not there. It's outside it. Yes, we have to go through a particular method to arrive at you know let's say you know mathematics but mathematics is not there in books. Right? So in the last 300 years we have developed a lot of mathematical concepts. These are the branches of mathematics you study. Number theory, which in your this thing is called number system. I think in autonomous syllabus it's not there. Real numbers? You have real numbers? Yes. Anyway, so real prime and all that stuff. Then there is algebra the part of which is linear equations which we will be dealing after this. Right? Algebra in algebra, polynomials linear equation, quadratic equation and inequalities all are in. Algebra then we also study something like we will be covering these things in this year. Minority theorem then principle of mathematical induction then you know what else is there. Theory of equations are there. All this is algebra. Then there is a full-fledged body of which is called geometry where circles, diagonals and all properties are related to that in this study. Then geometry evolved into something called analytical geometry. There was a philosopher in France called René Descartes I included last time René Descartes and he was a very famous philosopher. Cartesian coordinates come the name Cartesian comes from right Cartesian C-A-R-P-E-S-I-A Cartesian comes from the word René Descartes and this guy again you know this he is the guy who fitted mathematics into anything in the world. So for example if you see this beam, this line René Descartes will occur here if you see this strand there is a curve there is a mathematics all the shapes. Every shape he linked geometry to algebra through a body of knowledge called which you call as coordinate geometry but it's got an analytical geometry so when we study analytical geometry as well there is a very famous branch a very important branch of mathematics called property statistics where we also study something called the art of counting counting something without actually counting it so 20 people are there how many handshakes are possible you can't really go and count one handshake, one second handshake there is an art of counting it and we will study that and there is a very famous principle which we will be studying that is called principle in that component in itself. These things please hold your nerves are not there in your 10th grade syllabus but I think we have enough time to not only cover your 10th grade syllabus beautifully and I expect all of you to score minimum of 99% in mathematics and then we will also touch upon elements of calculus where we may be discussing functions differentiating continuity limits and then integration see if you guys are really panicking then don't do it because you are a counterpart you must be knowing those who are in math IGCC counterpart already do it you know that right so hence if they can do it and the idea is simply this those who will show promise promise and says I will see that commitment in you rest assured if we cover 11 grade courses of mathematics this year itself next year you will directly set it to a grade class you will just skip 11 grades we will make sure that happens and you are on the go on the way to let's say top 500 ranks in J very much possible if not under 100 that's my dream I want to get published my name should be published in time for video so you help me help me get on to time for video sometime take it for some what is that Kada paper Prajavani Prajavani also Prajavani also help me do that is that clear to you 28 weeks program of all the PRs it will be directed towards region of you are not PRM PRMO is a exam if you are preparing for PRMO don't sit for hour okay so this this preparation again only will be done by those who will be selected for entrance process we will tell you the dates and you have to write one exam and if you are really good enough so you will be around there NTSC classes will be covered through our regular classes as I told you by June your syllabus will be over mostly maximum in July NTSC exams happen in October late October long work and once we are done with our CPC 10th course we will be our exam revision for 9th grade so sadly stated care of math exam is open to all sorry math classes but it will be conducted online so lots of work sheets every week we fix our machine so please keep your slots free Saturday second half Olympiad classes will happen early morning Sundays so those who are really early morning 6 to 8 Sundays so 6 to 8 if you are ready to get up early we can start at 4 o'clock also so math is all over here okay so we ready to embrace all such things okay and yes I will make you put a lot of hard work this year so if you are having backache headache if you are going to multiple classes my request to you is hold it for this year don't play badminton forever play for half an hour play but don't play for 5 hours yeah you have to see I will be very very blunt about this you have to re-practice your training and once you start re-practicing you will see the results so that's my point my entire focus will be at least 5 people should crack RMO from this batch and regional mathematics all over here we are preparing courses for Indian national mathematics all over here so those who will be clearing up will be getting enough order for I&M as well after I&M anyway they will be getting care of by the cap so they don't have any responsibility so we will be helping you but that's nothing I want all of you to crack it I have a bottle bottle and I have kept it on my table okay the bottle is made up of glass and if you do like this it will break okay this is good enough for all of you okay so if you make a mistake do not clear NTSC then we do it so let me not put it in on camera so you understand guys it's very much global it's very much global people of your age have only done it and it's a very close end date program it cannot be easier than NTSC let me tell you you have wrong myths in your myths are always about NTSC so in your head I will tell you people tell you how to do it you have to walk the path I can't walk the path it's very much very easy all of you can crack it math is your strongest point you can crack 95 plus 100 we are even going to just imagine I am recording history lectures like me history just for you so that NTSC to check possible you guys don't want me to be there I have a see I want to die with this that I want to be there on newspaper for good reasons these are the exams not because you just go wait and kill someone okay promise if not done and you know possible or not anyone believes it's not possible don't tell me that you believe that it's not possible very much possible you only said it's not possible last time you were saying so as you are studying linear equations so we will go directly to work sheets which I went for 15 so we have so there are also activity sheets inside so this is for your observation skills okay you have to do that sheet follow sheet signature so there will be I have communicated that there are reports are automatically sent everything yes so that will be there already uploaded so please be there okay so the moment I press send the sms button so now it's all scheduled so automatically on that particular why are you pushing to say please send it isn't it and that's what the class is for okay so there are also activity sheets when you go back you please read them and then you will come back to me anyways next class we will discuss that I want to start with linear equations today very quickly we will be solving linear equations and we will be telling you our different aspects of linear equations so linear equations are this is the load type yes something called have you studied this yes what is the polynomial thanks for highlighting I don't know it's very much visible polynomial what is polynomial freedom more than freedom so any combination of combination means not plus or minus not of some kind difference but product of how do you give freedom that's what happens so I guess you have to take the more that leads to that here when the particle is down here the power is a big difference the setting the particle setting what happens if the particle is down it is going to change how do you make it to be linear is why why we need power see now down is clear So this is not my individual term, there are terms that are clear. What is expression? What is expression? Without an equal design. Well, I have not got an equal and all that stuff like that, right? More than one term added together. Now you say why can't I use minus subtracted together. Subtraction is nothing but negative addition, right? So added together. Example, x plus 3 is an expression, right? Blue x plus 5. Expression, yes or no? Yes. For 3 plus 5. No. It's an expression, yeah. It's a critical. You can actually simplify it like this. Right? Now. Expression is clear, right? Now what are we seeing? There are 3 components. 3 components, please understand. Let's just say when I say blue x, it is, actually I am saying 1 and this dot represents multiplication, not a decimal point here. So 1 is 2 x to the power of, 3 things are there. This one, this part? What is this part? So efficient. So efficient. Or constriction? Very efficient, very efficient. And what is this part? Easily. Next slide. Convection. Convection in algebra. In technology, the definition becomes change. Sine theta, sine x, sine y, all that. Right? So this is very good. Now. What is now a polynomial? How is, so we started with what? We started with tau. Then we get into the expression. Now we are talking about polynomial. Yeah. What is polynomial? Yes, no. Yeah. So polynomial is characteristic of polynomial. How is polynomial different from expression is simply this. In any expression, the power of the variable can be anything. Can be anything. Real number. Especially you cannot have powers, anything but integers or others. So what is a polynomial? How we denote polynomial is simply like this. When we need this function, I will go much deeper in that. But for the time being, you remember p and within basis, I am writing x. That means there is a polynomial whose variable is here to everyone. This is polynomial. This is equal to a naught. A naught is a constant. naught is just to put an index to understand. I will not use the a, b, c, d, e, f, g because there could be infinitely many terms in a polynomial. Right? But alphabets are only 26. So I am putting an index. 0, 1, 2, 3, 4, 5, 6, like that. So a naught plus a1 x to the power 1 plus a2 x to the power 2 plus a3 x to the power 3. And then we say a n x to the power x n. How many terms are there? Plus 1 terms are there. Is it n terms? Then an extra a1. So there are n plus 1. Don't, where I am saying, they are not comfortable, but for 19, understand. Wherever you see this sign, this is called? Sir, we have that. Real. That plus means? Integer. Integer. So I can say, it belongs to... See, actually in mathematics, we really don't use it. We also explain the use of that. But otherwise, we talk only in terms of integers. So you will say, it's all non-negative, non-negative. So this is how the technique is done. Non-negative. Where do you come? Down there, polynomial here. Yes, it also depends on poly means. Poly means? Many. Poly means? Many. Many. Where else do you come? You have come across this word, poly. Polygon. Polygon. Polytheme. Polygon. Polytheme. Polygon. Polygon. Hence polythemes club together. Polymerization. And hence we get polytheme. Many. Polygon. Polyominal. So polyominal is one and so forth. So everything is called polynomial. Clear? Clear? So this is polynomial. Now, I am interested in seeing... I am interested in visualizing the polynomial. Who made your life miserable. Hello, I got a great record in France. He came up with a very beautiful mechanism of visualizing mathematics. And he called it Cartesian target system. And hence now I am going to visualize visualizing. What two things do you need for visualization? And? What is that? And you need something to see. Yes. I need the tools. Take that. You can do it. You can do it. One is the picture. Four. And? Nice. I... Without the polynomial function. What do you plan? We set the brain. Yeah. And then write. And write. And write. Write. Write. Write. This is it. Come. Okay. Okay. How long? So... Yeah. Hey, where would you like to have dinner? There. Something to eat? Yes sir. Now what is that? Dolly pop. Dolly pop? Dolly pop. Oh, so what? What do you mean, candy? No. Any way is good. Any way is good. What do we need? Examiner. Why? What do we need to learn? Can we learn? Can we learn? Can we not? Some other language? No. Why? You read perpendicular. Why? Who says? No, he didn't say that. He never said it. Yes. But it doesn't say... Most popular and most standard form today is Cartesian. It's called right-hand orthonormal Cartesian coordinate system. What is it called? Right-handed orthonormal Cartesian coordinate system. Why is it called like that? Right hand. Right hand. Right hand. Right hand. Right hand. Right hand. Right hand. Right hand. Right hand. Right hand. Right hand. Right hand. And the coordinate system which we use today. We call it right-handed orthonormal Cartesian coordinate system. Every word has a meaning. Cartesian coordinate system. Orthonormal means usually numbers together. So hence, coordinate system is the system of this particular coordinate system. Now, why right-handed? Because... Now if you say x and y, we know that there are three coordinates. Authorities? Z. Z. Now whether z should come this way or it should be inside the corner. Sir, you are right-handed. For that, you mean? Some will give you zero x. Hence, it is called right-handed. Right. So whenever you get x... Sir, how then... If you take a negative x, this will be all the other x. Yes. If you take x here and y like this, then you put x, y, and the stress come. Oh my God. Stress come. Coming out of the board. Hence, it is called right-handed. We have to have a standard. Right. See, any-handed talks about something and then no one understands. Because it's not standard. Right? So, standardize your communication. So, I don't mind. I just keep putting the lines. So, x, y, z. Correct? Understood? Now, when this coordinate system is in the same corner as x, y, z. See, any corner is like that. That coordinate system is x, y, then x, y, z. X, y, z. Here, program up. And we have it so far. Now, what this rendered card set is, the moment you fix a coordinate system anywhere, what does it mean? Let's say I want this point to be my origin. And I define x. I define y. And I define z. Then each point in space can be uniquely defined using three numbers only. Yeah. Each point in space. All of your city. All of you have now a set of three numbers which define your position. Correct? That's the basis of definition of point in a partition coordinate system. So, we are now going to the third dimension right now. We will rest in all our analysis in two dimensions. Is that okay? So, hence, we are not considering, in your 12th grade, you will study 3D geometry as well. But right now, 10th grade. And if time permits towards the end, we'll go to 3D geometry. Right now, we'll do only 2D geometry. So, x and y. So, any point in that plane, Now, which plane is it? The plane of that arrow. This board. Right? Which is defined as x, y. Clean. Right? Because x and y are the time in which this plane. Any odd point that you draw, and you draw a perpendicular, means orthogonal system. And measure this distance. So, this is far by y. Far by x. And this point p is denoted by x, comma. Y. This is what our color, p. Because if you reverse x and y here, some other, another point. You reach another point. Correct? This is called Cartesian coordinate system. Clear? Everyone? This is point. Yes. Any doubt? Talk to me. Talk to me. Yeah? Any doubt so far? Anyone? Please talk to me. No. Okay. This is called apsica. Distance from y. Yeah. How far is this point from y? Axis. This part? x coordinate. Okay. How far it is from y part? y axis is, x axis is far? y coordinate. Is it clear? Clear? So, hence, this distance is x. So, here, x, comma, 0. This point. x, comma, 0. And what is this point? 0, comma, y. Is it clear? This point is 0, and it's missing all of this. It's now that you know how to, how to express a point on a plane. Now we'll see how to visualize a polynomial on a graph paper or a x, y coordinate system. Is that clear? And hence, all this discussion. Now, what was it? What was the topic? The topic was how to visualize a polynomial? How to visualize a polynomial? What is important? System. Let us start with a polynomial. Y is equal to, or rather, what did I tell you? I didn't talk about y at all. I talked about p of x is equal to, let us say, a0 plus a1 x. Okay? There could be other terms as well. a1, x, let me first talk about that. Okay? Now there are a few other important things you must know. n is called the degree of polynomial. So when n equals to 0, it's a p a0. That's it. You can't go there where x is 1. You can't go even there. n is 5. But you don't need to when you first, you know, first be aware of. Because most of it, yeah, so this is the theory. Okay? Now, polynomials can give it only one way. There can be all you will see, p of x comma y. So it is nothing but a1, x to start with. Then a2, y, plus a3, x, y. So on and so forth. And let's say something of that sort. Anything but positive, integer, or not negative, integer. Okay? And then, if now you have to calculate the degree of this, how do you calculate it? It's just good. It is? What is the degree of this? Whichever is highest. So whichever is highest, not the single variable, but both together. So if you see, the degree of this will be n plus n. Is it okay? Right? Any doubt? Please ask. So let's say if the polynomial is of this form, p xy is equal to a0, x, plus, how do we mistake numbers now? It looks easier for you. So let's say 1 plus x plus 2 xy. What is the degree? 1. No. 2. The degree here is 2. Why? Highest now. Highest is. So, when there is a problem with the degree that is considered to be 1. So you add all the power. So 1 plus 1 is 2. Now, 3 plus x plus 2 xy. What is the degree? 7. What is the degree? 8. It's not a polynomial, please. Why? 9. x has been our discretion only to a polynomial of a single variable to make your life a little easier. So, now we have to discuss what is p x? So, polynomial polynomial in p of x is equal to let's say a0 plus a1 x. Let me, you know, you don't expect, you know, just in thought. So, I usually am saying, I'm replacing this by y. So, I'm saying y equals a0 plus a1. y equal a0 plus a1 x. Correct? This is, now I want to visualize this. This is a polynomial. I want to see how it looks like. A or you know, coordinate plane. That's what is made by expressing, and this is what is the link between algebra and geometry. Which you have, okay? And algebraic stuff is now being represented in the, you know, I'm trying to visualize that. So, it's geometry and algebra are connected. Okay? Now, what does it mean? So, let us take an example and then understand with you. Let us say, I have y equals 2 plus 3x. What is the b1 polynomials are called? Linear polynomials. Linear polynomials. Now, you will understand why it's called linear polynomial. Okay? Now, let us draw a text and then I will write corresponding points. Let us say, the first point is 0 10. So, there is my xy. And I plot that to the ground. So, this is my x. This is y. Okay? Let us plot 0, 2. So, this is 0. Let's say this is 1, this is 2. This is 1, this is 2. But I have to go to 5 also. So, let's divide 0, 2. Where could it be? 0, 2. Is it the point? Yeah. So, I will change 0, minus the ground. Somewhere here, it will fall out. If you join such points, all of it, you will get a next, this expression is for linear. Clear? Clear. Why is it called linear? So, whenever you have, I add another thing. Let's say, y was 2 plus 3x plus z. Then what? Yeah. So, you are planning 3-dimension space. Yeah. Right now, x and y are playing. Now, the movement, you add another. It will come in 3D space. Okay? Linearity is clear. Clear? Now, in the, you know, in the ninth grade, you would have studied linear equations in one way or another. No. You didn't study linear equations? No. I studied linear equations. No, you studied two variables. What all did we cover in science films? Science films. Science films. Okay. So, let us say, because I am going to take this, we will learn now. Now, what I am trying to say is, now I am going to say if I equate a polynomial to a fixed value. Then, the polynomial expression gets converted into a equation. Now, we are talking about with any value, any fixed or variable value again. Then, the entire thing becomes an equation. It's like a beam balance. There is the polynomial on one side and there is the value on the other side. You are trying to balance it out. This thing is called equation. Do you get the difference between expression and equation? Any difference between expressions? What is the difference between expression and an equation? An equation is equal to an equation. Exactly. So, the expression never talks about an equality sign. Expression is just a combination of terms. Now, the moment you equate, the moment you have something compared to this expression and you say that this particular value is equal to this expression, it becomes an equation. So, now, our purpose of today's lecture is to understand linear equations. So, first of all, the polynomial of which degree we are talking about? One. So, we are not talking about anything of two and three and all that. That would be static and all that. We are not going to discuss that. We will limit our discussions to only linear expressions equated to something and becomes a linear equation. If the equation has only one variable, it can be linear equation in one way. Having two variables equation and two variables. Okay, what do I mean? Let us say and we take two different polynomials. First point is equal a0 plus a1x. This is and then I am equating this thing to, let us say, five. What is this? This is linear equation in? Okay, what do we do? a0 plus a1x equal to five. What are you equating? What? Let us say I have an equation in x and y where I am saying a1x or how it becomes a2 plus ax plus by plus c. Is it a is it a polynomial in two variables? Yes, sir. Equate into anything constant and here I mean you can see it becomes linear equation in two variables. Here here linear equation in two variables. If it has x plus by plus so let us say p of x comma y equals linear equation in two variables is clear? Why is it called linear equation or sharp? Because linear the word linear so because it's expression or when you visualize can now the thing is what is meant by solution of so there are there are couple of terms one is called zero of a polynomial. What is it called? Zero of a polynomial or roots of an equation. You get a difference? Yeah. I'm talking about zero of a polynomial. I will not I will not say zero of a equation. I will go zero of a polynomial. Means what is the value of whatever number it is x y z whatever which makes the entire polynomial reduced to zero that that value of polynomial is got the zero of the polynomial. So let's say if I have a polynomial px is equal to x plus five so what is zero of the polynomial? Minus five. Minus five. Because if you put x equals to minus five you will get the value of polynomial reduced to zero. So I'm saying phi of x x plus five I am interested in finding out zero of polynomial. What is the value? Minus five. So x equals minus five is zero of polynomial. It's also called intercept, right? No more r. No more r. Wait, wait for it. Now I'm saying plus five equals zero. What is this thing? What is this chord? This is a solution of polynomial. So what should I do here so that when I add five to it it becomes zero. That means if there is a linear equation in one variable it will have one and only one solution. Clear? Yes. Now next is what is meant by finding the zero of a solution of a patient in two solutions of go out. I have come back. It's really good. It's really good. It's a little bit tasty. Anya? Anya, Anya, Anya. Where? Right there. Over there. Over there. Over there. Over there. Over there. Are you guys understanding that? Yes. Clear? Yes. Clear? Yes. It's running. It's really good. I'll record again. So, when you have a system You're going to have three to four lines. No? Six times. Six times a week. Four NPS to my centre. What do you know? Teach linear equations in two variables again. Six times a week. How many NPS? Is it? If I give you a video I'm proud of Sir, you felt like you were in other school, right? Now we have to use the cash. I am telling you. So, the concept of sent him coins. So, you can find coins this year. And one coin is worth... See? That I have not revealed. So, you keep counting and you give your coins. Cash. So, that okay? Okay, I'll talk to them. So, I'll talk to them. Not letting go. So, it's slowly slowly. One, one cinema. I have promised I'll take you. Reward. Reward from me. It's 50 paise. 90 paise. So, good. So, will we solve the linear equations? Great. Okay, fine. Now, from the solution phase. You can work with the hand. But first we solve only linear problem. So, you should have linear equations. What is meant by solution? So, those values in this equation. Because this is going to be there. Hey! Ania, I'll throw you off now since last year. But I have been unsuccessful. This time, I'll cut you. I am right to make it a little wider. It will make it really painful. You can say this to my wife. Here. A x plus b y plus c equals 0. I have to find out values. That means x and y. It has to be an order repair. How do we do that? What? How do I do that? How do I find x and y such that? Let me say. Hey! I want. Don't. Don't. Don't remember. 2 x equals 5. Is it a linear equation? I have the solution to this equation. The solution is a pair of numbers. Not a single number. It's a pair of numbers. For x equals 1, y equals 1. Is that all? Or can we find some more? Sir, x is 5. So, x is minus 5, y is 5. So, one equation is x is 1. So, one solution is x is 1, y is 1. Okay, correct. Anything else? x equals minus 5. For example, minus 5. y equals 5. y equals 5. No? Yeah. Is it? Yeah. Minus 1 and... y equals 2. Is it correct? Yeah. Correct? x equals minus 1, y equals 2. No. No x. No x. No x. No x. No x. No x. No x equals minus 1 and y is... 3. Now, what I'm trying to say... Minach. Minach, right? What do you want to say? Minach. Minach. Now, what I'm trying to say is... We'll do here. Okay. What is the 26th equation? Let me. I'll explain. Which part? Which part? This part. 2x plus 3, y equals 5. So, who said... I have to say, if you put x equals 1, y equals 1. You get 5? Correct? If you put minus 5 in x and y, 5 in y, you get minus 10 plus 15. 5. Now, if you put x equals 2, minus 1 in y equals 2, then you get 5. If you put x equals 2, minus 2, so it becomes minus 4. If you put y equals 3, it becomes 9. 9 minus 4 is... What I find... The point is, do we have some rule in this mathematics? Or you can just do... Sir. Sir, for every value of x, there is... So, how do I find that out? What is the rule? The rule is this. You have to express one variable in terms of the... The other variable. And then, he randomly is feeding in one variable. Second bit. Sir, in this case, it can be x is equal to minus 1, y is equal to 7, 5. Yes. So, what will it be? So, can I write this as p, y equals y minus 2x? Yeah. All of you are with me on this? Yeah. So, can I write y equals 2ypx? Yeah. All of you agree? Yeah. All of you agree? Yes. No problem. Now, we need putting x value here to 0. I will be... Perfect. Now, put x equals to 0 here. Yeah. So, you get one solution as... Zero... Y is equal to 1. Y will be... 1. 1. If you put x equals to 1, this will become 5 minus 2 of 1, 3, which is 3 of 1, 3. This is 1 comma... Many many solutions. Yeah. If you have only one linear equation, just see the difference. If you have only one linear equation in one variable, how many solutions? One and only. One and only? One. But if you have only one linear equation in two variables, how many solutions? Many many solutions. Many many solutions. And what does it physically signify? That means that a line is made up of... Many many points. Many many points. Each point will represent one set of x and y. Right? Each point which is sitting on that line, its x and y coordinates are nothing but the solution to this equation. Right? Yeah. So, you see, if I have to, you know, let us say 2x plus 3y equals to 5, now I am trying to learn this. You will get 2x plus 3y equals 5 and y was nothing but 5 by 3 minus... 2 by 3. 2 by 3. 2 by 3. Okay. And then we have got some value 0 comma 5 by 3 and 1 by 1. So, it is, you know, to plot a line, two points are good enough. Correct? Only one line can pass through. Six in one. So, 5 by 3 is 1.67 for 192 by 3. So, let us, this one, this is 2. So, when it takes axis. So, let us say this is... I divide it. This is 1. This is 2. So, 0 comma 5 by 3. So, this is 0 comma 5 by 3 minus somewhere here. Yeah. Right? It is somewhere here. Right? Yeah. If I join it there, it is the line. Yeah. Right? So, basically, 2x plus 3 by y equals 5 in algebra represents this line. This is 2x plus... So, if you want to see this, a photograph of this is this. And all these points, if you somehow find all these points which are sitting on this line, find the x-coordinate, find the y-coordinate. If you see this value, let us say this was b comma q. This point has coordinates b comma q. If you fit in p in x and fit in q, you avoid p by y. Is that understood? Yeah. So, what is meant by solution? Two things. Value of those variables which will satisfy. Right? Or you find the coordinates of all the points represented or placed on that particular line. Is this concept clear to all? Yes, sir. So, hence, one line can have infinitely many points. So, hence, one line or one linear equation in two variables will have infinite solutions. Clear? Yes. The beautiful part is when you have two lines. Okay? So, let's say one line will have infinitely many solutions. But two lines together can have... They can have one solution? None? None. Yes, sir. Now, when we have another line, both of them will have a unique equation. Agreed? Yeah. Now, if they are intersecting, that means there is one and only one point which is lying on the first line of that intersecting point or the point of intersection. Then, I get a solution not only to the first but also to the second equation. Is that clear? Yes, sir. So, hence, in this... So, let's say this is one line. Yeah. And are they intersecting? No. And are the two lines not intersecting? No. They are parallel. So, two parallel which lies on both the lines simultaneously. Hence, we don't have any solutions parallel lines. Correct? Now, there could be another case where one line sits on... No. So, we choose length. Now, we are working on system of equation. Whenever there is system of equation, now we are working on system of... So, we will be talking about linear equation in two variables. B1y plus C1 equals 0. This is equation number 1. And A1, B1, C1, U2, C2. How do we not do that? It's important. If there was one, it can be 0. But both together cannot be 0. To represent that, we put this constraint. A1 square plus B1 square cannot be equal to 0. 5. Because if both A and B become 0, 0. This is not possible. So, one of this can be 0. But not both. Similarly, A2 square plus B2 square equals 0. But B1 cannot be 0. If A1 is 0, then B1 cannot be 0. If B1 is 0, then A1 cannot be 0. Same for the other equation as well. Is it clear to everyone? If A1 is 0, A2 is 0, A2 is 0, A2 is 0, or A2 is 0. This is very important. Understood, Goa? Yes. Now in this chapter, what are we going to do? Now we are going to learn how to solve one equation. Another equation is called as a equation. What is the graphical method of solving linear equation? Find the x and y coordinate of that point, you get the solution. Is it clear to all? What is the graphical method of solving linear equation? So, draw the two graphs of the point of intersection. You get the solution here. What is your name? Your name? Yes. So, you know how to solve graphically. So, what is the process? It takes 6 hours to learn. So, I was talking about methods of elimination. Then finding out the solution will be approximation. Let's say if the coefficients are irrational numbers. Let's say 0 to 13. Then you can draw the plots. First of all, you have group 7 and 13. So, plotting will not be able to precisely find out the point of intersection. So, your solution might be erroneous. There will be some percent or some bit of error in your solution. So, hence, we prefer. So, you did the graphical method last year. How do you touch that? So, we will be dealing with algebraic methods. So, I just, you know, you can solve the equation of this. Okay? What is the substitution? So, let me give you, let us take. So, we will be solving that. So, there is no problem. Now, when you discuss, let us say, solving, solving in here. So, first there is 2x plus 3y equals, let's say, 4. So, 2x plus 3 times x plus 7 equals 4. And everyone is minus 17. Clear? Yeah. Yeah? So, x is? y is 17. To do what? Let's put it back into the same equation. The equation back. Don't do that. You will get 0 to 0. It will not be, it will not help you. So, put whatever equation you arrive at from this equation. Then don't define this back into the same equation. Go to the other equation. So, you put it in the first. So, 2x comes as this. 3 times y equals 7. You will now simplify. Open this bracket. x equals 4 minus 21, which is minus 17. So, x will be minus 17 by 5. Clear? Yeah? Yes. It is a, is it? It will be, you know, you will get a lot of errors once in a while. So, best is, the advantage is, when you put x equals 17 by 5. So, it will be 18 by 5 minus, minus 17 by 5. Isn't it? Which is 18 plus 17. Which is 15 by 5. Which is, many a times, one equation might be correct. Yeah, sir? Yes. Yes. So, it will give you minus 5 plus 7. So, you know, you know. So, you put the values back and check. Right? So, this is one way of solving it. Substitution is very easy. Looking, but it's very lengthy. It will become very, my preferred way is either I use Kramer's rule or I go for elimination. It's much more simpler. What is elimination now? We will use the same equation. We are going to do, we are going to rectify the coefficients of x. I will take 5 minutes to finish this and then we will go for the problem solved. Anyone who is, who wants to leave? No. You want to leave? You can leave. What's your name? Stuti. Sir. Hi, hi, hi. I recognize them. Yes. Adarsh. Aditi. Aditi. Aditi. Aditi. Aditi. Aditi. Aditi. Aditi. Aditi. Aditi. Aditi. Aditi. Aditi. Aditi. Aditi. Aditi. Aditi. Aditi. Aditi. Aditi. Aditi. Ritu. Ritu. Ritu. Siddharth. Adri. Ritu. Siddharth. Adri. Siddharth. Adri. S shining. Adri. Santiran. Sand Bassit. Sandabh. Sand Musean. And So what happened? So what happened? There or not, there or not? The name was not there here. So it's Tutti and Luchita. Luchita or Luchita? Okay, fair. So you just need to give me your... piece of paper. Okay, you guys can leave. Right, so you're also leaving? Your name was there, no? You're also leaving? So those... So now we... I'll explain a little bit. Yeah, I'll explain a little bit. Okay, so... One, two, three. And then we'll decide what to do. So first of all, if you guys... minus 2i plus 4b equals 0 as a line... Yeah. Now, just like minus here and then delete it like minus here and you put minus here and then add. And this 3i equals minus 4 minus 2i times... a coefficient of x becomes 2x and you will ask why do you want to... Your chair is just... two minutes. Okay, good. So, I'll explain again. Who asked this? Yeah, so yeah. I'm going to divide by 2i. Because there is 2 here. So, divide. Once I subtract then they will get eliminated. So, this plus 4b from the other side becomes minus 4b. So, what I did was I wrote in that brain board. Let's do one thing. I'm giving you a series of questions. The questions now are solved by the time I just go and talk to you. It will become easier.