Okay, fantastic guys. So I formally welcome all of you to comprehensive revision program. So like every year, what we do is towards the end of the session that is from December last week till first week, we conduct classes on a daily basis, where all the subjects will be on maths and science subjects will be covering again, you know, repeating everything that we have done. And especially this year there have been some deletions so hence we'll be taking care of that as well. I hope you're all aware about the new curriculum so your school already would have told you and would have got papers according to the new curriculum only. So, you know, so hence keep that thing in mind and I'm going to share my screen now and just tell me if you are able to see the screen. All of you. Yeah, so I will. Yes, very good. So guys, as you know, the schedule has been given to you already. And yeah, just a minute uncle by any chance you're there in the group right now. Yeah, yeah.  Got it. Yeah. So, yeah, I made you co host so you can do what you can just let them enter one by one. 1520 minutes all the people who have joined any which way. Okay, folks. So, yes, so we are going to discuss quadratic equations first, and what will be the approach is this. Okay, so let's talk about it straight away. So, first of all, you must be aware of what is the code structure for class 10 mathematics from CBC. So you can see, you know, number systems and the weightage all all the seven categories and the way it has been given and this is the, this has been extracted from CBC dot n i c dot i n website of CBC. And there is some chat going on parallely so let me open that as well. Why? Yes, it can be viewed in this later so no discussion on the technical aspects of it all discussions related to this thing will be entertained is that fine. So yes. Okay guys, so number systems you can see six marks. Algebra 20 marks. And this is where quadratic equations will be falling coordinate geometry six marks geometry 15 marks geometry 12 marks menstruation 10 marks and statistics and probability 11 months. So, as far as the priority of the topic goes in terms of marks you can see algebra has the highest weightage. So here is where linear equations and polynomials and quadratic equations and all that, which we have studied will be there. So hence total math is 80. And let's set a target for all of us. Our target is 80 on 80. Right. That is what is called sent him right so everyone should be thriving for 80 on 80 and trust me it's not impossible. And 10th board papers are too predictable actually so you can know you can actually calculate what could be the question question number one question and things like that, but overconfidence also should not set in so hence, we will try our best and make sure that we get 80 on 80. The other 20 part 20% of the total is with your school. I hope you have done well all through through this year. You have done your whatever internal assessments were there you have done them well. So assuming that in 30 paper, we have to get 80 or 18 possible not possible. Let me see how much energy do you have possible or not. I can see a total silence in the group in the entire room. Yes, guys, possible. Yep, everyone should be thriving for the only agenda right so on 12th of January, Swami Vedic Ananda's birthday is there and he had very famous code so and you have to set an eye on on a target and just go behind it without thinking much about anything so if you have not been able to, let's say pop up well all this all this while throughout this year don't worry. I as I told you if you just restrict yourself to the pattern and the questions which have been given. And you solve all these questions once again, I am very sure.  Okay, so you understood the division division of, you know, mark so you can see algebra is the highest priority post which there is geometry so hence your triangles and Pythagoras theorems and similarity of triangles and circles and all that will be included over there. Trignometry including identity as well as what are the heights and distance application of tree normally. So hence, and this time around there is some change in the pattern of the question also let's discuss that straight away. So what is that. This is a question paper structure, you know, structure so that goes without saying that you have to read it at least twice you get some time to read all of that so that you are well aware what you are going to do. So as you can see there are two parts a and b. Okay, so what are part a and part b, and part a and part b have internal choices now the moment you see this, don't think that okay I can, you know, skip any particular topic. Why because you know that even if there is an or there is some kind of, you know, what do you say, choice the choice will be internal choice, it cannot be. You know, external choice. Okay, so sorry, as in sense you can't have choice between trigonometry and, you know, let's say triangle. Yeah, great. Now, let's move ahead so we were discussing question paper structure. And what was the question paper structure I told you there are two parts part eight part B, and there are internal choices so what does internal choice mean. So it will be let's say for example if there is quadratic equation. Question, so there will not be quadratic versus trigonometry choice. Okay, so hence there is no way you can skip any particular topic. So you have to either choose this. Oh, sorry, you have to, you know, be thorough with all the topics. Now, there are three sections, you know, sorry a consist of sections one and two, one has 16 question. Okay, so one has 16 question of one mark each now unlike previous year. Last year it was MCQ. So you would have seen this, you know, change in your question paper all of you just confirm if you have seen this in the recent pre-word example where yep, sorry. So you don't need not say anything in, you know, on the mic you can just type in so use your chat. Yep. So you have you would have seen that change. Now, internal choice is provided in five questions. Okay, and section two has four questions. Now this is a new type of section which has been introduced this year, where which is it says it's based on case study. Okay, now each case study has five case based sub parts. So these are all objective types only. And, and your examinee is to attempt any four out of five sub parts, any four. So keep that in mind any four of five sub parts. Okay, so be thorough with the instruction so that you maximize your case study will be how each carries how much marks. So each, each one is once four marks, right. So four marks each. So there are five. So four out of five sub parts you have to answer. So you can do the calculation total marks is 80 right 16 into one is 16, then four four for the 16. So another 1632. Correct. Then how many two of how many are there six question marks. Sorry, so this is 12. And what else 27 to 33 that means four plus seven. Sir, sir. Yes, sir. So the only the case study part in total is just 16 marks. Yes, there's not 16 months per question. That's what I'm one month. Yeah. So that's what I mentioned right 16 is the one marker objective 16 is the total four into four 16 and then two into 612. Okay, and then what 27 to 33 so seven questions of three months 21. So let me write it here 21 and then what else three questions of five months is 15. So 156. How much is this this is 46 and sorry 36 here. And what here it is 32 plus 1244. So 44 here and 36 here totally 80. Okay. So the other question which usually people ask whether we should apply a start from the beginning work from the end. Right now, this is something which is totally an individual choice until it's instructed, you can begin from anywhere.  So you know, I if I would have been here I would have definitely gone through the normal sequence. So I my preference is that so I will you know I will earmark so there are how many 16 plus 1632 marks right so 32 marks. One marker question are there. Right. So hence I would spend around 4045 odd minutes there so you can do the calculations 45 minutes for the objective one let's say. Okay, and then. So this is section a. Okay, and for section be obviously the rest of the time, but I usually spend 15 minutes or I leave 15 minutes towards the end for revision so hence I am left with 165 minutes in total. So if I'm left with 165 minutes in total. And then how do I divide 45 gone so I am left with 120 minutes or two hours exact for the other two marker three marker and five markers right so you can calculate so how many questions are there 21 to 20, 21 to 36. So there are 16 questions, obviously not every question will have the same amount of time. So what will you do for two markers not more than three minutes or not not even three minutes, three minutes only so hence how many 21 minutes 18 minutes for the three marker. Okay, so let me write that three marker three marker question, let's say on an average three minutes 18 marks there maybe you can take 18 to 20 something like that. Then three marker questions are 17 numbers I'll spend that's a five minutes each. So this is oh this is two markers. Yeah, so for three markers you'll have, let's say five minutes each so 35 minutes. And for five minutes you can take, let's say how many questions are there there are three, three only. So three question of five marks. So even if you take 10 minutes each 30 minutes more. So in total also if you add it will not be 120 so basically 10 board paper should be over by two hours two hours 15 minutes, and you have all the time to go again and again device. And that is very very important because CBSC at times is notorious for having some issues in the question when I was writing in my 10 word there was a there was one issue in one question, which we never knew and we spent unnecessarily huge amount of time, ending up with losing in there so hence keep that in mind.  So keep that in mind. Okay, now coming to. This is the syllabus now modified syllabus for quadratic you can see there are three subunits one is standard form of quadratic equation. That is one, all of you know this it is so common that we are, we have done it so many times then solutions of quadratic equations we have to, you know, only real roots so this is mentioned this is taken just copy paste from. Yes, three hours. Yes. So, yeah, any question. No, who's asking in any question there. So let me know if at all there is any shoes. Okay, solutions of quadratic equations, only real roots they have mentioned categorical. So there and using factorization and by using quadratic formula so you can be rest assured to be allowed to a large extent that anyways graphical solutions are not there. So between discriminant and nature of this is the third element. Now, if you see the sample paper I hope you are aware that CBC has released some sample papers. So let me show you what is there.  You can see that exam CBC website, so you can keep checking that for latest information don't go by here and say or what is the news are saying newspapers are saying no, go to the CBC website and see if there is a notification. So usually there will be a notification here. Can you see that this is the most authentic reliable source of any information related to your exam. So always bookmark this site and keep looking at this notification. Okay, fair enough. So examination related material this must you must know all these previous year question papers of CBC, you can see marking scheme for compartment exam question papers for compartment exam marking scheme for examination 2020 this is why you are you so you must see these these things so just a minute. Yeah, so what I'm saying is, yeah, so these, these are the, you know, things you should be taking care of, then marking scheme do look at marking scheme one so how marks are awarded will give you a good sense of how to go about it model answers again so we'll be discussing model answers anyway in this session and good that they have given that so that you can have a, you know, view of how exactly answer should be written is my answer writing skill matching with whatever is the demand of the board. So that you must be taking care of so lots of lots of information. So you can use that as well very good now. Let's go to. So next, next thing is here. So this is curriculum so all the latest revised curriculum is here so do have a look before you attempt any preparation so this is again another important side, then official papers of CBC. This is the sample question paper they have released. So you must practice them for sure and you know, even if you have seen the question no worries, take down the paper, take out a time slot three hours and then solve the question paper Okay. Deleted portions, everything we will be anyways discussing in our sessions but you must be aware of so this is where the portions are. So you can see this is for the class nine so don't get confused class 10th one is here. Right, so you can see in quadratic equation what is deleted you can see that here. So what is deleted in quadratic equation, situational problems based on equation reducible to quadratic equation this is not going to be there in your question paper this year. What is this, so situational problem is nothing what we say as or what we call as word problems. So there will not be any word problem where this form of equation are not there. So let's say to this kind of an equation will not be there this year. Okay, so anything which is requiring this will not be there. So that's you can, you can be assured of but then you know already how to solve this, you know how to transform such equations into a quadratic one, and then solve. Right, so give me a thumbs up if so far so good. Everyone is with me.  Yes, we will be discussing completing the square method so there is no mention of deletion of completing the square methods will be discussing that very good so keep the energy level high guys because you have to sustain for at least two more three more months. And if you know nothing goes wrong because last year students had to suffer literally.  So we will be revising the theory part quickly and then we will be going we will be doing what we will be doing our questions previous year questions only. Wait a minute again I lost the chat where is the chat thing. Yeah. Okay, so let's begin guys so what is this first the fall. First of all a polynomial of degree two. So we start with this only so what is. Yeah, so polynomial of degree two is called a quadratic polynomial so what is quadratic polynomial it is given over here. So what is any px is equal to a x squared plus the x plus see where a b c are real numbers. Right, so we are not going to take up any complex in this sport and put ABC are real numbers. An important thing is a should be equal to zero why because if is not if a becomes zero, then the equation is reduced to a linear form. This is very, very important so many a times this get this particular thing gets neglected always remember that is never going to be zero and x is a real variable so it's cannot take any complex variable over here. Okay, so this is the first thing. Let's go to the next one. Okay, so. What is the next one quick. So. Yes, let me do one thing yeah this will be making it. Yeah, so I will be using the yeah if px is a x square plus b x oh this would not be yeah so a x square plus b x plus C is not equal to zero is a quadratic polynomial and alpha is real number then you know that p of alpha a x square a a alpha square. Sorry for that. So hence p of alpha is a alpha square plus b alpha plus C. Okay, is known as value of the quadratic polynomial. So why do we need to understand polynomials because the next slide itself are in M. Yes, those were instead of raising hand just unmute and say. Yeah, anyone or don't do don't raise hand if you are not you're not asking any questions. It's a long back raise by mistake. No problem. Anyone has any doubt just you know unmute say and put your mic back on. Okay, so this is what we mean so hence if a polynomial is there what is a polynomial quadratic polynomial px, let me change the color of this pen so becomes easier becomes more. Yeah, so px is equal to a x squared plus b x squared plus C. This is my polynomial if you put alpha over there you'll get the value of the polynomial right this is a alpha square plus the alpha plus sorry, this is not square here. This is the square plus b alpha plus C is a polynomial quadratic polynomial right and let's take an example. So example is let's say px is equal to x square minus three x plus two. And let us say we are trying to find out p of one so p of one is one square minus three times one plus two, you all know this this is zero. So what is linked to the root of our quadratic equation we can see next let's go to the next slide. Yeah. So a real number alpha is said to be a zero of the quadratic polynomial so zero of a polynomial and solution or a root of a question right so zero of a polynomial zero of quadratic polynomial px, x square plus b x plus C is p alpha is zero. So any value which reduces the value of the polynomial to zero will be called zero of the polynomial and the same thing we'll see in the next slide, it is related to, let's say the equation. Now, now we are doing what we are doing px equals to zero so whatever polynomial quadratic polynomial we had so sorry. Yeah, so whatever quadratic polynomial we had so px was the quadratic polynomial and what was it a x squared plus bx plus C and we are equating it to zero. The moment we equate any polynomial to zero random entered the yeah who's random dude.  So any such miscreant will be thrown out of the uncle you're there. Uncle if you're available, otherwise I will do the honors myself anyone having. I am here I have seen and I have admitted as of now. No, there any person who is having any random name or anonymous people just bar them from the meeting remove them. Okay, got it. Okay. Summarily remove. No miscreants allowed. Okay. Yeah, so just remove them and give them a private warning once saying that you are not welcome and then you can remove them without any issue. Okay guys. So what I was saying is px equals to x square plus bx plus C is equal to zero is the equation now equation which we are going to discuss all parts of it always remember I will keep emphasizing is not equal to zero. And this is the definition of quadratic equation so we already know all of this so we'll not spend much of the time over here. A real number alpha is said to be a root of the quadratic equation, or it should be instead of root we've many times people interchangeably use the word root but then best word is to use a solution solution or root both are interchangeably use of the quadratic equation a x square plus bx plus equals to zero if a alpha square plus b alpha plus c equals to zero. So you know this we don't need to over emphasize on this. Alpha is a root of this if and only if alpha is a zero of the polynomial this so there could be a case where polynomial and quadratic equation so quadratic polynomial you have a chapter on polynomial and quadratic equation the question could be very very close. So, hence, you know they can use the concept of polynomials on quadratic equations and vice versa so be careful. Okay, now, so how to solve so one is factorization so now we are solving. So the first one is solving a quadratic equation so how to do first you already know in your when we discuss this chapter in our class we also discuss graphical method of solution of this thing but that will not be there in your paper for sure. Because in the sample papers or rather in the syllabus also it doesn't mention that so hence, it should not be there but you know how to, you know, solve our quadratic equation using graph tip, you know graph, so you have to plot. Yes, sir. Sir, is this meeting is for 40 minutes. No. Why are you worried about it. Yeah, just, just, you know, anything non-technical please resist from that, you know, so anything related to the subject matter please ask those questions only. Is that fine with all so that we can save time for each and let's respect each one's time and try to finish quickly. Okay, don't worry, your interest will be taken care of so even if it is suspended in 40 minutes why do you get a notice of something like that. Don't worry about it. Okay, so if a x square plus b x plus c is factorizable into a product of two linear factors, this is the best thing which we have known and we also call call this particular thing is splitting the middle term. Right, splitting the middle term. So if you and all that we are talking here we are talking in context of the examination. So in the examination papers, we have not seen for a long time that they have given a question and they are asking you to solve it. No, they in at max in your paper you'll have two questions to for quadratic one will be related to the nature of root finding k and all that. And that is one marker so usually the division is one mark plus three months. So that is what we have seen. So the marker definitely you they will not give you a problem to solve equation to solve in one mark they will ask you to find some k value for depending on the nature of roots and all that we'll see that. And there will be a word problem of three months, usually, or, you know, there is, you know, no four mark question so hence this is going to be the pattern most of mostly. So hence, you will be getting a word problem to solve or they might give you a question to solve also so be prepared. So they will give you splitting the middle term maybe or, you know, quadratic formula or completing the square and things like that. But anyways, so splitting the middle term all of you know so if you have x square minus let's say 5x plus four is equal to zero. So you know what to do you have to split this term, including the sign also so please be careful and what you do you do x square minus 4x minus x plus four is equal to zero. And then it becomes x times x minus four minus one times x minus four. And this is equal to zero. So you say this is x minus one times x minus four is equal to zero and this is what is called factorization into linear terms. So when you divide or factorize into two linear terms and then what you can do you can equate each of each one of them to zero. So you'll get x minus one is equal to zero and x minus four is equal to zero. Right and hence you'll get x equals to one and hence you'll get x equals to four. So this is one way of factorize. We'll see actual board papers that will make more sense. Let's go to the next one. Completing the square what is it. So let's do this for all of you once again so let us say we have a x square plus bx plus c is equal to zero. This is one of the methods adopted early on so hence we'll utilize the space here so I'm going to begin from here. What you need to do is you have to complete the square so how do I complete the squared so I can start with either multiplying the entire thing or dividing so I can multiply by a let's say I multiply the entire equation by a what do I get I will get a square x square plus abx plus ac is equal to zero. Right then now this is where the first element of mice completing the square is coming out. So this is you can write this implies a x mind you they will not ask you prove this technique prove the technique of completing the square. I don't think so you never know but I think I don't think they will be asking you such things. They will give you an equation and ask you whatever I'm writing is not able to are you guys is this thing visible to everybody whatever I'm writing on the left most part of my screen. Yes, so just just confirm I'm putting what do I write here what did I write. Very good and what did I write here. Great so that means the span expanse is okay and what did I write here. So hence everybody is able to see okay folks so I think is completing the square method in is in your portion so in deleted portions it's not there that means it is definitely there I showed you the website of CBC only deleted portion is whatever equations reducible to quadratic form is there those application problems will not be there only so that means everything else is there. Okay, so but having said that I've never seen a question which says that okay prove the completing or demonstrate the completing the square method know they will give you a question and ask you to solve. Yes, we haven't been asked we haven't been taught that in school either we haven't been any questions. Okay, so no problem but you know, since it is there as a, you know, hence I'm saying they will never ask you to adopt it. But just in case you have to, you know, there is nothing, nothing more dangerous thing here, so no worries. So you have heard of CZRH area as a whole the quadratic formula so how is that derived hence quickly both of both of them are similar. So hence no worries just check, you know, any basically add to your knowledge, even if it is not there. So what can I do I can say 2x, 2ax rather so you know see the manipulation I'm trying to do, and I'm separating this be out and dividing this by two. So hence, if you look at this term, 2ax times b by two, so it is same as ABx. Okay, and then plus AC let it be, or rather you know, what I'm going to do is instead of that I will add one more term here which is B square by four. And then subtract also B square by four plus AC give me a thumbs up if everyone is okay with this part. Is this clear? Is this clear? Yep, anyone didn't understand you can you know so what did I do I simply manipulated a bit so ABx is written like this. And then, you know, I just added and subtracted B square by four. The reason will be clear just in the next step and done. Hence it is nothing but ax plus b by two whole square. If you look at it, it is simply this. And on the right hand side I can take B square by four plus minus AC. I hope this is clear and this is where the entire thing ends. Right now I can solve this, you know, equation why because I have separated the variable on one side, and the constants on the other side. So hence what can I say I can say ax plus b by two is equal to plus minus under root. If I simplify this this is B square minus four AC by four. Right. And hence I can write ax is equal to minus B upon two plus minus under root B square minus four AC by two. This is what is also called quadratic formula. So x is minus B plus minus under root B square minus four AC by two. Right. So repeat after me. So x is minus B plus minus under root B square minus four AC by two way. This particular hero item is called discriminant D. Right. Discriminant. So you're discriminating against something discriminant. Right. So discriminant D. Right. And this D is going to create a lot of trouble. Why? Because under root square under square root anything if it becomes negative and then the real world gets, you know, in trouble. So hence that's that only takes us to the nature of it. So is this understood completing the square method is not a big deal. So you can simplify by a complete the square using a plus the whole square and that and separate the, you know, variable on the one side, all the constant on the other side.  Any problems so far in this slide. So far so good. Any difficulty so far. Okay. So finding the zero using the computer can be used quadratic formula. If, if the word completing the square is given, I don't think we'll be there in this year. If at all that is given, then you have to write all this. Don't use the formula directly. Is that okay. The completing square method is deleted from last two years. No, sir. I'm just saying. If at all, see, there is no, you know, in my opinion, many a times question do have it, but we haven't seen it in the last three four years. But I'm saying, if at all there is see we have to be ready for every exigency and hence we are also going to discuss the reducible forms as well, even if they have said, not there but you know, in some place it is applicable and you have missed it and gone. This is our risky thing to do. Friends, we will be seeing there is no problem in going a little bit extra and winning it over. Is this difficult by any chance? Don't look for cutting corners that okay, this is deleted. I will not do it. Something is, you know, you learn it anyways. 10th is not the last year. No problem. If you think it is deleted, don't study it now. So quadratic formula, all of you know how it is arrived at. We just proved it. There was a very famous Indian mathematics mathematicians. He also came up with, you know, the same formula. But instead of multiplying by a, we have to divide it by a in that equation. So what I mean is this. So he adopted this is equal to zero. So instead of multiplying last time we multiplied by a. So instead of that, he divided this by a. Obviously, a is not equal to zero. So we can do this game. If a was zero, I could not have divided, right? So hence it is simply x square plus b upon ax plus c upon a is equal to zero. And again, now follow the completing the square mechanism. Okay. If you do that, again, you will land up here. x equals to minus b plus minus root b by two. Where d is b square minus four ac. Discriminant. Many a times people get confused between a, b and c. For example, let's say the question is a minus b x square plus b minus cx plus c minus a is equal to zero. This is the equation. So don't get confused in this equation that okay, the same a, b and c. So here, the small a, which is this is not this a. So these two are different. Do you understand this difference? What is small a here? Small here. Yes. So as you say, so you can take, you know, either capital A or some px square plus qx plus r like that. So here the solution will be x is equal to minus b minus c, not only b. Now this b here is the entire thing here b minus c, right? So minus b minus c plus minus under root b square. So in this case, b minus c, this square minus four ac. So what is four ac? A minus b c minus a divided by two a. So twice a minus b. This is the solution for this equation. So don't get confused between a, b and c. So hence one way of doing it is you write this as a x square plus bx plus c is equal to zero. So this is the quadratic equation. So your solution will be x is equal to minus capital B plus minus under root capital B square minus four ac by twice a. Is the value of the coefficient taken when replacing the value of a and b and c? Yes. So basically this entire thing is capital A. Value of the coefficient only is a, b and c. a, b and c are what? Not only value, but the sign. Nine plus value both plus a, let's say for example, the question is, let's take an example, three x square or let's say minus three x square minus four x plus two is equal to zero. So what is a here? A is minus three. What is b here? Minus four. And what is c here? Two. Is that okay? Confirm. Clear? A doubt? Any doubt? Any doubt? Any doubt? Confirm. Right number eight. Okay, good. Now, nature of the roots of quadratic equation. Now comes the game. So why all this is required? So if you see x is equal to, we found out minus b plus root over d. In fact, minus plus minus root over d and two a. One thing, another one important thing you must notice here is this is plus minus. So quadratic equation. Always remember, how many roots can I have? Can I quadratic equation have three? Two or false? False. False. False. So they might ask you these kinds of questions as well. So which of the following statements is true? You never know. So be ready. So quadratic equation can, will always have two roots. True or false? A quadratic equation will always have two roots. True. A quadratic equation will always have two roots. Two roots. So false. It's true because quadratic equation will always be having two roots. If it is given quadratic equation have two real roots. Always. Then it is false. Understood? The play of words? No. So hence if the question is quadratic equation has two roots. Two roots. True or false? This statement is true or false? This statement is absolutely true. Every quadratic equation will have two roots. But in your grade, what have we done? We have restricted our study only to real roots. So when we are saying and if I change the equation or statement, Qe has two real roots. Always. True or false? True or false? Then it is false. Right? Why? Because we can have only one real root also or none at all. None at all. So hence be very, very careful. Be watchful. Awesome. Yes. Please. So what if a quadratic equation has two same roots? So then do we say it has one root or two roots? Two equal roots. Technically speaking, we have to say two equal roots. A quadratic equation will always have two roots. So hence what happens is in this case, one root is coincident on the other. So we always say the quadratic equation will have two roots. Yes. Depending on the D, this D, everything depends on this D. Why? Because square root of D, if negative, is not real. Hence if D is greater than 0, all of you know these are what? Two distinct or other word used is unique root slash solution. Mind you, it mean, yaha pe 0 nahi. This should not be there. This is wrong. Equality sign many times people add here. No, wrong. Then D is equal to 0. This is two equal roots. And third is D less than 0. We say two imaginary roots. So don't say no roots. Or say two non real roots. But in all the cases, there are two roots. Can we have two real roots? Yes. Obviously there are so many cases. Just now we proved. Is it it? The core is two real roots on x square minus 3x plus 2. Is equal to 0. Real or unreal? Unreal. All of you play video game. Unreal Engine. Yeah, have you ever tried Unreal Engine guys? No. Anyone? Lots of people. See, I knew Rn was going to answer that. Okay, cool. So real roots, right? Here, this is real roots. What are the roots quickly? Let's see without solving. What are the roots here? Who will say first? Quick. What are the roots here? X square minus 3x plus 2. Roots are? It's not time. Come on, quick. What are the roots? Ah, bhai. Quick one actually, right? Okay, so technical issues. Okay, never mind. So now I'm not going to repeat. Let's go to first question. Sample paper. One more. This is given in your sample paper. Can you see in the sample paper? Hence I said these are so predictable. For what value of K, the equation 9x square plus C, Ckx plus 4 equals to 0 has equal roots. Quick. Solve muddy. K equals to 2. Surya has solved it. But my dear friends, since it is one marker, just don't write. So how will you write it? So this is how you should write. This is how you should write. Oh, sorry. Wait a minute. This is now mark the steps. So you have to write like that for equal roots. For equal roots. What do you need to do? D is equal to B square minus 4ac must be equal to 0. Okay, so since it is one marker, so don't write the equation and repeat all of that. So hence B square. What is B here? 6k whole squared minus 4 times 9 times 4 must be 0. Am I right? So 6k whole square is equal to 4 square into 3 square. So 6k should be plus minus 4 times 3. So k should be plus minus 4 times 3 by 6, which is plus minus 2. Is it okay? This is how for one mark. Write it. Sir, the question can also have two equal unreal roots. Two equal unreal roots. No. No. Two equal unreal roots are not possible. Always complex roots are in conjugate form. Or you can have actually, yes. No, no, you can have how this one is, for example, x square plus 4 equals to 0. No, but here also no. Sorry, sorry, sorry. Mindset. You cannot have two equal roots ever. Equal unreal roots. Is that okay? Yes, in quadratic equation. For example, this, what is the root here? The root is plus minus 2i. Okay. So, again there are two roots, two complex roots. Is that okay, guys? Clear? Yes. Yes? Yes or no? Sir, next question. Sorry? Next question. Next question. Do it. Find the roots of the equation. x square plus 7x plus 10. Minus 1 minus 5. Yeah, you already solved it. How? Mental maths. Mental maths. Oh my God. Good. Very good. So, how will you write? This is again one marker. Sir, factors of 10 are 2 and 5. So, x plus 2 times x plus 5. Yeah. So, in the question paper, both 11th and 12th question were in or are in tendon. Signs are wrong. So, the signs of mistakes are now, they have started emerging now. That's sad. Okay, they go. x square plus, if this is one marker, what will I do? I will write x square plus 2x plus 5x plus 10. Always remember in one marker, they are not going to give you a very tough splitting the middle term. So, you can very easily split the middle term. And towards the end, I will give you one fail safe method of splitting the middle term, in case you got stuck also. Okay. So, this is and it is better to write this implies sign all the time. Okay. So, hence it is x plus 5x plus 2 equals 0. So, you can write x plus 5 equals to 0 or x plus 2 equals to 0. These are hygiene things. And hence x equals to minus 5 or x equals to minus 2. Okay, dear. Aditya, sir, could you do go to the previous question once? Yes. Here is the previous question below. What is the issue? Aditya, unmute and say if you have anything. Nothing, sir. I just wanted to copy it. Oh, copy. No worries. This will be... Oh, don't worry, guys. Try to solve parallely, but all the notes will be available on the same learnings portal. So, don't be, you know, worried about availability of all of these. The entire slides are for you. Don't worry. Next, quickly see different... This is one marker again. Find the roots. Sir, it's the same question. Oh, just slide copy. Okay. Never mind. Come here. Do this. So, for what values of a quadratic equation has no real roots? No real. Sir, why do I always get the science wrong while factorization? Avantika, the best way to do is when you solve it, put one of the values back into the equation and check whether you have got it right. Check the best part about equations is when you're solving it, you can always check whether that is correct or not by putting the value back into the equation. Isn't it? So, just simply put two back into the equation. It will take hardly 15 seconds to calculate. You will get to know whether the solution is correct or not. Fair enough. Okay. Okay. Next. Done. So, the values of a, again, one markers, you can't afford to invest more than a minute on this. One minute max. So, Mehta says a greater than one. Okay. How will you solve this? Again, step, mark the steps. So, again, CBC, as I told you, go to the marking scheme, see how marking scheme is there. Anything greater than three by 10. Achita. No, you have to write like this. A greater than three by 10 or something like that. Okay. So, hence, what values of a quadratic, what values of A? A. 30 AX square minus CX plus one has no real roots. So, you'll start like this. Even if it is for one mark, write the theory first. For no real roots, write this statement. You will get half a mark straight away. For no real roots, write D is equal to B square minus four AC less than zero. This is the condition. Am I right? So far so good. For no real roots, this has to be the condition B square minus four AC less than, you know. But since there is an A here, since there is A here, try not writing this. Why? Because people will think that you are trying to, this A and that A are same. So, you can stick to this. B is less than zero and then go straight. So, D is what? B square. So, minus six. Now, be careful when you are picking up the sign. B square minus four AC. Write, write, write. 30 A is A here and one. This must be less than zero. Have I done correctly? This is the equation. So, minus six square is 36. Minus 120 A is less than zero. So, hence be very, very careful that 120 A has to be greater than 36. Okay. So, A is greater than 36 upon 120. Right. So, this is nothing but three upon ten. So, done. Good enough. You can leave here. A is greater than three by ten. Right. Okay, guys. So, hence this bracket may S. Okay. Clear. Sir, I have a doubt. Sir, we got like lots of MCQs. No, no. For R and R, the autonomous paper may be a little different. This thing. No, sir. We have all a board pattern on there. But there is no MCQs in the sample paper given by, you know, R and R. R and R. R and R. R and R. R and R. R and R. R and R. So, there is no MCQs in the sample paper given by, you know, this thing, there was no MCQs. But anyways, if MCQ is there, then no problem. Happy. Right. Sir, only the case study based questions have MCQs. The rest are fill in the blanks. 60 questions are like, you know, you have to one marker. No MCQs. No choice as in no ABCD. Okay. So, next. For what values of quadratic? Same thing. Why is it competing? Okay, never mind. Chalo. Do this. This is three marker, guys. See, now here is the game changing. So, what I have done is I have taken all the question paper, previous year question paper or sample paper, this thing. And then every year, year by year, these are all 20, 20 sample papers. Then we'll go to 2019 question papers. Then we'll go to 2018 question papers. Then we'll go to 2017 like that. Okay. Do it. And quadratic equation, I don't think there will be a case study because a case study may, usually they will give either the coordinate geometry part of it, whatever they have given. So they will try to restrict it to one reason is or no, they can give you one question of four mark together. Then there will not be any other question on quadratic equation, but then I think polynomials will be there for quadratic, sorry, but the case studies and the menstruation will be there. Right. So, but possibilities, you can never say zero possibility. Yeah. So, but for, but I think for quadratic equation, they will go for one plus three only. So, hence if one plus three is there, then there's no case study. If one root of the quadratic equation is this, then find the value of P and the other root of the equation. That's a mistake people do here in such question, they will read here, read this also, but they will forget it. So my dear, what is the question, what is the way, learn how many, what are the demands of the question. So, keep this in mind and write somewhere. Two questions, two answers have to be there in this. Okay. So, hence mostly I have seen people will write this P successfully and because of, you know, your thought process says that, okay, there are lots of questions left to be solved and hence you will miss the other root of the equation. So, hence there are two parts of it. Remember now how to do it. So, in three marker, I will start. Sir, P is minus eight. Let me see. I will also solve. So, I will start like this. See in these question, please mark the steps. Sir, what should I do? No, don't get confused between, you know, the moment you are seeing this is an equation. There is no question of polynomial. This is a quadratic equation. Sir, I just told the answers to these questions. What? They just finished. Sorry, I didn't get you. I just told the value of P and the other root. That's it. I don't, sir. No problem. See, you might be able to solve it in your brain, but... No, sir. I didn't solve it in brain, sir. I solved it in book. Good. But you have to follow few protocols because your copy would have been, let's say, your would be corrected by someone in Andhra Pradesh, Tamil Nadu. You never know. It's better to, you know, write as per the demand of the board. Okay. So, 3x square plus px plus 4 equals to 0. I will start here. One root of the quadratic equation is this. One root. I will write one root and I will put a comma alpha is equal to 2 by 3. So, this gives an indication that you know alpha, beta, other. To find beta and what? P. Right? So, P, how will you... So, hence... Oh, my God. Oh, my God. Mute yourself. P into 2 by 3. Mute yourself, guys. Hello. I'm going to mute you. Hi, guys. Hello. You have... Hi, sir. Who is this? Yeah. Who is this? Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. No problem, but do not disturb. Okay. Yes. All of you. We want only serious people here. Okay. So, 3 times 2 by 3 whole square. Okay. And... Sorry. So, try not to do overwriting here. I'm unable to delete. Why? Because the... You know, I have to go for eraser and all that. So, hence, don't do this. This is not good. Not good. Let me write. So, if you are not able to erase, you simply cut. Okay. Let me write again. Once again. 3 times 2 by 3. Overwriting frustates the examiner. So, hence, please try avoid. And don't do this. Don't do this. These are all... This is all going to pull your... This thing impression down. So, hence, simple. If you have made a mistake, simple cut. That's it. No doing painting over there. Okay. So, this is 2 by 3 whole square plus P times 2 upon 3 plus 4. This is equal to 0. Okay. Now. So, hence, you can see this is 3 times 4 upon 9 plus 2 by 3 P plus 4. So, this is 3, 3. So, be careful with the calculations. So, 2 by 3 P or this is equal to 0 will be equal to minus 4 minus 4 by 3. Which is equal to minus 12 minus 4 by 3. Yes or no? Yeah. Correct. Guys. So, what is this? This is coming out to minus 16 upon 3. This is 2, 3, 2 by 3 P. So, what is P? My dear friends, P will be simply minus 8. This is one part. Do not leave the question and go to the next one. There is 2. One is done in the mind. You know, one is done. Other root of the equation. How to find out other? Now, you know that this equation is 3. So, either you can do by sum of the root root, you know, or you can now solve it. Minut 8x plus 4 is equal to 0. This is the equation. Is it? So, either what did I say? Either if you know the sum of root root. So, first spelling sum of root R o t root R o u t e. So, sum of root root is also possible. You can do that. Or, if not, then you need to do what? Solve this equation. So, 3x square minus 6x minus 2x plus 4 is equal to 0. Right? So, you write 3x and x minus 2. Correct? Minus 2. X minus 2 is equal to 0. So, this is x minus 2. 3x minus 2 is equal to 0. So, you can validate also. 3x minus 2 will give you this root 2 by 3. So, hence x equals to 2 and x equals to 2 by 3. So, write down. So, other root is equal to 2. That's it. This will fetch you 3 marks. Okay? Now, I have written it in two columns. You don't write it in two columns. Right? In one column only. Okay? Fair enough. Okay. So, next. This is done. This is again copied. I don't know why. So, every slide is getting copied twice. No worries. Oh, maybe some extra space for solving. Okay. Roots, alpha and beta of the quadratic equation are such that alpha minus beta is equal to 1. Find the value of k. Solve. This is 3 marks again. Will these slides be uploaded? Yes, my dear. Slides will be uploaded. Don't worry. And where can we access it? You will be able to access it in the same Learnist, my dear. Where you have seen the link. Anyways, roots, alpha and beta. All technical thing you can talk to us separately. Okay? If, yes, guys, k is equal to 3. Akshita, you are trying to find out a and b. Where is the issue? Find the value of k. The question is find the value of k. Nothing else. So, if you just write alpha and beta. This happens. And then you come back and then. Oh my God. I know this expression. I have been through this. Sir, will we get extra questions as well? How many do you want? We will give you. Don't worry. Roots, alpha and beta are the quadratic. How to solve steps? Roots of alpha and beta. Sir, alpha will be minus b by a. Yes. So, hence, here is a mix of polynomial and quadratic. Isn't it? So, you know that in quadratic equations or polynomial, right? If alpha, beta are the roots of this, then zero of the polynomial. So, you can also write alpha and beta are zeros. Roots of quadratic equation, if you saw above, these are now zeros of polynomial, polynomial px which is equal to x square minus 5x plus 3k minus 3. This is a polynomial. Alpha and beta are zeros of this if alpha and beta are roots of this. So, now what do I know? Alpha plus beta, in case of polynomials, if you remember, minus b by a, right? So, hence, write minus, then write minus 5 again, then divide by a, 1. Write all of this. Don't write directly 5. Show them that you know. Or if you have time, write this minus coefficient of x divided by coefficient of x square. This is what is alpha plus beta. So, hence, I get alpha plus beta is equal to 5. And alpha minus beta is given as 1. Now linear equations, 2 alpha, sum them is equal to 1. Please do not write this. Sorry, my bad. I also tend to, I keep doing this. So, write all of these 2. You write adding 1 and 2. You will get 2 alpha is equal to 6. So, alpha is 3, right? Write 3. Then I am again writing here. Don't write like this. Write in one column from, let's say anyone you want, 2. From 2 and 3. 3. You can write beta is equal to alpha minus 1. Hence, it is 3 minus 1. Hence, it is beta is equal to 2. Now, what is, find the value of k. So, how to find out the value of k? So, you now know product of zeros. Product of zeros.  Alpha beta is equal to c upon a. Or you can write constant term. Or you don't need to write this. You can directly hop on to what is constant term? 3 times k minus 1 by 1. But do write by 1 like that. So, they will know that you are aware of this. So, what is alpha beta group? Alpha beta is 3 times 2, isn't it? So, write down from here. Oh, where did it go? So, you write again I am because of lack of space. I am doing it here. Let me take it. Now, this is what alpha beta. So, write 3 times 2. Oh, that's happening. So, you write 3 times 2 is equal to 3k minus 3. So, 3k is equal to 6 plus 3. 9k is. Bolo dosto. Any difficulty? So, you now, you now learned what are types of problems? Can you see? So far, now the problems are either to find out the value of k, the nature of roots, or one root is given, find the other root and the other value, or you have to find out, let's say, using the sum and the product of root. But that anyways will take up in polynomials once again. So, hence I have not included that as a theory in this slide. Okay. Now, this is now previous here. 2020, actual board paper. If one root of the equation is this, is the reciprocal of the other, then the value of k is, right? Try. If one, this is one mark. So, again, you know, not that intense. You can again use the sum and product of roots, root. I'm waiting for your, you all to solve. R in tendon says k equals 4. Ranjini Ghosh says k equals 2, 4. Anyone else? 4, 4, 4. Okay. So, easy, right? How did you do? Or any other way? How do you do? Product of roots. Yes. So, hence why, sum and product of roots. So, product of one marker. So, I will directly write product. Don't write prod and all, rightful. Product of roots. Alpha beta is equal to, or you know, you can directly, sorry, you have two. Okay. No, you can, you should actually start from here. Let the roots be, let the roots be alpha and 1 upon alpha. If it is MCQ, so no problem. But if you have to write and write, let the root be alpha and root. So, alpha times 1 upon alpha is equal to simply write 3 upon k minus 1. Be careful with the sign. C by A. So, 3 by k minus 1. So, 3 is equal to, so, or one step more. Write 1 is equal to 3 by k minus 1. This implies 3 is equal to k minus 1. This implies k is equal to 3 plus 1, that is 4. Now, many a times people will make a mistake here. Error zone. Error zone is, what will you do? You will do this. That k is equal to 3 minus 1. And make it 2. And we will get 0. So, be careful, okay? So, calculation error is possible. Next. Now comes, no, this was four marker in previous 2020 paper. So, in this year, there will not be any four marker questions. But then, questions will be there. So, let's solve. And I have seen a trend that in the last three years paper, there has been one question on speed distance time. So, out of all different types of word problems, you can expect a question on time distance and speed. Okay, so, how will you solve this? A train covers a distance of 360 kilometers at a uniform speed. Had the speed been 5 kilometers per hour more, it would have taken 48 minutes less for the journey. Points to be careful of. So, this unit is, friends, I normally say unit alarm. So, the moment you see minutes and you should be aware that, oh, there is. So, don't write 48 directly. People write equations using only 48, no. That will be one thing. Find the original speed of the, and always keep an eye on what to find out. Original speed. So, what will you, what will be the variable? Oh my God, situation problem using transformative equations are in, are in single. You will not get equations where in you have this kind of equations. Is that okay? But rest all are very much there. Is that okay? Yeah. So, any equation of this form, which is reducible to quadratic form will not be there. But any equation which is directly into quadratic form will definitely be there. So, this is deleted. I showed you, right? You can check on the website also. Only this is deleted. So, all done. So, in these questions usually I would have drawn a diagram. Okay. How simple one like that. Okay. I would say this is a, this is b. And I would say this is d is equal to 360 km. Okay. So, original speed is R n tendon has already solved. Okay. Had the speed been five kilometers per hour more. So, we'll start with saying this. See now I'm, let's focus on the steps guys. Let the original speed of the train be v. I am comfortable with v. You can take u because I'm a physics person also. So, I usually take v. You can take ux whatever, right? No problem. So, whichever variable you are comfortable with. Let the original speed of the train be v. Now you have to directly jump on equations. So, you can and also let the time taken by train at v or you can, you can write, you can write. Let the time taken by the train to cover 360 kilometer at v kilometers per hour. So, here I do mention this very, very important per hour be TRs. TRs. Okay. My pointer nib is little wider. So, hence don't, you know, these things are hence at times it might not be that clear. Never mind. Okay. Now, what do I know? So, you can say very easily v times t is equal to 360 point number one. Right. Next is how do you write? You write at the speed being five kilometers per hour more. So, v plus five into t minus obviously for 48 minutes less. Now, 48 minutes is 48 by 60 my dear friend. That is important. And this is again 360 no doubt about it. That's it. These are the two equations. The moment you write two equations, you will get 50 percent marks. Now, you have to just solve for eliminate t and solve for. Right. So, what will you do? So, you can get a quadratic like this. So, you can, you can from one to from one and two expand this you'll get vt plus five t vt plus five t and minus 48 by 60 watt four by five. So, minus four by five v and minus four. Okay. Everybody is doing. Okay. Everybody is able to solve. And this is equal to I am writing vt itself from one and two. I have written here. So, I don't need to explain this from one and two. Is that both are 360. Is that correct? Below those to any problem. I hope this is cool. Correct. Correct or not? So, vt and vt will go and you'll get a relation between t and v. So, you'll get five t minus four by five v. And you can write this as four. Okay. Now, you can use. So, hence, you know, eliminate t. So, t is nothing but five into 360 times v. Don't eliminate v because you want v minus four by five v is equal to four. This is the final equation. Now, something can go. Yes. Four. Four. And you're 90. Right. So, right. Clearly, and then 450 by v minus v by five is equal to one cross multiply. I'm writing it here. Right. I'll start from here. So, you'll write. Yeah. So, what will this be? So, 450 into five. I will write 450 into five minus v square is equal to five v. Directly, I wrote. Right. And then this is v square minus five v minus 450 into five. I'm purposefully writing like this. I don't want to calculate unnecessarily. Because anyways, I have to split the middle term. Okay. So, 450 into five is nothing but 45 into 50. Here is why I kept it like this five v is equal to 45. Sorry. Minus 45 into 50. I hope this is clear to everyone and this is the catch. So, hence v square 45 times 50. You can write 45 plus 50 v. Sorry. Minus 50 v. Am I right? Minus 50 v plus 45. Oh, plus five was that. Oh, did I mistake? Do a mistake. Ah, sorry. My bad. That's why I forgot. Correct. Thanks for highlighting. Thanks for my writing. So, this is then it will be cut. Then write v square plus 50 v. I was writing like that 45 v minus 45 times 50. Sorry for lack of space. Okay. Now, you have to solve it.  So, hence write both of them and I will write directly v is equal to 45 v is equal to minus 50 kmph. But, but you have to also write one more step. Where do I write? I write here. So, let me put a star and write here. Okay. I will write since write these steps also since velocity cannot be. Or speed. Sorry, not velocity. Speed can't be negative. Hence, hence the desired result. Right like that. Clear. This should be this methodical and I will show you how people have done it in the past. Board has also released some sample model answers. You'll see that. Clear. In solving part there is no issues, right? See, most of you are anyways are aware of how to solve the catch is how to present it now in a proper way without messing up with data and getting it as per the norms, whatever they require. Clear. Give me a thumbs up. What happened to the energy level? People. All good. All good or not? If there is an easier way of doing this, please, you can. If you have easier way, you can do that. No problem. You can always, always, always. If you think if you're confident, there could be many ways. No problem at all. Okay. Is it, is it fine? Next. Next. This is a previous question again, previous year paper, board paper 2020. 2020 or sometime back. Okay. Do it. This again. Can you see that? Another question from actual board paper. And again, speed distance time. Same, same way. So who was saying there is another method of this thing? I forgot the name. Yes. Yes. You can, you can tell us your method so that everyone else is also enlightened. There's a basically in this speed distance time question you draw table of speed distance and time. So if you could draw it, it will be. Yes. Why not? Let's say, let's draw speed. Wait. This is speed distance and so speed distance and time table. Okay. Yeah. Original. Okay. So in this case, we take the time to be X. X. D is 360. Yeah. So speed will be. 360 by time 360 by X. Yes. And time of the flight increase by 30 minutes. So X plus 30 by 60 X plus point five X plus point five distance. The same. Yeah. The distance is 600 in this case. Oh, I'm sorry. Oh, I was thinking of the previous question. Wait. So 600. Oh, I could have done the 600. Yeah. That's very easy. Let me delete only painful. Painful technique is their pointer of a laser. Now pointer options. And I hope I've got the same color. Yes. So this is 600. And cut this and 600. Huh. No. So speed in this case will be 600 by X plus point five. It's average speed for the trip was reduced by 200 kilometers per hour. So. And time of flight increase. Yes. So what? Yes. Tell me. So this is 300 by X plus point five. 600 by. Oh, this also is 600. Sorry. 600 by X and. 600 by X plus point five sir. In the second. 200 kilometers per hour minus 200 kilometers per hour. Correct. Yes, sir. This will be the table. If you were going for the table route. Yes, sir. Next. Speed, the new speed with the second data will be 600 by X plus point five is equal to 600 by X minus 200 and we simply find the answer. Come again. 600. 600. 600 by X plus point five. No, what are you waiting because our speed is 600 X minus 200. 600 by X minus 200. This is the new speed. Yes, sir. It's average speed was reduced by 200. Yes. So 600 by X minus 200 is equal to 600 by X plus point five because that was the time taken rate. 600 by X minus 200 is the new speed. 600 by X plus point five. Why will that? Oh, okay. Like that. Okay. X plus point five. Okay. Yes. Correct. Yes. Simplify and we get it. Yeah. So basically, you know, the thing is Andrea, the thing is, sorry, who is this? This is a trick for, let's say, MCQ based questions. We also adopt this, but in this case, my advice to all of you would be go for equation writing, you know, that, you know, write two equations and solve. My advice would be that because usually we have seen that in the model answers as well that you can, you know, so that you reduce your risk of losing if at all, there is a possibility. This is perfectly right. No problems in this. So you get the picture. It is same thing, you know, you are doing the same thing. My dear. How? What are we doing? We are doing this. In this case, what would have we done? We would have done this. So my original speed is V into T. Right. Now in this case, original duration has to be signed out. So T has to be signed out. Okay. Now V into T is equal to 600. The second equation is V minus 200 into T plus 0.5 is equal to 600. Right. These are the only thing which I have. Yeah. And then you equate the same thing. So you write what VT plus 0.5 V and minus 200 T and minus 100. Correct. Check once again. Whenever you do, keep checking. So VT 0.5 V minus 200 T minus 100 is equal to VT. Cut. And now you will get 0.5. Now V has to be eliminated. So write 600 by T minus 200 T. And this is what you are also saying. Correct. Is equal to 0. This is the equation. This is the quadratic equation. Final quadratic. Solve for T. Okay. So 0.5 into 600 is 300. And thankfully we can eliminate all the 000. You can see 0, 0, 0, 0, 0, 0.5 into 6. There is an int to sign here. It's simply 3. 3 upon T minus 2T minus 1 is equal to 0. So hence my dear friends, where do I write? Let me write here. No. So put a star, star. Don't do this star, star thingy there. Okay. Write in one column. V as clear as possible. Now this is because one slide is there. So hence I'm utilizing the space as much as possible. So 3 minus, multiply the entire equation by T. So you'll get 3 minus 2T square minus T is equal to 0. So hence 2T square V careful plus T minus 3 is equal to 0. Did you all get this equation? So there is marks till this level also. Right. All of you confirm. Did you get this equation? Yes. So this is very easy to solve 2T square plus or plus 3T minus 2T minus 3 is equal to 0. You can club these two and these two. Right. So hence 2 common, 2T common T minus 1 plus 3 common T minus 1. So hence you'll get, again this implies 2T plus 3 is equal to T minus 1. Hence T is equal to 1 or T is equal to minus 3 by 2. But again you'll write what? Since time can only be positive value. Time can have positive value only. Hence T is equal to 1R. Don't miss the unit. So let me write it. Do not miss the units. You'll lose half mark otherwise. Do not miss the units ever. Aditya asking if we get a negative speed as one of the solutions we neglected. Yes. In 10th grade mathematics. Yes. Answered. Okay. So speed cannot be negative in our case. So this is done. Now next. Same. Same again. So hence we are not going to, you know let's not spend time over there. Right. You'll be able to solve this. Will they ask velocity? No. No. So you are not going to have any vector related stuff. You know. So they will talk in speed distance and time. This is speed distance time problem. So they are not going to ask you if the ball is thrown up. What will happen? Yeah. So those things are. So everything, whatever it will be happening will be happening in a straight line in one direction only. So don't worry about these things. Okay. Next. Will you be able to solve it? I'm not going to solve now. You know the technique. You know the pattern. Right. Step by step. There is no hurry. Take two, three minutes. This is a good, you know, practice of let's say finding how much time you're investing in such questions. So try to find that question time also. So do it methodically and tell me the answer. Let's see whether everyone is able to solve it. Quick, guys. Waiting for your response. Take time, but similar type of questions. Speed distance time. Sir. Yes. Done. One train is the original. One train is. My headphones have fallen down. Speed. Speed of the slow train is 30 kilometers per hour. And speed of the other train, fast train, that is 40 kilometers per hour. Oh, so Arayan and Monish are getting different answers. Check. They just really check, check, check. Two people are getting the same answer. So Monish, my dear friend, without units, it will be very safe. Don't say sorry to me. I am not going to give you marks. So be careful. There is no hurry. Yes. CBSE 10th grade paper can be finished in two hours, 15 minutes, 20 minutes max. So relax. Now you know what kind of questions are there. So it's always an advantage. Yes, guys. Shall I solve? Akshita is saying 40 Kmph. Anish, which is for 5th. But units also have to be mentioned. So I hope you have mentioned that in the notebook. Okay. Is 50 40 correct? Do I need to solve? Anyways, let me solve. So fast train takes three hours less than a slow train for a journey of 600 kilometers. So again, we'll start with, let us say, let the speed of slow train. So we'll take your language. Slow train be X Kmph. Okay. Like that you start. And a fast train takes three hours less than a slow train for a journey of 600 kilometers. If the speed of the slow train is 10,000, it's given 10 kilometers less than. So the speed of fast train is equal to X plus 10 Kmph. Okay. Find the speed of each train. So we have to X and X plus 10. We have to find out again each train. So don't only find X and V satisfied speed of each train. It says now it takes three hours less. So who will take less obviously faster train. So X plus 10 speed into P minus. My dear friends. Mute, mute, mute. Guys, you have to mute yourself. Yeah. Yeah. Thank you. Now. Yes. X plus 10. And what? We have to also assume. So he's right that whatever you're using is a variable. So he is. Or you can write time taken time taken by. Slow train is equal to TRs to cover 600 kilometers. Right. So hence the same type of question presented in a different format. That's it. So instead of the same train, they have given you two trains. That's it. Otherwise. The entire scheme is same. So X plus 10 T minus three is equal to 600. And X into T is equal to 600. Same two equations. You'll get one equation and I'm not going for the entire steps. So these two equations and table method, whichever you find. Well, so X is coming out to be 40 km pH. So you can check whether it is correct or not. And the other one is what X plus 10 fast train. How much? 50 km pH. All of you are getting this or any trouble. Then I will have to solve full equation below. I else will go to the other type of question varieties of question. Okay. Clear guys. Yes. So far so good. Slide number 22 or 26. 22 is the information. 26 is the slide number. All of you clear. Give me a thumbs up. If everybody is there, what happened to all of you? Oh, energy level should be back. Come on. Yes. Thumbs up. Thumbs up. Say why, why, why, why all of you will type. Why, why? So let me see the energy level. Energy level should never be down. Always, always up. Okay. So good that you are enduring this long, long hours, but our job is bigger. Okay. Do this. Now nature of root in three mark setup. Find the values of K for which the quadratic equation this has equal roots. So they are getting nature of roots and solving a quadratic equation, both together. So that's the beauty. So both the concepts are checked in the same question. Please solve and let me know. So you have done it already are in good others also. Achita is also done. Okay. So everybody is getting minus three comma five. Okay. So find the values of K for which the quadratic equation has equal roots. So again, you'll start with where did the pointer go. So for equal roots, what do we, we will have, we will have again be very, very careful a plus one B square minus four a k plus four C one is equal roots. That means B must be equal to zero. You can if you want discriminant, you can write discriminant because we are not writing what is the K and all that. So discriminant equals to zero B square minus four is zero. Correct. This is the equation. So hence now is there any possibility without expanding? Yes, there is a possibility without expanding also. So you can write K plus one whole square minus four K plus one and how much three no minus four. So minus 12 am I right minus four minus 12 minus 16 correct. Correct. No. This is one equation. Okay. So hence what can you say about it? Six into two. Hence this implies, see I never expanded it. So you can solve like this also. So six K plus one plus two K plus one minus 12 is equal to zero where my variable is K plus one now. Okay. Where did the 12 come from? Okay. Let me solve this and I'll come back to where this, you know, basically what I did is they go. Okay. From here. This is K plus one whole square minus four K plus one plus three is equal to zero. This is what is given. I simply put a bracket around it and expanded. So this is K plus one whole square minus four K plus one minus 12 is equal to zero. All okay. Am I right? Why did I do that? I didn't really need to expand. I don't need to expand now. Why? Because now it is simple here. What do I do? I'll get, I'll get K plus one common within brackets K plus one minus six. Then here I will get two six or two common and K plus one minus six is equal to zero. So K plus one minus six is common, which is K minus five. This means K minus five K plus one is equal to zero. Correct? So hence K is equal to five or K is equal to minus one. Okay. You got minus three. Oh, well, sorry, my bad, my bad, my bad, my bad. K plus one plus three, my bad. Yeah. Oh. Eraser. Eraser. So now back to pointer K because this two and this one will get added up. So three. Correct? So K is equal to five and minus three. You could have expanded also. So if it is appearing to be dangerous, when you expand it, you'll get K square plus one minus four K minus 16 is equal to zero. This is another way of doing it. Your way method. So this is K square, then minus four K plus two K is minus two K and minus 16 plus one is minus 15 is equal to zero. And again you will get K minus K minus five K plus three K minus 15 is equal to zero. So K minus, sorry, this is K square. So K minus five plus three times K minus five is equal to zero and you'll get the same thing. K minus five K plus three is equal to zero. Okay. And hence from here. Same. Right. So both ways you can solve. Okay. Next. Again, this is one mark. See, one markers are usually about finding the value of K. Oh, sorry. X equals to three. Either one root or nature of root. So X equals to three given. Quick. No time. No time. This will be like 30 seconds, but with full awareness. Don't rush. Don't rush through. Okay. Cool. Right. Do I need to solve this one? I don't think so. Good. Next. One mark again. For what values of K, the roots of the equation X square plus four X plus K are real. So hence if you mark the word here, the values. So there will be more than one value on it says plus minus. No, it will be inequality on it. It will be an inequality. No. So the moment it is not equal to zero. The moment nature of root, not, or they're not saying they're not equal, not equal. The moment it is not equal, then you know it has to be greater than less than things like that. Okay. Let's solve. So values of K, the roots of the equation are real for real roots and it is one marker. So for real roots. Now I can see people are doing the same mistake. Everyone. Lots of people are doing the same mistake. One mistake is now here D will be greater than equal to zero. First mistake. People are doing only D greater than zero. Equal roots are also real understood. So first mistake is this. It says values of the roots of the equation are real. Now real may only real is given nothing else. So real roots are D greater than equal to zero equal roots are also real. So this will be the criteria not and not D greater than zero. Right now. So hence if this is there then B square that is four square minus four times one times K must be greater than equal to zero. That means four square is greater than equal to four K. Right. So four is greater than K or K can be less than equal to. Okay. You can one marker. We don't need to check. So hence if you do if you adopt this, Anish is saying greater than equal to minus four. No, there is no K square term Anish. Why are you saying minus one on that B square minus four AC simple is greater than equal to zero. Why will there be two? There is no power on K. Okay. So hence. Okay. Next. They will not ask you for quadratic inequalities. Rest assured. Hmm. One marker again. Similar question. Reciprocal roots. Not clear previous question. What happened? Arayan. Unmute and ask. Arayan, can you unmute? Hello. Yes, sir. Can you hear me? Yeah, yeah. Go ahead. What happened? Sir, K coming being at the same time, how can it be greater than four? And at the same time be lesser than four? Where did I write? I say K is less than equal to four. Both are the same. Both are the same. Just see. This is not appropriate way of writing. So hence I turned it like that. Okay. So it is about. Sir, why isn't it an appropriate way of writing? Variable is, so they're asking about K. No. So you say, do you write three is equal to K? Are you right? K is equal to three. What is K equals to? So three. What is greater than or less than equal to? K is equal to three. So hence what I'm saying is it's not that you lose marks. But you know the, you know, so they're asking about K. So tell K is greater than or K. Or this statement means four is greater than equal to K. But they're asking, what is K? So K is less than equal to four. These are the values of K. Hence. Okay. Next. Solve. These are the three. Why are only three, three, three? Flight number 30. 23. So three. Find the value of K for which the roots of the equation are reciprocal of each other. So reciprocal roots, let the roots be, let the roots be alpha and one upon alpha. Therefore, product of roots, product of roots is equal to alpha into one upon alpha is equal to what? C by a K upon three. Right? So this means one is equal to K upon three. So K is equal to three. Okay. So I just tell you a trick. If there is a reciprocal roots C, they are A and C have to be equal reciprocal roots case. So I have a doubt if I, if I write a first one, first of all reciprocal always keep in mind. In case of reciprocal roots, A will be equal to C. A and C will be equal. Is that okay? Has to be. Yes. What happened? You were saying, sir, I doubt if, if you write minus 4K is greater than equal to minus 16, you get K is greater than equal to four for the previous question. Is it where I have a doubt if you write minus 4K, where is that? If you write minus 4K is greater than equal to minus 16, you get K, no, no, no, no, no. Oh, I understood your problem. They go. My dear, if X and Y are positive, let's say are positive. Okay. Then if let's say X is greater than Y, X is greater than Y, let's say X is greater than Y, but minus X will be less than minus Y. So inequality flips when you cancel negative sign both sides. Understood? So if minus 4K is greater than equal to minus 16, then if you cancel minus minus or multiply by minus sign, what will happen? 4K and then equality will flip. Understood? So if you can't cancel mine, it's not equation. So you can cancel minus minus from both sides. No. If you can't cancel minus from both sides or you're multiplying by minus 1, both sides, inequality will change. Less than will become greater than. Why? Because if 5 is greater than 3, minus 5 is less than minus 3. Right? And vice versa. So minus 7 is less than minus 3, but 7 is greater than 3. So if you're multiplying by minus 1, the inequality flips. Okay? In case of inequality. Got it? So do not cancel negative factors in inequality ever. If you're cancelling, change the inequality. Okay? Less than becomes greater than, greater than becomes less than. This is done. Next one. Now we are doing something which are, now this is, this used to come a little while earlier. Recent years, 2018 onwards, we haven't seen an age related problem. But then I thought probably you must be also thorough with all of this. So try this. The father's age is three times the sum of the ages of his two children. After five years, his age will be two times the sum of their ages. Find the present age. Hence I have eliminated the digital problem and all that. Why? Because it includes one upon X plus one upon X minus one, you know, things like that, which is so wherever that is there, I have kind of not touched them reducible form. 45 years. Find the present age of the father. How do you solve this one? Let's quickly. Yeah. Science. One is, let the age of the father of the father. This is what I wanted. So the X. Okay. Father's age is three times the sum of the ages of his two children. Okay. Three times the sum of the ages, let the sum of ages of children be why? So don't think that, okay, since it is a quadratic, so I can't use Y and all, no use both, no problem. So let the sum of ages of children be why? So what is given? Father's age is three times. So X is equal to three times Y.  Now here you need to be careful. Why? Because sum of their ages will be after two years, after five years, ten will be added because both will have five plus five each. Correct? Am I right, guys? Below. But this becomes a, oh, this is not a quadratic equation, I believe. This is not a quadratic equation. Oh, I put linear one. Okay, no problem. You can solve it. So it was appearing to be a, but how? Okay, never mind. So X equals to three Y. Yeah, yeah, yeah. So anyways, that also you can. So what was needed? X. So don't eliminate X, eliminate Y. So X plus five is equal to two X upon three plus 10. So you'll get X directly from here. Multiply the entire equation by three or you can first simplify. Two X by three plus 20 and then multiply by three. So I'm writing here, this will be three X plus 15 is equal to two X plus 60. So X is equal to 45. Oh, this was a linear equation. Sorry, no problem. Anyways, we'll cover linear equation later. So let's go ahead. Do this. A fraction becomes one by three when two is subtracted from the numerator becomes one by two when one is subtracted from the denominator. I think this is also from the linear equation part of the remix. Because X and Y you will be taking. Okay, wait a minute guys. Just a minute. Leave it for some time. This is again plain left 30 minutes. Less than similar type of question. Again, 2018, you can see speed time distance. This is to 2014. So try these, this one. Some of the squares of two consecutive odd numbers is 394 find the numbers. This is quadratic. Some of the squares of this is little earlier. So in 2014, just in case they come back with these, then what we'll do off late. We have not seen that show the father and the child age question again. Yes, I will just let me solve this and then come back to that. The sum of the squares of two consecutive done guys. Studying by very less people. Now responding. Hello. Board. Total board. Come on. Come on. Everyone is getting 30 and 15. Everyone is getting 30 and 15. Only few answers. Actually also 1315. Everyone is getting 1315. Okay, so let the numbers be. Squares of two consecutive odd numbers it is. So let the numbers be 2n plus one and 2n or rather. Why? Why this? Simply. No odd. So it is better. So let it be like that. So let me. You could have taken n plus. Then that will be for even. Wait a minute. Point or options. Eraser. Okay. Okay. Let the number be 2n plus one and 2n plus three. These are odd numbers. Mix of real numbers as well as or a bit of knowledge of real numbers. Anyways. So 2n plus one, 2n plus three. What is that? They are saying some of the squares of the two. So 2n plus one whole square. Plus two and plus three whole square. This is equal to three, nine, four. You have to find out 2n plus one and n plus three. Don't just find nn. Be content with it. No. So let's square it. So this is four and plus three. Plus one. And this one is four n square. Plus two. Three. Six. Two. 12. Yeah. Four. 12 and plus nine. Is equal to three, nine, four. Check again. Four n square plus two times two and four and plus one. Four n square plus two, two, four, three, 12. And plus nine equals three, nine. So correct. So hence we have. Four n square plus two times two and four and plus one. Four n square plus two, two, four, three. So hence we have eight n square. We have 16 n. And we have one plus nine 10 and that is minus three eight four. Correct. Did you all get that? Not all of this. Oh, no, I mean we can cancel this also, but only four. So this is two n square. You can take two n plus one and two and minus one either. I just took like that. No problem. So, no, it will not go by eight. It will only go by four. Is it? So eight n. Sorry, four n. And this will be minus 90. Oh, it will go by eight also. Sorry. So 24, four, six is equal to zero. Hi, it will go by eight. Sorry, my bad. So anyways, it will be n square plus two n minus 48 below the X plus one since X. So why can't we take directly X and X plus one? No, X and X plus one are not two consecutive odd numbers. I didn't. Correct. X and X plus two will work. But then I am trying to, you know, just avoid the even number is if possible, hence, otherwise you can try X X plus two or X minus one X plus one also two consecutive odd numbers is given. Right. No, sir, I'm saying if you just take X and X plus two then. Yeah. So hence, since it is given odd numbers, so hence I have taken. Yes, sir. No, sir. I'm saying that's why we need to take two n plus one to ensure it's an odd number. If we just take it as X, it can be odd or, you know, in that case, in that case, actually you will see you will end up getting this equation only. See why? Because the solution is fixed. So at the end of the day, it will be like that only. So because if we, because the root, if it is even, let's say, then solution doesn't exist. So it will, the solution will come out as odd numbers only. Understood. So and just to, you know, make sure that they are asking for odd. So we are giving odd. That's it. Otherwise you can take X six plus two. What will happen? Check.  Yes. No, you have to actually take two n plus one like that. Make sure that they are odd. You can assume X to be an odd number, but not every X will be an odd number. So hence this is 100% odd. No questions asked. Right. So hence don't take X because X can be even as well. How do you ensure that X is not even? Yes or no. So you make sure that whatever your variable is or whatever you are assigning this to, that is odd. So it is odd. No questions asked. Okay. Now, did you get this equation or not? Am I, am I getting the right equation 22 and minus 48? Yes. Okay. So eight six or 48. Clear. So n square plus eight n minus six n minus 48 equals zero. So n times n plus eight minus six times n plus eight equals zero. Should not be any relief here. No problem should be there. So hence it is n plus eight and n minus six is equal to zero. So n could be minus eight or six. Yeah, good. Okay. Good only you can get it directly also because it's fixed. That's what I'm saying. 394 will work only for this case and hence you're getting it. But just to satisfy the demand of the question, nothing like that. Not that you lose marks there are in. Don't worry. Okay. So here. So what will, what are the possibilities now n is equal to eight seven is equal to minus X and is equal to six. So hence the numbers are two times eight minus five minus 16 plus one, that is minus 15 and minus 13. One pair is this. And the other pair is 13 and 15. Okay. So go back to age problem. There are lots of age problem, obesity, blood pressure. Which problem do you want to go back to? So yep, sorry for the bad joke. This one. Okay. Yes. For that, they will have to move forward. We have to even we have to move forward. Who has to move forward? If they want to go back to age problem, they can't. They have to move forward to go to age. Oh, age problem. That was a bad book. I was talking about my age problems. So I am having all of these. You know, so the, the, I'm getting bored. Why the hair on my skull is getting white. Gray hair. All these are age problems. Yes. Sorry. You were just having some discussion. Okay. All right. So that's my question. But it. Okay. So where were we? This is done. Hi. One geometry problem also you cover here. Then I will show you the model. One. No, is this six days problem? These ones, six days problems I have just mentioned here. You just practice once, but it is unlikely to be asked. Why? Because the moment, if there is a, you know, equation, which is reducible to quadratic form is there, assuming that they're not going to ask such problems, but you must practice it. But anyways, before that, this one, geometry problem. Do this geometry problem. There could be a geometry problem as well. Yep. Folks done. Okay. So. Shortest side and diagonal and. Okay. Cool. So let's see. Diagonal of a rectangular field is 16 meters. In such cases do draw a diagram. Representative diagram. Label it also. Now what diagonal of a rectangular field is 16 meters. So, right? This is 16 meters. And. Oh, sorry. 16 meter more than the shorter side. Okay. So let this side be X shorter side. So this will be X plus 16. The longer side is 14 meter more, more than the shorter side. So mention everything on the. This thing only find the lens of the sides of the field. Lens of the side only. So you don't need to give the diagonal. Where are they asking? Find the lens of the sides of the field only this much. Yes or no. So we will invoke Pythagoras uncle. So you will say by Pythagoras theorem. By Pythagoras theorem. X plus 16 whole squared equals X plus 14 whole squared. Plus X squared before you go for solution check once more if you have written the equation correctly. So the diagonal of a rectangular field is 16 meter more than the shorter side. X plus 16. If the longer side is 14 meters more than the shorter side. So X plus 15. Correct. Then find the lens of the sides of the field. So you have to find out X and X plus 14. Simple. So square them up. So what will happen? You will get X square plus 256 plus 32. X is equal to X square plus 196. Right. And what 28 X plus X square. Okay. So one X square one X square. Fantastic. Now what? So take this on one side.  And if you are very prone to errors, you write like this. Minus 256 and minus 32 X. So that lesser error. And then also let me write here. This is equal to X square minus 4 X. And my dear friend how much? So 56 plus 4. That is 60. Am I right? Right. Or not? Yes. Good. So 60. 4. So X square. And it has to be what? Yes. 10 plus 6 is 60. So X square minus 10 X plus 6 X minus 60 is equal to 0. So hence it is X X minus 10 plus 6 X minus 10. So hence X minus 10 is equal to 0. X plus 6 is equal to 0. So X is equal to 10. And X is equal to minus 6. So you can write since what this thing is not possible positive only right. So hence your dimensions are 10 and 24 10 and 24 other dimensions. Okay. So we'll quickly now go to. This is important. This is actual board paper. You know how people have written. Can you see that? This is one of the question. Let the number of books bought by the shopkeeper BNC. Total money spent. Very clear. No. This thing here. Still it is, you know, very legible and very clean neat and clean step by step. And see when they are striking it off strike with one line like that. Right. So that people are very clear where to do what the answer has been properly highlighted. Okay. Since they both they are writing since number of book is a whole number it cannot be a negative number. It cannot be ignored and equals to 16 number of books bought by the shopkeeper is final answer like that. And this is one marker. So for one marker, what are they done? They go. So given they have written so much given quadratic equation is this where a is going to be able to see on all that. I think this is not that much required. Okay. So you can directly go to for no real roots. And the person has highlighted this. So you can, you can, you can avoid this. It's okay. Even if you don't write it, it's okay. But this is just to show that, you know, this discriminant must be less than zero. So hence B is less than zero. Even if you start from here for one, one marker, it will make sense. And hence they have got. Yeah. So this is X equals to K minus nine. So K should be less than minus nine. Right. These are all not required. Okay. You can stop here. Right. So this is how the model answers should be. So what are you learning from here? Neatness is definitely a factor here. Whether you agree or disagree. Neatness. Second steps.  Right. So that will be very, very clear always.  So summarizing mean you highlight that this is what you are wanting. Here it is. Okay. So you can just highlight the answers. Highlight. So there's a thumb rule, which I usually say highlight the answers. Highlight the answers. There's a thumb rule for writing an exam, writing a theoretical exam. The more you make the life of the examiner easier. The more she or he will make your life easier. So hence don't put the examiner under stress. Don't let him or her mind data from your notebook. Then you are risking your, let's say scores. Right. So hence make it easier for them. They will make it more easier for you.  And that's how, you know, so you're, even if you're very, very smart, very intelligent, but your writing and presentation is. They think that the child is insincere and hence usually at the human tendency, it would be that I will, you know, so for example, this need handwriting, even if the person is not very intelligent, let's say, even if then I am getting an impression that, okay, good, sincere work and sincerity does bring some, what do you say, brownie points. So hence, yes, obviously your knowledge has to be expressed properly. First of all, you must have that knowledge that is there. But after that also, if you're not presenting it properly, because it is going to anonymous unknown stranger examiner, it's possible that they miss, miss read you possible. And hence you can lose here and there, half a mark and our target of 80 upon 80 is lost. So I hope I made myself clear. Any question guys. So you now know the pattern of quadratic equation portions. You know what kind of questions are going to be there. You know that there will be one marker and a three marker. You know how to approach what type of problems will be there. So hence we say that it is at the end of the day, when you will start writing marks, it will become so predictable that you will think that, okay, enough. Well, any question, any problems, all this will be uploaded. The video will be uploaded on YouTube and the link will be shared in that Learnist course. So from tomorrow on, I'm sorry, my channel name, my channel name is Centrum Academy. So what you can do is you have to enroll there in that Learnist course. There is where every link, every resource, every PDF, every text, every document, every video, everything will be there only. So you don't need to, even there is a drive link, it will be posted there only. So tomorrow's class onwards, so those who have been with us, there will not be any link shared in the group, guys. So you can just come at 3.30, log in there, and you can start attending the session. Is that fine? Okay, so... Sir, one more time, can you say that from where we can get this video link? Centrum Academy YouTube channel. So those who are anyways, those who have joined the course on our Learnist platform, they know how to do it. Those who have not, they can approach me or Ankursar, or in our, you know, you take down, you know, you have to, those who have not registered themselves for this, they can go to Centrum Academy's website here. Those who have not registered, please register, because then only you'll be able to access the, what do you say? Let's say some material. So here is our website. This one is Centrum Academy's website. Now here, what you need to do is, you need to go to this enquiry admissions, and here you fill the form up, and here you please highlight the course comprehensive, this one, revision program X10. Then only we'll get to know that you are interested in this program. Okay. So everyone who's interested can just do that. For those who have just explored our program and then want to join this. Okay. So I hope it was useful session for all of you. It made some sense to you. Keep consistently attending the sessions. Keep writing the mocks. As all of you know, the 10th graders, NPS students, and our students in Centrum Academy, they know that you have a, come on this Sunday. So those who want to write, come on, can write, come on. Those who want to write mocks, can write mocks. So that freedom is with you. You can write both if you wish. Okay. Guys, so thanks a lot for your time. I hope for the initial glitch which was there because we were never expecting so many people to be attending it. I hope the momentum is continued. Keep working hard. Our best wishes is always with you. We will be working with you as well. Any issues, any problems, please reach out to us as quickly as possible. And we will be helping you with that. What about NTSC mocks? All of them are there. So keep writing. Every week NTSC, CBSC mocks will be there. So keep writing. There is no end to that till you finally write NTSC paper. So this Sunday we'll have Cam on the NTSC mock and the CBSC mock. They are available. You don't need to write all of them together. They will be available for a good amount of time. So don't worry. So write one, whichever you want to write. All options are open. You write whichever one you want. Is it okay? Yes, sir. And it will be available throughout the week. So write any... And anyways, now this is a holiday season. So you can utilize your Monday mornings, Sunday mornings, all that. To write as many mocks. Always remember, the more the number of mocks you'll write, the better your performance would be. There is no doubt about it. Okay? So friends, see you then. See you tomorrow with Ankur sir. We'll be taking up electricity tomorrow.  Once again, Target Centrum is the theme. So all of you should get Centrum. In Centrum, get Centrum. Bye-bye. Take care. Thank you, sir. Bye-bye. Thank you, sir.