 can we start? Anyone else wants to join? I hope all of you are here. So let's begin the topic is quadratic equations are going again with the same similar type of problems. There will be questions with traps at times. So please read carefully and do not rush through the questions at times it would require some knowledge of your previous grades. So it is not directly straightforward questions. It will be requiring some trigonometry, some linear equations, some previous grade identity knowledge and all that. So and if you don't really understand, let's say any question, any ways we explain it. So all of you ready and here we start. Read the question carefully. First question on your screen guys. So all the questions will be related to some of roots, nature of roots, very rarely solving a quadratic equation. So yeah, read the question carefully and then attend it. Don't rush through. Even if you are taking a lot of time, don't worry, we have 100 seconds. So keep yourself focused. Last time, many of you made mistakes by not reading the question properly. So there will be similar type of questions today also. Yeah, so take time. No worries. There is no try to finish within 100 seconds. That should be the target. That's it. Accuracy 100%. Accuracy should be 100%. All of you try for 100% accuracy. Speed will automatically come as you gain more and more practice. This was, I think, very simple question. Typically, you will see only three terms in a quadratic equation, but I have purposefully split the middle term. Okay. Okay, so the first answers are here. Lot many people have, oh my God, so much of trouble in this. Why did people write it? Okay, two possible values. Some of two possible values it was. Yeah, it was very, very simple question. It was quadratic equation in a quadratic equation. So here is the first explanation. Right? So, yes, how many of you could not solve it? What was the trouble? How many of you just put an X over there so that I could know how many? Okay. Okay, never mind. Didn't get enough time. Didn't get enough time for this. Oh my God. See, the roots of the quadratic equation are equal. So, the quadratic equation actually is this. Is it it? So, hence, roots are equal means a plus 8 whole square, that is b square minus 4ac. Discriminant has to be 0 minus 4 times 4 times 9. Correct? Am I right? This must be 0. Right? So, hence, if you expand this, you'll get a square plus 64 plus what? 16a minus, how much? 16 times 9, 144 is 0. Okay. So, a square plus 16a and this much is minus 80. Am I right? Did you get all of you got this equation guys? Right? Then what? Sum of two possible values of a. Sum of two possible values of a. This is a quadratic equation. So, two values, let it be alpha and beta. So, sum of alpha plus beta is how much? Minus 16. How much time did it take? Right? So, it was hardly, you would have, you don't need to solve the equation also. From here itself, you can say alpha plus beta. So, two possible values of a will come from here. One is alpha. Let's say another is beta. So, sum of roots alpha plus beta is minus b by a. So, minus 16. Anyone didn't understand? This was not at all time consuming, is it? Hardly 30 seconds, more than enough. If, yes, you have to do calculations properly. Clear? Any doubt in this? Any doubt in this question? Right? So, only 13 people got it. That's really concerning. What is there to be aware of type of problem? So, what is the question? The question will not be straightforward, find the sum of the roots. Right? So, you have to infer sum of two possible values of a. What is two possible values of a? Roots of a in this equation. What are the sum of the roots? Minus 16. Right? Okay, next question, guys. So, first leader board. Only 13 people have got it, right? That's not cool. Good. So, Pratik. Okay. Aniruddh, Ananya, Daksh, Aniket, Shreyas. Who's Wolf-Mister? I will throw you out. Identify yourself first. Who's Wolf-Mister? Name please. If you are part of this group, then I need to know. Otherwise, I will just bar you from this. Anyone is Wolf-Mister in this? Yeah? Arain. Arain is Wolf-Mister? Arain, you're there? Okay, no problem. So, you are Wolf-Mister, right? No? Okay, next. No problem, Arain. So, read the question carefully and then solve. Patience would be required. Do not rush through. No problem, Arain. Come on, solve. Okay. So, I think you would have got it by now. Very straightforward. Very straightforward. If you know the concepts, very, very straightforward. Okay. Very good. So, let us see who has done what. So, still people are not that confident. People are arbitrarily marking the answers. Yes, 1 by 8 is the correct answer. Right? How many of you could not get it? Mention. Okay. Okay. Yes, all of you, please register your, this thing, so that I could know. And hence, okay. Only five, six people didn't get it. Got it right. Okay. Psi, Rishita. Okay. Others, there are 64 people. Come on, guys. I need everyone's response, so that I could make an assessment. Okay. Okay. Okay. Okay. Never mind. Chal. Let me, let me solve this question. Quadratic equation x bar plus mx plus n has roots twice those of x bar px plus m. So, let us say this particular quadratic equation has roots alpha and beta. So, what will happen? Alpha plus beta is equal to minus p and alpha beta is equal to m. Correct? Now, the roots of this equation are two alpha, two beta, isn't it? So, twice alpha plus beta is equal to minus m and twice alpha twice beta is equal to how much n and you have to find out p by n value. Right? So, if you divide these two equations, what will you get? 1 by 2 is equal to p upon m. And if you divide these two equations, you will get 1 by 4 is equal to m upon m. Correct? And then you simply multiply these two equations. You will get 1 by 8 is equal to p upon m. Isn't it? Did you understand? So, what is the answer? Just play of words, nothing else. Did you get all of you who mentioned x? Did you get the answer? Right? Yes, Magna. Yeah. Anything else here you want to understand? Clear? No problem. Right? Shall we move ahead then? Okay. So, let's see the leaderboard. So far. Yes, who is this? Sir, I had some internet issues. So, could you just please repeat the answer? Yes. Explanation, sorry. This is a question. But that equation x square plus mx plus n equals to 0 as roots twice those of x square plus px plus m equals to 0, where pmn are non-zero real numbers, we have to find out the value of p by n. So, I had assumed alpha and beta as the roots of the second equation. So, alpha plus beta is negative p minus b by a. Right? Alpha beta will be c by a, that is m. Now, in the second equation, the roots are twice. So, 2 alpha plus 2 beta sum of roots of the first equation. This one will be minus b by a, that is minus m. And product of root 2 alpha 2 beta is n. Right? And then I simply divided first these two equations. I got half is equal to p by m. Then I divided these two equations. I got half, 1 by 4 is equal to m by n. I multiplied both the last equation and I got this. p by n is equal to 1 by 8. I hope this is clear. Let's go to the next question. Look carefully. Options are a little confusing, deliberately made. So, hence, please check each one of them. Last five seconds. Okay, let's see if this was, oh my god, this is hard. This is really depressing. Why? Why was this done? Why? Why? Why? Why? How many of you could not understand the question? Could not do it. Shardulish. Okay. Intimidating, but easy. Yes. Time. Oh my god. That means concept. This question could be solved within 10 seconds. Yes. Search for 0 x square. Yes. You had picked it well, but then it was very simple question, guys. Not a parabola. For parabola, what is the, you know, parabola, when do we get a parabola when you have an equation like a x square plus b x plus c? This is the first line in the introduction of your quadratic equation. Right? Yes. Isn't it? So, a should not be 0 for a parabola. If a was a is 0, then gone. That means coefficient of x square must not be 0. Right? So, which out of all, which one is, you just check the coefficient of x square. That's it. 3x times 4x is 12x square. Here minus 2x and 3x is minus 6x square. And here it is 4x square. Clearly this is not adding up to 0. So, this is ruled out. Right? Let's say second one, 3x and minus 4x. Right? This is where you will get the quadratic terms. So, minus 12x square. Then x into 3x minus 3x square. And 2 and minus 2 is minus 4x square. Again ruled out. They are not adding up to 0. Yes. Ananya, listen. Yeah. So, what I'm saying is you have to just check the coefficient of x square in all the four. In whichever you see coefficient of x square to be 0, that's not a parabola. That's not a quadratic polynomial. Isn't it? So, I was doing one by one. So, if you see the third one, it is 12x square. Then these two multiplied together will give you 6x square. And this together will give you minus 6x square. Again not equal to 0. So, all three were ruled out. D was the only option. Check. 3 into 4x 12x square minus 2 into 3x minus 6x square. 3x into minus 2x minus 6x square. Total is 0. How much time? So, these all questions are concept based. Many people would have started blindly multiplying and then trying to find out, isn't it? Did you not do that? All of you would have started multiplying all the factors, all the terms here. And that exactly was the intention, right? So, that you fall for this waste time. And so, hence, what is the learning? Learning is for any quadratic equation to exist, a must not be 0. Right? Only then you will see a parabola. Yes. Tempting to multiply. That's what the intent of the question was. And you fell for it. Okay? But the result is pathetic. Only 80 people got it. That means you need to work on these kind of, you know, so go by basics. The normal tendency is whichever way I know I will move in that direction. That's a wrong strategy. Okay? So, good. So, there is some shuffle now. Next question, guys, on your screen. Think about the steps before you attempt the question. Next question. This question will also test your nerves. So, let's see how many of you keep yourself cool while solving this question. Only knowledge will not help. You know, to keep yourself cool, relaxed and focused. If it is a 100 second question, it has to be solved within that. That means it is possible to solve within that time. So, if you are going through a longer route, then you have to learn that, you know, there are some intelligent ways of solving. It will take time if you get distracted by unnecessary calculations. There are lots of parameters which have been given uselessly. Still people are struggling. Okay? So, see, your MMP are useless quantity. They are not required. Not required at all. You have to find out Q. What is Q? Here is Q. Q is product of routes of this. Isn't it? Q is nothing but product of the given route. What is the product of the given routes? Routes are alpha plus 1 by beta and beta plus 1 by alpha. If you multiply, you will get alpha beta plus 1 plus 1 plus 1 by alpha beta. That is, and what was alpha beta? Alpha beta was routes of this equation and their product is 2. So, 2 plus 1 plus 1 plus 1 by 2. 9 by 2. Done. So, those who are, who would have started with okay, alpha plus beta is M okay, alpha beta is 2, then you would have some then product and all that. What's the point? So, hence, all these questions are indicating yes. So, it will be, you know, the questions will be made in such a way that it will take you to, you, how many of you have heard of Chakravju in Mahabharat? Chakravju? Have you heard of Chakravju? What was the specialty of Chakravju in Mahabharat? So, the strategy was very clear. Only Arjun, what knew how to, you know, crack that question. So, he was deliberately distracted away from that, right? So, when Acharya Drone set that up, one of the foremost strategy was that to distract Arjun towards somewhere else, some other place and he was taken away by some Yodha, someone and only Abhimanyu knew how to enter. He did not know how to come out, right? So, these questions are like that. It will be distracting you, okay? Unnecessarily, you will be falling for that. That question will, you know, what do you say? Poke your ego as well at times, okay? And that is where you will make mistakes, right? No. So, hence, all these exams questions will test your temperament, okay? Right? So, hence, learn that. Don't get distracted. Be focused on the demand of the question. Then only you will be getting it in a shorter amount of time. So, this practice has to be done again and again. Next question. See, if I have given question in, you know, to be solved in 100 seconds, then it will be solved within 100 seconds. Because I framed the question and solved myself before I put it in front of you. Okay, now who's venture or what? These are people who are really venture. Who's venture, anyone? Who has by mistake got this name or there is some? Who's venture? Anyone is venture or there's no venture? Venture out my dear friend. Okay, never mind. Menti ojinam. What is minti ojinam? It should be done easily, I believe. Good, fellas, good. Okay, okay, okay. So, last three seconds. And I hope this time you should have got it correct. All of you, most of you at least, most of you, but then still less than 50% are getting it correct. Who could not do this? This was plain and simple. Very, very easy problem. Right? What happened in this? What happened in this? So, let's say alpha beta roots. So, clearly alpha plus beta is equal to 2a and average, average value has been given alpha plus beta by 2 is 1. Right? So, hence, you will see if you solve these two, you will get a is equal to 1, a is 1. If a is 1, then x square minus 2x plus b is equal to 0 and 5 is one of the roots, then 5 square minus 2 times 5 plus b is 0. So, if you solve this equation, you'll get b as minus 15. Right? So, there's another way like you do need to do all that. Now, what is another way, sir? Sir, you know alpha plus beta by 2 is equal to 1 and we know alpha is equal to 5. So, beta is equal to minus 3 and then you just multiply alpha into beta is equal to b. So, minus 3 into 5, so you get minus 15. Why? My God. Don't you think that much is too much? So, I, you know, anyways, you can do that as well, but then you don't need to find the other root. No need. Right? So, hence, you know, sort of, if you think that is making steps easier for you, you can do that. But simply, you got a as 1, deploy it, put the value of 5, x equals to 5, you got it. Okay. Anyways, your idea is to get it correct. Okay. So, good. Next question. Oh, before that, the leader, the leader of Sibodas. Okay. So, venture is coming. Who is venture by the way? Anyone known person? Who's venture? Is this someone amongst us? Anyone? Who's, who's this guy? Oh, Avani is venture. Oh, okay. So, you should mention that. Okay. Cool. Chalo next. Next is good Avani. Put your name, no. Oh, no problem. Chalo. Next question, guys, on your board. No number theory. Common sense. Common sense only. Common sense based question. Mathematical common sense. Good. That majority have got it right. It was very, very easy question. One of the roots has to be 2. Isn't it? Some of roots is, some roots are prime, alpha and beta are prime numbers and alpha plus beta is 91, which is an odd number, right? Alpha and beta is odd, alpha plus beta is odd, right? And alpha and beta prime. So, hence 2 has to be there. Only even prime is 2. Correct. So, 2 plus 89 is equal to 91. So, 89 is the second root, right? So, twice. Now, what is the question? Find the product of the roots. So, 2 into 89. 178. Hello. How was that? So, every time you will not be given to find this value, this is for robots and calculators, not for humans. Okay. So, please be ready for such questions, right? Common sense question. How many of you got it correct? Okay. Cool. Very good. Nice. Nice to see that. Next. No problem. Now you know how. Oh my god. Oh my god. Okay. So, careless mistakes. Very good. Nice. Next question. Yes, Aryan, that way also you can do. Yes, Meghna, I will show you after this, after this question. Do not mess with calculations. Please mess with calculations. Okay. Solution time and I am sure most of you would have got it correct. Oh, what has happened to you today? Most of you are getting it wrong. You need more and more, more and more practice, more and more. It was again very simple. What is there in this question? Both roots have to be negative since all coefficients are positive. I did not understand. Is it fine if we say both roots have to be negative since all coefficients are positive? Yes, you can. Okay. So, some of the reciprocal of the root, roots of the equation. So, you know, there are two ways of doing it. One is identify the signs of the coefficient. All are positive. Okay. All are positive. All are positive. That means alpha plus beta is negative. Correct? Alpha plus beta is negative less than zero. Yes or no? And alpha beta is greater than zero, right? Because all the coefficients are positive. If you reduce it to the quadratic equation, all the coefficient will be positive. Correct? So, alpha beta is greater than zero. Alpha beta is less than zero. That means both alpha and beta are individually less than zero. So, right? So, sum. So, one by alpha is also less than zero and one by beta is also less than zero. So, hence, one by alpha plus one by beta will be less than zero. Right? So, hence, only option negative is this. So, you can solve like that. That's, did you understand the logic? Since we, once again, once again, important tricks to solve. Right? So, alpha and beta are the roots of what, this is the quadratic equation. So, if you really want to convert this into quadratic equation, it is this x square plus 20 21 x plus 20 21 is equal to zero. This is the quadratic equation. Yeah, that's what I'm, I'm, I will solve both ways. This is the equation. Now, all the coefficients are positive. That means alpha plus beta is simply minus 20 21 by 2020. And alpha beta is 20 21 by 2020. Right? Now, the formal way of doing it is you have to find out some of the reciprocal of the roots. So, you have to find out one by alpha plus one by beta. That is alpha plus beta by alpha beta. You have to find out this. So, that means you have to simply divide this. And if you simply divide this, you'll get minus one. So, answer one minus one. It can come from here. Let's say I don't want to do all of that. So, this method is understood how to find out reciprocal kassam. Yes. Now, let's say if I don't want to write anything, if I were an exam, then what would I do? I will see that all the coefficients of the equation, all the coefficients are positive. That means some of the root is negative and product of root is positive. That means both the roots are negative. If both the roots are negative, their reciprocal will be negative and some of their reciprocal would be negative. So, hence clearly all these are ruled out. These are ruled out. So, this is the so without even solving or touching or you know, you can just, you know, do it mentally. Got it? Both ways. Yes. Clear? Everyone shall we move ahead? Now, tell me all of you sincerely. Did we use anything which you have not learned so far? Have we have we discussed any question? Have we taken up any question which you guys have not learned, but still the result is this, which shows a very terrible picture. It will be confusing. Why do you think it will be a piece of cake? Competitions are like that. They want to pick those students who are stable, intelligent, knowledgeable and very good temperament. Understood? So, hence these are the factors which are being tested. So, please work on that part as well. Okay? This is a skill. Over a period of time only you will be able to develop. So, next question. Again, pay attention, focus and okay. Nice. So, next question guys. Aniket, you have a ghost in your device. Guys, there is some, okay. Now, this is some issue in this question, but anyways, so I'll give you, I will give you a, so I'll give you a, what you said. All of you will get marks here. There is some issue in this question, I believe, but let me see your attempt. Time problem, no problem. Everyone gets marks here, because there was some issue. Now, I would like to definitely understand those who have marked D. What made you mark D? Yes, anyone can unmute and say why D? Those who have marked D, how many of you have marked D and why? Random selection, okay. All are positive if X is 1. I guess D, 14 marked. Okay. Now, there was slight this thing. So, what I'll do is wait a minute, but let me explain the question first, then you will understand. None of the options are correct in this. Why? Yes, I didn't, why? Explain. Let me see. How did you figure out that none of the options are correct? Yes, explain. If you know, then intuition, if you know, then intelligence, as in all that, also intuition. And then I just substituted X for 1 in all the equations and all the problems. Coming up with positive rules. You substitute, I mean, all of this equation, it's coming out to be positive on it. The other idea was that if it has, it only has roots, if it doesn't have roots, then it will either be only positive or only negative. And over here, A and C have opposite signs. So, B square minus A C will always be positive. So, there will always be two real roots. So, there will be at least one negative value. One negative value, okay. Anyways, the idea was here to demonstrate the power of completing the square. Okay, so one, actually one statement is missing in this. In fact, there was, yes, one small element was missing in it. Anyways, but all of them, if you see X to the power 1010 whole square plus 256 is 16 by X to the power 1010 whole square. Then if you see this is minus 30 can be written as minus 32 plus 2. Is it it? Yes, if you do that, this will become X to the power. So, the question was this, which of the following expression is never negative for any values of X? Okay, the question was, so this wording became little, so the question was which of the following is never, but then there will be three options. Okay, anyways, so tell me which of the following are never negative for any value of X? Which of the three? All the three are never negative. So, you know, so the question has to be worded in that. Do you see that these are all greater than zero? All of them. All 123 is greater than zero. Why? The completing the square fund will give you that. So, X to the power 1010 first one is this minus 16 X to the power 1010 whole square plus 2. This was the first one. Then second one was X 1010 minus 19 X 1010 whole square plus 2. And similarly, third one was X to the power 1010 minus 21 X to the power 1010 whole square plus 2. All of them are always greater than zero. So, the question should have been worded like this. Which of the following expression can be negative for some values of X? Like that. So, in that case, this was negative. Why? Because if you see the fourth one is X to the power 1010 minus 923 to the power X 1010 whole square. And this is minus 6. So, the fourth one is this. So, hence, this can be negative. This D can be negative sometimes when let's say, when this quantity is zero, then D becomes negative. So, hence, D is the only option which can have negative values. ABC can never take negative values. Did you get this question? This understanding is clear, at least. Right? So, hence, if such questions come, now you are prepared. Right? So, what is the meaning? So, A, B and C. So, hence, the question could be this. What is the minimum value of FX? A, can you tell me what is the minimum value of FX? Yeah, I'll repeat once again. So, if you look closely again, let me explain like this. So, if you see, the first expression is this. This one. First expression is simply this. X to the power 1010 minus 6. We add by X to the power 1010 square plus 2. You can check. This one is the first expression. Second expression is this. Right? And third one is this. All of them, all of them are greater than zero. Yes or no? Why are they greater than zero? Because we have a square term and a positive term added to it. Did you get that point? Guys, did you get this point? It's a square term as well as 2 is getting added. So, there will not be any value where whatever be the value of X, these three will always be greater than zero. Right? Is that clear to everyone? Now, can you tell me what is the minimum value of FX? Minimum. Don't use formula all the time. What is the minimum value below? No need to. 2. Very good. What is the minimum value of B? 2 again. What is the minimum value of C? And what is the minimum value of minus 6? Got it? So, when you complete this square, squares are always greater than equal to zero. So, you get the same question could have been asked. How do you get the minimum value? What is the minimum value of any square or in tandem? What is the minimum value of any square? Zero. Right? So, hence, if something is something square plus 2, what is the minimum value of something square plus 2? When something is zero, then something square plus 2 is 2. Correct? So, that's how you have to find out minimum and maximum values. We have explained. Anyways, I have to just tweak this. Just give me a second. I will cancel this question. Question number 8, it was right. Was it right? Yes. So, question number 8 and what was the 14? Yes. So, I am resetting the result. This slide reset. So, I have reset it and it has been now. Okay. So, this was the previous leaderboard. Now, all of you can just, you know, don't do that anything. I will skip. So, leaderboard is not changed. Question is removed. Next question on your screen. Do this. Oh, we are running slow today. Quickly. Don't guess. Options are very close. I have purposefully made the options very, very close. Don't guess. Don't try to calculate also. Okay. I hope you have not followed the trap. There is a trap in this question. And 3, 2, 1. How many got it correct? Oh, nice. Few people did get. Few people could not complete. Time was less, isn't it? Time was less for many. Okay. So, this was the question and there was a negative sign here. So, you know what to do. So, write it Y is equal to this or X equals to this. So, there was a negative sign. So, I hope you have taken care of that. So, X equals to this. So, yeah, here itself it is. I can just see what I am doing. Am I right? So, follow this. Yes. So, X minus 2 or 2 minus X is equal to 1 upon 1 plus 1 upon 2 plus 1 upon 2 plus like that. Sorry. Am I right? 2 minus X is this. So, that means 2 minus X is equal to 1 upon 2 plus 2 minus X. Isn't it? Over. Game is over. So, 2 minus X is equal to 1 upon 4 minus X. You have to solve this quadratic equation. So, that means 2 minus X whole square. Sorry. Not whole square. 4 minus X is 1. How 2 minus X? Look carefully. Once again, I am doing. Please let me know if you don't understand any of the steps. So, take it. Yes. Once again. X is equal to this. There is no problem. So, X minus 2 is minus 1 upon 2 plus 1 upon 2 plus 1 upon 2 plus. Right? Right? Isn't it? Is that okay? So, 2 minus X if you reverse, you will get 1 upon 2 plus 1 upon 2 plus 1 upon 2 plus. Right? Right? You can do that whichever way. So, you don't need to do separately. You will get X directly Rn. So, this is 2 minus X is equal to 1 upon 2 plus. This entity is what is being repeated. So, can I not write this as 2 minus X below. Clear? Right? Now, game over. So, 2 minus X is equal to 1 upon 4 minus X. So, 2 minus X times 4 minus X is 1. So, that means X square minus 6X plus 8 is equal to 1. So, X square minus 6X plus 7 is equal to 0. And from here directly, X is equal to if you solve minus V that is 6 plus minus under root V square that is 36 minus 4 AC that is 28 divided by 2. Right? So, that is 3 or let me write 6 plus minus under root 8 by 2. This is 6 plus minus 2 root 2 by 2 which is 3 plus minus root 2. But it will not be 3 plus root 2 because 3 plus root 2 will take it beyond 4. But this entity is 2 lesser than 2. This entity X is less than 2. So, what will be the value? 3 minus root 2. Isn't it? Go away. Clear? Everyone? Did we deploy anything which we have not learnt in this question or in our quadratic equation? Now, you are getting the message loud and clear. Message is loud and clear. Right? So, life will not be straight forward in the message. You need to brood over each and every question to get it right. Good. So, next question guys. Easy one. Revision of previous topics is also necessary. So, hence you will see application of previous topics in this question. I hope you remember your basic trigonometry rules. Trying options will not help because there are no standard angles in the option. I feel those nine people who answered before sir could finish his statement. Let them, let them. Okay. Good. 20 people got it. Rest of the people have forgotten their trigonometry basics. Very, very simple question. It was right. So, 9 sine to the power 4 theta plus 6 cos theta I will write as 6 minus 6 sine square theta minus 5 is equal to 0. Right? I hope this is clear. So, this is 3 sine square theta whole square minus 2 times 3 sine theta plus 1 is 0. So, that means sine theta 3 square theta minus 1 square is 0. That means 3 sine square theta minus 1 is 0. So, sine square theta is 1 upon 3. So, sine theta is plus minus 1 by root over 3. Right? Sine theta is what did we find out? Sine theta is plus minus 1 upon root 2. Yep. Now, cos theta will be under root 1 minus sine square theta which is 1 minus 1 by 3 which is root 2 by 3. Correct? So, tan theta is sine by cos. So, 1 by root 3 divided by root 2 by root 3. So, 1 by root 2. Right? Clear? No problem? Okay. So, let's quadratic equation application in autonomic application in quadratic equations. Oh my god. Avani is cracking. Very good. Shallow. Very less gap between the top two people. Okay. Nice. Shallow. I have given 120 seconds in this question. So, you have lots of time. Please solve. There are two minutes in this question. You should be able to solve within two minutes. Then you must be able to answer this Aranya. This type of questions are too many in your Adi Sharma book here and there. I just tamper with the values so that you get trapped. The values will be little this thing. Body of the question will remain the same. Aniket, if Menti was sad, then he would have been sad for everyone. You have a haunted PC. How do people answer so fast? They press the key immediately. One of them. A, B, C, D. Spray and pray. Right? Trial and error. No. Why trial and error? Not at all. Oh my god. Two minutes also. 52 people only solved. Oh. Yeah. Most of you got it. That's nice. Cool. I'm really happy. So, this was you don't need to follow your methods and steps here. One fourth of the herd of cows. Let's say the herd of the moment you see the question, you start writing. Herd is X. That is what total number of cow they're asking. Right? So, let it be X. So, one fourth is in the forest. So, how much in the forest? X by four. Twice the square root of the herd has gone to mountains. So, twice root X mountain and 15 are on the banks of the river plus 15. This must be equal to the herd itself. Is the equation clear? This is a quadratic equation and then the question is boiled down to or you just need to solve this equation. That's it. Yeah. You can eliminate also. No problem. Yeah. So, hence you can see which of these, you know, so clearly it has to be X has to be integer, right? X has to be integer. I will solve, but just in case you want to click quickly. So, X has to be a perfect square. So, these are ruled out. Okay. Yes. Either 100 or 36. So, you can check. This will suit. So, this is So, you have to do what? Multiply the entire equation by four. So, you will get X plus 8 root X plus 60 is equal to 4X. Okay. Now, keep the root at one side and take it to the other side. So, 3X minus 60. This is the, right? Okay. Yes, style and error definitely works. So, in your exam in NTSC, don't really go for formal. If you can solve in some shortcut, please do. Okay. Then what? So, 64X is equal to 9X square plus 3600 minus 66360X. Am I right? So, this equation is 9X square on 360 and 64. That is minus 424X plus 3600. So, this is equal to zero. So, I have to solve this one. Right? Now, 9 into 36, I have to break. 9 into 36. So, 424 I have to get somehow. I have to break this in this fashion. So, yeah. So, 324 and 100 works. Right? So, 9X square minus 324X minus 100X plus 3600 is zero. Yes. Yes. Right? So, hence what do you need to do? You solve it. You solve this equation. You will get 100 by 9. One fraction term. Let me solve only. So, let's say it is 9X common, X minus how much? 9 into 36. Yes, 36. And then 100 common minus 100 common X minus 36 is zero. So, this is 9X minus 100 times X minus 36. This is zero. Clearly, X is 36 because X equals to 100 by 9 will not satisfy this equation. Why? Because it's a fraction. And number of cows cannot be a fraction. So, answer this. So, 120 seconds for more than enough to solve this problem. Good that most of you have done it. And the easiest problem of the evening before that the leader. Oh, Avani is on first position now. Very good. Next. Last question of the evening. Very easy question. Last 10 seconds, guys. Last 10 seconds of the quiz. I hope everybody has marked their answer. Okay. So, time's up. And okay. Cool. Both of you are correct. This is a very simple question. Right? Right angle, diagonal given, right? 15 meters. So, let's say the difference between length and width is 3. So, if width is X, it is X plus 3. And you have to find out X square. Right? You have to find out X square. So, hence, you know X square plus X plus 3 whole square is equal to 15 square. So, this implies X square plus X square plus 9 plus 6X minus 225 is 0. So, 2X square plus 6X minus 216 is 0. Okay. So, 216 and oh, so 2 can go. So, this is X square minus plus 3X minus 108 is 0. So, this can be X square and 12 times 9. So, 12X minus 9X minus 108 is 0. Correct. So, hence, X X plus 12 minus 9X plus 12 is 0. So, X plus 12X X minus 9 is 0. So, X is either minus 12 or X is 9, but minus 12 cannot be the solution because length cannot be negative. So, X has to be 9. So, X plus 3 is the length. So, 12 is the length. So, area is 12 into 9 or not 8. Very simple. So, A was the answer. Okay. So, last. Finally, the bell goes. Okay. Okay. Of me, you missed it. Oh, okay. Very good. Aniket Gupta. Congrats once again. So, Momita, Shreyas and Anvi are all in Dhanagar. Then, Omesh from Radhaj Dhanagar, Sharduli from Kormangala, THK. Who is in front? Wait. Let's see once again. Okay. So, Ananya from CHK. Very good. So, these are the rankings. Okay. Ananya, Omesh, Sharduli, Akshit, Pratik and Anish Bharatwaj. Lots of Indian people do from THK, PPS East, Akshit, Sharduli from Kormangala, Omesh from Radhaj Dhanagar. Very good, folks. Very good. Congrats. So, see you again. So, learning is again whether you are loose. Did you learn something? Did you, you know, you have to explore beyond your regular curriculum or whatever we have done. Questions will be difficult at times. So, friends, I hope you enjoyed the quiz. Let's meet again next time. Bye-bye.