 start. All of you ready with pen and paper? Okay. Okay. So here I am starting the quiz and three, two, one, go. So the first question is on your screen. If you are not able to see the question, then I can show it on the tab as well. So just let me know. If x is the root of equation sine square x plus sine x minus two equals zero, then please read the question carefully. Read the question carefully. Again, I will keep on reiterating this because you will be trapped there. So questions have been set by me with some, you know, things in mind, your vulnerability is in mind. So hence, take your time and solve it. So in today's quiz or centermar, we'll have two types of questions for the application of trigonometry questions. The time is two minutes and you need to learn how to solve that question in within two minutes. We'll discuss that at length because in exams like NDSC, you are supposed to solve it like that only. Or in fact, within a minute. Okay. So hence, we will, I have, this is the first quiz. So hence, on trigonometric application, I have given you two minutes, but later on, we'll reduce it. Okay. So we have some 20 more seconds and some 20, some around 15 people have not voted, I believe. There are 67, 68. How many are there? 70 around. Very good. So here is 3, 3, 2 and 1. Cool. So time's up. Let me see how many of you got it correct. Okay. So 21 of you and see, this is what happens. So, you know, even question numbers will be lesser, but not everyone is able to solve this. X, please. How many X and what happened? This was so, so easy a question. This was mix of your ninth grade identities as well as trigonometry. So questions will not be that straightforward. There are two approaches to it. Either you can figure out through value. Yes. So if you see X is equal to 90 degrees, satisfies this equation. So X equals to 90 degrees. You know, in such question, if you are in a competition, you can try value or values out. So the standard values will be the solution. You know, there are only, yes, four, five standard trigonometric ratios and mostly it goes around 0 and 90. So you could, you can, you know, immediately check whether it is working like that. Or you can factorize. Yes. As Aniket is not mentioning. So what would be the, don't need no need to. So, yeah, so quadratic is one way, but many people haven't done it. So you can always read the middle term. So middle term splitting is the best. Formally, it should be done like that. Signs square X and you can split the middle term so that it becomes a minus sign X plus two sign X minus two equals zero, right? So hence you will get sign X common and this is sign X minus one and then from your two common sign X minus one and this is equal to zero. Okay, so hence what will you get? Sign X minus one into sign X plus two is zero. Okay, clearly sign X plus two will never be zero. Why? If you see this is sign X is equal to one or sign X is equal to minus two, right? So this cannot be. No, can we, can't be negative. Why? It can be negative, but then this is out of bounds. Sign X lies between minus one and one. So this is gone. So X sign X is one. When is sign X one, when X is 90 degree or pi by two radians. X is 90 degrees. So hence you put X equals to 90 degrees. Anyways, now you will get what? So X is 90 degrees. So only tan X is not defined in when X is 90 degrees. So hence you would be expecting that X equals to zero degree, 30 degree, 60 degree and 90 degrees will be given as options. No, here is the twist. So it will take more time for you to mark it correctly. So these are the questions will be set like that so that even after you have solved the question, they will still expect you to be very, very careful. Okay, so you could see the difference. There are no direct options. Okay, let's see the leaderboard. All of you got this, you know, those who had asked for the explanation. Okay, very good. So that is okay. Very good. Next, shall we move ahead? Okay, any any doubt in the first question? No doubt. Okay, so let's go to next question. Again, again, please read the question carefully. Read the question very, very carefully. It is there so you can always try values or you can go formal way. Enough time is there. You can see and mark actually, but if you are careful. Okay, so some 60 people have already marked it. These are called trigonometric equations. So don't get panicked by the terminology. You already know how to solve equations. We have simply replaced the variables by trigonometric ratios. Nothing else changes. Okay, so here we are done with the second question. And okay, so okay, not bad, but then still people are there who are struggling. Okay, very good. So, yep, this was fine. Right, you could, you know, so why, why many people have marked it too? So these are the people. These are the people who have not seen that I have not put a equality sign here. Okay, so 90 could not be the solution. So many people, how many of you did this mistake? You have counted 90 degrees as well. Right, right. Now you understand, right. So this is why I keep on saying that please read the question carefully. I have purposefully not given this equal to sign here so that you get trapped. And this is what the objective is that you please learn how to, you know, how to behave in a competition or, you know, you know, test like this. So hence, very simple. It was actually, if you see, again, two sin theta cos theta minus cos theta is zero. Right, since cos theta is common, so cos theta common and two sin theta minus one is zero. So this is factorization you have learned in grade nine, right. So now there are two factors product is zero. That means either cos theta is zero or two sin theta minus one is zero. That means sin theta is half. So theta is cos theta is zero. What does theta mean theta is 90 degrees and sin theta is half. That means theta is 30 degrees. Okay, but theta 90 was not acceptable. So this is ruled out only this one. So hence one, right. No, that will, don't, don't divide by cos, don't divide by cos q. That means, you know, you got the right answer, but in a different way. Why, why it is not cos theta, it is not advisable to divide by cos theta. Why? Why, because if cos theta by chance becomes zero, then it is, yeah. So if you are already aware that cos theta cannot be, you know, zero because it is less than 90 degree, then you are on the right track. Okay, but if you divide, so in an equation, never cross, never cancel two factors from LHS and RHS in an equation. Never ever do that. Again, never ever cancel common factors from LHS and RHS. Make sure that they are not zero because you will miss one root over there if you cancel that. Okay, yeah, if you know that, then it's fine, Laksh, but you know, I know most of you will not, you know, pay attention to those minute details and hence said. Okay, anyways, so let's go to the leaderboard. So be alert now again. So hence alertness pays. Please be alert. Okay, here is the next problem. Okay. Again, read the question carefully. It looks like people are more comfortable now. So they have taken only 25 seconds. So many people have responded already. So last 25 seconds, this was fairly simple. If you have done the calculation, carefully. So last five seconds now. How many X? Anyone who says X here? Okay, so here is very good. So I can see lots of people have got it correct. Nice. So what was the problem in this question? Okay, never mind. So let me explain it to you. So this was the question. Which are the following is the solution to the equation for cos square theta minus three. So what is cos square theta? Cos square theta is three upon four. Right? So hence cos theta will be plus minus root three by two. Okay, cos theta is plus minus root three by two. Yeah, so hence if you see 30 degree is what the solution is 30 degree cos 30 is root three by two. 390 degree is as good as 30 degree, isn't it? 390 degrees is 360 degrees plus 30 degrees. Okay, so 390 is as good as 30. So this also will satisfy and and theta five by six. Right? So that is also a radian five by six radian is also is the solution. So all the above can be said as the solution. Is it okay? Fair enough. Since I had not described what theta is, whether it is in degree or radians. So all the three could be yes. So those who had put in X, now is it okay? This is for radian. So those who have solved, you know, or gone through the coursework which we have floated, you would have known it by now. And in the in the test papers of trigonometry also this type of questions were there. Okay, yes. Actually, you don't need to, you know, solve all this. So whenever all the above are there, you know, it's a hint enough, good hint that, you know, so check 3390 makes sense, five by six makes all are same angle. You don't need to solve also. All of them were same angles. So you could have done this way also. Okay. So fair enough. So in the next CV leaderboard, okay, so here is the leaderboard next. Very good. No problem. Next. Okay. Okay. Now solve it. Don't get distracted again. You'll have to work it out. Read the question carefully. Come on. Good. Good. You are taking time. No worries. Accuracy has to be maintained as 100%. Even if you solve five, six questions, make sure that all of them work correct. Or you just let it go. You don't need to mark all the questions. So last 10 seconds. Last 10 seconds. 14 people got it correct. Okay. So hence I think A and D are the same thing. Yeah. So I think there is some, yeah, this was a question anyways. So cost square minus sign five is minus one by four. So did you solve it or did you put the value? Did you solve it or how many of you solve it? Okay. So hence those who could not solve, right, cost square five is one minus sign square five minus sign five minus one by four is zero. Right. So this is what you can see here. So hence it is if you take everything on the other side, you'll get sign square five plus sign five plus one by four minus one is zero. So sign square five again, you have to do what? Spread the middle term. So no need to put quadratic formula and all that minus three upon four is zero. Okay. So hence it is four. What happened? Surya saying something. So hence now what will it be? Yeah. So hence four sign square five plus four sign five minus three is zero. This is the equation. Right. This is the equation. Now split the middle term. So you know how to split it. So four sign square five plus six sign five minus two sign five minus three is zero. Okay. Okay. Right. Now what? So four sign square five. Right. So you can take common. What will be common two sign square five and two sign, sorry, two sign five and two sign five here. Then plus three. Correct. And minus two sign five plus three is zero. Okay. Now what? So you can say two sign five plus three times again irrelevant chatting. You have to then leave. I'm telling you leaderboard is not the final agenda here. Okay. So is that clear? If you are not happy, people are advised to leave. No problem. So stay calm and solve problems. Okay. Minus one is zero. Okay. So then sign five is what will be sign five minus three by two and what will be sign five again? One by two. Okay. So which one is correct or both are correct? Which one should I pick? Both are correct or one only one is correct. Yes. So this is correct. This is correct. Why is this not correct? Because sign five cannot be less than minus one. Right. Sign five cannot be less than minus one. So sign of any angle, sign theta is always greater than equal to minus one. Always remember. And also sign theta is less than or more. So with that, this is ruled out. Sign five is half. So five will be how much 30 degrees. Okay. So in this case, tan theta and cos five. Both are correct. Right. Both are correct. Won't this take a lot of time considering the time constraint on the question? No. No. It will not take because these are the questions which are typically asked in NTSCs and exams like that. So you have to be fast enough. Obviously I am going slow. I am going slow here. Right. I am going slow here. The other way of it is as I as you know that it was very obvious. You can you can start from here to go to this. So you can inspect the inspect the options. What options they are talking about? They're talking about 30 degrees. They're talking about 90 degrees. They're talking about 60 degrees and 30 degrees again. So you have to check three. Right. Okay. So when you solve it, see I am going slow. Why? Because I need to explain to each one of you. Yeah. Which one? Which one? So Rishita is saying something. Yes, Rishita. Once again, is this explanation clear how to solve trigonometric equations? See, I'm doing nothing. Guys, whatever we have learned in previous grades, I'm just utilizing that. Okay. And then so let's say this kind of a thing is happening in your NTSA exam. Let's say you are, you know, if you are getting two answers correct, so you can use your best judgment to mark it. No problem. And it will be taken care of. Last three steps. Okay. So last is this clear? So I factorize the given expression factorize the given expression to get two sign five plus three and two sign five minus one equals zero. When two factors are product is zero, then each factor could be zero. So in equating each one of them to be equal to zero, I got sign five as minus three by two and sign five is half. But this cannot be acceptable because sign five cannot be sign of any angle is never less than minus one. Sign value revolves or, you know, oscillates between minus one and plus one. So all the values of sign will be between minus one and plus one. Hence this is ruled out. So only sign five is half. So hence possibly there was an error. And yes, this was the targeted last answer. So we'll skip the leaderboard and go to the next question. All of you are ready with the next question. Okay. Ready. Fair enough. Any other doubt in this guys? Please ask. Any other all clear? All clear. Yeah. Is it okay? Okay. Next question. Next question is forget that leaderboard. So this you have to find out the value. What is this value equal to? What is this value equal to? Times up. So we had, okay, here are lots of people got it wrong. Why? Guess it. Guess it, I believe, right? How many of you could not get this? No time. Okay. Let me see. Let me, let me, let me show you how to deal with such problems. Okay. There go. First of all, the moment you see minus three, minus three. Okay. It is very, you know, if you see, it will take less than 30 seconds to solve this question. The questions will be typically set like that. Okay. So immediately what you do is you need to reciprocate one by minus two sine a by root three plus two costs a whole cube, isn't it? And then this is plus root three. See, how am I solving now? I'm doing it fast so that you can understand that it can be done within no time. Now you have to take, you know, what common common denominator. So this will become one minus two, sine a and one plus two sine a, right? Cube, cube, and then plus and I'm ignoring the denominator for the time being because I have a hint that these are simple numbers. The denominator is not going to play much of a role. So see, and then I'm doing three plus two costs a and cube, correct? If you see, this is one minus a minus v a plus v form. So this is nothing but one minus four sine square a whole cube. And this one is three minus four cost square a, right? And now what is convert any cost or cost or sign into the other thing, right? So hence you can write one minus four sine square can be written as one minus cost square a correct whole cube plus three minus four cost per a, right? Whole cube. Okay. Now see, what is this? If you look carefully, this numerator is one minus four plus four cost square a whole cube plus three minus four cost square a whole cube. These are, if you see what is this one minus four you have to do like that minus three here itself. Now do you do you see these are additive inverse of each other? They are additive inverse of each other. It will become zero over problem is over within 20, 30 seconds you'll be able to solve, right? So answer is zero. So hence I was explaining how to do it faster now and now I'll explain the process, right? So it can be very well done in no time. Okay. Now coming back to the explanation. Okay. See, that's why I keep number of questions lesser so that everyone can see. Now what did I do? I just reciprocated because there is minus three over here. I think this is not a problem. And then this is what I got after reciprocation. And then I did common denominator. So if you do common denominator, whatever is denominator, forget it. What is numerator? One minus two sine a cube and one plus two sine a cube. Okay. And then this one root three minus two cost a cube. And for this want of space I'm writing here root three plus two cost a. Now don't you think these are a minus b a plus b form? Okay. So if that is so that's where I started doing that. So hence a minus b a plus b will give you a square minus b square form plus a square minus b square form. Both are cube. So I'm not touching the cube at all. Now if you look carefully, so what did I do? I converted sine square into one minus cost query here. So I got one minus four plus four cost square a whole cube. Yes, sir. Who's this? Yeah, so what I was saying is, shall I start from the beginning? So once again, reciprocated reciprocated and then took common denominators, common denominators taken and then use the identity a square minus b square form. You will get this expression dot this one at the end of the day. You do it once again and you'll get it three minus four cost square a whole cube. And if you watch carefully, this is four cost square a minus three whole cube plus three minus four cost square a whole cube. Is it? If you see they are additive inverse of each other, they will add up to zero. Why? Because this item can be written as minus four cost square a minus three whole cube. So this one added to this gets you zero. So hence you don't need to even look at the denominator and the answer was zero. So this is how it will be really time consuming and confusing. Yes, life will be like that. They will not give you piece of cake all the time. Okay. So you are and there will be people. It is in this lot where 22 people could do it. My concern is six plus 31 plus three. You have to work on this through a lot of practicing. So 31 plus six, 37 plus three, 40 people, right? So hence those who have done it, I congratulate you because you you are able to solve a complex problem in lesser time. So here is the leaderboard which I will not be more interested in. I will be interested in who is solving it correctly again and again. So good. Lots of people have done it. That's nice. Okay. Next question. Again, take do fast, but always keep track of what you're doing. Okay. Here is a very simple question identity. So hence solve. Good that people are taking time and solving. That's really nice. Okay. So how many of you lagged time? Very few people got it correct. Oh my God. Why? So was this a guess? Was this a guess? Why could you not solve this halfway and guess? How many of you guessed? So don't guess when there is a negative marking guys. Okay. Don't guess when there is a negative marking, right? It's better not to mark than to right? Never mind. So now you know what KVP wise are meant, who was who's meant for KVP wise and other exams. Now you're getting the flavor. Okay. Right. So someone has, okay. So of course sir is suggesting substitution. That's also fine. But then yeah, many times substitution may also lead to two solutions, but never mind. So let's solve it. Okay. And it was again, fairly simple. If you would have, I had purposely removed this bracket. Right. So this was what you had to do. Okay. One plus tan square a what is the identity one plus tan square a is below one plus tan square a secant square. Right. So secant square a and and then if I would be doing it, I would not be doing step by step in the exam where in I will be writing common denominator and all that. Immediately I will write this. Tell me if this is correct. Is it correct? So I will not go step by step in this. So hence what will I do then? Second square is common. Then what is it one plus again tan square a plus one by tan square a. Right. So what do I get? I get secant square again a secant square. So total will be secant 4a. Right. Again a secant square from here. So secant square, secant square, secant 4 divided by tan square. Period. End of the game. So this is nothing but cos square a by sine square a which is tan one upon tan that is and this is secant 4a. So what do I get? I get, what do I get? I get secant square a times sine square a. Right. Right. So hence it is one upon cos square a sine square a. That is what I am getting. Correct now. So if you see this is nothing but one upon cos square a can be written as one minus sine square a and game over. Right. So this is one upon sine square a minus sine to the power 4a. This is what and correct. Got it. How much time did we take? Not even one minute. Okay. So it's very much possible to solve. Right. Very much possible to solve. So now hence you know all those who are aiming at again reiterating this funda. This is exactly will be the scene in a exam like KVPYJ, NTSC and all that. Taste it now guys. Practice it. Isn't it practice for all of you? Okay. So hence what did we learn? We learned that it is very much possible. Keep yourself calm. Don't worry about the leaderboard, leaderboard and that doesn't happen. Right. Yes. So please focus on whether you are able to solve the problem here because it's not a competition between 64, 50, 70 odd people, lakhs and millions of people will write the exam. Where do you lie? So leaderboard automatically will be taken care of if you are careful and conscious. So next question be ready with the pen and paper. Do not panic. It is solvable and very much doable. Okay. Never mind. So here we go for the next question. Start. Do not panic. Complete the question. Even if let's say time's up. Complete the question. At least you will know what the solution is. Next time you will remember that hey, this is how it has to be solved. Still 35 seconds left still can be solved. So I will tell you what is there, when you solve identities, how you have to keep in mind whatever results you get. Normally the tendency of all of you is what I have observed. You just check whether the answer key matches or not. No one ponders upon any question that can we just kind of make this result yours or ours with that. No one does it. Hence, it becomes difficult. Now here is the, okay, so friends. Okay, so hence here lots of people have gone wrong. Let me explain it. It was very simple. Again, I will not. So I remember this. What would I remember? I would remember that this is nothing but simple. 1 minus cos theta by 1 plus cos theta. Anyone has any problem in doing this? Right? Anyone? Right? And then same here is again 1 plus cos theta by 1 minus cos theta. So I'm not going to write there in the exam that CKD is 1 upon cos and take common denominator and all that. No. You have to remember this. When you remember this, then what it's simple. Again, the common, what you can do is you can, what have we discussed? If there is a root, go conjugate way. Then we use secant theta minus 1. I need to solve this. Secant theta minus square theta. Yes, you can do that. No problem. You can do that. Yes. So hence conjugate is the key. So if you conjugate it and I have done it so many times that I know when I'm conjugating, so I'll get 1 minus cos theta on top, direct. And here it will be sin theta. Have you done this multiple number of times? Yep. 1 minus cos theta by sin theta and this one, this will be again 1 plus cos theta by sin theta. If you have done this multiple number of times and you can see the answer. Yep. Anything. So what I'm saying, see, this is by practice that I know that if under root 1 minus cos theta upon 1 plus cos theta is given after conjugation, you get under root 1 minus cos theta whole square divided by 1 minus cos square theta, which is 1 minus cos theta divided by sin theta. So if you remember this directly, so you'll save time. So always remember under root 1 minus cos theta upon 1 plus cos theta is 1 minus cos theta by sin theta. So that will help you. And then then after this no-brainer, so cos theta will go only 2 upon sin theta. So hence, 2 cos you can think of. Right. Whichever way. Yep. How many of, how many got it correct? Anyways, we are timing, time is running, running out. Never mind. Good. Let's see the leaderboard. Very good. Keep it up. So this is the next. Okay. Angpur sir, you're, you're taking one position. That's not correct. You are eating, you're taking one position of these people. Anyways, next question, guys. Start. If sin square theta plus sin theta is 1, then cos to the power 8 theta plus 2 cos 6 theta plus cos 4 theta is equal to. Problem, Aniket, learn. Learning is important. Winning or losing is not important here. We will talk about winning and losing when we get to that point. Don't worry. See, go practice, practice in your own paper and pen. Three, two, one. Yes, go. So how many? Oh my God. So lots of people have got it correct. How many X? How many could not solve it? Tell me. X, X, X. Yeah, a lot of, a lot of you have not been able to know what is, guys. See. So here is the explanation. I have partly solved it so that it saves time. So what you can do is sin theta is 1 minus sin square theta from the equation C. So it is cos square theta. So what do I get? Sin theta is cos square theta. This is the given part of it. Okay. Not really. You don't need to square and all. So, you know, you can get it directly from here. So sin theta is cos square theta. Now, you don't deploy this here. So you'll get this equation, sin to the power 4 theta plus 2 sin q theta plus sin square theta. And take sin square common, you'll get sin theta plus 1 whole square. This is what it is. Okay. So this is what it will get reduced to. Now what you can do? What can you do? You can say, so what will you do? You will do this, that sin square theta can be written as 1 minus. Did I get something wrong? Okay. No. So did I get something wrong here? Sin to the power 4 theta plus 2 sin q theta plus sin square. Sin square is common. Sin square theta plus, yes, it's correct. Right? No. So, okay, fair enough. Now, sin square theta was 1 minus sin theta. See, from top 1 minus sin theta. Okay. And so I replace this sin square theta from this. So sin square theta is 1 minus sin theta. Right. And this is 1 plus sin theta whole square. Correct. Then if you see 1 minus sin theta is equal to 1 plus sin theta and 1 plus sin theta. This is the final expression. And now I use this. What is this? 1 minus sin square theta. So I'm writing here 1 minus sin square theta times 1 plus sin theta. This is what is left. Now 1 minus sin square theta is again sin theta, guys. See, 1 minus sin square theta is sin theta from here. So sin theta and this is 1 plus sin theta. So what is it? Open it up. You'll get sin theta plus sin square theta. And what is sin theta plus sin square theta? It's given to be equal to 1. So answer is 1. Right? Is that okay? So you have to do. Understood. The explanation is understood. Explain again. Okay. No problem. See. Okay. So what I did was explain with this color. Yeah. So sin theta is cos square theta whole square. Right. From this equation. Given equation. Because you can write this as 1 minus sin square theta is equal to sin theta. If you rearrange it. So 1 minus sin square is cos square. So cos square theta is sin theta. Remember this. So 1 cos square theta is sin theta from here. And then you just deploy simply code. Why you lost? Just a minute. Don't worry. Someone can help Varsha Victoria. So what is the expression? So cos 8 is cos square to the power 4 theta. Isn't it? Like that. So cos square theta is sin theta. So this will become simply sin 4 theta. Okay. Sin 4 theta. Sin square theta. Whichever other if you found it easier. No problem. Sin 4 theta plus 2 sin cube theta plus sin square theta. Right. This is what the expression would be reduced to. And then to take sin square theta common. So what is left? Sin square theta plus 2 sin theta plus 1. And what is it? A plus V whole square. So sin square theta into sin theta plus 1 whole square. That's it. And then what did I do? Again I deployed sin square theta from the equation. What is sin square theta from the given relation? Is 1 minus sin theta. So I ended up doing like that. Whatever came in my mind immediately because I have to save time. So this is what I got. Sin square theta is replaced by 1 minus sin theta. And then I took 1 plus sin theta from it. Borrowed from here. So it becomes 1 minus sin square theta. And this becomes 1 plus sin theta. And this square term is gone. Because I borrowed. And then 1 minus sin square theta is or again or rather 1 plus. Yeah. 1 minus sin square theta. Did I do that? Yeah. Correct. Yeah. So 1 minus sin square theta. I again wrote it from here. So if you see 1 minus sin square theta. If you see 1 minus sin square theta is sin theta. So I simply wrote this as sin theta and then multiplied whatever was left. So I got sin theta plus sin square theta. Now sin theta plus sin square theta was given to be equal to 1. I ended up getting the same thing. So 1. Okay. Clear. We still have four questions left guys. Okay. If it's still not clear, I would suggest just go through the recording once again. Now. Now. Now. Diagrams will be required. So be ready. Application of trigonometry questions are to follow. And I will show you how to do it fast. Very fast. Okay. Okay. Next. Quickly draw the diagram as fast as you can. Label it and then use the information of 45 degrees. Still people are maan. Guesswork. So this time I have to identify who all are guessing. Maybe. Maybe. Yes. Maybe. Still solving. Very simple. If you know. If you draw the diagram quickly, it will be very, very simple. Three, two, one. Not everyone could answer it. No problem. So that means you're struggling. More practice is required. Answer is A. That's correct. And many of you have got it wrong also. Now let me check. Tell you how to quickly draw the diagram and solve it. I've already drawn it. But for your understanding, let me start from fresh. So yeah. So the moment I see this, I will, a man is standing on the deck of a ship which is 10 meter above water level for 10. Then he observes the angle of elevation of a top of a hill. So there's a hill. Then angle of elevation 45. So join. So this is 45. Put it X. So this is X. The angle of depression of the base of the hills. Okay. Fair enough. Depression. So this is 30. Okay. This is 10. So this is 10. Close. Come on. 10 30. Don't even write 10 30. 10 30 is one by root three. Remember it. So it is 10 upon X. So X you got X is 10 root three. X is 10 root three. What is that caliber distance of the hill from the ship and the height of the hill? So X is known. So now you can rule out X is 10 root three, 17 point something, right? So only one option is there done. So you can solve so quickly, right? And now explain how to solve it methodically. Is this method clear to everyone? How did I solve quickly within no time? So that's how you have to do quickly draw diagram and it could be rough also. Don't worry about the neatness over there in the exam in such kind of exams and then quickly the C. Oh yeah. So I quickly drew the diagram and then if it was 45 degree, I use this. The moment it is 45 degree. Okay. Once again. So this is how it is to be done. See man is standing on the deck of a ship which is 10 meter above water level. So first line 10 meter, write it like that. He observes the angle of elevation of the top of a hill. So that means there is another hill like that and top of a hill. So this is the angle of elevation, which is given to be equal to 45 degree in a right angle triangle. The moment anything is 45 degree. The other has to be 45. So it's nice. Firstly strangle. This is done. Now angle of depression of the foot is 30 degrees. It's given right angle of depression of the base of the hill is 30 degree. Calculate the distance of the hill. So you have to find out x first and then 10 plus x. You have to find x and 10 plus x. What is x? So if you see this diagram 10 upon x is 10 30 in this triangle. You have to look very fast 10 by x 1 by root 3. Remember the values. Don't really forget that and then gone. If you forget it gone. So x is 10 root 3 done exist and root 3 and that is first thing. So hence check the option there itself. So 10 root 3 is root 3 is 1.73. So 10 times it 17.32. So one is one up. The distance of the hill has to be 17.32. Only one option is there which is having and you can see even if you have to calculate the height of the hill, it is 10 more than x. So see 27.3. So hence a done. Okay. That's how you have to do it. Next also we are going to deal with similar type of question. Draw diagram quickly and keep keep in mind the data which is given. Keep in mind the data which is given. Very good. Next question is next question is here on your screen. Come on guys. Take your time. It will sometime will be lost in reading. Keep track of 45 degrees. The moment elevation 30 degrees. So you remember the tan values. Tan is only required and I have given you two minutes for this. So don't worry. Two minutes are there. Two minutes is good enough to solve this. How many could get it? Let me see now. Still a lot of people are making mistakes then or you are guessing right. So quickly see how to yes. I will explain. Don't worry. See now observe all of you. Don't write x. I know I have explained. Don't worry. I'm explaining. Listen to what I'm saying. Okay. Pay attention to the way I will be reading the question and then drawing the diagram quickly and then jumping on how to always remember in height and distance. Majorly it is sign and tan or tan only most of the time and see how I'm reading and solving simultaneously. See this is a question. Amit is standing on the ground. So quickly draw a ground and flying a kite with a 100 meter of string. So this is 100 meter at a 30 degree angle done right 100. Then another boy is standing on the roof of a 10 meter high building but you have to read it and is flying his kite at an elevation of 45 degree. Very good. So keep that in mind. So there will be a guy who is flying like that 45 degrees. So keep in mind only both Amit and Nitin are on the opposite side. That is what I was looking for. So hence if this is a kite this person is here at 10 meter height right now and he is flying kite like that. So this is a kite both are together. Find the length of the string that the second boy must have so that the two kites meet. So let's say this is L diagram done quickly and then you're do the needful right. So what is 45 degree guys. This is 45 brilliant. If this is 45 you know this is L by root 2. This is L by root 2. This has to be in your mind as sine theta L cos theta right. If that is there then what this is 10 okay then what you have to find out what do you need to find out L only. So L by root 2 yeah. So now if you see 100 sine 30. So this length is 100 sine 30 is equal to L by root 2 plus 10. Okay this is the equation I got and that's it. Sine 30 is half 100 becomes 50. 50 is L by root 2 plus 10. So 40 is L by root 2. So L is 40 root 2 done. Answer is 40 root 2 meters A. Yeah so hence this is how how much time did we take? No time. So almost like 30 seconds maximum 30 to 45 seconds. Good enough time to solve such problems. There will be definitely 2 to 3 problems of application of trigonometry at least in NTSC and then one step ahead in KPY definitely. So hence this was very simple question again what did we do? Read the question very fast without losing the so you know the usual idea is just keep circling the data also it's very very important at times. So mining the data right. So immediately I saw 100 meter okay length of kite so always keep this in mind. Length of kite that means it is hypotenuse we are talking about string 30 degrees is the angle of elevation then another boy anything standing on the roof 10 meter high so this is 10 meter from 45 the moment it was 45 automatically it was L by root 2 which came in my mind. I deployed it immediately without going into that. So all this is part of the game. So this you will not be able to do if you don't solve the problems which you do regularly in your assignments without consciousness right. So hence you have to see that if 45 degree is there if L is the hypotenuse the other two sides will have to be L by root 2. So I saved my time over there and then the moment I got L by root 2 and I got 10 and I know the hypotenuse I know the perpendicular is sine theta times the hypotenuse. So I use that otherwise normally people would have done sine 30 degree is equal to L by root 2 plus 10 by 100 this also you could have done no problem and then solve it for L you get 40 root 2 clear fast and effective good understood next next this all is not so don't think that I have you know mastered it over a you know few days it takes a lot of time and lots of repetition to achieve this. So hence we kept keep on you know emphasizing on do as many questions as possible and learn from the questions also don't only just remember there is theory there is problem okay next similar very easy very very easy. So hence these type of problems would have been discussed okay let me see how many of you got it okay good number of people but still not not many people could not again see the pace how do I do and how do I so angle of elevation of a cloud from a point h meters above oh lake is q that is theta actually I hope this is not this has not disturbed you so I should have told you that oh no this is in my this thing in your question it was theta only right in your question it was theta only because I had corrected it yes so theta and h meters above a lake is right theta so hence what the angle of elevation of a cloud so this is cloud and it is elevated from a point h meters above the lake so this is lake h theta right and the lake is on the ground obviously oh gonna have to draw reflection of the lake is 45 degrees so let's say this is the reflection and this is 45 degrees clear clear diagram is clear so this is the lake ground I'm going slow now this time and h is there so let this be h and let's say you are finding capital H always keep the variable which you want like that okay h and then and then here here it is this is also h sorry it is actually playing with two colors so hence let me use this yeah so this is also h because reflection has to be you would have studied in physics reflection you know image distance is equal to object distance now over so 45 degree actually helps a lot why because if this is x this is x then x is this this is this one is h so don't you think I can write x is equal to h plus h why isosceles triangle so this solves a lot of purpose and then what is this length h minus small h correct so h minus small h divided by x now divided by x is equal to tan theta right now instead of this I can write x as h plus h tan theta now you just need to solve this one so quickly solve h minus h is equal to h tan theta plus small h tan theta and you have to find out capital H right so keep that in mind so capital H here so 1 minus tan theta is equal to h common from here and this is 1 plus tan theta so my dear friend h will be small h times 1 plus tan theta actually you don't need to write till this step the moment you see the previous step your mind should know that this is the hypotenuse is not x no this I have not I have not treated that as I have treated this is a rectangle right so this is the going in my mind so this is x so this is also x okay yep so hence the option is this so hence I was saying that you don't need to go to this level here itself many people would see this and mark the correct answer so what was this 1 plus 1 minus tan theta by 1 plus tan theta this one so did I get the author other way around I found out something other h I think what was the question the height of the cloud capital H only had to find out now so h 1 minus tan theta is equal to okay wait a minute wait a minute did I do some wrong calculation or something I messed it up maybe so x is h plus h angle of depression was 45 degrees so s h plus h is x that's fine so h minus h divided by x is tan theta so x is h plus s so h plus h tan theta that's fine yeah so hence h and 1 minus tan theta is equal to 1 plus tan theta yes so it should be 1 plus tan theta by 1 minus tan theta what is the correct answer or did I mess somewhere up I don't know wait a minute just let me check once again h h minus h on x is h plus h is x so h minus h is 1 plus h tan theta plus h tan theta so h 1 minus tan theta yeah it's correct why it is coming like that 1 plus tan theta capital is h is 1 plus tan theta by 1 minus tan theta it should come correct h meters yes so I think the answer no I have to okay so maybe the oh b is correct no oh no no no the other other a is correct guys sorry a is correct but only four people I marked it that is surprising only four people mark that or we are doing something wrong here I think we are only four people anyways let's see never mind let's go to the next question the final one answer is a sorry so not a few people have marked it but these people are wrong so Shobita Aiden Momita Surya you are wrong sorry never mind let's go do last question okay so in last 15 seconds for all here is the final tally oh cool so people have got okay so again let me explain this quickly angle of depressions of the top and bottom of 10 meter tall building from the top of a multi-storey building right and so hence this is multi-storey building okay angle of depression of the top and bottom of 10 meter tall so this is multi-storey building and this is the another building which is 10 meter tall and the same depression right angle of depression so this is quickly this is 30 degree obviously it cannot be 60 this is 60 now that's how I quickly draw the diagram so you know that these are alternate interior angles so you can mark it directly 60 and this is 30 okay this is 16 30 actually this was given as 60 degree fair enough now 10 is given so hence you have to find out what two things find the height of the multi-storey building so many a times this itself will give you but then they are smart they are giving two options with the same thing so hence we have to be little you know cautious so h is the total height this is 10 then definitely this is h minus 10 this portion is h minus 10 okay now let it be x x immediately take the you know and then then what then solve h minus 10 by x is tan 30 don't write tan 30 and all write value directly then second second triangle h upon x is tan 60 is root 3 so you know now h directly root 3x or x directly what was to be found out both h and x are unknown never mind so hence if you divide both of them this is two equations what will you get h minus 10 by h x x cancelled is 1 by 3 why because 1 by root 3 divided by root 3 is 1 by root 3 times root 3 in the denominator so 1 by 3 so fast you have to do it so this is 3h minus 30 is equal to h so 2h is equal to 30 so h is 15 h is 15 so hence two options b and c are correct which one so now you have to kind of distance between so x also you have to find out so x is h by root 3 so that is 15 by root 3 meters so which is which one 15 by root 3 yeah so 15 by root 3 is 5 root 3 meters isn't it rationalize quickly you can do it mentally also so 5 root 3 so which one is correct c yeah okay either understood did you understand so we are not c well i am c don't go by my speed and all so i have been doing this all this my life that's not the agenda agenda is are you minutely observing so what are the learnings of the day learnings of the day is whenever you are solving problems also there will be lots of insights generated lots of insights for example if you see i'm not i'm very comfortable in dividing many people will be taking a lot of time then i'm very comfortable in the values right so 10 30 i know in the back of my this thing this is 1 by root 3 10 60 root 3 so you have to be very very thorough it happens only when you do multiple number of times and then carefully you know so i am not writing steps like 3 into h minus 10 is equal to h into 1 and all that that will eat up a lot of time so you have to learn how to do it x how is x is x plus h where question number 11 where how is x is equal to 45 degrees it was given no so this is x so this triangle this triangle is isosceles right so if this height is h if this height is h and this is capital h so capital small h plus capital h will be x okay yeah so you don't need to do sign in for 45 momenta 45 degrees do not complicate unnecessary the moment 45 water mouth watering the moment i see 45 degree in the question the first thing which strikes my mind is the triangle is isosceles so this is our trait which we have developed over a period of time so you have to develop the same thing at the moment you see 45 oh equal the moment you see 60 360 degrees that means oh sides are equal equilateral triangle so hence all of that has to flash in your mind like the moment you read it and hence it saves time it happens not because you are extraordinarily intelligent or something no it happens when you do similar type of problems n number of times and hence repetition repetition repetition i keep shouting i keep encouraging all of you that solves many many questions okay and methodically that is so all this will take some time for you to develop don't worry so we solved 12 questions only but the learnings are immense learnings are point number one read the question carefully many of you lost the question number two above because you didn't see that less than equal to science this one this one so reading question saves saves your you know effort read question carefully be you know the other thing like cancelling and all that is not you know allowed all the time then many a times through angles itself you can figure out that you did not need not solve this and your answer was all the above in this case this is another trick then in this one what did we learn you have to yeah this one actually there was a problem here so value pudding is also helping this one i didn't care about denominator numerator itself serve the purpose you don't need to panic in this kind of question next next one all identities quick and fast practice period no big deal then here remembering that secant theta minus one can be expressed as one minus cost theta and one plus cost is like that and conjugate kafanda if you when you are solving it you remember these uh results remember remembering is not a bad thing so hence if you remember it saves time okay now this one yes little complicated little tricky but then if you stay calm it will be done right so we have multiple ways people sorted multiple approaches that's really fine that's great here drawing is very essential drawing is very essential why because it saves a lot of effort in terms of equations and all that so always in heightened distance problems or any application problems draw diagrams quickly it gives a lot of clarity to your mind and look at the same time on the options look have a look on the options maybe the options are helping you to solve okay so hence be mindful of that as well correct okay now and don't fear with you know and the other thing was that 45 degrees the moment I saw 45 x and x should be fast lightning fast like that okay and again read the question note the data wait for you know some other I was just waiting for whether the people were on the same side or the opposite okay opposite side and the moment I saw opposite I jumped over my seat and then drew the diagram I was kind of when I was doing it I was always having this kind of a diagram in my mind so this was going on so it could be either like that or like that so hence the moment I saw opposite I jumped on that okay then here reflection problem so physics concept is also deployed over and above that 45 degree again same isosceles triangle drawing neat diagrams will help and always remember in exams like all these right yes you know many at times they will give you very less amount of stationary so you have to learn the art of doing it you know in a very small amount of space over and over again so in one sheet there will be around 62 you know so now though it's online so it's not a problem but then stationary could be an issue and even if they give you an enormous amount of stationary that's not a good you know way of because then you will be all over the place we have to manage it in a very small amount of space so all this comes through practice my dear friend and the whole intention of the centrum hour and other things is that you get enough practice enough sense of problem solving leaderboard and all that is okay fair enough you get some position here and there you know just take it with a pinch of you know what I say you know don't really take it very seriously it's good that you are you know it gets gives you a good benchmark so hence oh wow so we have Suman and who else so okay no Shobita from I and R wonderful very good Shobita keep it up Shobita is so whether this is actually true or not doesn't matter let's celebrate Shobita's success followed by Prana of stress all top 10 for that matter anyone who had participated you really did a good job guys keep an eye on problem solving that's the whole intention okay yeah anyone wants to say something yeah someone was saying something okay see you guys so all of you keep it up and then next week as I have promised no problem our target is to restrict it to nine only aren't so we will try and see see you know explanation takes a lot of time don't worry and next week as we have suggested we will do social science quiz so be ready guys for the next year next week's geography quiz okay bye bye take care