 screen. All of you just confirm on chat. Yes, very good. So quick protocol repeat. So for yes, everybody should write yes. So let me see if you know. Yes. Why, why, why, why, why? Just type why everyone let's see how many of you are present and energy is full. Okay, so don't be sluggish in the session. If you feel sleepy, take a nap, come back again. No worries. But, you know, try to be active and present all this while. Okay, it only is to energetic. Okay, never mind. So let's begin guys. Okay, for if you didn't understand anything, you have n. And if you want to repeat, want any explanation re explanation, so you can go for R, whatever. So what what is the other protocol you have been following? So for uncle followers, a great set of protocol protocols, er, explanation required. All right. Good. Yeah. So here is one. Okay. Okay. Andy here. Very good. Chalo. Okay, fellas. So there are a few more people joining in with everyone. Got it. Okay, cool. So you know the protocols anyways, so will not be a problem. So you now know that your board exams are going to be in the month of March, not in the month of February, right? So don't be complacent about it. Okay, so guys, once again, repeating the, you know, so you can see the slide. Let me make it full screen. Okay. So here are lots of people are still joining in. Wait. Okay, fellas. So can you see the screen in which I have again highlighted me? Let me close this one. I have highlighted the class core structure of class 10 once again. So if you see last time we were discussing this 20s in this 20s of total of algebra, total of algebra. Okay. Now, and out of that quadratic was having a one plus four one plus three cup quota. Okay, so one one marker and mostly on the finding the value of k depending on the nature of roots or finding the other root if one root is given or some other variable to be found out and three is three marker would mostly be a word problem. Actually, is this pen, pen color okay? Or this pen is okay? Any what color is okay for you? Any color, any pen color is okay. No, but it should not be glaring on your eyes. Anyways, so okay, fine. So now we are going to discuss today trigonometry again. This is 12 marks. So what is the pattern here? So if you see, there is one and one. So two one marker will be there. And then another two marker is there. So one two questions of one mark, one question of two mark, one question of three mark and one question of five marks less. So two plus two four three seven five three. This is the breakup for trigonometry. Now you would have guessed what would be this five mark for? Can you guess what is this five mark for? Yeah, application of trigonometry. So this is heights and distance. So there is no way. So you know that one five marker is in heights and distance h and v. Okay, so this is heights and distance. What about these ones? I'll also talk about that there are some deletions. Now this one, this one here and this one here are mostly based on t ratios. Or again, the other thing which is left is not proving. So proving usually will be either here or here. In the sample paper, there is a two mark identity proof, three mark again, a small h and height and distance question. And one, these two one markers each are of t ratios or standard angle 30 degree, 45 degree voila, right? So you now know how to go about it. So this is mostly either trigonometric identities. Okay, or another smaller application of h and d. So in the, the, in the sample paper, they have given two heights and distance eight marks of heights and distance only. So obviously, you know, it could be mixed, mixed in, you know, from mostly I am expecting a three marker and there will be an option here. So best part is in the sample paper or is given, not here, not here, here, here, there's an R. So, so you now know the structure. So be very, very clear in your mind. Okay. Now, this is deleted, removed from for this year, 2021. So what is that motivate the ratios, whichever are defined at zero degrees and 90 degrees. So you don't need to apply 1090 sine zero and things like that. I will show you, Surya, what is the sample paper, you know, like, so that you can get. So if you show it, you let me show it. This is the paper. Okay. So you can see the, yeah, there is an R in h and d. So you're lucky. So there is an R here. Yes. Have you moved to the next slide? No, no, no, this is a question paper sample paper available on CDC's website. So you can see there are question number 34. So have you opened it or is it not available? Is it not, you know, you're not able to see it on the screen? Oh, not visible. Oh, oh, I see. Okay. Wait a minute, because I'm sharing only the wait a minute. I will change the settings. Give me a time. Give me some time. Okay. So anyways, so I have anyways that in the slides, I'll come back to, so let's come back to, you know, let's come back to the slide. So what I'm saying is, yes. Now PPDs are visible to everybody, right? No problem. Right? Okay. So now this is there. So we talked about this one. So I was, I was talking about this, that signs all the standard angles of zero degrees and 90 degrees are they are saying they have removed trigonometric ratios of complementary angles. Now this is something which should not have been removed. But okay, no problem. In next year, 11th grade, you will learn anyway. So hence, sign of 90 minus theta is cos of theta. So this will not be asked to you. So any application of this kind will not be there and heights and distance, no deletion. So, right? Good. Next. So again, we will keep reiterating the structure so that it is well established in your mind. So what are there? There are, you know, two parts, A and B. Part A is objective base, part B subjective, long answer type. Part A, again, there are 20. Or what do you say? Yeah, yes, 20. So 16 questions are there which are one marker, 16 of one marks, you know, this, then there are four case studies of four each. Okay, four questions on each. So you have to answer four into four. So this is a four into four. Am I right? 16 again. So these will be all also one marker each because there is case study, there is objective base case study. So hence, so 32 marks is coming from the part A. Okay. And you have a choice here, four out of five subparts of each case study you have to answer. 21 to 26. Again, question number 21 to 26. Six in numbers. There are two marker each. So two marks. So total is 12. Then 27 to 33 are seven questions. So three marks each. So hence it is 21. And finally, three questions 34 to 36 are five marks each. There is no four marker this year. Last year it was there. And hence 15. So total is 8020 comes from your school. And hence center. This is how the breakup is. So you must be clear to throw with this. Let's go do. So now let's revise quickly. So what is the basic ones angle cut definition. So we'll not spend time on it. So you know what an angle is. And so this is a basic definition. An angle is considered as the figure obtained by rotating a given ray about its end point. So this is an angle just customary. This is an angle, right? All of you know inside outside of angle, nine grade, you've studied all of that will not deliberate more time on it. Initial position O is called the initials, you know, this is O and OB is called the so OB and initial and final positions of the side. Okay, next. So the measure of an angle is the amount of rotation from the initial side to the terminal side. So you're not going to definition of all of that. But you know, anyways, theta is nothing but arc length. So you might have studied this divided by radius, right? Where theta is in radians. This is the definition of angle. This is called angle measure method of defining an angle. So again, not very relevant for say, or with respect to your upcoming board exam, but next. So I need not reiterate this and many a times you have this acronym as well. What is that? So, okay, so you, everyone knows this, but not a very good way of, but for 10th standard, okay, you know, you, you can write like that sign is opposite upon hypotenuse cos is adjacent upon hypotenuse and tan is opposite upon adjacent. And the reciprocals of this are cosecant, secant and cot. So I don't think anyone here would be having any issues. So this definition is good enough. Yep. And this is given, let's say in case of for your 10th grade, we have restricted the theta to be between what theta is going to be equal to zero and less than equal to 90 degrees. So we are going to be in acute angle range only. Okay, that's what we learned. This is, let's say, oh, this is a, this is B. And in terms of this is perpendicular, this is base, and this is hypotenuse. Now definition of perpendicular and base depends on the position of theta. So for this theta, the base is OA. But if you define the angle like that, the base will become, if this is the angle five, the base will be P or AB. So hence, depending upon where the theta is, you can define or, you know, you can define all the P ratios, right? So it's more confusing. So don't, don't go for it. So whichever is more helpful to you, remember that. Okay. Okay, is that fine? So this is all these are the three or six ratios. So you know that sine is what sine is reciprocal, sine theta is sine theta is one upon cosecant theta. And cos theta is equal to one upon one upon secant theta. And tan theta, tan theta is one upon cot theta and vice versa. So you know this, right? Let's go to the next one. Okay, so this is what they were talking about, which we just discussed. So we'll just run through it. Thanks. Value of sine theta and cos theta never exceed one. Always remember. So it's an added information, though there are no more questions related to the maximum and minimum value of the trigonometric ratio here. But you always remember that sine theta is always sine theta is always less than or greater than minus one and less than one. In your case, since theta is between zero degree to 90 degrees, so you know that sine key value will be zero less than equal to sine theta less than equal to one. And for cos, it will be again zero less than equal to cos theta less than equal to one. Both sine and cos will be between zero and one if the theta is acute angle. Okay, this is what. And for tan, it is, what is the minimum value for tan? Zero again, less than equal to tan theta. And here, less than infinity. So this is something like, you know, it is tending towards infinity, very huge, very big amount, very big number. So these are the general limits or values between zero. So the range of values of sine cos and tan. And accordingly, you can see that cos secant theta cannot be between zero and one. So cos secant theta, if the theta is acute, is greater than equal to one. Secant theta is again greater than equal to one. And cot theta, cot theta will vary similarly like tan only, cot theta is zero, cot theta. I hope this is clear. Though there are no questions directly related to this, but having such knowledge always helps in checking whether you have done correctly or not, whether let's say you're proving an identity and it is coming out that after all that proof, you're getting sine theta greater than one. So you can, you can tally your answer, check your answer that, hey, it seems to be little dubious. So these are some tricks to remember. And these will help you in your process. Okay. So this is not there, but I have mentioned that knowledge anyways is not bad. So this is deleted. So they are not going to ask you any question related to this, but just in case it can aid again as a checking mechanism. So in sine 90 minus theta is cos theta and cos 90 minus theta is sine theta, tan 90 minus theta is cot, cot 90 minus theta is tan, secant is cosecant and cosecant becomes secant, right? So you can change the trigonometric ratio by changing the angle. And hence here, co means complementary, complementary to sine. And hence you can, you can see that 90 minus theta, what are called complementary angles anyway. So theta and 90 minus theta are complementary angles. Hence sine, cosine, secant, cosecant, tan and cot tangent. So hence you can relate to the theory like that. But anyways, now that unfortunately it has been deleted. So let's skip. And this is the standard table. You must know, and there are no questions related to proving how do you get, so all of you know how to, how to prove. Let's say if at all by, you know, I have never seen that question. But if someone asks you, prove that sine 60 is root three by two, how many of you will be able to do it? Why, why, why, why? If sine 60, if they're asking prove that sine 60 is root three by two, how many will not be able to? So I, you know, anyone who will not be able to. So for them, I can, you know, maybe I can do this very quick proof for all of this 45. So you don't need to worry because yes, I will show why not. Here it is. Oh, how do you know? Let it be not like that. So a, b, c. And let us say this is a 45 degrees a. So this will be root two a without any doubt by Pythagoras theorem. Correct. Any doubt in so far in this diagram, how did I write a root two a any problem? So this is 45 degrees. So I saw some strangle. So behind the scene, this is happening is equal to let's say this was my H. So H square. So H is clearly root two by a. So you can write now. So what will be sine of 45 degrees? So sine of 45 is opposite by hypotenuse. So a upon root two a is one upon root two. Yeah. So likewise, you can find out all the other six ratios are in any problem. Now what about the 60 degree wall apart? So let us say we draw a equilateral triangle like that. And we drop a perpendicular from here. So you all know that this angle is 60. Let us name it also. So oh, wait. Why am I so this is a. So a, b, c, d. Okay. This angle is 30. Okay. So tell me guys, now if this is a, this will be a by two bd, right perpendicular drop from the vertex of a equilateral triangle will bisect the base. Correct. So hence, can you not find out it? Let's say this is p. So what do I know? So p square plus a square by four is equal to a square uncle Pythagoras in action. So what is p square then? So p square is a square minus a square upon four, which is thrice a square upon four. Is it? So what's p? Appendicular is root three upon two a. Correct. Now we have to find out sine 60. Tell me from this figure, sine 60 will be a b upon a without doubt. What's ad? That's p. p is how much root three by two a upon a hence root three by two. Yep. Taking side as two a makes it easier. Take two a if it, you know, if it makes easier, but don't while, while proving those since it is proving, so it will not incur any error, but many a times if you take any factors with the size, what will happen is there is a chance that you miss that two coefficient and end up getting a wrong result. So if you're careful, you can manage. You can take two a four a 10 a, whichever works well with you. Okay. Likewise, you can, you can find out sine 30 as well. So sine 30, where is 30 degree? Here is 30 degrees. So sine 30 opposite. That is bd. That is a by two divided by hypotenuse. That is a is half. So likewise, you can design, but unfortunately, the zero degree is they are saying they are not going to ask you, but then you must know sine theta, sine 90 is one cost 90 is zero, tan is not defined, rest all you can anyways. Yep. Many a times it happens that people forget the values, forget the values in terms of, you know, what is sine, what is cost. So hence there was a trick I gave you during that also. So you can remember like this. So for sine, let's say if you write zero degree here, you write 30 degrees here, you write 45 here and 60 here just in case you're totally, you know, nervous, you forgot the values. Don't worry. What you can do for sign, simply write zero upon four root over. This is first. This is one upon four. This is two upon four. This is three upon four. And this is four upon four. This is a sign. Now for cosine, you need to go in the reverse order. So this is zero upon four. This is one upon four. This is two upon four. This is three upon four. And this is four upon four. Right. And tan, this is sine. This is cos. And then tan, you can reproduce the table within no time. So you know the tan is nothing but ratio of sine and cos. So this will be simply zero. This will be root over one upon three. This will be root over one or two by four upon two by four is one. This is root over three. And this is not infinity. This is not incorrect way of writing it. N defined, not defined. Okay. And then the rest three columns you can reproduce by just taking the reciprocals. Clear. So root over zero by four, one by four, two by four, three by four, four by four. That's it. So and then one column for sine. If at all, if at all you have forgotten it can help you regenerate the data within no time. Any doubt so far? So far so good guys. Yes. All right. Everyone went to sleep or what? Where is the energy? Hello people, are you there? Yep. Okay. Sir, you are too boring. Hence sleepy. So hence, please get lost. Type's feeling is coming. If yes, I can't help you have to bear with me next. Okay. So done. Right. So for the T ratio, this is good enough because complimentary angles are not there. The life is easier. Let's go to sample paper 2021. Start solving sine A plus cos B is one. This is one marker. So out of two more marker, I told you. This is the first one. Sine A plus cos B is equal to one. A is equal to 30 degree and B is an acute angle. Then find the value of B. So find the value of B. Hmm. So stress is saying 60. Yes, there is absolutely no nothing to, you know, what do you say? Think here. So sine A. So how will you write? So someone is now asking, do we have a class today? Just a minute. Let me answer this guy. Just a minute. Yeah, no problem. Hmm. Okay, guys. So sine A plus cos B is equal to one. So very, very clear. So how will you write? That's very important. Writing, because losing even half mark is not good here. So because we are a what was our aim by the way, guys, what is the aim? What's the target? What's the target of? Yes, the target is symptom. In symptom, get symptom. Okay. So, you know, we have to target 100% only. So hence, not a single mark should be spared. Now, sine A. So A is equal to 30 degrees. Therefore, I would have written like this. Sine A plus sine B is equal to one. So sine 30 degrees plus sine B. Sorry. Hello, someone is saying something. Sir, is cos B, sir, not sine B? Oh, I myself is not, I am not getting symptom. That's so bad. Okay. Now, but I will use eraser. Now I know how to use eraser in this thing. PowerPoint also. Okay. But the pen nib is little wider. Wait a minute, why it's green now? What was that pointer options? Okay. Thanks, page of three. You are getting sent my blessings to you. So sine A plus cos B is equal to one. So sine 30 plus cos B is one half plus cos B is equal to one. So cos B is equal to one minus half. And it's better to write this symbol as well. So this means cos B is equal to one. Sir, in the main board exam, do we have to write cos B is equal to one minus one half? Like can we skip a step and write cos B is equal to half? Deco, what did I say? I said I will write like that because I will not leave any stone unturned. Now, as far as time is concerned, I told you that three hour time is more than enough for your board, baby. You would have realized it by now. Are your papers lengthy? You know, when you wrote the pre-board, did you feel that, okay, this is my God, so many questions. How do I cover? So when you have a lot of time, why don't you help the examiner a bit to give you a sentence? So B is indeed 60 degrees, sorry, 60 degrees. Okay, don't forget to write small, small things. For example, many people will write. So error zone, please remember these things. Here is where they lose marks. So they will write theta is equal to or whatever it is, B, B is equal to 60. This is not correct. Maybe the examiner will, what do you say, ignore the error, but frankly speaking, this is not correct. So you have to put the degree sign over here as well, right? So things like that are small, small things, but you should be careful. Okay, I hope it is making sense. Next one, guys, this is another question, another one marker, root 3 sin theta minus cos theta is 0 and it has been given, they have, they have not put any equal to sign here. If you see, this is not there. This is missing, okay. So just 60 implies it is in radians. Yes, Rn, correct. But there also, until and unless you write RAD, RAD is the abbreviation for radians. Okay. So root 3 sin theta minus cos theta is 0. Find the value of theta very easy again. So how would I write it? I would have written it root 3 sin theta is equal to cos theta. Hence, hence. Now what do I say? I say sin theta upon cos theta is equal to 1 upon root 3, correct. This implies you can do that. Why can you do, in this case, you can do that, but otherwise you can't do it. Why can't you do it? Because here is the reason. Reason is cos theta is not equal to 0. So hence, you can divide the equation by cos theta. If it was 90 degrees, let's say there was equal to sign here, you could not have done that. So they have categorically mentioned that theta is not 90 degrees. So since cos theta is not 90 degrees, so cos theta cannot be 0. So you can divide the equation by cos theta. Otherwise, this state, this step would have been wrong if you divide by cos theta. If there is a possibility of theta being 0, 90 degrees. So wait a minute, eraser. Okay. Now, okay. So hence, sign by cos is tan. Obviously in one mark, they are not going to test your. How are we supposed to show the exam that we know the step is wrong when cos, when theta is 90. So hence, you know, so when I divide, so I know that division by 0 is not allowed. Right? Right? Yes, you could have used, yes, stress, that's good. But then there also, you're dividing by sign, no? So sign must not be 0 then, stress, clear. So you have to mention then, sin theta is not equal to 0. Because in mathematics, dividing by theta, so here, here basically, what are you doing? You're dividing the equation by dividing the equation by cos theta. Dividing the equation by cos theta. So hence, I would have, again, guys, there is someone whose mic is on unmute mode. Please put your mic on mute. Thank you. Yeah. Now, so tan theta is 1 by root, 1 by root 3. So theta is equal to since it is given 0 to 90 degrees, hence theta will be simply 30 degrees. Because in the next grade, you will see that if this restriction is not given, there could be so many values of theta. Anyways, so far, this is good. Okay. Any doubt? Well, next, let's go to this one. Now, this is the question which came in board paper last year for two months. Okay. The rod AC of a TV disc antenna is fixed at right angles to the wall AB. And a rod CD is supporting the disc as shown. So you can see. So you have seen this kind of an arrangement in your dish antenna. When you go to your terrace, you will see the dish antenna is supported on this kind of a beam, right? So there is this structure, which is fixed on wall, something like that is there. So that's what they are trying to show, right? Anyways, so hence what are they saying? If AC is 1.5 AC, so hence how would I have done this? I would have done this. I would have drawn the figure first like that, which is relevant for me. So C, A and B, I think B is not required, I believe. Yeah. And the rod CD is about to take it if AC is 1.5. So this is right 1.5 here itself in the diagram and CD is three three meters, right? Find tan theta. So this angle is okay. So for the first answer, tan theta. So for that, you need to find out tan theta is nothing but AC upon AD, right? Right? So hence tan theta is equal to AC. AC is given 1.5, but AD is not given. So hence what you can do, use Pythagoras theorem, do mention this right angle and then this is nothing but under root three square minus 1.5 square, right? So hence this is 1.5 upon. So 1.5 is common here. So 1.5 can be taken outside and hence this is 4 minus 1. Am I right? Tell me, am I right or not? All of you agree to this? So this 1.5, 1.5 will go. So this is 1 upon root 3, okay? So tan theta is done. Now secant theta plus cot theta. So you can hear from here itself, you can say if tan theta is 1 by root 3, then theta is equal to 30 degrees, no doubt about it, 30 degrees. Then you can find out mark the question number very, very clearly. So 2, secant theta plus cosecant theta, then write secant 30 degrees plus cosecant 30 degrees. Secant 30 guys, how much value? Secant 30. Come on, quick. Let's see. Who will write secant 30? How much is the value for secant 30? So someone said yes, 2 upon root 3, very good. And what about cosecant 30? cosecant is 2, perfect. So hence 2 common, 1 by root 3 plus 1. Ideally, you should rationalize the denominator, okay? So hence you write 2, 1 plus root 3 by root 3. Hence it is 2 root 3, 1 plus root 3 by 3, okay? Tell me, fair enough. All clear? Any doubt? When you write the answer, try to rationalize the denominator. Don't leave in your rational form. Ideally. Come on. But if you're, let's say if you're falling short of time, then don't, you know, you can stop here also. Here. But if you have, I told you in math paper you'll have enough time. So don't worry. Come on next. This was previous year sample papers, previous year question papers. So one mark, a triangle ABC is right angle at C, then the value of secant a plus b is secant a plus b is. So right angle at C. So a plus b is 90 degrees. So how, since it was an MCQ, so you don't need to solve it, but let's say it is, it is asked in a one mark of form in the current format. So what will you see? Say a triangle A be right angle at C. So a plus b plus C is equal to 180 degrees by angle some property, angle some property. So D not defined. And G. Wait, wait. Let me solve. Yes. So a plus b plus C is 180 degrees. So what is a plus b then? A plus b plus 90 degrees is equal to 180 degrees. So hence a plus b is how much? 90 degrees. So that means we have to find out secant 90 degrees is one upon cost 91 upon zero. So not different. But people, there are a few people who went for zero also. Why? So you now know how errors will be in your examples. Be very careful folks. Next. So I've eliminated all those questions which have complementary angle component to it. Sin theta plus cost theta is root 2 cost theta theta not being 90 degrees and find the value of tan theta. It is very, very simple actually. So what will you do? You have to simply divide by cost the entire thing. So hence I would have done sin theta plus cost theta is equal to root 2 cost theta. You want tan, isn't it? How what is tan? Tan is sin by cost. So let's get cos sin by cos. So divide the entire thing by cos theta. And God is helpful here. So hence it is tan theta. And you can again divide by cost theta. Why? Because cost theta is not equal to zero. Why? Because theta is not equal to 90 degrees given. They have mentioned. So tan theta plus 1 is equal to root 2. So tan theta is equal to root 2 minus. Okay. But this is these steps should be written only when then you know if there is no mcq types which is mostly be the case this year. Okay. Do this. Given that sin alpha again, I think this is of one mark. So given that sin alpha is root 3 by 2, this is all very, very simple ones. So these are like, you know, you can predict. So I told you right in the beginning of the session, you can predict your board question paper. So this question, question number 6 will be your trigonometry question. 7 will be trigonometry like that. So sin alpha is root. So how will I write it? If it is a one marker, sin alpha is root 3 by 2. Therefore, alpha is equal to 60 degrees. There and cost beta is equal to 0 degrees. Sorry, 0 or don't take it 0 degrees. 0 cost beta is 0. So beta is equal to 90 degrees. Okay. So beta minus alpha is equal to 90 degrees minus 30 degrees. Sorry. What is that 60 degrees? 60 degrees is equal to 30 degrees. So, okay. So I am not asking also whether it is understood or not. Do this again. This is based on second type now, based on standard angle. So you must be very, very thorough with your table. Or I told you, if you forget it, just recreate the table quickly by that 0 for 30, 45, 60, 90 root over 0 by 4, 1 by 4, 2 by 4, 3 by 4, 4 by 4. But anyway, if you remember nothing like it, sin 60 is, so I would have written it root 3 by 2 whole squared plus 2 into tan 45 is 1 minus cos square 30 is root 3 by 2 whole squared. So this goes, so answer is 2. So these are like freebies, freebies. One, one marker, freebies. Bonus. Do this. Sin A is 3 by 4. Calculate, seek in A. Now here, you can adopt both methodologies. Either you can go for a triangle mechanism, where you draw a triangle and do a base perpendicular hypotenuse or you can use what? Identities. So there are two ways. Method one. One is draw the triangle. Draw and get free mark A, B, C, A, B, C. Draw the triangle quick. Oh, I am sorry. This is sin A. A is this, this angle is theta. Now 3 by 4. So this has to be 3 then and this has to be 4. So what will this be? Simply 4 square minus 3 square 16 minus 9 root 7. Don't forget the root. So what is the question? Seek in A. Seek in A. Is equal to first hypotenuse over base. Seek in A will be hypotenuse over base. So what is hypotenuse here? Root 7 divided by, sorry, 4 divided by root 7. Is that okay? This is one. Another method I told you, you use identities using. Sir. Yes, madam. Aren't we supposed to rationalize the root 7? You can. In trigonometry, yes, yes, I already suggested so you can do that. But I am just willing to just show you the method. You can definitely, please rationalize everywhere possible. So it should be 4 root 7 by 7. You can do that. Okay. So this is one. How to use identities? So sin A is given. So you can find out cos A from this. So cos A you can, you know is nothing but under root 1 minus sin square. Okay. So what is 1 under root 1 minus 3 square by 4 square. Which is again under root 16 minus 9 divided by under root 4 star. Which is root 7 by 4. So this is cos A. So secant A is equal to 1 upon cos A is equal to 4 upon root 7. Either way is perfect. Okay. Or hence it is equal to 4 root 7 by 7. Next. Last year this was 2 more. I think this is 2 more. Sorry. It is for 2 more guys. So change it. Oh, sorry. I can't change it here. Anyways. Do it. Figure 1. PS is 3. QS is 4. PRQ is theta. And PSQ is 90 degree. PQ perpendicular to RQ. And RQ is 9. All information is there. Evaluate tan theta. So people are smart. Very good. Oh, this would be one mark only. It is very, very simple actually. So how to do this? You know, in one mark don't draw the diagram like that. If this is too cumbersome, you can directly write. You can write PQ is equal to under root PS square plus QS square. Which is under root 3 square plus 4 square. Which is under root 25, right? Directly is equal to 5. Okay. So this is 5. Now tan theta. Again, directly write tan theta is equal to PQ upon RQ. So hence it is 5 upon 9. Done. Very good. Next. Again, tan alpha is 5 by 12. These are all previous year question guys. So yes, you know, we will, we will solve all different types of problems so that you are thorough. Tan alpha is 5 upon 12. Find the value of secant alpha. Okay. Meghna has already done it. Come on guys. Hey, thumbs up. The teacher gets demotivated if you are, you know, anyone sleeping? Tell me. I'll tell you a good story. I watched a good movie yesterday. So I'll tell you about that if you are feeling bored about it. Maths. No, sir. Math is too boring. How do you even enjoy maths, right? Guys, how many are in that frame of mind? Maths too boring. My God. I would get rid of maths. Maths is cool. Why? What's the temperature? And Munish Reddy is saying maths. Yes, maths is boring. Aditya says 13 by 12. Maths is 13 by 12. Okay. Never mind. Let's solve. So tan alpha is 5 upon 12. Again, two roots. Find the value of secant alpha. Right? So again, the triangle root you already know 5, 12, 13. So this will be a Pythagoras uncle triplet. So hence it is A, B, C. So let's say 5 and 12 and alpha. Hence this is 13. I don't need to prove it. One marker. I will not prove it also. secant alpha. You can actually, you should in the exam. So hence you can say AC directly is equal to under root 5 square plus 12 square, which is 13. Now secant alpha, secant alpha is equal to hypotenuse by adjacent 12. Done. Or if you have to use the identity, then you can write second option method 2. So you know the relation between tan and secant. Secant square alpha is equal to 1 plus and square alpha. Use this identity, which is 1 plus tan square alpha is 25 upon 144, which is 144 plus 25 upon 144, which is 169 upon 144. This is secant square alpha. In this case, though it will be plus minus, but since alpha is between alpha in all your, on all cases, your angles. Anyway, in this, this is a triangle. So alpha is an acute angle. It can be more than 90. So it can't be negative. So hence it is simply 13 upon 12. So don't write plus minus. Don't show overactivism and write plus minus 13 by 12. No. Only plus 13 by 12. Why? Because it's a, in your grade, we are considering alpha to be between 0 and 90 degrees. Okay. Shallow. Do this. Sin x plus cos y is 1, x is 30, y is an acute angle. Oh, we just did it. See, can you see the repetition? This is a previous year question again. So questions are repeating in a different language only. That's it. We solved it above. Isn't it? We solved, the first question was the same. So now you can pick the pattern, right? So different, different years, same question in a different language. Beautiful. Sometimes it could have some, some time in Sanskrit. Okay. So you will, you will see it will change and nothing else changes. So sin x is sin 30 degrees plus cos of y is equal to 1. This is where I was talking about getting sent, sent them. So half plus cos y is equal to 1. And they just finished said whether I should just skip that one minus half part. Can they mark wrong if you write plus minus? Because it's technically correct. Just not there for our grade. Again depends actually, very subjective. Mathematics teacher should not, but then again, please understand, there's a marking scheme given to the examiner. Okay. So there is very less scope to deviate from there. Because they have to standardize this because otherwise what will happen? The same mistake someone else would have made somewhere else and he would have got a lesser mark and he would have got better mark. That will be like injustice. So hence, we will try to restrict to, sorry, this is cos y. But you know, it's not that serious that, okay, now you are going to get less, you know, get less mark. If I were a teacher, I will not deduct marks definitely. So cos y is half. So why is 60 degrees? Okay, never mind. But the problem is you must be understanding that whether, whether secant can be negative also, which that is not thought to you. So hence, it is assumed that you don't know it. And hence, you are ignorant about the negative sign here. Hence, okay, do this. Got a plus one by got a is two, then find the value of part square a plus one by got a see all these questions can also be asked into marker format. So it's don't think that all of them are one marker, whether multiple questions down there in the slides. So done six. Yeah, so simply you need to do what? Square it. Square both sides. So got a plus one by got a whole square is equal to two square. Six, yes, how come six, make now four, how come four, all of you are making small silly silly errors. So caught square a now I will teach you identities algebraic identities of ninth grade. Don't let me do that, folks, please. Is it a little right all all the steps, it doesn't cost you that much time, but it saves you good amount of marks. So got square a plus one upon got square a plus two is equal to four. Hence, got square a plus one upon got square a is equal to right. Don't say sorry, my dear, you will feel sorry as in, you know, you will feel back when you don't get sent them. So getting out at 99 hertz, you know, Satyam Tendulkar's story. So when he was going for his hundredth century, it took a lot of time for him to actually get and he got so many of chances. Yes. Okay, no problem, dear. Put your mic on mute. Okay, calm down. Chalo next question. Now mix of now three is three marks. I hope you noticed it. This is where other topics get, you know, this three marks. Yeah. Oh, so now I told you, right? It will be, you know, easier. Questions will be easier. But you will do mistakes and lose marks. That's the only way people lose. People do not lose marks in 10th grade math paper, because they don't know it. Mostly they know because they lose it because they make some overconfident may overconfidence may error. So that's that should not be the case. Chalo this is mix of two topics, trigonometry and linear equations. So three marks. Finished. Done sir. So a is greater than we then find the values of a and b 75 by 205 by two. Okay. So tan alpha plus beta is one. So how would I solve it? Tan a, sorry, a plus b is one. So a plus b is 45 degrees. Correct. And tan of a minus b is equal to one upon home three. That means a minus b is 30 degrees. Okay, this is equation number one, equation number two, we'll write one plus two. That will eliminate b. So that will give you two a equals 75 degrees. And so a bar over 37.5 degrees or 75. And b will be 45 degrees minus a, which is 45 degrees minus 37.5 degrees, 7.5 degrees. Someone says 105 by two make now what happened here? 105 by two will make it 50. Why? Why is 105? Where is 105 coming from? 40 plus 45 plus 30 became 105 or what? Anyways, chalo please take care of your errors guys. Errors. So you know, all in all these questions so far, none of the questions you didn't know. All the all the answers are coming wrong because you're making some silly mistakes. Next. Anji. So now let's take up heights and distance, you know, topic. And you know, there are only two concepts here. Unfortunately, in CBC board, we do not have bearings, but that is also very interesting topic. So, but we discussed this in our classes. Anyways, so there are two angles. So this is line of sight. Line of sight. Right. And from here, trigonometry is trigonometric identities will be taken up as a separate session dice. If you had noticed the, noticed our schedule, there are two sessions on trigonometry. So in first, we will do the basic one and the heights and distance in the second one, some more heights and distances and the what do you say? Identities. Okay, chalo. So this is X degrees. So the line of sight as the, like the diagonal ones, right? The middle one is horizontal level. Oh, line of sight is horizontal line of sight. What I mean was when you are, so all the references from this horizontal line of sight. So I, I, I usually in the, so what I mean is your eye in usual case will be this, right? So you have to measure your angle from this line of sight. So let us say some normally you don't keep looking up or down, right? You look when your head is straight, what happens to one of the, what is the, you know, line that is the horizontal line of sight from here we measure this reference line. This is actual line of sight when you have tilted your head to that angle. This is also actual. So in this case, it is lower than horizontal line of sight. So this is, this will be called depression. The angle gets depressed. This is elevation. Angle gets elevated. Okay. So you have to raise your eyes from the horizontal line above that line that is called angle of elevation. So for example, when someone at this, what is that lighthouse kind of a thing is having a site here on the airplane. Okay. And here on the ship, the other one. So X here is X degrees. X is angle of elevation. So all the measurement of the angles will be taken from the horizontal line of sight and Y is equal to angle of depression. Okay. Folks, the best thing is to learn through applications. So let's start. This is sample paper, three marker, right? 2021 sample paper, three marker problem. What is the question? If the angles of elevation of the top of the candle from two coins, distant ACM and BCM and A is greater than B from its base and in the same straight line from it are 30 degrees and 60 degrees, then fight the height of the candle. I hope you understood question. So what is happening? If the angles of elevation of the top of the candle from two coins, distant ACM and BCM, so hence somewhere here is A and A is greater than B. Sorry. So this will be B and here is A. This total is A. So this is B minus A something like that. I hope this is clear. So you have to find out height of the candle. So typical height and distance question. Right? So what do you need to do? So I will draw a diagram first. So what I'll do, this is my A or I will name the, I will not name it like this. I will name it like this. So this is B. This is A. This is C and this is D and I will join this and join that as well. I would have written like this. This is B and please be careful A from its base and in the same straight line from 30 degrees and obviously this angle is going to be 60 degrees. This angle is going to be 30 degrees. This is third three marker question. Okay. And this is H. Very simple. What do you need to find out? H in terms of B and A. So oh, find the height of the angle. This is ridiculous. Top of the two points, distance A centimeter and B centimeter from its base and in the same straight line. So you don't need B or A for any one thing is going to help. Have I read it wrongly? The angles of elevation at the top of the A and N from two points straight distance A and B from its base and in the same straight line from meter 30 and 60 degrees. Then find the height of the angle. When angles are given, A and B is given. You have to find out the height of the candle. Simple. So hence H by B is tan 30. Is it? So they are not saying whether to find out this is some you know hurriedly created problem by anyway. So H by B is tan 60 which is root 3. So H is simply B root 3. So maybe the question was find the relation between A and B. Let's say let's modify the question. Find the relation between A and B. Find the relation between A and B. Then probably it's more interesting or find the distance between B and A. Yep. Anyway, so what do we do? So again, so in that case, H upon A is tan 30, which is one upon root 3. So you have H is equal to A by root 3 and H is equal to root 3B. Both are same. This implies A by root 3 is equal to root 3B. This implies A is equal to 3B. So this could be. Excuse me, sir. Excuse. Sir, I think the question being asked is to find the height of the candle in terms of A and B. So that's what we did. H is B by root 3. I can't incorporate A just like that. No, sir. We can actually because we know that H by A will be 1 by root 3. We can take root 3 as reference to equate the two. So H by B will be equal to A by H. So H square will be AB. But there is no such mention that you have to find it in the terms of A and B, right? So why will I assume that you have to do it in A and B? They asked us to give in A and B. Or sorry, they have never asked anything. So I give the height whatever is given from that data. Is B given? Yes. Is root 3 given? Yes. So that is if you try to fit it into it, maybe you can. But then again, there is a relation between A and B sense. I don't think you need to really do much about it. Okay, so it's fine. You can leave it like that. Or if at all you wish, let's assume that whatever, who was speaking anyways. So if you see, you can try to fit root 3 if you want to. This is nothing but B times root 3 is nothing but root over A by B. This way also you can do. But then I don't think there should not be any issue limit here. Anyways, so this is not very, the language is not very great in this question. Now come to the game. This is the game of yes, you have to mute your mind guys. Yeah. Now the two palm trees are of equal height. Mute your mind. Read two palm trees are of equal heights. This is again given in the sample paper. And are standing opposite each other on either side of the river, which is 80 meter wide. So your diagram will be crucial. So while you are doing, I will also do parallely five marks. You'll have to spend time. You can take three minutes, four minutes, even five minutes for that matter. So there are three answers they are demanding. So how would I solve it? First, read the question twice. Five marker can't take chance. Two palm trees are of equal height highlighted and are standing opposite each other on either side of the river, which is 80 meters wide. Highlight the units and data from a point O also, which is O instead of T. From a point O between them on the river, so between them on the river, the angles of elevation of the top of the trees are 60 and 30 degrees. Find the height of the trees and the distances. So what all height of the trees and distances of the point O from the tree. So there are three data, three data, three answers. Keep it in mind. Okay. So I've drawn the representative diagram. I have labeled it also. So now you can start working on it. So hence you can, if you wish, you can label it as X and this as 80 minus X. So this is very, very small looking diagrams. I'll redraw it with pointer options eraser. I'll erase everything and draw once again a little bigger in size. Okay. So because make diagrams, you know, good enough to illustrate everything in one shot. Thank you. So what I would have done broader and bigger like that. So this is point O. Okay. This is point O. Let us say this angle is 60. Obviously the closer it is higher will be this is 30 degrees 60 and 30. Yes. Now this point is P. This point is Q. This point is R. This point is S. You can mention it as H. You can mention this as H. You can mention this as X and this as 80 minus X and write equations and solve. So always tan will be helpful. Toyant 60 degrees is H upon X. This implies root three is equal to H upon X. Okay. This is the equation one, or you can also write, you have to find what? Anything X and H both are required. So H is equal to root three X. Write it as one. Number the equations also next triangle. So tan 30 degrees and it is always good to write in triangle P Q O and then here in triangle P sorry, R S O tan 30 degrees. You please write, you know wider space over there with lots of space between two letters. I don't have much space here. Hence I am writing it in a very compact manner. So tan 30 is clearly H upon 80 minus X. So this implies one upon root three is equal to H upon 80 minus X. This implies 80 minus X is equal to root three H. Correct. Okay guys. So either you can eliminate X or eliminate H first. Anyways, both are required. So what you can do is you can replace H first, let's say. So let's substitute. So 80 minus X is root three. We very, very careful while doing all of this because this is a five marker. We can't afford any steps. So I am writing everything, everything which I can write 80 minus X is equal to three X. This implies four X. Again, when you write such steps where you're adding a subtracting variable, please check once again three and this minus X will go on the other side will become four X. Very clear good. 80. So this implies X is 80 by four and check. You might make mistake. No worries. Keep checking. So X is 20. So what will be 80 minus X? So you now know this is one and what is H? H is root three X. So 20 root three and never forget to write right. Am I right? No, no, no, not needed. Unless they say you can leave, or if you have no need to calculate until unless they are saying that can take root three as 1.73, then that means they are expecting you. I would have left it anyway here. And if you wish, if you have time again, so this is how the second run, third run is done while you're revising. So if you have time, you can, but then be careful while you're doing all of that because you might end up doing your own calculation and losing marks. So hence, but here you have to clearly say three things, three items were to be given. So write three items. So height is equal to 20 root three meters. And you can shortly write Q is equal to 20 meters. If you don't have time, you can write like this also OS is equal to, but make sure that you are writing all of them 60 meters like that three data you gave. Okay. Clear so far. So, so good guys. Any doubt? So we have done one full run through of the entire this thing with all the different types of questions which are, which have been asked in the previous few years as well as the current sample paper and the last year board paper. So let's go and solve more here. This is another one. This is thankfully they are going to give you or here. So the first part you saw here and the second is or either you do the previous one or this one. So the angles of depression of the top and bottom of a building 50 meters high as observed from the top of the tower or 30 and 60 find the height of the tower and also the horizontal distance between the building and the tower. Always remember in five marker, there will be no single answer. There will be more demand of the question because obviously they are heavy five marker, right? So there will be more than one answer more than or not of the same question, but there will be more questions in the same one question. So more demand will be there. So always keep it in mind, keep in mind more demand there is going to be more demand. So hence always remember that there will be two, three answers. Keep checking in five, five marker. So what all height of the tower? This is one then and also the horizontal distance. That's it. Two answers will be coming out of this question. So let's do it. Do it guys. Take your time. Deep breath, relax, solve, solve to get 100% accuracy. You will have a lot of time in the paper. Many people struggle in understanding the question itself. So how to go about it? So see they are talking about angles of depression. That means we are looking at from some height of the top and bottom of a building. Now angle of depression of top and bottom of a building 50 meters high. So let's make that building itself first. So this is building. The best part is I can, you know, it encourages a lot of drawing. So unfortunately, there is no such tool here. Why is there no full box suite? So building again. Yeah, never mind. So tower has to be higher than the building. This is edge. This is capital H. Okay. And the angle of depression. So angle of depression. Who's standing? Where is someone standing here? From here is looking at both the height of the building and the top of the building and the bottom of the building. Right. And this is the ground. Okay, let's name it. Let's say AB is my tower. CD is the, sorry, AB is the building CD is the tower. Right. And this is what they're asking. H is equal to 50. It's given. Okay. Now I am, I'm writing it very, you know, you know, very wrong manner, but I don't have space here, but you guys will be having space. So don't do this kind of a drawing over there. Okay. Wait. It is because this doesn't look good at all. So don't do that. So make it as neat as possible. Okay. Now what is asked for this X? Right. What is given? This angle is given. How much is it? 30 degrees. And how much is it? Last one is, sorry, this one is 60 degrees. So mention that there. Draw this perpendicular, let's say E. Then this angle is 60 degrees. And this angle is 30 degrees. That is what is the diagram all about. Now you have to find out X and capital H. Okay. So obviously you have to write equations, equations, what all this height is H minus 50. And this also is X. So write that. Okay. Now let's write the equations. Equations are first. So write down in triangle CEA. In triangle CEA, we have H minus 50 upon X is equal to what? Tan 30. Right. This is the first one, isn't it? So hence simplify this one when you get, this implies H minus 50 is equal to X by root 3. Tan 30 is 1 by root 3 multiplied by X. This is H minus 50. So hence you will get H is equal to 50 plus X upon root 3. One equation. Second equation will come from the other triangle CDB. Okay. What will that be? In this triangle, H upon X is tan 60. Tan 60, you have to remember the right values. Otherwise, you will mix that up. 30 becomes 60 and gone. So always remember tan 30 is smaller than tan 60. Right. So hence in 1 tan 30, it will be on 1 by root 3. Tan 60 will be simply root 3. So from here, what do I get? H is equal to X. Now I can equate 1 and 2 to get X first. Anyways, X was also needed. Or you can do the substitution and solve for H first, whichever way. So you can say from 1 and 2. Since it is a 5 marker, so you need to have more elaborate steps as well. So 50 plus X upon root 3 is equal to X root 3. This implies 50 equals X common root 3 minus 1 upon root 3. So you get X first here. So I am going to be lacking the space. But anyways, so X times root 3 minus 1 upon root 3 is equal to 50. So this means X is root 3 times 50 upon root 3 minus 1. And you have to rationalize the denominator. So 50 root 3 into root 3 plus 1. And when writing directly, you can write the steps as well. Divide by 2. So this is 25 root 3 root 3 plus 1 is my X. Error? Some error? Did I do some error? X is 25 root 3. Oh, you are getting some different answer. Wait a minute. Let me see. I will check. I will check. The equations are correct or not. H minus 50 upon X is tan 30. So X by root 3. Yes. So X is 50 plus X root 3. Shouldn't it be 3 minus 1? So 2. Denominator is 2 only. Yes, I mentioned 2. I mentioned 2 only. Let me check. I might have done some mistake. H by X is root 3. So H is X root 3. Fair enough. So 50 plus X root 3 is equal to X root 3. So 50 is equal to X common root 3 minus 1 by root 3. So which is root 3 minus 1 by root 3 times X is equal to 50? I don't think. Is there an error? Everyone is getting different or where? Someone can unmute and see. Sir, it's 3 minus 1 by root 3 X is equal to 50. Where? Which step? Sir, last line just when you start. This one, no? X is, this is correct till this point it is correct? No, sir. That exactly. Oh, here is the mistake. What happened here? Oh, sorry, my bad. So I just, I have put an extra root. Thanks, thanks, thanks guys. This is what happens. So hence, five marker please check calculations, multiple number of times. Wait a minute, first of all. Where is the eraser? This is a little troubling. Yeah, it's here. Correct. I'm sorry, my bad. Then X will be, oh, so let me delete everything from here. Thanks for correcting guys. Abhisco, let us, so I was also thinking it was coming to cumbersome. Anyways, so now that we have known this, so what do I do? Let me go to pointer options, pen and hence this exactly is 2. So 25 root 3. Very good. And how do I know whether that was correct or not? If I would have, you know, so one is you look the ratios, look at the ratios because they are coming out to be, you know, so 25 root 3 is somewhere around 3740 ish, 40 ish number, right? So in the other case, it would be something different. So then that should have, you know, rung the back. But anyways, good. So X is 25 root 3. So what is the h? h is equal to X root 3, I mean 25 root 3 into root 3, which is 75 meters. Don't, right? Is that okay? Tell me. Think I should have. Is that clear? Okay, let's go to the next one. This is actual board paper 2020. There, there was no five markers. They all had maximum was four marks. So, but then the same question. Now here, a root 3 is 1.73. So please convert. I would advise one thing to you. Aditya, you asked me a question whether to convert. So now let's, let's keep it off. I don't know what have been instructed in school, but try calculating up to two decimal places. Anyways, is that okay? All of you just give me a thumbs up. Now that they have any anyways mentioned this, that means they are expecting the final decimal answer. So hence try converting the final answer after rationalization into two decimal places answer that will help because the question itself here in this they have categorically mentioned. Okay, show. Do it vertical tower stands on a horizontal plane and is surmounted by a vertical flat staff of height six meters. At a point on the plane, the angle of elevation of the bottom and top of flat staff are 1345. Find the height of the tower. This is third category question. So you saw a river, there is a point between it. Then you saw the typical height and distance question where one building and one side, there are two angle of elevation. This is third one. Something on some, some flat staff over a building. This is actual board paper last year. RAR house is 2.5 height of tower one. Oh, it's a previous one. No, this is Oh, find the height of the tower. Okay, 8.21 8.19 different answers. Decimal places manter two decimal points. Anyways, see four marker. So if you are able to solve a 45 marker question in two three minutes, that means you have a lot of time. So I'll keep on reiterating. You'll have a lot of time. So you'll have a minimum of 15 to 20 minutes to revise the paper. For God's sake, please revise the paper because otherwise you will end up making some mistakes. Okay, so you can be, you can decide your pace. Anyways, the mocks will give you the what do you say? Right kind of exam setup. Anyways, so vertical tower stands on horizontal. How to crack this vertical tower is there. So let's draw a travel before that. As I told you, read it twice. Vertical tower stands on a horizontal plane and is surmounted by a vertical flagstaff of height six meters. So there's a tower on top of it. There is a flagstaff whose height is given at a point. So the six meter is height of the flagstaff not of the tower. Obviously six meter towers are not there in our country. At a point on the plane, the angle of elevation of the bottom and the top of the flagstaff of 30 and 45 respectively, find the height of the tower. So let me draw the tower first. And tower has to be longer. This is the point where the flagstaff would be there. So this is my flag, let's say. Okay. And they are saying that top and bottom of the flagstaff angle of elevation is given. So this is the setup. So let's say AB is the tower and AC is the flagstaff, which is six meters high. That's given. Okay. What else is given? 30 and 45 degrees. So clearly this is 30 and this has to be 45. Okay. Now what? What is to be, find the height of the tower. In this case, they are asking only one thing, find the height. Okay. Is there any other question? Many a times in the second line also they are, they give the question. So hence read once again at a point. The angle of elevation is this, this, find the height. So make sure that only one thing is asked for I. Okay. So we have to find out this height H. Let us say this is D. So now again in triangle, careful. ABD. ABD. Oh. AB, that's a D. This point is D. ABD. What will happen? H by D. H by small D is 1030. Okay. So hence H is D 1030. So D by root. Okay. This is one. Right. And secondly, in triangle, which one? CBD. CBD. Either you can adopt isoscelesca GAN or you can adopt trigonometry, isosceles triangle, no? CBD is the size list. Why? Because it is 45 degrees. So clearly this has to be 40 degrees. So you can directly write or you can use, since it is a trigonometry question, I will go for that. So 6 plus H upon D is equal to 1045, which is one. So let me write now here. As suggested last time also, don't write in multiple columns. Write in one single column in your, there is no dirt of paper in the board exam. So 6 plus H is equal to D. Right. So H is equal to D minus 6. This is equation number 2. Once again. Yeah. So D. So H is equal to 6 minus. Now, sorry, D minus 6. H is D minus 6. Now, equate 1 and 2. So from 1 and 2. What is this? D upon root 3. See, I will eliminate H because I don't want. Sorry. What did I want to eliminate? D only. No. Oh, no. Then this is the wrong step. You should write D in terms of D. So from here, you can write D is equal to root 3 H. That will help you in one step itself. Okay. Now here, D is anyways 6 plus H. So you don't need this step. Write this to only. Okay. So from 1 and 2 again now. From, oh, I have already written it from 1 and 2 root 3 H is D and 6 plus H is also D. So root 3 minus 1 is equal to H is equal to 6. And hence H is 6 by root 3 minus 1. Okay. But you have to solve. So let me solve here. Some space, some space. Yep. So rationalize first. Don't try to divide here at this stage. Rationalize is better. So this will be 6 root 3 plus 1. No doubt. And below it will be 3 minus 1. That is 2. So this is nothing but 3 into root 3 plus 1. Am I right? All of you got the same thing. So hence, this is 3 times and now deploy 1.73. So 2.73. That means this is 9, 1, 8 point meters. Right. I was getting some other values. Yeshika, how did I, how did you get 196? Someone also mentioned 21. Who mentioned 21? And how did you get stress? 8.21. From where? Oh my God. No, no, no. Don't take that. No, no, no, no. Don't take 1.73. Do whatever. See, they have mentioned 1.73. Oh, sorry. Why do you need to take something else? Nothing. Whatever is given, just abide by that. Okay. Next up. This is again previous year paper. Do it. From a point on the ground, the angles of elevation of the bottom and the top of the tower fixed at the top of a 20 meter high building are 45 and 60. Mind the height of the tower. Now, this is, this happens in many of the apartments. They will have this mobile tower, you know, these days. So you can measure the height of the tower by this or height of the building. Arian has done it already. Nice. Check, buddy. Now, who is this 87025961915? Guys, if you have, you know, beautiful names, why don't you use that? So who is this phone number? 8, 7, something, something. Please use your name. 8, 2, because you are risking your identity if you are, you know, putting a phone number there. Okay. So 20 root 3 minus 1. Okay. Very, very good. Lots of people are already done. So from a point on the ground, there has to be a ground, but first read the question. Oh, what happened? We went to some other. From a point on the ground, the angles of elevation of the bottom and the top of a tower fixed at the top of a 20 meter high building are this respectively. So what is the question? Very simple again, straightforward. There is no challenge to this. So this is my building, capital H. This is my small edge of tower. And here are two points on the ground. Many times, they can give you on the opposite side and all that. So only diagram will differ. Otherwise, not much of a problem will be there. So, you know, so in this case, both are on the same side. Right. So the angles of elevation of the bottom and the top of the tower fixed at the top of 20 meter high. So this is given already 20 meters. Building are 45. So let me write name A, B, C, D. And this is 45. And this is 30. No, sorry. Oh, this is 60. My bad. This one is 60. And this is 45. So again, the moment you see 45, mouth watering. Why? Isosceles triangle. So a lot of life becomes much easier. Okay. Anyway, so you will write now in triangle A, B, C, you will write H plus H, H plus H upon, let's say this is D, small D. D is equal to A, B, C, no A, B, C to tan 45. One. Oh, sorry, tan 60. This will be bigger one. 60. My bad. So root 3. Fair enough. So in this case, H is equal to D, doesn't apply or doesn't appear from the figure. But anyway, so hence H plus H is equal to root 3D. Equation number one, you can write like that. And then in triangle D, B, C, 1 plus plus H, where? No, no, no. It is H only. Sorry. On Rilla. Okay. H is not 1 plus plus H. It is H plus. Sorry. While writing fast, it happens. Okay. So take care there also. Okay. Now H plus H is root 3D. And in triangle, what? DBC. DBC. What do we have? H upon D is equal to tan of 45. One. So clearly H is equal to D. Okay. Now, so what do you need to find out? Height of the tower. So what do you need to eliminate? D. So hence I can write from, so I have to find out small h. So from one and two again, I'll do the rituals. So this is two. So write from one and two. And since capital H is 20 meters, you can write 20 plus small h by D. D is capital H only. So 20 is equal to root 3. So 20 plus H is 20. So H is what? 20 common what? What? Rd 3 minus 1. And meters. So let's try to find out the value 20 minus 1.73 minus 1 meters, which is 20 into 0.73 meters, which is equal to 641. 14.6 meters. Okay. Okay. Next one. Now aeroplane. Good. Solv it. You should not miss. These are very easy low-hanging notes in your board paper. In the exam, can we take the value of root as 1.73? Yes, you can. See, whenever you are in doubt in exam, whether I can do or not. So whatever is there in your mind. So for example, you are assuming something right down there, assuming root 3 to be this much, like that. Okay. This is so value wise. So you must remember, you can't assume root 3 to be 1.4. What I'm saying is, let's say anytime in doubt, you can write your assumption there in the paper. Find the speed of the airplane. Okay. 200 meter per second. Angle of elevation of an airplane from a point on the ground is 60 degrees. After a flight of 30 seconds, the angle of elevation becomes 30 degrees. If the airplane is flying at a constant height of 3000 root 3 meters, find the speed of the airplane. Okay. So here is my airplane. Airplane is moving away from the point or coming closer to the point. What do you think? It's moving away or coming closer? Or in both cases, it will work. We had this question in view. Yes, it has to be away. Okay. So here is my plane looks very good. And it is moving away. Obviously, hence the angle of elevation is going down from here, which was 62 here, which is 30. Drop the perpendicular, join these. So A, B, C, D, E. This is h is equal to, right here, it's a 3000 root 3 meters. What is to be found out? Basically, you want to find out AD, distance AD, isn't it? Then divide by time, you'll get the speed because it is constant speed. So now again, right triangle, you have to find out AD. So you now know this is also h. Okay. So let us say this is x, fair enough, x and this is y. Let us say these are the things. So triangle ABC. In triangle ABC, what do I know? H upon y is equal to tan 60 degrees. Okay. This is equation number one. You can simplify h upon y is root 3. This implies h is root 3 y or y is h upon root 3. Right. Okay. Now next, I don't need y, hence I am writing in terms of y. Okay. In triangle DEC, triangle DEC. Right. What will this be? H upon y plus x, h upon y plus x is equal to tan 30. Okay. This implies h is equal to y plus x divided by root 3. This implies root 3 h is equal to y plus x. Now from here, I will get x. Why? Because y, I know this is equal to, you can write root 3 h is equal to y. y was h upon root 3 plus x. I need x. Right. So what is x guys? So this implies x equals h common root 3 minus 1 upon root 3. Okay. So this is nothing but again the same calculation. So x is equal to 2 by root 3 h. Am I right? This is x. So now I am writing here. Okay. From here. So what is speed? Speed is equal to distance by time. That means x divided by 30 seconds. What is x? x is 2 h by root 3 divided by 30. Okay. Now h is given 3000 by root 3. So that means this is equal to 3000. So 2 times 3000. Last step only I am doing the final calculation. Divided by 30 root 3. So root 3 root 3 will go. This is 100. So this is 200 meter per second. Isn't it? Very good speed. Okay fellas. Make sense? It is moving at 18 by 58 for 520 kilometers per hour. Anyway 540 kilometers per hour. Okay. So far so good below. Any problem? Let's see the energy level. So I know we have already crossed one hour and 45 minutes in this session. So I know you guys are you know bearing for these long hours but can't help guys. So hence let me see how many of you are there. Yes. Yes. Yes. Yes. Yes. Yes. Why why why why why? Show your presence by just typing why so that I can also understand how many people are still there. Okay. So good that a lot of you are maintaining the patience. Okay. So these are typical same type of you know these are again this is last year paper. So now here if you check this is 1.732. A man in a boat rowing away from a lighthouse. These are typical you know find the speed of the boat in so do this. Okay. Here the catch is please be careful. So a man in a boat rowing away from a lighthouse 100 meters high. Okay. Two minutes to change the angle of elevation of the top of the lighthouse from 60 degrees to 30 degrees. Find the speed of the boat in meters per minute. Solve. This is previous year question. Worth 4 marks. Our different different answers are coming from Akshita, Akshita saying something else, Meghna saying something else. AB is height of tower BC is distance from boat to tower 57.8 oh sorry where did it go yeah 57.7383. Different different answers. Meghna is saying 6.1. Meghna what's that answer? 6.1 is okay my turn to solve. Okay. Man in a boat rowing away from a lighthouse 100 meter high. So that has this is 100 meter high 100 meter high lighthouse can you imagine what kind of length it is 100 meter high takes two minutes to change the angle of elevation of the top of the lighthouse. So it was here initially. So the boat is going in this direction. And here don't go by the diagrams little use you know proper scale and pencil while drawing it. So in five markers you should do that ABC and now D what has happened this has happened. So hence it has moved from C to D correct and H is given 100 meters and so you have to basically find out this X to find out the speed and this takes two minutes to travel right. So meters per minute anyways it has to be calculated so I don't need to change the units. Okay now let us say this small distance is D and now these angles are known these angles are known this is 60 degrees this is 30 degrees. Okay now quickly in triangle in triangle ABC ABC tan 60 so 100 upon D is equal to tan 60 degrees is equal to root 3. So this implies D is equal to 100 upon root 3 meters. So D is now known now in triangle ABD 100 upon D plus X so D plus X is tan 30 degrees which is 1 upon root 3 this implies 100 root 3 is equal to D plus X. Now you can find out X from here how D you already know so deploy it so 100 root 3 is equal to D D is 100 upon root 3 plus X again so from here X will be 100 common root 3 minus 1 by root 3 this is the third time we are doing this calculation. So you must be now remembering it anyway so this is 100 into 2 by root 3. So 200 by root 3 isn't it? So mute yourself okay 100 root 3 right so now 100 into 2 root 3 so hence this is nothing but 100 or 200 root 3 by 3 so I have already rationalized X. So now I am calculating here because of positive space I am writing here so speed is equal to X upon 2 minutes 2 minutes is the time taken right so hence it is 200 root 3 by 3 into 2 meters per minute all of you got that that part so this is 100 root 3 by 3 did you get this part all of you 100 root 3 by 3 but you have to convert this so this is nothing but 173.2 by 3 this is easy rather than dividing do this now answer is nothing but so I am writing here so answer 3 5s are 15 23 7s are 21 22 7 3 right since it is 3 digits so you also should write but it is good enough 5 27 7 5 23 7 so 22 7 meters per fair enough and you must also add the what units okay cool so now see we have I will anyways lots of I have compiled lots of other questions as well so probably we will not be able to do all of that and it is typically now repeating so what you can do is when I upload it these are all previous your questions so my request to all will be do solve all of them okay and even before your exam you go through all this this particular slide these slides you don't need to do anything further so in my opinion you have done so many times so you can keep all the notes handy just solve these questions to get some enough practice and you know these are the model answers you can see let me I can zoom oh no I got so this is um you know the uh what is it heights and distance question see how what are the points to be noted over here so they have used rulers scale scales and rulers are used so no free hand drawing no free hand drawing right so what this guy has done is ab lighthouse is 100 meter high c c is boat one cd is both two you know labeled or to find cd or distance between ships see how precisely or very you know in a concise manner he has mentioned you know and in one particular shot everything is here every information here only so an examiner would be happy to see this because it shows one clarity of mind so you are clear what do you need to do clarity of mind so hence it will attract my attention so this guy knows where to go and hence very you know in a very very clear way you have represented it in the right then see uh you don't need to do this I think this you can avoid so tan of acd or you can mention your tan 30 directly also is so that will help you then same you know if this definition is not needed not needed that much this is good enough then uh you know but though they have done in parallel you can see there are two columns they have this guy has maintained but I would advise against it I would say if your school teachers have uh encouraged you to do like that so then please do it otherwise don't do it two columns one column one top of the other like that so x is 100 meters okay and then um because what happens is if you write in two columns the examiner doesn't know what to where to go first what to do wow you know are they interlinked how is that so hence give him give him the direction to evaluate so lead him to one particular direction so hence you can see very clearly and given root 3 is 1.732 written so d is equal to this 73.2 meters and did this kind of thing so this you don't do uh all these decoration part which is there they are highlighting if I were him I would have simply underlined it simple don't do all don't spend time on making square you know what do you say boxes and all that if you have a lot of time left towards the end and you are done couple of times revision now you have enough time to go through and decorate your answer paper do not use multiple colors and all that that's not advisable you can simply use the pencil to differentiate it from the pen maybe and then keep underlying so I in my answers I used to underline twice to show that this is the answer I don't need to go for all this box making thing no you can avoid all of that without impacting your scores so hence this should be the way of writing any questions in uh from your side also I would recommend that you know you can feel free to post questions in your respective group guys and in my opinion if you before the exam for the trigonometry part all these questions I think around 25 odd questions are already there in this you know in these slides if you just go through the solved answers and check the questions and solve that will be good enough do not try to practice so many things together this is just before the exam I'm saying before the math exam you have you don't need to do everything once again you just flip over these slides and there's a high probability that the questions would be in similar nature very very high probability is it okay so what I'm going to do is anyways I'm going to give you these questions you can solve and you can you know these are anyways available on the CBC's website this slides also we will upload and in the next class what we'll do is the rest of the you can solve this and whenever we have the next class we'll have just tally the answers maybe and then we will take up trigonometric identities okay so that will solve more of identity based questions so I will pick up identities questions from the previous year papers will come here and solve all of them so that you get a fair enough idea how that is done is that fine so we'll stop here guys so thanks for your time and I hope this is making sense to all of you are you guys do you think this is useful so these are everyday you know meeting for two hours and solving problems previous year questions so you do do let us know your how do you feel about it and what else do you want so do we do anything else we can do from our side to help you please feel free to reach out and you should all refer to Ardisha but not needed for boards not in your talk not needed so yeah anything else any other question you have please ask me and you know please be connected and those who are who have joined the CRP WhatsApp group you guys can post your queries in that group as well so that just to reiterate the announcements tomorrow three options uh you know uh three options of writing exam NTSC slash mock test of board slash chemon so you can try anything chemon link will be active for till fifth of january so you know during the week whenever you have time please go through and make sure that you attempt is that okay any other any other thing i can help you with guys tell me or you can reach out to me individually as well okay uh no CRP group is for those who have not joined sentence formally so or sent all sentence rules uh will anyways be receiving their information on your respective groups don't worry CRP group is all those invitees who are joining from other schools okay fair enough so see you again on i think next class for mathematics would be on tuesday there will be first chemistry class on monday do attend that and i hope all of you have the schedule with you keep keep checking the schedules okay bye guys and take care and enjoy your weekend then stay safe okay bye signing off now in ncr takes good enough for boards um yes to a large extent so you can restrict to that and whatever questions previous year questions if you you know do we usually say that rs agarwal is a good enough now this thing for boards because they have wide coverage of questions so you can do that so you know uh i would suggest you can keep rs agarwal as the mean level so and then you don't need to deviate from there for the board exams and whatever questions previous year questions solve all three four year previous year questions for every subject is more than enough very much recommended obviously previous year papers we will all these questions which we are discussing our previous year questions only vanish so uh very very very very important very very relevant because as i told you by the end of february you will be able to predict your own board papers mark my words okay bye bye take care see you