Alright, so this is the first problem for the day. Let's start with this. We have to find the value of x which satisfies this particular equation. I'll be putting the poll on. Almost one and a half minutes gone. I have got one response so far. Good, good, good. Now responses are increasing. Six people. Almost two minutes gone. By the way, many of you are still joining in. So good evening, everyone. So we are up again with the problem solving sessions for your upcoming semester one exams. This is the first problem we have started with. Okay, so let's wrap this up in another 30 seconds. Five, four, three, one, go. Okay. Only 15 of you responded out of which 12 of you have to say option B. Okay, let's check whether B is right or not. So first of all, what's this cause of cot inverse x? So let's write it down in terms of x itself. So normally cot is treated as base by perpendicular, right? So it will become x by hypotenuse. Hypotenuse will be this. Okay, so the first term over here will become x square under root one plus x square under root. Okay, x plus under root of one plus x square. Next is sine of cot inverse x. So again, cot inverse x means x by one. This is your, this is your base. This is your perpendicular. So it'll be one by under root of one plus x square. So if you combine these two terms, so the given LHS will become, the given LHS will become, is my voice proper now, Shraddha? Yeah, in between, I think the internet was slightly weak. Now it is proper. Okay, thanks. So this is going to become x square plus one by under root of one plus x square whole square, which is 51 by 50. This clearly means this clearly means x square plus one is equal to 51 by 50. So x square is one by 50. So x is going to be plus minus, plus minus one by five root two. So any one of these two answers which is sitting over here that is good enough for us. I think one by five root two is sitting over here. Option number B is definitely right. Okay, well done. Most of you got this right. Any questions? It is still breaking. Let me check. I am on Geofiber. Yeah, that's fine. Should not be breaking actually. Yeah, sure. Fine. Any questions? Okay, I'll switch off my camera for something because bandwidth issues may be there. Okay. Now is it proper? All right. Should I move on to the next question now? Next. Sign of cot inverse one plus x is cos of tan inverse x. Then x is which of the following. Pole is being relaunched. Same strategy you have to adopt here as well. Use your reference triangle. One minute gone. Only two people so far. I can give 45 more seconds from here on. Five, four, three, one. All right. So this time participation has drastically improved. 51% of the people have replied and most of you have said option number D. D for Delhi. Okay, let's check. So here look at the first expression.  Okay, so cot theta is given to us as one plus x. So if you make a reference triangle, if you make a reference triangle, cot theta is one plus x means base by perpendicular is this. Okay. So this is automatically under root of two plus two x plus x square. Am I right? Square of this plus one under root. So sign of that angle theta. So this is your theta angle. So sign of the angle theta that would become one by under root of two plus two x plus x square. Okay. In a similar way, let's say I call this angle to be five. Let's say I call this angle to be five. So tan five is x. So make a reference triangle. This approach is very well known to us. By the way, this was theta here. This was five. So this is your perpendicular by base. So this is x. This is one. So this is under root one plus x square. So cos of five, cos of five, that would become one by under root one plus x square. So as per the question, these both expressions are equal to each other. Okay. So from the question point of view, these two are equal. These two are equal. So one by under root of two plus two x plus x square is equal to one by under root of one plus x square. So that clearly means this is equal to one plus x square x square x square gone. So two x is equal to minus one. So x is equal to minus half, which is option number D. Okay. D is absolutely right. Well done. Any questions, any concerns here? Any doubts? Can I move on? Next question. Simple one. Poll starts now. Something wrong with the poll. Super easy. Idly should not be taking more than a minute for this. Excellent. Very good. Five, four, three, two, one, go. Again, in same lines. Okay. Most of you have voted for option number C, which is 15. Let's check. So on the same lines, tan inverse two is seek inverse. Can I say root five? Yes or no? So treat this as, treat this whole angle as theta. So again, when you make a reference triangle, tan theta is two. Tan theta is two means this is two. This is one. Okay. So this hypotenuse will become root five. So seek theta is hypotenuse by base. So it will become this. So seek of seek inverse root five square you have to do. So that's nothing but root five square. Okay. So that's five. That's five. Similarly, cot inverse three is cosec inverse root 10. Okay. Again, let's call this as some angle five. This is five. So cot five is equal to three. So this is three. This is one. So this is root 10. So cos cosec of five will be root 10 by one. So the same angle is cosec inverse root 10. So cosec of cosec inverse root 10, the whole square is going to be root 10, the whole square. That's nothing but a 10. So your answer is five plus 10, nothing but 50. Simple question. Option number C is correct. Any question anybody has? Should we move on? Okay. So let's take something from continuity differentiability chapter. The whole is on. That's a good question guys. Should not take so much time. I can only, I can only see five of you, six of you responded so far. 45 seconds more. Yes, it's done. Yes, yes. We can implement that also in the previous question. Correct. Okay. Let's start the countdown. Five, four, three, two, one, go. Okay. 17 of you have participated in this poll. Out of 17, 12 of you say option B. Let's check. See, look at each of the options. If you take a number less than zero, let's say I take a minus one. So your function will become cos of one by X divided by X when X is not equal to zero. Okay. Now, when you, when you're trying to find out, let's say the left hand limit, okay, what will happen? What will happen? This is a bounded quantity. Please remember, this is going to be a bounded quantity. So this is somewhere between minus one to one. Correct. And this is going to be a very, very small negative number. So it will be some, some fixed quantity divided by a very, very small negative number. So this result will go to minus infinity. And on the other hand, if you find the right hand limit, the same is going to give you something which is plus infinity. So of course, your upper and the lower limit, sorry, the left-handed limit and the right-hand limit will not be equal to each other and forget about it being equal to the value. They're not, they don't exist first of all. So this option cannot be my right option. Okay. If I take something greater than zero, let's say I take a one. So X cos one by X, I'd already discussed with you that this is an auxiliary function, but this guy arrests that oscillation in the neighborhood of zero. Okay. So do you remember this was done in our regular session of continuity and differentiability? So what happens that when you evaluate the limit of this as X tends to zero minus, again, this is a bounded function. This is somewhere between minus one to one. So this is of the nature bounded into tending to zero. So overall, this will be zero only. This is not infinity. If there was an infinity here, then I would say it is still an indeterminate form. Okay. And same will happen with your right-hand limit as well. So when you try to find the right-hand limit, that will also be a bounded function into something very close to zero. So that will still be a zero and that is equal to the value of the function also. Okay. So anything greater than zero is going to work. Even if it was X square, X cube or whatever number greater than zero, this is still going to be satisfying the fact that left-hand limit, right-hand limit both will be zero each and that will be equal to the value of the function. So B is the right option so far. Equal to zero, it is not going to be continuous at zero because it will be oscillating, which I already discussed to you. And because there's an equal to zero over here, this also cannot be your answer. Okay. It should be greater than zero strictly. It cannot be equal to zero also. So option B is right. And I think some of you got confused and mark D as well. What do we need to do if we want this? Okay. Very good question. So a follow-up question is what should be the value of K says that this function is differentiable at zero. So first of all, K should be greater than zero for it to be continuous. Okay. That is the necessary condition. Let us see what is the sufficient condition. So now if you want this function, so let's find out F dash zero minus. So that will be limit extending to zero plus F of zero plus H minus F of zero by H. Okay. So F of zero plus H means F of H that will be something like this. Okay. And very important thing is that this value should match with, oh, I'm so sorry. I think I had written zero plus here and this is zero minus. So for zero minus limit extending to zero plus F of zero minus H minus F of zero by minus H. So this is, this will be minus H to the power of K cos of one by H minus zero by minus H. So both should give you the same result. Both should give you finite and the same result. So here what will happen? It will become H to the power K minus one into cos one by H. Okay. Now see this guy should be greater than equal to zero. Yes or no? Greater than equal to greater than equal to zero or greater than equal to one. See H extending to zero plus if you want this answer to be finite. Okay. Aditya, what should be the value of K minus one? Greater than equal to zero. Can it be equal to zero? If it is equal to zero, it will become cos one by H. Correct. What are the limit of cos one by H as H tends to zero? It's oscillating, right? So this quantity should be a quantity which should be positive. Correct. Some positive value will be, should be there. Sorry. Greater than equal to zero. Am I right? Aditya, what is your route? Sir, it can't be equal to zero, right? Why? Any positive value that comes here, H will be on the numerator, right? So this will become zero. So K minus one, K should be greater than equal to one in this case. In fact, it should be, sorry, it should be purely greater than one. Yeah. Purely greater than one. Yeah. Got it. Because equal to one will make this get, getting, this will get cancelled and then the limit will not exist. Yeah. Same here also. You have minus, you have minus H to the power of K minus one, cos one by H. So K minus one should be greater than zero. So of course, the same criteria will hold true on this as well. Ideally, shouldn't it be infinity? What shouldn't be infinity? Kinshukh, unmute yourself and talk. What is your doubt? Power should be anywhere greater than one, which includes a very high power also. Why should the power be infinity? It should be a power greater than zero because you want this H to exist in some form in the numerator so that it arrests the oscillation of this fellow and make it as a finite limit. That is the concern here. So that can only happen when K minus one is greater than zero. Okay. If K minus one is equal to zero, that means K is equal to one, then what will happen? This will result into cancellation of H and H, giving you cos one by H. Limit of cos one by H as H tends to zero is an oscillating value. It's a value somewhere between minus one to one. This will also result into the same thing. So you cannot compare the two results. So you can compare the two results of your left hand limit and right hand limit only when you have certain trace of H left on the numerator to arrest the oscillation of this guy. Makes sense. Can we move on to the next one now? Sorry, I think poll was on. In case the poll is on and it is blocking your view, please bring it to my notice ASAP. This is an easy question. We have already done this type of question during our regular sessions. Last 30 seconds. So all of you have got your admit cards from the school. How is the exam happening? It is happening in the school premises only. Okay, great. And external examiners will be coming, right? Okay, all right. So five, four, three, two, one, go. Okay, so there has been a split of the votes between B and D, but B of course has got double votes than D. So there are people who think none of these is also an answer here. So first of all, let us understand here. What is the point of discontinuity of f of x itself? f of x is discontinuous, discontinuous at x equal to one. So this itself will also be the point of discontinuity of f of f of x also. Okay, now let's evaluate f of f of x that is one by one minus one by one minus x. So that will give you, if I'm not mistaken, one minus x by minus x or something, but x minus one by x. So this function will be discontinuous at x equal to zero. Okay, so this will also be the point of discontinuity for f of f of x as well. f of f of f of x as well. Okay, so these two points are already my points of discontinuity. Okay. So if I just have to take a call based on this, I would go for B, but let us also check out what is f of f of x equal to. So in this, of course, if you put your x as one by one minus x, let's see what do we get. So this is going to give you an x divided by, okay, they will all get cancelled. So this doesn't have any point of discontinuity. So overall, these are the two points where your composite function f of f of f of x will be discontinuous. So option number B is correct. Many of you who answered D, you probably would have taken a call just on the basis of this, which is not correct. Okay, because the function is not defined. The function is not defined at one. So you can't put a one in the function. The function is not defined at zero. Right. Yes or no. So you can't put a zero because f of f of x, that is not going to exist. Okay. So these are also included under the points of discontinuity of that final function as well. Is it fine? Any questions? Can we move on? Simple question. Something wrong with the poll relaunch. Want to continue? Can you see the poll now? Two minutes gone. I can get 30 more seconds. Okay, five, four, three, two, one. All right. Very less than 15 of you have only responded. Most of you have said option number C. Okay, let's check. So the first statement, dy by dx, it's dy by d theta divided by dx by d theta. So dy by d theta, that's very obvious. It's going to be minus a sin theta. And this is going to be a one minus cos theta. Okay. So this is broken down into half angles. Right. And this also can be broken down as two sin square theta by two, which is clearly, which is clearly nothing but negative cot theta by two. So first statement is definitely right. So B cannot be a wrong option. Sorry. Yeah, this cannot be the right option on because it says only two is true. And neither one or two is true. D is also gone. Okay. So people who have voted for a, sorry, B and D. Okay, they're wrong. Now let's check whether only one is true or both one and two is true. Let's check that. So for that, we need to differentiate this guy also second one. So dy by dt. So for dy by dt, we can use our quotient rule. So that will become, I'm just doing it directly over here into derivative of sine cube minus and derivative of this is going to be minus two sine two t upon two under root cos two t four divided by is it fine? This and this will get cancelled. So on simplification, this is going to give you three take multiple. Can you see my screen? I'm audible. Okay. Yeah. So this is going to be three cos two t. And this is anyways, sine square t cos t. Okay, here it will be plus and this will be sine two t into sine cube t. So you can write it as two sine to the power of four t into cos t. Is it fine? One more simplification I can do here is I will write this cos two t as one minus two sine square t. Are you able to see my screen now? Thank you. No, screen not visible. I'm audible. Hello. Just give me a second. I'll check my internet connection. Is it unstable? Can you see my screen? Is it fine? Okay. So if I check this out, this is going to be three sine square t cos t minus four sine to the power four t cos t upon this. So I think a bit of simplification is, you know, involved over it. Okay. So let's say this is my result for dy by dt. For dx by dt, let's do the same expression. So for dx by dt, dx by dt is we are differentiating this expression. So that is going to become or I think I did dx by dt. Sorry. So I was thinking how the same expression is done. Yeah. So let's do dy by dt now, dy by dt. So dy by dt will be, okay. This also we have to simplify in a similar way. So I'm just writing the simplified step version of this. So this will be minus three cos two t cos square t sine t plus plus cos cube t. And this is going to be again two sine t cos t by cos two t under root of cos two t. Okay. Now this time here you write this fellow as two cos square t minus one. So when I do that, I'm going to make this replacement over here itself so that we save a bit of our simplification time because the problem is not difficult, but it is taking some time to simplify. So when I do that, I will end up getting plus three cos square t sine t and minus six plus two will give you minus four, minus four cos to the power 40 sine t. Okay. Yeah.  So what is the ratio of, let me call this as one and let me call this as two. So two divided by one. Two divided by one will only leave you with, just make a barricade here. Yeah. Two divided by one will leave you with three cos square t sine t minus cos to the power 40 sine t upon three sine square t cos t minus four cos to the power, sine to the power 40 cos t. Now we can cancel off, we can cancel off one sine from both numerator and denominator. Okay. We can cancel one cos from both numerator and denominator. Okay. Now what is this formula? The numerator formula is clearly the formula of negative cos three t, isn't it? Because cos three t formula is four cos to t minus three cos t and denominator is the formula of sine three t, which means it is negative cot three t. So is the question set us saying that this is negative cot three t? Yes. Means option or statement number two is also correct. So both one and two are going to be the right options. Okay. But yes, this question comes with a bit of simplification involved and is slightly time consuming. Okay. So try to write a cos in sine in abbreviated format. Don't write it like SIN, COS, etc. That will take a lot of your time. So try to write in an abbreviated format. Okay. But again, don't do it if you have not practiced it so far. Don't start experimenting things in the examination part. Can we move on to the next question? All is on. Simple question. Okay. Can you check that out? Done? Okay. Most of you have gone with option number A. Okay. Let's check. Let's check.  If you differentiate it twice, it becomes negative a cos x plus b sine x. Yes. So n could be two. So a option is definitely correct. But some of you are also claiming that even six is going to work. Okay. We will verify that as well. So triple derivative is going to give you negative, sorry, triple derivative is going to give you a sine x plus b cos x. Fourth derivative is going to give you a cos x minus b sine x. Fifth derivative is going to give you negative a sine x minus b cos x. And sixth derivative is going to give you negative a cos x plus b sine x. Yes. Or even option c is correct. Okay. Now, here, many times, normally we discuss the nth derivative formula with the students. So nth derivative of sine of a x plus b. Okay. I'm not forcing anything, anybody to remember anything. But if you could just keep this in your mind, it might make your life simpler in many situations. Okay. And nth derivative of cos of a x plus b is a to the power n. Same expression, but with terms of costs. Okay. So this is something that I would like you to note down and keep it may save your time provided you are differentiating multiple times. Okay. So in our present situation, in our present situation, what is happening is you want this cost to become negative costs. And you want this minus sign to become plus sign. So n equal to two ways basically works fine over here. Provided you have put your a as a one and b as a zero. So do that. So a to the power n will become a one. Okay. So this formula, in case you need it, I mean, I'm not saying that it is absolutely important to know these formulas. In case you need it at some places where you are performing a higher order, a very high order derivative, then you can keep this formula in your mind. Okay. But yes, in the present example, both a and c options are right. Is it fine? Can I move to the next question? Let's take this one. Oh, a function is defined from R to R as this. Find the value of a so that this function is continuous at zero. Definitely I'll share this recording. No issues. I've been already doing it. Okay. Five, four, three, two, one, go. Okay. Most of you have gone with option number, option number D, D for Delhi. So it's just a question of limits actually. So for the function to be continuous, your value of the function at zero, okay, that is your a must be equal to the limit of the function as x tends to zero. So your a value will be limit extending to zero to sign x. And this is just to sign x cos x by two x cos x. So this two will go off.  Take a sign x common one minus cos x upon cos x and x. Okay. So this is as good as a tan x by x. This is as good as this is as good as tan x by x. And you can just split this limit as two separate limits. So this is anyways one, but this is a zero. So your overall answer is zero. So option number D is going to be right. Is it okay? Any questions? Any questions? Any concerns? Next question. Oh, yeah, Prakul, I remember you having, you know, doing this question exactly. Now, I remember that. So how are your exams going on? ISC people? So far so good. Paper has been coming easy. Okay, good as of now. No problem with the use of scribe. I mean, I hope there's no delay and all time till four. Till what time to what time is the writing time for ISC people? Kinshuk, here M is an integer. M is any integer. Okay, I can give you 30 more seconds. Simple question. Okay, five, four, three, two, one, go. Awesome. 100% people have said option number C. Okay, let's check. So it is very obvious that this term should be, this term should be of the nature. Correct? Right? A multiple of pi plus pi by two, because at pi by two, if this becomes discontinuous, then it is periodic. The whole function is periodic with pi. So at every m pi plus pi by two, the function will start facing infinite discontinuity. In other words, x by x plus one is m plus half. Right? You just take a reciprocal of it. So one plus one by x is equal to two by two m plus one. Subtract a one. Okay. So if you subtract a one, it becomes one minus two m by two m plus one. Take the reciprocal, it becomes option number C. Oh, sorry. Option number C. Is it fine? Oh, good. Good, Prakul. Should we move on to the next question? It's a one-minute question actually, lesser than that. If you know your properties of differentiability well, basics, basics, Prakul, paper will come easy. Okay, you just have to control your emotions. So practicing harder problems will not, what you need to practice is calming down your nerves and trying to read the question properly before answering. Okay. It is applicable to all of you here. Don't go for tough problems. You're not preparing for J advance here. Just keep your basic understanding clear. So last minute, I would just request you to just look through your inverse trig formulae because that is something which is slightly property heavy. And application of derivatives concepts, you can just practice a bit because that is the heavier of all the chapters. And just a bit of formula for a number of functions, bijective functions, how to find domain, range, etc. Beyond that, you don't have to worry too hard. Practice too hard. Okay, five, four, three, two, one, go. All right. Most of you have said option number eight. Now see here, this is a function which is like this actually mod x square is as good as x square. So this is a function which is continuous and differentiable. This is a function which is continuous but not differentiable at x equal to zero because this guy has an issue. Okay. So overall, a sum of a differentiable and non-differentiable function is non-differentiable only. But if you still want to make it differentiable, that is what the question is saying, I want to make it differentiable, then I don't want this guy to be there. So if this guy is not to be there, I have to, I should choose my a value to be zero. Okay. b could be anything. So a should be zero, b could be any real number. So option number a is going to be correct. You don't have to worry about doing anything else. Is it fine? Any questions? Should we go to the next one? Poll is on. Oh yes, these special brackets are gif's. This bracket is the gif function. Okay. If you feel multiple options are right, please give me a response on the chat box as well. Okay. Should we discuss it now? Five, four, three, two, one. Okay. I think if more than one option is right, then this poll has no meaning. Okay. So see, they're all talking about neighborhood of two. So let us redefine the function close to two. So if you talk about very value, which is very close to two, let's say less than two, I would say, I would not say zero to two. I'll just say less than two, slightly lesser than two. So lesser than two, gif of this will become a one. So that will be as good as an x. Okay. Equal to two, if you check this function value will be, or let's say greater than equal to two, also you can take simultaneously. This function value will be just x, x minus one. Okay. Now, is this function continuous at x equal to two? If you check the left hand limit is two, right hand limit is also two, right? Because this, when you put, it'll get a two. This also, when you put, you get a two into two minus one. So this is definitely continuous at x equal to two. Is it differentiable at two? So for finding the left hand derivative at two, you just have to differentiate this and put x as two. By the way, there's no scope of putting anything. They'll get a one. And if you differentiate this, you will get two x minus one. And here if you put x as two, you will end up getting a three. Okay. That means it is not differentiable at two. So option c is correct and option b is correct. So bc is going to be the right option. Simple. Oh, I'm so sorry. Anusha, any special reason for finding it differentiable at x equal to two? Then what happened? x equal to one? Okay. Maybe this kind of functions will not be asked in board exam, but just wanted to check your understanding with respect to min function, min max function. I'm putting the poll on, but if you feel that there are multiple options, right? You can give me a response on the chat box. See, let me tell you again, my questions will be of a slightly higher level than what you will be getting. Okay. So this is because we are practicing it. We can't practice with easier questions. Have you seen player practicing for their cricket match? They will not practice from 22 yards. The bowlers will bowl from lesser than 22 yards for the batsman at least. Okay, Aishwetish. Okay, Aditya. Aishwetish wants to change his answer. Okay, five, four, three, two, one, go. Okay. 10 of you have only voted out of seven of you say option a. Okay. Fine. See, first of all, min function or max function, whatever special functions you're given with respect to min max, you make both the graphs. So mod x, x plus one is like this. Okay. And mod of x plus one is going to be a graph like this. So I've already told you for mod of x plus one, you make the graph of, you make the graph of x plus one first, which is going to be like this and remove the part of the graph, which is on the negative side of the x axis and reflect the graph, which is on the positive side like this. Okay. Now our graph will be the minimum of the two functions. So minimum of this blue and green one is the green one till you reach zero. And after that it is all, both of them are same actually. So it is final graph is actually a straight line y equal to x plus one. So it is differentiable everywhere option number a is correct. b is not right. Is it always greater than one? I don't think so. It is going down in the negative direction as well. Okay. So option only a is correct. Is it fine? Any questions? Any questions, any concerns? Are there any special reason for marking b and c? Oh, you plotted max by mistake. That's the thing I'm worried. Many of you will make a mistake because you read the question incorrectly, not because you don't know it. Yes, all is on. Okay. I've got one response so far. Okay. Last 30 seconds, just two of you have voted so far. Okay. Five, four, three, two, one go. Okay. So C and D have got the vote so far. And most of you say option number C, C for Chennai. Let's check this out. See, first of all, we need to know what is the function actually. Okay. For that, there is a simple approach here, which probably you would have used several times to replace your x with pi by two minus x. So that sign and cost interchange their positions. So when I do that, this is what I get. Okay. So on replacing x with pi by two minus x, we get that. Okay. Now think as if you have been given, you have been given something like this. Two a plus b is x and two b plus a is pi by two minus x. Okay. You have to find your a and b. Okay. Let's say I want to find my a. So what I'll do here, I'll multiply this with a two. Okay. Whole thing. And then we'll subtract the result. Okay. So when I do that, I get three a, two b, two b will get cancelled and this will become, if I'm not mistaken, it will become three x minus pi by two. Correct. That means three a, a is nothing but sin inverse x, sorry, sin x. Okay. Which means f of sin x is x minus pi by six. Now in order to make this, in order to make this x, you can change your x with sin inverse x. Okay. So again, replace your x with sin inverse x. So that will become f of sin of sin inverse x equal to sin inverse x minus pi by six. In short, you have got your function. This is your function. Okay. Now the question is asking us for the derivative that is clearly one by under root one minus x square. Is it fine? So option number c is going to be the right option. Any questions? Okay. Now try your hand on this. It's of the similar type. So those who could not get the first one, try your hand at this problem. One response so far. Okay. Last 30 seconds. Five, four, three, two, one. Go. Okay. b is the most chosen response. Most of you say it's half. Let's check. Again, the same story will change our replace your x with one by x. Okay. So when I do that, it becomes this, this becomes f of x and this becomes this. Okay. Again, think as if you have written three a minus two b is x and three b minus two a. So you can just write it minus two a plus three b is one by x. You have to solve for a thing like that. So multiply this with three, multiply this with or two. Correct. And add them. So when you add them, this will become five a minus six, six will get canceled. So this will become three x plus two upon x. So your a is one fifth of three x plus two upon x. Okay.  Now go for the derivative. Find the derivative at two. So the derivative here is going to become one fifth, three minus two by x square put x as two. So that is one fifth, three minus two by four. So that is one fifth, three minus half. That's nothing but one fifth of five by two. That's a half option number. Option number b is definitely right. Well, is it fine? Any questions? Easy question. Find the maximum value, find the maximum value of this expression. Very good. This question came in your pre words. Okay. Okay, five, four, three, two, one, two. Okay. And a four will only 13 people responded. And most of you have got option number c. Let's check c is correct or not. So let's take all this as y. So you have to take a log on both sides to the base of e. Okay. You can write it as negative x ln x. Okay. Differentiate both sides with respect to x. So it's this. Okay. Now put it to zero. Putting it to zero means putting this as zero, which clearly means ln x is equal to minus one. So x is one by e. Okay. Since this is the only thing and they're asking you which of there is the maximum value and none of these is not there, then probably putting this will give you a maximum value only. Don't waste your time checking the double derivative unless and till there is a none of these or unless and till you get one more option or one more value of x. So your answer is one by one by e to the power of one by, which is e to the power one by definitely option number c is right option. Is it fine? Any questions? Can you comment on the minimum value? See a function need not have a minimum actually. It's not necessary that there has to be a minimum existing. See this function, this function x to the power one by x. I don't know whether you have seen the graph of it or not. Okay. The graph of this function, if you draw, it basically starts with one over here goes high and then after that it starts diminishing. Okay. So when it is still falling, it doesn't have a minimum. Are you getting my point? It's like something which is rising. What is the greatest value? It is still rising. Infinity is not the greatest value as I already told you. The global maxima or the global minima, they are fixed values. You cannot claim infinity and minus infinity is to be global maxima or global minima. So it is still falling. So global minima doesn't exist. No tending to zero is not a global minima. That's the problem. Why should one be the minima? It has a lot of values lesser than one also. But this value is still dropping. That's what global minima is not defined. It is still falling down, down, down, down, down, down. It is approaching zero. Agreed. But you can't claim that the global minima is equal to zero. Exactly. Global maxima and global minima concepts when we were talking about global maxima and global minima, if the functions, if the function limiting value is beating the maximum or the minimum value occurring in it, then that particular absolute extrema is not defined. So let's say x square function. I give you a simpler example, x square function. It's global minima occurs at x equal to zero. But what is the global maxima? It's not defined. It does not exist because it is still rising. Okay. And there's no fixed value that you say, okay, this is the global maxima it attains. Same is happening with the case of this function one by x to the power x as well. Next one, if the parabola y is equal to f of x having axis parallel to the y axis touches the line y equal to x at one comma one, then which of the options is correct? Two minutes, two and a half minutes gone. I've only got two responses so far. Five responses I can give maximum 45 seconds more. Okay. Five, four, three, two, one. Very, very few people have only seven of you have responded to this. And out of that four of you say option B. Okay, let's check. See, first of all, a parabola whose axis is parallel to y axis means it's an upward opening parabola.  So what are the things that is given to us in the option f0 and f dash 0?  So at least they're trying to say something like 2b plus c is one. This is trying to say 2c plus b is one. Okay. This fellow is trying to say again, 2c minus b is one. And this fellow is trying to say, let me not write here, let me write above the option, 2c minus b is one. And this fellow is trying to say 2b minus c is one. Okay, let's see which of them is correct. Now, a couple of things to be utilized here. First of all, the function touches this line at one comma one. That means the derivative of this function when x is one is actually one. Right. Which clearly means 2a plus b is one. Okay, let's call it as first equation. Second thing is the function will pass through one comma one. That means f of one is one. That means a plus b plus c is one. Okay. Let's call this as two. Now let's subtract one and two. If you subtract one and two, a minus c becomes zero. That means a is equal to c. If you put this back in the first one, you get 2c plus b equal to one. So which option says 2c plus b equal to one? 2c plus b equal to one. Option b says it. I hope you can read it. I've written it in yellow. So option b is right. Okay. Any questions? Any concerns? Can I move on to the next question? It's opening upward, downward doesn't matter. It's just a value is going to decide it, but it is of this nature. That is what I can say. So Aditya, it could open downwards also because I've never claimed a is positive. The equation of one of the tangents to the curve y is equal to cos x plus y from x line between minus 2pi to 2pi inclusive. That is parallel to this line x plus 2y equal to zero. Is which of the following poll is on test for the options. Okay. Please act as per the options because they say equation of one of the tangents to this curve. Okay. 45 more seconds I can give for this. 5, 4, 3, 2, 1, go. Okay. Very less people have voted this time also. 11 people only. Okay. 11 people and out of that seven say option number b. Okay. Let's check. Okay. First of all, a line parallel to this should off be, should be of this nature, right? Of course, all of them satisfy the same criteria. Now you want that line to be parallel, tangent to be parallel to this means having a slope of minus half. So first of all, let's find dy by dx. So let's differentiate both sides. So when you differentiate both sides, okay, this is what you end up getting. And now you have to put your dy by dx as a minus half because this is the slope of this line. And this is going to be half. That clearly means sin x plus y is one. Okay. So you can also say that cos x plus y is zero. That means x plus y is a or multiple of pi by two. Okay. So let's say I take a, yes, that's what. So this is going to give you y as zero, right? That means your x is off this nature. Now take any x, which as per the option is going to work out. I think let's try taking pi by two first. So if I take x as pi by two and y as zero, that means I want to find the tangent at this point. And let's say this is the tangent. It must satisfy pi by two comma zero. That means C is pi by two. So it falls in line with our option number B. Okay. So you could take any x of this nature. There could be so many x depending upon your n value, but I decided to look at the option B and decide to take n as zero. Okay. If you take n as one also, you'll get pi by two comma zero. In that case, your answer will become x plus two y equal to five by two. So there could be so many tangents. We want one of the tangents. That is what the question says. Okay. So option number B is going to be the correct option. I think we have done this question before. Okay. We'll skip it. This function is increasing in which of the following intervals. This question can be solved in less than 30 seconds actually. Okay. Should we stop in another 30 seconds? Five, four, three, two, one, go. Okay. End of poll. Most of you have gone with option number B. See, I would like you to recall the concept of increasing decreasing functions where we had talked about how do we decide the increasing decreasing nature for composite functions. Okay. So let us say I have a function F. I have a function G and I have a function F or G. Okay. Let's try to decide if the function F is increasing and G is also increasing. What could you comment about F or G? Will it be increasing or will it be decreasing? Increasing? Correct. Good. If this function is increasing and this function is decreasing, then what can you comment about F or G? See, G is a decreasing function. So if I put, let's say X1. Okay. And X1 is supposed to be lesser than X2. Let's say. Right. Then what will happen? This guy will have, if this G is a decreasing function, this guy will have a higher value. So this guy will have a higher value because F is an increasing function. G is a decreasing function. So for lower inputs, G will show higher values, isn't it? So if F gets a higher value, so this will be G of X1 will be more than G of X2, isn't it? So F of G of X1 will be more than F of G of X2 because F is an increasing function. So if you increase the output, sorry, if you increase the input, what is going to happen? Your output is going to fall down finally. Yes or no? So this is a decreasing function, no? How you people are saying increasing function? Shittich, any special reason for calling it as increasing function? Sinchen? Anusha? Okay. If this is a decreasing function and this is an increasing function, then what will happen? Let me make some space over here. If F is a decreasing function and G is an increasing function, what will happen to F of G? Decreasing only? I don't know if you agree with this. Okay. If this is a decreasing function and this is also a decreasing function, then what will happen to F of G? Check it out. So let's say X1 is less than X2. Okay. So G of X1 will be more than G of X2 because G is a decreasing function. And F of G of X1 will be lesser than F of G of X2 because even F is a decreasing function. So what is the nature here? It becomes increasing. It becomes increasing. Okay. Now, we will use a part of it in our problem solving.  And the question setter is also asking the interval where this entire thing is an increasing function. So basically, we are catering to this need. So if your F is increasing and F of G is also increasing, that means we need to tell the examiner that even G should be increasing in that interval. So in which interval do you think sin X plus cos X is increasing? Now, this function is written in a harmonic form like this. Right. So this is a function which will be graph of sin X shifted pi by 4 to the left. Okay. So pi by 4 to the left. So this graph looks like this. At 0, it becomes 1. It goes surpasses. So this is your minus pi by 4. This is your pi by 4. Okay. And this is going to be 3 pi by 4. Okay. And this is going to be a pi by 2. Add a pi by 2 here. It will become 5 pi by 4. Anyways, so out of these intervals, out of these intervals, which is the interval where the function is increasing, is it pi by 4 to pi by 2, pi by 4 to pi by 2? No, it is falling. A cannot be my option. Answer. Okay. Minus pi by 2. Okay. This is going to be minus 3 minus 3 pi by 4 if I'm not mistaken. Correct. Yeah. So minus pi by 2 will fall somewhere in between. So minus pi by 2 to pi by 4. Is it increasing? Yes, it is increasing. Option B is definitely correct. 0 to pi by 2. Pi by 2 will be somewhere here in between. Okay. So 0 to pi by 2. No, it is showing both the nature. It cannot be my option. Minus pi by 2 to pi by 2. Again, it is increasing and decreasing both. It cannot be my option. So simple. Only by looking at the graph of root 2 sine x plus pi by 4, you can answer your question. Is it fine? No need to go for any kind of a derivative or not. So we'll take a small break over here. 601. We will meet exactly at 6.15 pm with further more questions. I think we have to touch upon matrix determinants and your concept of linear programming problems. So we'll take that up on the other side of the break. So see you in 10, 15 minutes. All right. So let's take this next question. Suppose alpha, beta, gamma are real numbers such that sine alpha, sine beta, sine gamma, none of them is equal to 0. Then delta, this determinant cannot exceed which of the following values. I'm relaunching the poll. So two responses. So far almost two minutes gone. Okay. Last 30 seconds. Five, four, three, two, one. Go. Okay. So I can see a confused response between A and B. Both of the options have got equal number of votes and one person has responded with option C. See, first of all, what we can do is we can take, we can take sine square alpha common from the first row, sine square beta common from the second row, sine square gamma common from the second row that will leave you with a standard determinant which you have actually seen before. Okay. So if you recall, you have seen something like one x, x square, one y, y square, one z, z square kind of a determinant. Am I right? And for such cases, we have also learned that I'm just recalling that concept for you. So for one, one, one, x, y, z, x square, y square, z square, the value of this determinant becomes x minus y, y minus z, z minus x. Okay. So you can use same thing over here. So it is going to become, it is going to become cot alpha minus cot beta, cot beta minus cot gamma and cot gamma minus cot alpha. Okay. Now, just a quick recall of trigonometry, cot A minus cot B is actually sine B minus A. Okay. Let me just write it down in an expanded way. Oh, sorry.   So why I decide to, you know, take up this particular identity because each of these terms here, for example, this term, you can write it as sine beta minus alpha by sine alpha sine beta. This will become sine gamma minus beta again, sine beta sine gamma.  Okay. So sine square terms will start getting cancelled. This gets cancelled. This gets cancelled. This gets cancelled. Okay. So basically, you're left with a product of three sine terms. Okay. You can, yeah. Okay. So you can write it as product of sine beta minus alphas. Okay. So this will always be less than or equal to one. It cannot exceed. It cannot exceed one. Okay. So option number one is correct. Or option number A is correct. Is it fine? Any questions? Any questions, any concerns? Next one. If A square plus B square plus C square is minus two, then the polynomial f of x is off which degree? Yes. Two and a half minutes gone. Last 30 seconds. Okay. Five, four, three, two, one. Okay. Take a call. Even if you're not able to solve it, take a call, take a guess. Okay. 10, 11 people have responded. Most of you say option number D. D means degree two. See here, first of all, in order to utilize this information, let's do this information, let's do this operation, add every column to the first column. So that clearly gives us, if I'm not mistaken, one plus take x common x square plus B square plus C square plus two. Okay. Yes, I know. Same here also. It will give you one plus, again, A square plus B square plus C square plus two. Okay. And one plus x again here, A square plus B square plus C square plus two. Okay. And let this be as it is. Be careful about the use of brackets here. Okay. Now, this is anyways going to be zero because you have been already provided that this is negative two. Okay. So this entire first column will reduce to 111. I'm making that change here itself. I don't want to write this again. Takes a lot of time to replicate the same data. Now you can use the fact that all the elements of one of the columns have become a one. So you can start doing this operation R1 as take any one of the rows. Let's say I take the third one and subtract it from all of them. Okay. So this will become a zero. This will become a zero. This will become, if I'm not mistaken, x minus one. Okay. Zero. This will also become a one minus x. Okay. And this also x minus one. Now expand with respect to the first column. So you will only get this into this and this into this, which is actually a second degree equation. Okay. So you'll end up getting negative of negative that is this whole square. So this is your function. So degree two, absolutely right. Option number D is correct. Any questions? Any questions? Any concerns? Yeah. Next question. X and Y are real numbers. Then this determinant lies in which interval? Almost two minutes. Okay. Last 45 seconds. Last 30 seconds. I can give max to max because already three minutes have gone. Five, three, two, one. Okay. A split between almost A and C options. Both of them have got almost equal number of votes. Okay. Let's check. See, let us do this operation here.  Cos x plus y is cos x cos y minus sin x sin y. Correct. Okay. Cos x is always sitting here. Sin x is always sitting here. Okay. So let me do this event. R3 is R3 minus minus cos y R1 and plus sin y R2. Okay. So what will this do to this expression? This will create a zero here. And thankfully, if you see this expression, if you multiply this with negative cos y, it will cancel out the negative sin x cos y here also. So it will create a zero here as well. Okay. Check it out. Okay. And this will create a sin y minus cos y here. If you want, you can expand it. If you think it is a shorter way to expand it and do it. But this directly gives you, if you expand with respect to R3, this will give you sin y minus cos y times plus minus plus minus plus. Yeah. Cos square minus sin square, which is minus minus sin square, which is one. So now if you have a function like this, the max value of this function is root two. The min value is negative root two. We have already done this in our trigonometry. So sin y minus cos y max value is root two. Min value is negative root two. So the interval in which your delta will lie is minus root two to root, which is clearly option number A. Is it fine? Any questions? Should we move on to the next one now? Yeah. If one plus A111 is equal to zero, A is not zero, B is not zero, C is not zero, then this is equal to. Pole is on. Can you all see the pole? Now it is on. There is a small lag in the appearance of the pole. Don't you think this should be 2C actually? I feel like this should be 2C. Oh no. That's fine. Okay. Last 30 seconds. Just four of you have responded. Five, four, three, two, one, six of you out of which, sorry, seven of you have responded out of which four of you say option C. Let's check whether C was correct or not. See, take A common from row number one, B common from row number two and C common from row number three. That will leave you with this expression. Yeah. Now do this activity R1 transform to this. Okay. So when you do that, you will end up seeing one by A, one by B, one by C plus three appearing in the first row. Correct me if I'm wrong. Okay. The rest of the elements will be as they are. Now take this guy, the first row guy common from the entire row number one. So this whole thing I'll take common out. Okay. So you'll have one by A, one by B, one by C plus three. Now, since you've created one, one, one in row number one, you can always go for the operation C1 is C1 minus C3, C2 is C2 minus C3, because we want to create enough number of zeros before we start expanding it. So this will be zero, zero, one, one, two, one by B minus two, minus two, three plus one by C. So when you expand this, you'll end up getting minus two plus four, which is two. So it makes it two ABC, one by A, one by B, one by C plus three. And this entire result is given to you as a zero. That means this whole thing is a zero. But ABC, oh, I'm so sorry. ABC are non-zero quantities, if I remember. So it is only possible to get a zero with these guys not being equal to zero when this is equal to zero. So one by A, one by B, one by C is negative three. Okay. So Janta, you were wrong. Option number B was correct. Most of you, I'll share the result again. Most of you went for option number C. Okay. Aditya, which option you got for this one? Okay, let's take some relation functions question. Simple one. May have seen this question a number of times while practicing relation functions of class 11th, maybe 12th also. Okay. Let's stop this in another 30 seconds. Five, three, two, one. Most of you have chosen A, but there is a strong conflict between A and D both. Okay. Now, first of all, is it a reflexive relation? Because yes, I feel that this will be a part of, sorry, this will be the part of the relation for all A belonging to real numbers. So it has to be reflexive. So reflexive, undoubtedly. What about symmetric? If this is belonging to the relation, then it obviously implies this will also be true, which means B comma A will also belong to that relation. So it means it has to be symmetric as well. It has to be symmetric as well. But is it transitive? That is something that we need to figure out. Now I would request all of you to think of an example of such number A, B, C. And I'm sure you can do it where A comma B, or you can say A is related to B by that relation. B is also related to C. But A is not related to C. Think of an example. Aditya, your A is also related to C only. Okay. Yes, a lot of examples are coming up, by the way. You can think of an endless number of examples. Okay. So one example, which I liked from Bever, he said, let A be minus three. So minus C is related to minus one because yes, one plus A B will be positive. Minus one will be related to half because one minus half is positive. But minus three will not be related to half because one minus three by two is negative. So it is clearly not transitive. So there is no, there is no scope of option D being correct. So option A only is correct. Okay. This anyways was ruled out because it was symmetric and it was reflexive. So A and D, the confusion was there. Now D is wrong. A is the right option. Is it fine? Any questions? Another easy question. I want the answer for this in 30 seconds. Poll is on. Simple. Okay. People have already responded. So let's stop. Five, four, three, two, one, go. Okay. F O F of this means you're first putting one minus root three into the function. When you're putting one minus root three into the function, the function is going to see any rational input coming. So I'll throw one at you. Okay. Now you're showing one which is a rational input. So this will give you a negative one. Option B is definitely right. Am I audible? Okay. Okay. Some issue with the net today. I don't know what is wrong. Anyways, we'll move on to the next question. If the function GF is defined and is one one, then which of the following option is correct? This is a theory based question. I think we had already discussed this in our functions chapter early in this year. Okay. Five, four, three, two, one, go. Oh, see the response everybody. Confusion confusion everywhere. Four people saying B, four people saying C, four people saying D. Okay. A quick recap of that theory, even though we're not going to the details of it, if there is a composite function, okay, and this happens to be one one. Okay. If this function is a one one, then this guy must be one one. This guy must be one one. Okay. And if a composite function is on two, then this guy must be on two. Okay. Just remember this theory. Very, very important. So if GF is one one, then F this guy has to be one one. So option number D is correct. Okay. So this question can be asked in either format. They can say FOG is one one. Okay. Then which of them should be one one? This guy inside function, the function which is fed inside. If that is not one one, there is no chance FOG can become one one. Because for two different inputs, G can give you the same output. So F will also give you the same output. Right. So it cannot be a one one in that case. Similarly, if FOG is on two, then F must be on two. That's very important. Could you give example where GF is one one, but G is not one one. GF is one one. G, O, F, this is one one. And you want okay. And you want G to be not one one. G is not one one. Not one one. Okay. What example we can have for this? No, Pradhin, his question is G, O, F. G, O, F is one one. Okay. And G is not one one. G, O, F, G, O, F is one one. Okay. Fractional part you want to use. You can make, you can make a function. I mean, not a, maybe not a well defined function, but you can just try to make a function. For example, let's say G is a function which follows this kind of a mapping. Okay. So let's say I just take some dummy numbers and let's say this is A, B, C. Okay. And let's say the mapping is F is a function. F is a function which gives you one and three. Well, let's say alpha beta. So in this case, in this case, if I do G of F of X, okay. So if I put an alpha G of F of alpha, it will become G of one. So that will give you an A. And if I put beta, it'll give me G of three, which will only give me a B. So this becomes a one one function. Will this example suffice? I mean, you need not always have an example coming from your daily life functions. What if beta gave two in F of X? Now, if beta were, if beta were giving you a two, let's say, then what will happen? This will become two. And G of two and G of one, both will give you A. Okay. So G of one and G of two will both give you A each. In that case, F of G will not remain a one one. Right? That's it. Okay. So I think you're trying to, you know, you're getting confused with here. If this is a one one, then F has to be one one. Please understand it. So I'm not trying to say if F is one one, then G of F will definitely be one one. I'm not trying to say that. No, that is not a, yeah. So Siddhij has another example. G of X is X square. And this is root X. Okay. X square root X that will give you a one one function and yes. Correct. So that's another example, which Siddhij figured out. He said X square for this and root X for this. Okay. So G of F is going to give you an X, which happens to be one one. Okay. But G of X happens to be X square. Okay. Now here also we have to be very careful. When you talk about F of G of X, your domain is decided by this guy. So X should be greater than equal to zero. Correct. So by definition of the function X square seems to be positive. I mean, not a one one function. So G, I didn't get that. What is this? G of F you wanted to write? Hmm. Technically, this is a one one function. And as per him, this is a many one function. See, I mean, you can always define, you can always define G to be from R to R. Let's say. Okay. And this function is like this. So in this case, it becomes a many one function. Okay. And he's defining F from R plus to R as root X. Okay. Now the question center asks you to find out what is this? So this function will be G of F will be X, but for X belonging to R plus. So the actual definition of G was from R to R. So it is a many one function. Correct. So this criteria is fulfilled. Whatever you are saying, it should be not one one, which is fulfilled. All right. So there could be so many examples. Let's move on with more problem solving. F of X is sin X plus cos X G of X is X square minus one then G of F of X is invertible in which domain. Okay. Let's close this in another 30 seconds. Okay. Five, four, three, two, one, go. Okay. Most of you have gone with option number B. Let's check whether B is right or not. See G of F of X assuming that G of F of X exists. So that will become sin plus cos square minus one, which is actually sin two X. If this function is invertible, right? It can only happen when two X lies between minus five by two to five by two, then only it will be one one. And of course you have to, you know, check for the arranged part of it. So X has to be between minus five by four to five by four. In fact, in reality, it could be any, an interval where the function is not allowed to take a turn back. So as per the options, I basically found out that this is the most exhaustive answer, which happens to be option number B. Okay. Next. Let's take some questions from simple question minus five by two to five by two will also work. Okay. Let's check. So this is your graph off. Okay. Sine two X. Okay. This is zero. This is five by two. And this is minus five by two. Right. How come you're saying it is going to work? It is not one one here. It is not one one here. Sine two X graph. You'll get almost zero marks if you do that. Yes. Yes. Should we discuss it now? Guys, this is again a simple question. You just have to find which region is the, see this line is obviously given to you two X, a four X minus two Y equal to minus three. Now just check origin. Does it satisfy this condition? A zero minus zero is less than equal to three. Does it satisfy it for that matter? Minus three. So this is not a part of your question also. Yeah. Does it satisfy this condition? Zero minus zero is less than equal to minus three. No. Right. You only have to look at the inequalities which are based on this line. Okay. Don't entertain A and C at all. B and D, one of them will be your answer. So does B, you know, entertain? No. Only D entertains that inequality. I'm so sorry. They're saying that a region which is shaded, right? So zero. Yeah. It should be the opposite. Yes. So once again, small. Yeah. So zero satisfies this inequality. D one. So this should not be your answer. So the answer should be B actually. Correct? D, right? The inequality which is not satisfied by zero, that is going to be your answer. Correct? So zero is less than equal to minus three. Correct? This inequality is not satisfied by zero. So B should be your answer. Right? So B is the right option. Yeah. B is only correct. B is only correct. Simple question. Okay. Let me solve the poll. I think most of you have got it. Okay. Five, four, three, two, one, go. Okay. So most of you, in fact, all of you have said option number B. Let's check it out. See, this is going to be a simple question. All you need to do is put these points in your objective function. Okay. So for this point, for zero comma eight, you end up getting minus 32. For four comma 10, you end up getting minus 28. For six comma eight, you will end up getting minus 14. For six comma five will get minus two. For five comma zero, you'll get 15. And for zero comma zero, you'll get a zero. Okay. Out of which the minimum is occurring at zero comma eight. This is your minimum value. Option B is correct. Okay. So these are all easy questions. Poll is on. Yes. Two people have responded so far. Okay. Five, two, one. Okay. End of poll. Most of you have said option number C. Most of you have said option number C. Okay. Now see, x plus y less than equal to seven, it has to be lesser than this line. Okay. And x plus two y is lesser than equal to 10. It has to be within this line. Okay. So obviously our desired zone is going to be desired zone is going to be this zone. What are the corner points? Zero, zero, zero, five, seven, zero. And this will be the meeting point of these two, which is going to be, if I'm not mistaken, four comma three. So we'll have to check what is the value of the objective function at these points. So p value at the points. So I think the minimum of them will come out to be zero, zero will give you a zero.  So this guy is the maximum of all 35 is the answer. Oh, Kiran. Okay. Most of you are going to lose marks because you will be in a hurry to just zip past the question. So read the question properly. That's very, very important. Okay. Let's take this question. The parameter on which the value of this determinant does not depend. You can do an expansion here because I think you should be getting technometric identities when you are expanding it at every stage. None. Just five of you have responded. Okay. Five, four, three, two, one, go. Okay. So seven of you responded out of which four of you say option B. All right. So let's discuss it. So if you expand with respect to, expand with respect to row number one, so it'll give you one, this into this minus this into this.  So one into this, okay, minus A.   So overall, if you see the first expression is just sine DX. Second expression is sine 2 DX and third expression is again sine DX. So there is no, there is no, you know, P seen anywhere. But yes, you can see A, you can see D and you can of course have X. So only it is independent of P. So it does not depend on P. Does this find any questions, any questions, any concerns? Last, find the value of K. I've already told you how to solve these kinds of problems. So it should not take you much time. Five, three, one. See to solve this kind of a question, I've already told you don't have to, you know, go beyond a simple act over here, just put your values of the ABC as something, for example, put something which will make one of the, you know, values easy to calculate. Don't put such a value which will make, you know, both sides become zero. So this will give you minus one, two, two, two, I think again a minus one, two, two, two, minus one. And this is going to be 8K. So expand it. So this will be one minus four, which is minus three. So minus two, minus two, minus four, which is minus six and plus two, four plus two, six. So that's going to be 20, how much? 1224, 24, 27. Have I done some calculation error? Just check. Minus two. This is minus two, minus four. Oh, sorry. This is 27K. So this is equal to 27K. So 27 is equal to 27K. K has to be one option number B. Is this fine? Any questions? Thank you. So with this, we come to an end. Tomorrow we'll have the last round of revision. Post that. We are going to meet only in December last week. Okay. Keep practicing. We'll meet again tomorrow. Thank you. Bye-bye. Good night.


I don't know about you guys, but I do find it odd that in our oh-so-modern world, it's actually out of the ordinary when something isn't wrong with you. I think everyone has some type of dysfunctional family. The only difference is how dysfunctional that family is. I mean, the only time I've ever seen somewhat happy people is the Amish, and they abide by a lot of these principles in their natural way of living. They also don't watch the news and they ignore the government. Now you can correlate various studies to each of these aspects, but we're going to give it simple layman's explanations. When you look at the brain especially, it's heavily impacted by environment and diet in development through all stages of life. This could be the need for omega fatty acids as both a baby and adult. We tend to blame everything on genetics. Use it as a blanket cover for all of our issues. Just like they blame you know what for unemployment, small businesses closing, the reason we're locked inside, when it's obviously deeper than that, far more to the story. I've worked very hard on past videos for all of these topics, so I will list them at the end. But in the meantime, if you guys could please sign up at frank-defano.com to stay in touch with my daily videos through the newsletter. My personal experiences related to this are definitely the driving factor behind my YouTube channel. When I look back on you know the last 10, 15 years of my life, all of that kind of goes towards making sure other people can be happier and really get things out of life that they deserve. So these are in order of what I deem as important through my personal experience. You know if you're still having problems and you haven't addressed all of these things then it's hard to not relay back to these initial ideas. And I made a video that might be very similar to this, I think it was titled The Six Basics of Health and those really do apply to mental health especially. So first we need to eliminate Wi-Fi and EMF, reducing the environmental electric and magnetic fields including radio frequency which is basically Wi-Fi. The amount of oxidative stress ultimately damage caused by unnaturally high levels of radiation in our modern environments far exceeds the damage that can be caused by any other poor lifestyle factors.  Somewhere free from these unnatural, modern, high frequency fields. Basically keep your cell phone on airplane mode as much as possible, check on it throughout the day, turn off all devices at night including your cell phone and make sure you aren't sleeping next to a smart meter or circuit breaker. I've done some videos the past week explaining this in depth and the Amish don't use any of these devices. Ironically the past few farms that I visited had new towers set up right next door so these poor Amish are getting fried by their local officials. Next up we have our gut microbiome and I think there have been more and more studies and news articles even hitting mainstream revealing the importance of our gut and how we feel. But I experienced this first hand through my health problems from iron overload these past two years to an extreme extent. So when you have any sort of bacterial problem in your stomach ideally you want to remove the bad and replace it with the good. Unfortunately with some of the issues I was having I had to take a lot of antimicrobials like clove oil and think of those as almost antibiotics. I'm sure some of you have taken antibiotics and had very, very specific feelings when you took those.  I was feeling awful, not motivated, just really horrible. In contrast when I stopped taking antimicrobials and consuming probiotic foods like kefir the opposite happened. I felt happy, motivated, excited and we know tribes in the past indigenous people were always happy around their food and there were several statements about these people eating fermented foods with joy you know literally smiling when stuffing their faces with rotten meat because it made them feel happy. There's a physical process happening in your body when you're consuming that live bacteria and this is much easier said than done but I do have a video summarizing the gut microbiome and not to say you have to take those extremes to fix this you know heavy antimicrobial protocols reincorporating probiotics you know it might just be a matter of eating slightly better foods. Eating sometimes referred to as earthing you know being grounded and electricians people that work with those types of devices might be familiar with like a grounding plug or a grounding rod and that's so when something gets electrocuted it goes back to the earth because if it doesn't go back to the earth whoever touches it is getting zapped really bad very dangerous. And this is something I've seen other people subtly mention without explaining I remember that fake natty vegan bodybuilder Nimai Delgado you know walking on a beach talking about how good he felt grounded you know people even say being in touch with nature but these people never actually explain what they're physically doing or why they're doing it. The earth is full of electrons which act as antioxidants in the body when you aren't barefoot or in natural material contact with the earth like a Native American wearing moccasins you aren't receiving those electrons so most people are inside homes all day wearing rubber sold shoes when out and about if you go out in the woods you know even in your backyard and just touch a tree you will actually feel a surge of energy because that tree is connected deep into the ground with its roots and people are so used to being ungrounded and you can't really ground indoors it depends on your neighborhood because there's a lot of like unnatural kind of Wi-Fi interrupting that as well but if you go out to a park or hiking with natural shoes that's ideal you know barefoot on the beach if you're near the ocean and even in some cases sitting on the ground in your backyard I think I'll have some grounding devices available for you guys soon but you can check out the video I did I think a year or two ago now vitamin D and not just vitamin D we're also referring to any nutrient imbalance access or deficiency when people are deficient in vitamin D3 in particular they tend to be low energy depressed and upon taking it even as a supplement they are sprung to life it can be difficult in our modern world to get adequate sun you know hours and hours and hours per week something like an omega fatty acid or B vitamin deficiency also leads to certain behaviors vegans are a good instance you know start going psycho with memory problems you could also have like a copper zinc imbalance you know almost everyone in the general population has some sort of dietary imbalance some degrees of deficiencies and excesses those extremes can definitely cause certain mental states overall you have to be conscious of your past diet and make sure to improve it by consuming higher quality animal and plant foods in the correct ratios so lastly I didn't really want to put exercise because I know how can be really tough for some people to even get out of bed some days but it is important to mention that exercise will physically make you feel good by producing antioxidants and there are a few things kind of relevant to this that can also be done you know if you are in a certain stage of your life where you don't want to exercise the overall point is to have something you look forward to that makes you happy or driven every day you know we mentioned the Amish not watching the news you want to disassociate yourself with the negative as much as possible you know it could be a hobby even something as simple as exercise just make sure you enjoy doing it and if you don't enjoy doing it you know maybe try listening to a podcast or watching a movie or something you do enjoy at the same time and if you're really in a tough spot and you have to do something you despise it all day you don't have to be negative you know you could absolutely hate what you're doing but you could think hey let me you know force myself to be happy because I know at the end of the day I'm gonna get to lay down in bed and rest and relax and for me when I was younger this was kind of video games because I would just you know play video games all day you know literally 10 12 15 hours a day I should have I should have stuck with it because those Twitch streamers are doing a lot better than me now but that's a whole different story so just try to whatever you can do something to make yourself feel better even if you have to kind of like get in your own head and just force yourself to be happy put a smile on your face and you'll definitely start seeing improvements in all aspects of your life so hopefully this helps some of you guys out and a lot of the time you know it really is things people knew but just needed a little reminder of and then they can keep implementing them and being consistent so if you guys could please you know like the video drop a comment down below as I mentioned earlier definitely sign up for my newsletter on Frank dash defano.com and if you do want to check anything else out all of my businesses are also available on that website I guess thank you guys for joining me today videos as usual every day so I will see you guys tomorrow


My name is Alitha Kirshe and I'm the Regional Director for Climate and Development Knowledge Network, CDKM, and I'm also Chief Executive Officer of LEED Pakistan. I facilitated a session on challenges faced by donor agencies in supporting and funding adaptation in different parts of the world. There are many messages dealing with increased need for donor cooperation or collaboration with each other, tracking investments in adaptation, making systems more transparent, more accountable, more efficient, generating more data, especially gender segregated data in order to ensure that we are reaching out to all the marginalized groups or affected groups. Then there was also a lot of discussion and discourse on such aspects of making it more inclusive, more pro-poor, somehow converting into a bottom-ups approach. I think the most important ask was that in funding adaptation, the donors need to be able to distinguish between adaptation and maladaptation. They need to somehow have mechanisms in place to have better understanding of these issues and communication with each other.


Welcome to the second tutorial of the first week of the deep learning course. So what happened last time? The first one, and hopefully the most important, is that we get to know one another. Welcome to the deep learning course. We get to think about the pram of playing a teller, because intuition is always what deep learning is built on. We thought about how game playing can be expressed, and we saw how one can use the deep learning system to estimate what value. And then lastly, we built simple game playing agents. All these are important pieces, they're necessary pieces, but they aren't enough if you want to think about deep learning systems that really are good at playing games. And I should have a slight apology here. There's going to be a long development towards one goal, which is the alpha zero situation that is atypical for this cause. But I thought it was important to have that in the first lecture here, because it really gives you the overall view of things. So what will we do today? We will start with the ideas from last time, and we'll push them forward. And we'll focus on ways of planning in the future, not just the value right now on the board, but thinking about the future. Now, there's always many futures, and in fact, the future has the structure of a tree. And that's where we'll introduce Monte Carlo Tree Search. And finally, we will arrive at alpha zero. The thing why this is so cool is it's in a way one of the coolest deep learning systems that exists in the world. And yet you will see that in a way it's rather simple. It's a relatively small number of ideas that when combined in just the right way, produces very strong gameplay. And we will have arrived there within a week with the hope that you will see the overall structure of what we are doing before thinking about all the details. But first, before we dive in for today, let us recreate what we were last time and make sure that we remember what we were talking about.


What's good Josh boros back again with another video to run talk about a Hall of Famer having some Disparaging words about LA night man This has been all over my social media And I want to talk about it because I thought it was very interesting What this Hall of Famer had to say so we're gonna get right into the article man I'm pretty sure you guys have heard about this So I definitely wanted to give my input on what was said So Kevin Nash had something to say about LA night and the only one that sees an absolute ripoff of The rock can be very interesting see exactly what he had to say because that definitely didn't sound positive We found one of those critics LA night was responding to in interviews last week On the latest episode of his click the podcast WWE Hall of Famer Kevin Nash and his co-host Sean Oliver discuss among many other things Who might follow Roman Reigns as wrestling's next big star the phrase the next rock or Stone Cold gets used?  He said yes stick it The only thing he didn't do was turn them up like sideways. Jesus Christ. What does everybody? What does everybody have amnesia? There's some there's nothing original.  Oh, man. His promo.  He's the fact that they said called him that big sex He's on his take on LA LA But I'll offer a couple pieces of context before we get into the pitchforks and torches Before we get out the pitchforks and torches. Nash has been known Times of fun on his pot even about subjects that he may that that he may be Shouldn't his mocking of night strikes me as a Steve Austin and I gave up Watching AEW for lint then AEW wrestlers need more reps is what I'm saying The opening stretch of this episode is a story about house Stone he Sean Waltman and Scott Hall got from an edible before taping a click theme episode Of WWE table for the three back in 2016 and that says he's hopped up on that liquid THC before launching into that story So see point numeral uno now feel free to talk to us about diesel squoats LA you can listen to them yourself here. Well, look, I think that's kind of a cop out to be honest with you It's really a cop out if that's how he feels. That's how we feel I'm not knocking him for his opinion, but I have to give some Criticism there first things first.  He sounds like a young rock. So I get that The infamous promo that he's talking about is the promo they had before money in the bank and him pretty much You know running down the competitors, especially Logan Paul ahead of money in the bank And I thought it was nice. You can say oh, yeah, he did the whole sticky sideways and and shove it You know I'm saying I mean, yeah, you can say that's kind of a rip off or what the rock used to say But it's not like he says that every week. That was just in that Moment in that situation and if you want to be honest here, we gotta be honest here Wrestling has always been known for people ripping off of other people gimmicks or taking inspiration from other people's gimmicks If you want to be if you want to be completely honest here Let's let's keep it a buck a lot of things that we've seen for our faith from our favorite wrestlers have came from other wrestlers before them I'm just be honest A lot of these catchphrases and sands and and how people their mannerisms They're usually inspired by somebody else. So if he's inspired by the rock what you can tell that's fine But he's still LA night is still his own person if we're going off of Kevin Nash's logic Then at the end of the day Everyone having this cool one-liner catchphrase that can be a derivative from something from someone else from something else Like the yes chance is derivative from the what chance? Him having that LA night. Yeah, the yeah chance that can be derivative from a what chance? Are we supposed to sit there and knock him for it? If it gets over Then why not? I'm sure people have compared him to the rock before but guess what if he's over. Does it matter?  So I Don't know. Maybe it gives me that. Oh I'm kind of stuck in my old ways I don't really see nothing special in this guy type attitude and to be honest with you Wrestling has always had that especially with backstage politicking There's been a lot of people that didn't feel like certain people, you know Should be there should be at the top of the card. It's always been like that and Is there's no there's no denying it that Kevin Nash has been one of those people that will voice his opinion on if he felt like someone was a big star or not The truth even when he was wrestling behind the scenes so Am I surprised by this by Kevin Nash? No, that's his personal fending. He doesn't see the hype. That's fine It's not meant for him to see the hype. It's meant for WWE and everyone else that Goes to these shows to see the hype So if we're going off of Kevin Nash's logic, then there's not really there's barely any original thoughts and ideas and Wrestling altogether Because if you want to be honest your favorite wrestler got inspiration or took something from somebody It made it their own So just because he's that set that shove it line, I don't know if I'm gonna You know ridicule LA night because of that. I'm not I don't know if I'm gonna go out LA night Oh, he's trying to be like the rock. No, he's trying to be like LA night So comment down below. Let me know how y'all feel about Kevin Nash's comments I think you know, he's entitled to his own opinion, but I think it becomes a Slippery slope if you were for going off of his logic of no one coming up with an original idea when most of wrestling Has been forged or some of the best moments and best characters in wrestling has been forged from somebody else's inspiration, I Don't know but let me know how y'all feel about that down below But I appreciate all the love and support you guys have shown on the channel go to 150k And I'm still young speedy you to wrestling champion world. Appreciate y'all kidding me. See y'all next one. Peace


Because everything tell you guys through Hollywood, through books, through music, through our entire culture is that white men are bad. White men, they're bad, they have to be pushed to the back, they shouldn't be listened to. But what that's done to your generation and even some of those little older than you and those definitely younger than you is it has created hopelessness in many of these guys. Hopelessness, they're lost, they grow up in a broken home or maybe just a really bad home. No one is there for them as they're growing up. They spend time hours and hours alone, which is what do they do with their time when they're alone? They don't have an after school job if they're in high school. Maybe they probably don't play a sport. So they're spending hours and hours time alone which turns them to all kinds of bad things. Porn on the Internet, reading crazy stuff in chat rooms and God knows who's in there and saying what they're saying. A lot of time playing video games and I know people don't like to demonize video games, but I do think that maybe has a role there for certain people.  OK. So I'm going to first have to preface this conversation with a citation needed.  In fact, I know I've never faced discrimination specifically because I'm white.   Mom, I'm just kidding. If you're watching this, by the way, I hate porn. It's disgusting. But how do you explain that?  You just make it up.  So what is, first of all, what is their definition of discrimination for white men? What are some examples that they provide us with? Now, there's no concrete examples, no anecdotes, but she does say one example, which is pop culture, I guess, priming people to think that white men are bad and pushing them behind. And really what it sounds like she's saying implicitly is diversity because we see less men and we see more women and people of color. Well, then that must suggest that white men are inherently bad.  If greater representation for other people feels like oppression to you that says more about you than the other people. But that's the only example that they provide us with here. Oh, well, yeah, men are getting pushed back in culture. OK, you're not you're not providing us with any examples here. How exactly is this anti-white message covertly being delivered to us? Is it subliminal? It just she says things and she doesn't think them through. It's just her entire view of the world is not, you know, driven by empiricism or facts or data.  Now, the other problem with this is that she suggests that there's something inherently wrong with porn and video games. I mean, everything is fine in moderation, of course. But when it comes to video games in particular, there is a study that was just released that proves gaming not only boosts brain activity, but it leads to better decision making skills. I mean, there's a plethora of benefits with gaming, not to mention it helps people to relax. That's why I love video gaming. It helps me to relax. You know, I enjoy it. But according to her, it's inherently bad.   Play one. Maybe you'll like it. Maybe you'll talk less and learn something from video games. Perhaps history, right? Perhaps, you know, you need these decision making skills. So maybe you should play some video games, Margie Taylor Green. I'd recommend Elden Ring, but I don't think that you'd be capable of beating that. Not a lot of people are. I did, though, but not to brag. Anyways, I get caught up on video games. She also said something that stood out to me. She said that another consequence of anti-white discrimination is reading crazy stuff in chat rooms. Oh, like QAnon. So look, what she's saying here, like if you are a young man who feels isolated, I think that you could plausibly become radicalized by some far right message board or, you know, subreddit or something of that nature. But she herself has become radicalized by message boards or Facebook groups, perhaps, because she's a QAnon or a former QAnon, but perhaps potentially a current QAnon. So I mean, I love how she says that that's a negative when she's been radicalized in the same exact way. If you go back to her videos before she was a member of Congress, she talked about how Q was a patriot. She was a she was an actual Kool-Aid drinker when it comes to Q. Now, this interview is really bizarre to me. I've got to point that out. And let me put on my tinfoil hat for a moment because the individual who interviewed her is an 18-year-old college student. And on his YouTube channel, he has 990 subscribers. And his very first video is a lengthy interview with Tucker Carlson. And just a couple of weeks later, he already scored an interview with a sitting member of Congress. That's kind of weird. So the question is what right wing billionaire decided to like wish this person into existence with a lot of money, who has the connections that got this person prominence when they came out of nowhere. That's that's one thing that I think we need to keep an eye on because right wing funders of these media outlets, they prop up these outlets. Like I'd love to know how well a Charlie Kirk or a Ben Shapiro would do without all this money behind them, because people on the left, we actually have to survive without the funding of billionaires, believe it or not. Most of us don't even do advertisements. But yet these right wing people, they come into existence out of nowhere. And they're interviewing members of Congress and prominent members of right wing media. And that's a little bit sus to me. Now, one other topic that they touched on pretty prominently throughout the course of this entire interview was Christian nationalism.  It's always something that's going to be entertaining to me. But what's really an issue is the promotion of Christian nationalism. And Marjorie Taylor Greene on numerous occasions has promoted Christian nationalism, and this is a dangerous, fascistic ideology. Now, I do want to address that because I think that's more important. I don't have another clip for you because I don't want to promote this right wing YouTuber who's new to the game too much. But I do want to draw your attention to somebody who knows about this. And that individual is Benjamin Dixon. Now, he published an article on his substack titled The Futurist Progressive that goes into great detail, explaining the dangers of Christian nationalism. And I can't read all of it because it's really in depth and long. But I do just want to share a couple of paragraphs because this is by far one of the most comprehensive pieces on Christian nationalism that I have ever read. And it directly ties Christian nationalism to white supremacy and the history of white supremacy. And it's very informative and I would highly encourage you to check this out. But just to give you a little bit of a taste, let me read a few paragraphs here. Benjamin Dixon writes, with no regard for the unconstitutionality of their political crusade, this coalition of Christian nationalists is executing with precision a 50 year agenda to make America Christian again and to gain the power to punish sins with the force of the law. The case of Nebraska teenager, Celeste Burgess and her mother, Jessica, who are currently awaiting trial for violating the state's ban on abortions is an ominous shadow of what is to come. Each faction of today's conservative movement has fallen in line with the theocratic agenda to dominate every aspect of American society on behalf of the God of Christian nationalism from the judicial activism of religious sell its on the Supreme Court to the violence of the January 6th insurrection, conservatives, whether loyal to Donald Trump or to the Republican Party, have all made clear their intentions to rule America despite democracy. God is simply a justification for their political ambitions. Christian nationalism has been for white men in America, what religion was for European rulers throughout Western civilization, a dependable tool of justification for the political ambition of powerful men.  It is in the name of Christ that hundreds of thousands were slaughtered throughout the crusades and the inquisitions. Religion was used to justify the enslavement of Africans in America. When black Christians prayed for deliverance from slave masters, they were praying to be delivered from the power of white Christian men. Our rational humans to follow this religious sect down a race to the bottom simply because disgruntled white Christian men feel entitled to control society, even without a majority. Are we to be guided by men who cannot live up to their own moral standards, but are so offended by addressing individuals by their preferred pronouns that they feel democracy must be overthrown and will stop there. I don't want to read too much because I want you to read the article for yourself. I'll link to it down below. But yes, this is all really, really important. And it speaks to what Marjorie Taylor Greene is trying to tackle here. I don't want to downplay this idea that men today, maybe perhaps they feel as if, you know, they're isolated, they're alone. Social media does that to people because this is a phenomenon that's happening, right? The rise of incels is something that I think is really a serious thing that we all need to take, take notice of. But to address this, that doesn't mean that you impose Christian nationalism or theocracy on this entire country. There are underlying factors that lead to this current state of young men. And it's not just young white men. It's a lot of people in this country who are suffering under our current neoliberal, late-stage capitalist regime. That's exactly what's driving a lot of this problem here. But for individuals who are very simple-minded, they think, well, you know, you know, I see that there's problems in this country. So therefore the solution obviously is Christian nationalism. The problems that we face are obviously the result of God being driven out of the public sphere. In fact, that's what the host who interviewed Marjorie Green said at the beginning of this interview. And Marjorie Green was just grinning along, loving it because she also agrees. You know, these individuals, they don't believe in applying a policy solution to public problems other than just saying, well, the solution must be God. Our education system is falling apart. That's because prayer was taken out of schools. What health care in this country is in shambles? Well, that's because we're not praying enough. I mean, the solution to everything for these people is more religion. And if we just have God intervene, their perception of God in particular intervene in every area, every facet of life, every institution. Well, then that's what's going to make things better. But that's delusional. Religion will only make things worse, especially this disgusting fascistic Christian nationalism.   And I'm always here to laugh at Marjorie Taylor Green.