 Let's get started with semester one problem solving and the first question that I would like you to start with is a relational function question. So here's the first question for you all. Question is if e has one, two, three, four and f has one, two, then the number of onto functions from e to f, there you go, all is on. Functions from e to f is one minute max, one and a half minutes that depends upon how many of you have responded to that particular question. No problem. Let's get started with problem solving. Okay. 10 people have responded. 30 more seconds. I can give. Sure. Okay. Five. Four. Three. Two. One. Go, go, go, go, everybody. Okay. Just take a guess. Okay. So 24 of you have responded to this question and maximum people have said option number eight. Okay. Now this question can be solved in a very simple way. First of all, the total number of functions. So if there is a function from e to f, okay, then the total number of functions you already know this will be number of elements in f to the power of number of elements in e. Okay. So in this case, it will be two to the power four, which is 16. But a function will not be onto if any one of the elements of f is left out. That means let us say I have a situation where all these elements one, two, three, four, either all map to one. Okay. So this will not be an onto function. Or they all map to two. This will also be not not an onto function. In fact, it will be an into function. So what is into function onto function onto means none of the elements of the core domain is left unmapped. That means every element of f is the range or is participating in the range. So these two scenarios will be ruled out. Okay. So subtract two from the result. The answer is going to be 14 option number. Okay. So if you just link it to your understanding of, let's say, permutation and combination concept. So let's say a has n number of elements. Okay. And B has let's say our number of elements. So basically what do you want to do you are distributing and distinct balls. So finding the number of onto functions is basically the number of ways ways to distribute to distribute and distinct objects. Among our different groups says that no blank group is allowed. No blank group is allowed. Okay. If you recall the number of ways to do this. We had derived in our permutation combination chapter is R to the power and minus RC one R minus one to the power and minus RC two R minus two to the power and plus RC three R minus three to the power and plus R게 the data till starts becoming zero in case you want to use a formula. I'm not saying this is a compulsion to use in case you want to use a formula. So as far our scenario, this is for this is two. So this is too to be powerful minus two C one. This is two minus one to the power of four. This all will be 0 0 I mean not required for us. So that will be 16 minus two, which is 14 again. In case you want to go for a formula. Any questions any concerns but however the fastest way to solve it is by realizing that total number of functions is 2 to the power 4 minus 2 will be because if all of them map to let's say one or all of them map to do that two cases will be removed. Okay. Next, f of x is gif x square gif x plus one minus three, then which of the following option is correct. relaunch the poll. By the way, they should have actually mentioned the core domain actually, because then only we'll be able to you know talk about okay let's say the core domain is all the numbers. Let me make this function R to R. Just 11 of you have responded one and a half minutes gone. Okay, let's me begin the countdown five, four, three, two, one. Okay. So if you see in this question, many of you don't read all the options. And most of you have gone for one of the options immediately looking at the first option see here. Most of you have said option number D, but I can see four of you have also said option number A. So this is a message to those four of you. Whenever you have like both and we correct know you should always give both options are tried. Okay, that's why it's very important to scan through all the options, even though you have figured out that one of them is correct. Initially itself. Now if you look at the second option number be it says the function is zero for infinity many values of x, which is correct actually. If you put a one, you get one plus two minus three, which is a zero. If you put a 1.1 also. So 1.1s gif is one so one square only one gif of 2.1 is a two. This will be a zero also. So correct. So 1.11 also it will be zero only so there will be so many such values. Okay, so there will be infinite team any values of x for which f of x will be zero. Okay, so that f of x equal to zero. So it is definitely a many one function. Right, so option B is correct. Option A partially is correct. Now is it an into function. It is not going to be an into function because all the expressions involved over here are integers. So this answer will always be an integer so range will always be integer. Okay, and because of that, many of the real values will not appear in the range and hence it is an into function. So A and B both are correct actually. And since you have to choose only one of the options, you would go for option number D. So D is absolutely right. Well done. Can we move on to the next one? Next. All starts now. Set of all values of P for which this function is bijective. R to R, we can consider one minute gone. Okay, last 15 seconds. Five, four, three, two, one. Go. Okay. Good response. I could see one of the options which is option number B getting most number of votes. 85% of you say option B. Now this function, of course, would be a continuous function because it is made up of a polynomial and a trick function. Both of them are continuous. So if you're, if your inputs are from R to R, your output will also be from output will also be taking care of all the real numbers. So it has to be an onto function undoubtedly. Now for it to be a bijective function, it has to be one one. So how do we ensure it's a one one function? One one functions are monotonic in nature. So if you see the derivative of the function should either be greater than equal to zero or less than equal to zero cannot be both. Okay. So if you want it to be completely greater than zero, that means P plus cost X should be greater than equal to zero. In such cases. Now remember, this is for all X. So in such cases. Now remember, this is for all X. So in such cases, your P should be a quantity, P should be a quantity greater than equal to one. Can I say that? Okay, because if cost X becomes negative one, P should be such that it is able to make that positive still. So it should be greater than one. Right. Or, or means union. Your P plus cost X should be less than equal to zero. In this case, if your cost X becomes a plus one, P should be such that it brings everything to the negative side. So P could be lesser than minus one. So the union of these two, the union of these two, so either P should be from minus infinity to minus one, or it should be from one to infinity. Any questions? Any questions, any concerns? Okay, so option number B is right. Next, if a function is defined from R to S as sine X minus root three cost X plus one, and this is on to then suggest which of the option could be your interval S. All starts now. One minute gone, half a minute more I can give you five. Four, three, two, one. Okay, most of you have said option number A. Okay, let's discuss this out. See, as you can already see here, we can convert this to a harmonic form. So for harmonic form, you multiply and divide with the two. Okay, something like this. And this is clearly the formula of sine X minus pi by three. Okay. Now, for it to be an onto function, S should be the range of the function. So this should be the range of the given function and range of this function is going to be max value is going to be three minimum value is going to be minus one. So minus one to three will be your answer for the S value. So option number A is correct. Is it fine? Any questions here? Good. Next one. My mistake. Taking some questions. Can it be BNC as well? Okay, let me just check. Okay, now see, yeah, that's a very good question that you asked. It could be BNC also, but normally what happens is we always should mark the most exhaustive options. Okay, you should always go for the most exhaustive option. Even though you think multiple options could be correct in those situations, which might happen because of the question setting glitch, but you should always choose the option which is the most exhaustive. If BNC, F won't be a function. Once again, F won't be a function. Why? Okay, there could be some real values for which you don't get. Ah, yeah, there could be some values for which, yes, yes, yes. But it may happen that it could not be a function. Correct. Okay, anyways, I'll put the poll on. I'll put the poll on for this. Just give me a second. Re-launch. Yeah, find the domain of this function. Simple question. Everybody should get this. My coincident may be Prakul. Some of the questions may seem familiar, but not all the questions. There is some person who is constantly marking a wrong option. If that is a sadistic pleasure, please let me tell you, it doesn't affect me. But if it is a conceptual error that you are making, then sign to learn from that mistake. Okay, let me stop the poll. I think most of you have already voted. Anybody who wants to vote can vote in the next five seconds. Five, four, three, two, one. Okay, as you can see, option number A has got almost 89% votes, but some person has marked B and D as well. Okay, let's check. So this is a pretty simple question for the domain of the function. Sign inverse. Okay, let's write it like this under root of this should be between minus one to one, but you know that under root quantity is always a positive quantity. So this boils down to this. Okay, which means X minus one should lie between zero to one. That means X should lie between one to two. Option number A is correct. Is it fine? Any questions? Any questions? Any concerns? Can I go to the next one? There are some faulty questions, I think. My mistake. Just give me a second. Yeah. Let's take this one. Find the domain of this function. Cost inverse log of X square plus five X plus eight to the base of two. Re-launch. Last 20 seconds. What for cool? You went and told that the question that was given was again J level question. Sometimes you find the question is too easy. Sometimes it's a question that is too hard. Anpanma ma'am was telling that you went and tell it was J level questions that I'd asked. So what do you're oscillating? Okay, we'll solve the poll now. Five, four, three, two, one. Go. Okay, most of you have got with option number B. See, paper will be set as per the requirement of the boarding paper. You should not be judging the paper. It is easy, difficult. If it is easy, score. If it is easy, get 80 on 80. Who is stopping you? Okay. Please do not pass on expert comments on board level papers. Okay, this is easy paper. If it is difficult, everybody will struggle. But normally papers are not made difficult in the board exam. Okay. They are, they are meant for being solved by 99% of the people. Okay. So log of X square plus five X plus eight. This should be between minus one to one. That means two to the power minus one. You can write it like this. Okay. That means half. Now this one, this basically gives you two inequalities. This and this. Okay. But if you see X square plus five X plus eight, if you write it, it'll be X plus five by two, the whole square. And I think it'll give you 25 by four, 25 by four. And to make it an eight, I need a seven by four. Okay. Now this is anyways greater than half. Okay. This is anyways greater than half. So this will be true for all real numbers. Seven by four is more than half. Isn't it? So seven by four is of slightly less than two. Maybe. Okay. So you have to take care of the limiting. I mean, the restriction that it should be less than two. That means X square plus five X plus six should be less than equal to zero. Which means this should be less than equal to zero. In short, in short, in short, your X should be between minus three to minus. Sorry, minus three to minus two close interval, which is option number be clearly. Is it fine? Any questions? Any questions, any concerns? So normally we take the overlapping situation for these two situations. So whatever you get from here and whatever you get from here, we take the overlapping of the two, because it's a case of a simultaneous solving of those two inequalities. Next. Let's take this one. All is on. Easy question. Boss off something is half. And that something is to tan in 06 find the value of X. It's less than a minute question. I will not give you more than a minute. Okay. Five, four, three, two, one. Okay. Okay. Most of you have gone with option number B, almost 83% of the class. That's absolutely right. So here to tan inverse X will be actually pi by three. That means tan inverse X is pi by six. That means X is tan pi by six, which is one by root three option number B is right. Okay. Some easy questions also you can get. Don't worry. Is it fine? Any questions, any concerns? All right. Let's take this question. Choose the incorrect statement. Choose the incorrect statement. Cost range wasn't given. What do you mean by that? Cost range was not given. Let me go back. Give me a second. Okay. After this question, we'll go back. Okay. Last 15 seconds. Five, four, three, two, one. Okay. Most of you have again gone with option C. C is the right option because C. This is, this is definitely a scalar matrix of order one. So this is a correct statement. So this is true. This is also a scalar matrix of order two. Correct. This is also a scalar matrix of order three. But the question says it is this is order three is not a scalar matrix. So this is the incorrect statement you have to find out. So this is your right answer. Okay. So moving back to the previous question. I think Siddhant had a small doubt with respect to that. Or you figured out your mistake. You have to act according to the options. If there's an element of doubt, whether there is a multiple answers, then you have to act as per the requirement of the options. Next one is a square matrix says that a square is equal to a, then I minus a cube plus a is equal to which of the following. Awesome. There are people who are actually answering this. Well, like in, you know, 10 seconds, five seconds like that. That's a good speed. Oh, this was asked for the test. No wonder why. 36 seconds and almost 12 people have responded. Okay. Can we close this in another 15 seconds? Three. One. Okay. Okay. End of poll. Option A has got almost 83% votes and then option number C. See a square is a. I didn't put in matrix. I've already told you face square is a cube will also be a four will also be and so on and so forth. Isn't it? So now this is in this case you can apply a binomial expansion because I and a operations are commutative. IA AI are both commutative. So that's per the expansion. This will be minus three a then you will have plus three a and minus a cube that will also be a. So this plus a. So this, this cancels this, this cancels leaving you with an I option number. Is it fine? Any questions? Next. So it's a rapid fire round. I was doing. Yeah. This question done. This question. A and B are of same order. Then a B transpose minus be a transpose. This is a. 20 second question. I want immediate response on everybody. Correct. This is from the property itself that we had done. Five. Four. Correct. Three. Two. One. Go. Okay. 21 of you out of 24 say option number a. That's absolutely right. We had already seen this that if you have the transpose of this. This follows a B transpose whole transpose minus be a transpose whole transpose law of reversal will be applied. So it'll be B transpose transpose a transpose. This will be a transpose transpose B transpose. So it's B a transpose minus a B transpose. Which is negative of our given matrix. Okay. And hence, let's say if you had this as a matrix X, X transpose became minus X. So it's a skew symmetric matrix. Is it fine. Let's move on to the next question. I'm going to do down. Let's do this. So there's a product of three matrices. This into a into this is I. Then a is. Which of the following. All is one minute gone. I can give you one more minute. Five. Four. Three. Two. One. Okay. So this is one such question where I have not got enough responses on the 16 of you have voted and most of you say option number eight. See in this question, most of you approach the problem by choosing your A to be some ABCD. Okay. I'll carry on. Okay. But let's say I write it as B AC is equal to D. And I want to know what is my A. So to remove this B, I have to pre factor multiply both sides with B inverse. So this will become an I. Okay. And to remove this C, I have to multiply with C inverse on both the sides. Okay. So this will become an I now. So A will be nothing but B inverse. DC inverse. Now finding inverse of a matrix is very, very easy, especially when the matrix is a two by two. So B was this two, one, three, two. Okay. So what is the inverse of this matrix? First of all, one by determinant will be one only. Adjoint will be just switch the position of the leading diagonal and change the sign. Okay. So nearly what will be C inverse? C inverse. The determinant is minus one because nine minus 10 is minus one. Again, switch the position and reverse the sign of these elements. Right. So the answer to your question would be two minus one minus three, two multiplied with one, zero, zero, one. Okay. Anyways, you realize that anyways, this is an identity matrix. This is an identity matrix. Correct. So this will be as good as this is not into existence at all. Okay. Multiplied with C inverse. C inverse is nothing but three, two, five, three. Okay. So it's six minus five, one, four minus three, one minus nine plus 10, one minus six plus six, zero, option number is correct. Is it fine? Now it all depends upon how convenient you are with whatever way you are taking. I mean, I'm not imposing this, but this could be one of the methodologies you can adopt to solve the question. Can I go to the next one now? Any questions here? Anybody? All right. Next one. Okay. Let's take this question. Poll is on. Two minutes gone. I can give 30 more seconds. Five, four, three, two, one. Yes. Right. Aditya. So most of you have said option number a, but the number of people who have responded is quite less. Only 15 people have responded. Okay. Let's check this. So when you do a square, let's see what do we get? Because as you can see all the expressions here are actually powers of two only a square, a four, a six, et cetera. So let's see what do we get when you, when you write a square. Okay. So this multiplied with this, I'll give you a one. This multiplies with this. That will give you a zero. This with this will again give you a zero. This with this will again give you a zero. This with this will give you a one. This with this will give you a zero. This will, this will give you a zero again. This with this will give you zero. Sorry. This with this will give you a zero B minus B zero again. This with this will give you a one. Oh, so you get an item. That means a to the power four is also I a to the power 6 also is i a to the power 8 also is i and so on and so forth isn't it. So if you see here this is i 2 i 4 i so all together this sum should be equal to 7 i and whichever option says 7 i directly or indirectly will be your answer which is clearly option number a can't be b can't be c can't be d okay a is the right option is it fine any questions here can I move on to the next one next one is this is the sum of symmetric matrix b and skew symmetric matrix c what is b in this case again within 30 seconds I need the answer see some of you are answering within 15-20 seconds also means it is easy it is worth doing in 15-20 seconds yeah this this option has been very you can say designed in a very foolish way I would say okay five four three two one go okay most of you have said option a absolutely right so the only one option which is actually a symmetric matrix is a right these are all they are not symmetric matrices right but if you want to solve this question in a faster way I have already told you the trick just copy the leading diagonal as it is six to one then add these two divide by two and distribute it like this so take a mean and put it in the two positions whose mean you have taken okay similarly 9 plus 5 14 mean is 7 7 okay then 7 and 3 mean is 5 5 so option number a is correct in case of skew symmetric what do you do let's say the question was what is the skew symmetric matrix let's just see then what will you do copy 0 0 0 in the leading diagonal I'm just telling you how to save time okay please please take this into your consideration then subtract these two okay and divide it by two so this will be two so eight minus four now which one to be taken as eight it is always the the one in the row position this to be taken minus this so you do a i j minus a j i by two and this is to be written in this position here it will come negative of it please understand okay similarly 5 minus 9 by 2 which is minus 2 and there will be a negative of this coming in this position getting my point similarly 3 minus 7 by 2 that will come here and negative of this will come over here is this fine how to get the skew symmetric matrix also fast if a is this 10 b is this and b is the inverse of a fine alpha simple question should not take more than a minute okay five three two one okay end of poll only 11 people got this right I mean they are saying the same option and I'm assuming that since most of you have replied to that it is right so b is the inverse of a so can I say if I do a and times 10 b that should give you 10 i because it is as good as this right this is right so you multiply it and equate it to 10 i 10 i is this okay but see don't have to you know worry about all the elements see where is alpha involved so alpha will be involved if I multiply the first row with the last column and I could equate it to zero so two minus alpha plus three is zero alpha is five problem is sort question number option number a is correct okay don't try to equate everything unnecessarily you just have one variable to figure it out okay and most of you would be like sitting and you know getting four plus five plus one 10 so how does it matter 10 equal to 10 fine does it help you to get alpha no so why unnecessarily you know find all those things see which element in which multiplication alpha will be involved so you can either take you know first row and the last column and compare it with this fellow that will that will basically feature in alpha and if it feature in alpha you can find the value next okay let's take this one b is a non-singular matrix a is a square matrix then determinant of b a b inverse simple touch and go should we start the countdown five four three two one go okay end of poll most of you have said option c which is definitely right in this case see basically it works on two important things determinant a b is determinant a determinant b provided both a and b are square matrices and determinant of a inverse is determinant a whole inverse okay so if you apply this so determinant of I mean why am I writing dt I could have written this simple so this could be determinant of b inverse determinant of a determinant of b and this is one by determinant of b which basically cancels off with this giving you determinant a as the answer which is option number option number c next question which of the following is the inverse of this given matrix max I can give you one and a half minutes for this so a good practice of finding inverse also okay five four three two one okay so in such kind of questions go for you know the scamming part here most of you have said option number c see first of all if you check multiply multiply the first row with the first column over here you'll get a three minus eight plus six that is one will this give you the same check three plus 12 plus 10 this doesn't give you that so everywhere where there is this thing involved that will be rejected so b and d are gone for a toss okay but now if you see option option a and c both have the first thing common correct the last also common so just go for this multiplication okay with this one so this row with this column is it giving us a zero let's check so nine minus 12 plus 14 this is not giving us a zero so this cannot be my option is it fine so only option c is correct don't sit and evaluate it completely even if you want to please don't do that how did you eliminate b and d so this multiplied with this did not give me a zero oh sorry did not give me a one i'm sorry did not give me a one got it hurry up why should it because they are inverses of each other so their product should give you a one zero zero zero one zero zero one a b is equal to i know so let's say i take this as an a this is my b okay this should give me i by the way just to revise this fact everybody knows how to find it inverse by using the shortcut method which i discussed with you i'll repeat it once again just for the benefit of uh everybody and revising the theory so the approach was we used to copy the first row and the second row and then copy the first column and second column okay score this off score this off okay so criss-cross criss-cross criss-cross criss-cross criss-cross criss-cross this cross this cross okay so one minus zero write it like this okay then minus two plus four minus two plus four is going to be two write it in this position similarly minus two write it in this position then again two write it in this position five write it in this position minus four write it here uh two plus one three write it in this position and four plus three seven write it in this position and three minus eight which is minus five write it in this position so this becomes your adjoint okay so anything proportional to this anything proportional to this that will be your answer but by the way this thing is exactly matching with this so don't even go for finding the determinant because your inverse will be this into some proportionality constant which is your one by determinant okay is this fine so this is your adjoint of A so A inverse is one by determinant adjoint of A everybody knows that I think determinant will come out to be one in this case can we go to the next question now next poll is on if A is a non-singular matrix of order three and determinant of the adjoint is determinant to the power n find the value of n I can give you maximum 15 more seconds super easy question two one okay 19 of you have said option number A which is definitely right so please make a note of this again a quick revision of this part determinant of adjoint of A is determinant A to the power of n minus one n being the order of that given square matrix if you have determinant of adjoint of adjoint of A it becomes determinant A to the power n minus one square and this can be extended that means if you have adjoint m number of times okay so there are m number of adjoints then the answer becomes determinant A to the power of n minus one only n minus one has to be raised to a power of m okay this is the most generic formula so it can be applied here also it can be applied here also so this is the most generic formula can I go to the next question which one of the following statement is true is it five four three two one go should this take the best call take the best take the best option possible okay so I can see 20 people are circling okay 23 of you have voted out of this 17 have gone for option C see first of all a non-singular matrix doesn't have a unique inverse a non-singular matrix will always the inverse is always unique so this is a false statement determinant of a non-singular matrix is zero no it is for a singular matrix it is zero so wrong statement if A transpose is equal to A then A is a symmetric matrix and symmetric matrix are supposed to be square matrices okay so this is to a certain extent correct and the last one you already know that determinant of A into a joint of A would be nothing but determinant of A into determinant of a joint of A and that is nothing but determinant A into determinant A to the power n minus one so that will give you determinant A to the power n actually not this so option C is the best possible option is it fine any questions any questions any concerns do let me know C is not correct if A transpose is A so let's say m by n transpose if you say they are equal that means m and n should be equal so it becomes square matrix no what is the issue in that then see I'm not saying if A is a square matrix then A transpose is equal to A so it's a it's a unidirectional logic it's an implication if this is true then A is a square matrix is it correct or not whether that square matrix is symmetric skew symmetric how does it matter to us why are we reading the converse of it read the read this mathematical statement if p then q why are you reading if q then p next time to take some determinant problems also if p lambda to the power four plus q lambda to the power three is this find the value of t find the value of t simple question poll is on I should get this answer within 35 40 seconds not more than that oh 30 seconds gone only one answer I got so relation functions inverse trig matrices determinants continuity differentiability I mean including differentiation and application of derivatives and linear programming problem that is all you have for cvc people anything that I have missed out yeah application derivatives are counted okay yes should I start the countdown very few people have answered five four okay which is a three two one go okay now why less people responded to this is the reason why is the reason because many of you went for actual evaluation of the determinant you could just put lambda as zero on both the sites that will leave you with a t so those who got it fast it is because you had put lambda as zero in the in this equation so zero minus one three one two minus four minus three four zero you just had to evaluate this determinant that is going to be your answer so it will be one minus 12 and plus three four plus six ten so it's going to be 30 minus 12 which is 18 option number b is correct is it fine any questions here next I think I have missed some good questions we'll take them up first yeah we'll take this one okay a inverse is given to you which of the following option is correct do you think it's a multiple option I don't think so it should be a multiple option but if it is do let me know okay some of you are giving me multiple options okay fine if you feel it's a multiple option then give me a response on the chat box okay 20 more seconds I'll close the poll five four three okay Aryan dashmika three two one go okay great so end of poll b is the most you know chosen response let's check whether b was correct or there's a multiple option correct so first of all what is determinant of a inverse this you can expand with respect to the last column so minus two minus two plus one which is minus two plus one which is minus one so determinant of a inverse is two which means determinant of a is half okay and determinant of a and a transpose both are same so this cannot be correct this cannot be correct fine now determinant of a joint of a would be determinant a to the power of n minus one which is one by four this cannot be correct right so that leaves you with option number b but is b also correct let's let's verify it so a inverse is one by determinant a into a joint of a okay so you just have to multiply determinant a to this given matrix and that will give you the adjoint of a so half multiplied to this matrix half multiplied to this matrix definitely gives you this okay so this is okay option number b is correct but really surprised to see most of you got it wrong not most of you means the response was not that great on this yeah in fact most of you no not most of you which is seven of you got it wrong out of 16 next something wrong with my something is wrong with okay so if this determinant is equal to this expression find c value find the value of c full is oh one minute gone and one person only responded slow in this question see it doesn't mean that it's a school level question you'll use your school level concepts only to solve it there is this is an objective paper nobody's going to question you in what approach you took so you can use whatever you know j concepts or engineering intense exam concepts you have learned to solve this question nobody's going to ask you how did you solve it okay i'll be stopping this in another 30 seconds because the response is very less on only eight of you have responded five three one okay so just 16 of you have voted out of which nine say option number d see let's say this is some function of x can i say we see value will be f dash zero can i say that that means if you differentiate this determinant and put x as zero that will be a c value okay so differentiation happens differentiate differentiation happens one row at a time okay sorry slip off my pen yeah so differentiate the first row first so you'll get one five zero and another you have to put zero so minus one minus one eight zero zero zero so this anyways goes for a toss now first so put a zero minus one zero seven differentiate the second row and put a zero so two x will be zero this will be one this will be zero and this will be zero zero zero gone for a toss then third row differentiate you'll get two three zero and zero zero here will give you minus one zero seven minus one minus one eight so all you need to do is evaluate this determinant and your answer will be there let's choose our last row to evaluate it so plus minus plus so two this will be seven and minus three so this will be minus eight plus seven one have i missed out anything you let me know or minus one sorry yeah so 14 plus three 17 is the answer option number d is there how you decided see this is the f of x expression right okay so if I want to just find out c let's say if I differentiate it once I'll be left with three x three x square two b x plus c and I want these two to vanish so I'll put x as zero that's how we get c isn't it had I wanted only d I would have put x as zero something like that so you have to see you have to see which approach is giving you the faster result rather than manually expanding it of course manually expanding will also give you the answer but if you are you know let's say having a time crunch why to do all these things no but why that means you have not practiced enough determinant questions where the differentiation is involved so this kind of concepts are very well known to get these coefficients derivative putting some special values all those stuff three means I didn't get you what is this three what is that three prequel what are the three determinants oh you you didn't attain the determinants class on differentiation differentiation of determinants go do one thing not ask for you go and watch that video if you're absent for a class then the teacher can't help it okay so this is the question which says the value of lambda says that the given system of equation has no solution all is on done only four people have responded one minute gone okay five four three one two minutes gone only 12 people voted out of which seven say option number D solving the equation part is there no for your syllabus CBC I think for ISC it is not there but CBC is there right yeah it's there yeah so the first first in the foremost thing that you would be writing is that this determinant should be zero okay now determinant equal to zero could also lead to infinitely many solutions but this is a necessary condition okay so if you just evaluate it you get minus two lambda plus one lambda plus one and four times one plus two which is three which is zero so minus four lambda plus two plus lambda plus one plus 12 is equal to zero so you end up getting three lambda is equal to 15 so lambda is five lambda is five but that is not in the option okay let me check if I missed out anything so minus one lambda plus one and this is oh this is minus four okay last type of mistake that anybody should do okay so this is three lambda is equal to minus nine then sorry about that so lambda is going to be minus three so I was lucky that there was no five options sitting there else I would have gone so this is the necessary condition so hence okay so this may also give you infinitely many solutions but the necessary condition has to be satisfied which is coming from lambda equal to minus three so last question on determinants then we'll move on to aod and sorry continuity and differentiability yeah the value of this determinant is which of the following poll is on simple should be within like 10 second question people who could take my guess 10 second have already started giving the answer five seconds okay last 15 seconds I can give you so those who want to poll please do so five four three two one go so 18 of you only responded out of which 14 say option number d d is absolutely right it's zero actually because if you add r1 as r1 plus r3 you'll end up getting 2x 2y 2z okay and just take a two common from the first row if you take a two common from the first row you'll end up getting the first and the second row identical and that will make the determinant collapse option number d is correct or put x yeah correct put x equal to y equal to z because none of the options here will basically cater to the fact let's say each of them are one one each so only option d will cater to the requirement maybe we can take one of these properties seems to be good who knows are you not this one somehow my system is reacting very slow today see I'm pressing the down button it is not moving only now I got the point maybe the bluetooth of my keyboard is far off technical issues one second poll is on bluetooth has gone very far off yes easy question five second question just to make you revise the concept it is like no I'm not trying to test you here so a is a square matrix of order four i is a identity sorry i use a unit matrix then which of the following is true okay five four three two one go okay hundred percent vote has gone to option number b so we already know this property that determinant of k a is k to the power n determinant of it n being the order of the matrix so this cannot be your answer this can be your answer and this is minus one to the power of one so this is actually an order four so minus one to the power n means minus one to the power four this cannot be your option and there's no such property like this determinant a plus i is determinant a plus i so option number b is only correct see now my keyboard is working I think you can give for there was one property based question yeah this n b are invertible matrices invertible square matrices of the same order which of the following is not correct that means wrong option you have to pick up this also is very simple four three adit that is conjugate ab conjugate is a conjugate that's all obey law of reversal whether it is a transpose of conjugate whether it is inverse whether it is transpose whether it's adjoint okay all right so I'll be stopping the poll now most of you have got it right option c is the wrong statement and hence the right option so law of reversal is valid for adjoint also okay and adjoint a transpose and adjoint of a whole transpose will give you the same so that means this is also a true statement but this should be actually n minus one not to n minus one okay so this is a wrong statement and here see adjoint of a is adjoint of b is what adjoint of b is determinant b into b inverse yes or no so when you put adjoint b in place of b what do you get determinant of adjoint b into adjoint b inverse adjoint of b inverse now adjoint of b is determinant b to the power of n minus one okay what about adjoint b inverse how do you find that out so adjoint of b is determinant b into b inverse so adjoint of b whole inverse is going to be this correct so you are going to put determinant b to the power of minus one times b which is determinant b to the power n minus two into b however i'd given you a direct formula also for this when i was you know giving you the property so this is also a true statement now don't be happy that you figured out c if you were not aware of either of b c or d because you never know a question could have been framed on those as well so let's take it as a learning whichever part we were not aware of how is first option a true law of reversal is applicable to adjoint of this thing so okay see are you convinced that ab inverse is b inverse a inverse are you can are you aware of this property at least okay now inverse of a matrix is one by determinant ab into adjoint of ab similarly this will be one by determinant of b adjoint of b one by determinant a adjoint of a and if n b are square matrices then this term will be adjoint determinant a into determinant b and since they are invertible they are non-zero they can be cancelled off and that's how you get adjoint ab is adjoint of b into adjoint of it law of reversal holds value all right now we'll take i mean now we'll go to continuity and differentiability enough of matrices and determinants let's go to the poll is on okay almost one minute gone just seven of you okay five four three two one go right end of poll most of you have gone with option number c it is continuous for all x but not differentiable at zero okay let's check it out so let us redefine this function first of all so when x is negative this will be x e to the power when x is negative this just becomes a zero so it is just an x okay when x is equal to zero they have already given it is zero and when x is positive it will be x e to the power negative two by x okay now is it continuous let us check it out at uh see left hand limit at zero and right hand limit at zero both both are equal to zero each am i right why because left hand limit anyways you put zero in this okay right hand limit if you see this quantity this quantity as x tends to zero plus this is a negative infinity quantity so this anyway is tending to zero okay and x is also tending to zero so overall this entire expression will be tending to zero so continuous everywhere that is fine so option number a and d are ruled out now let us check the differentiability issue so left hand derivative is one undoubtedly what is the right hand derivative so for right hand derivative i will follow the first principles here because i feel more convenient with that so it will be zero plus h minus f of zero by h so f of zero plus h will be h e to the power minus two by h minus zero by h limit h tending to zero plus this h and this h goes and again as i told you when h tends to zero plus this is minus infinity so e to e to the power a very large negative number will be zero so left hand derivative is one but right hand derivative is zero so it is not differentiable at zero so it is continuous for all x but not differentiable at zero very good answer c is the option c is the right option is it fine any questions here anybody next isc people i just spoke to the resource people they are telling that the best of the two section will be taken into account so i think somebody was asking me whether i can attempt let's say one of the section properly and take a guesswork in the other section you can do that you can do that okay i'll give you 20 more seconds please wrap this up in the 20 more seconds five one okay d is the most chosen response let's see you can see here that the question checker basically is interested in knowing in the neighborhood of x equal to one point okay so let's redefine this function when x is less than one when x is equal to one and when x is greater than one less than one you know gif is going to become a zero i'm just talking in the neighborhood of one okay don't say sir less than zero can be minus hundred also right so i'm just interested in neighborhood of one don't go and redefine it as minus hundred and minus five lakh not required in the neighborhood of one that means slightly left and right of one i'm interested in okay so when x is slightly less than one this is going to be minus one by x minus one and when x is slightly more than one this is going to be zero is it is it continuous at x equal to zero if you see the left hand limit is one right hand limit is zero value of the function is zero so it is discontinuous at x equal to zero sorry x equal to one so if it is discontinuous at x equal to one there is no point doing any further check because it is going to be non-differentiable also because it is discontinuous option number d was correct everybody should have got this right is it fine any question any concerns here so only in the neighborhood of x equal to one you will make this analysis don't start defining it from from very far off intervals is this fine let's go on to the next one simple question but conceptual question if you're making mistake here means conceptually something is flawed okay people have answered this already x is f of t so there's a curve which is parametrically defined you have to find d2 y by dx okay i'll stop this in another 15 seconds oh there are people who are opting for option a also Ramakrishna five four three two one go okay thankfully most of you have gone for the right option which is option number b in this case so please understand double derivative doesn't mean double derivative of y with respect to t by double see get these wrong notions out of your mind it's high time you start you know realizing that this doesn't work like this there is no such so please do not do this so double derivative means what double derivative means you have to first differentiate first find dy by dx and from there you have to differentiate it once more again with respect to x okay so dy by dx yes this can be said no doubt about it this is fine this is absolutely correct so dy by dx will be g dash t by f dash t okay now you have to find the double derivative means you have to differentiate this with respect to x okay so as per our chain rule you first differentiate this with respect to t because it's a function of t into dt by dx into dt by dx means you are actually doing this derivative of this divided by dx by dt so derivative of this with respect to t will follow quotient rules so it is f dash g double dash minus g dash f double dash by f dash square but dx by dt is also f dash so there will be one more f dash okay so it'll become f dash let me not write okay that's fine since I've written one by f dash t whole cube okay so option number c is correct option number b is correct is it fine any questions next if a function is defined like this the points at which f of x is continuous number of points at which f of x is continuous whole is on this ad actually done in the theory session also yes modification on the reclet function okay should we should we solve the polo five four three two one okay slightly disappointed to see the result most of you have gone with c answer is actually it is just continuous it is continuous only at one point that is half so limit extending to half minus okay now half minus whether the number is rational or irrational over here okay it doesn't matter whether you use x or one minus x okay both of them will give you half of each same will be limit of extending to half plus whether you use one minus x both will be half of each and at half it is definitely half so half is the only one point where it is continuous so janta was wrong in this only four people got it right out of 13 who voted it is not a direct lead function it is similar to that direct lead function is where you are hopping between two different values which cannot be made the same for example this is five this is seven which cannot be made the same here the value for which x and one minus x will become same at that value the function will become continuous only that's the only value where the function will be continuous so it'll be discontinuous everywhere other than the value for which both of them will give you the same answer so all you need to do is figure out that value and which was half in this case so it is continuous only at that point half is it fine any questions any concerns but cool what is happening chalo is okay that you do all your mistakes now only so that stakes are and the stakes are low yeah derivative of this with respect to cost inverse very simple question should not take you more than five seconds to answer this i mean idly in the school and all they don't worry about the interval actually but people have bounced upon the option there is some person who is invariably putting b as the answer i don't know why whether the person wants to just remove that poll from his view so yeah maybe he's trying to experiment that if i do all bbbbb what is my chances that i will you know pass in this invariably you'll you'll fail actually if you do that okay five four three two one go yes option is option that is correct is a because this is two two cos inverse x if you are aware of the formula so two cos inverse x derivative with respect to cos inverse x is going to be two only so option number okay so let's take a break right now because we have solved almost 30 questions continuously 10 minutes break would be fine okay let's meet in 10 minutes time i think i have forgotten to put the recording on yes five four three two one go okay so end of poll most of you have gone with b and c combined i would say because eight of you have said b and c okay but maximum you know polls have gone to a okay so let's check what is the result for this see you want to know where the function is decreasing right so derivative should be less than equal to zero okay so it'll be secant square x secant square x minus four okay now you want this to be negative that means secant square x should be less than four okay by the way uh that means secant square secant x should be less than two correct but this guy is always more than one correct so it has to be between one and two minus two to minus one correct is it fine so this be this basically means that your cos x should be between half to one okay half to one uh half to one if you see half to one comes from zero to five by three okay and this means this means this means uh cos x should be between minus one to minus half am i right have i missed out anything to let me know okay minus one to half is this interval also or what about that because one to half it will be from this interval okay now this will go into two pi by three to pi interval so that i'm not interested in because in none of my intervals i have actually crossed pi okay so the option which is correct from the provided option is where your x has to be between minus pi by three to pi by three exclusive of both of them so option number a is going to be right is it fine any questions here any concerns here do let me know so good next one simple question poll is on f of x is x by sin x g of x is x by tan x x is somewhere between zero to one exclusive of zero then which of the following option is correct very simple question okay five of you have responded almost one and a half minutes gone okay five four three two one cool okay uh only nine only only 12 of you responded six of you say a and b and six of you say c okay so it's very obvious here that you know we have been always talking about these kind of functions x by sin x see when you're very close when you're positive when you're positive x is higher than sin x okay when they're very close to zero they can be approximated to be equal to each other almost but when as x grows this will be growing at a faster rate okay so this overall function will be an increasing function see and if you I mean normally if you predict the graph of it it will be going like this it will be going up like this okay and tan x graph will be coming down x by tan x graph will be coming down like this so this will be the graph of x by sin x and this will be the graph of x by tan x so this is a decreasing function this will be a decreasing function because this will increase at a much faster rate as compared to this okay so either they say f of x is increasing somewhere or they say g of x is decreasing somewhere but the problem is in the option one they say both are increasing that is not correct both are decreasing that is not correct yes this says f of x is increasing this should ideally have said f of g of x is decreasing but it says increasing so this is also wrong so option c is the only right option okay if you try to do it by your derivative concept does it give you a definitive result or something section c a section b part we will be doing it tomorrow towards the end okay not today we will not we don't have much time to do all those part yes is there any is there any person who tried to solve this by using the actual approach or finding the derivative in this interval did give you any kind of a definitive result anybody who did this exactly you will get stuck if you try to do it by that method it didn't it will not help you out so i basically have to rely on your bit of your understanding about the relationship between sin x x and tan x next on which of the following intervals is the function f defined by this is strictly decreasing again a simple question poll is on you have to look at the options and then take a judgment okay so don't take a judgment just by looking at the function last 30 seconds five four three okay brokel one one yeah please vote everybody so i have got almost all the options been touched upon but d has got most number of votes see the derivative of this function would be 100 x to the power 99 plus cos x now it can't be 0 to 1 because 0 to 1 everything will be positive so it cannot be strictly decreasing in that interval because you have to find out strictly decreasing okay what about pi by 2 to pi see pi by 2 to pi this guy will be negative that is fine but this guy will be super high in value because pi by 2 itself is 1.57 so you're talking about some value between 1.57 to 3.4 ish sorry 3.14 ish to the power 99 oh my god that will be very very large value actually okay so it will be much larger than this so this cannot be your answer what about 0 to pi by 2 0 to pi by 2 anyways this is going to be positive and this is going to be positive so it cannot be decreasing there also so this also cannot be your answer so only possible answer is none of these i mean there's no such value interval here for which the function is going to be decreasing so why people are changing their answer why prakul what made you change it to see all of a sudden actual interval that depends on the question so see it's a mix of a polynomial and a trig function it's very difficult to comment on that so when such question comes they will probably provide you an interval but if you see x to the power 100 okay let's try to just graph it up let's see what happens when we graph it up what was it x to the power 100 minus sin x or plus sin x yeah plus sin x so it is actually decreasing for a very small value of x i mean see it from here it is increasing only okay but it is falling for a very very sharp very very small interval of x very difficult to figure this out it is falling i mean no doubt it is falling but it is falling for a very very small interval of x okay where probably x to the power 100 where it probably 100 x to the power 99 has been dominated by cos x that interval and it is very close in the neighborhood of minus one somewhere in the neighborhood of minus one minus one radiance let's move on next question find the angle of intersection of these two curves whole is on okay most of you have gone in fact all of you have gone with one of the options no wonder and people have done this many times as okay good should we then close the fold in the next five seconds five four three two one go okay there's there's still b stairs b stairs are guest workers most of you have gone with c by the way which is 90 degree okay let's check so don't sit and waste time finding the point of intersection just find your general expression for dy by dx for both the curves okay so you can use minus doh f by doh x doh f by doh y for both the curves this is for c1 let's say this is your c1 curve so it's negative of 3 x square minus 3 y square upon doh f by doh y will be minus 6 x y okay that actually results into x square minus y square by 2 x dy by dx for curve to minus doh f by doh x will be 6 x y upon doh f by doh y minus 3 3 x square minus 3 y square so it's actually 2 x y by x square minus y square with a negative sign yes their product for any x y where they meet will be definitely a minus one okay so option number c is correct is it fine any questions let's take another one find the range of this function one and a half minutes gone only three people have responded to this question okay i can give one more minute by the way option c and d an option i mean all these options they will all differ from each other in the brackets see the use of the brackets the figures are same okay five four three two one go okay most of you have gone with option d okay let's discuss it out whether d is correct or not see what is the range of the what is the domain of the function domain of the function is more than two but less than four right so this is the domain of the function okay so basically we find we are finding the least and the greatest value of the function in this interval okay uh so let us first find out where is the maxima or minima point coming up for this function by the way this function will always be a increasing function am i right will it be increasing function always what do you say okay let's let's check i think i i don't think so we'll be you know able to comment on this so let's do one thing let's find out where is the where is the stationary point occurring for this so two by two under root x minus two oh there's a minus sign coming in sorry i didn't realize there's a minus sign so now we cannot comment so it's one by under root x minus two one by two under root four minus x okay anyways for stationary point dy by dh should be zero okay for stationary point whether it's a maxima or minima let's figure out where it is coming out to be zero so this means one by under root x minus two is one by two under root four minus x so 16 minus four x is x minus two so 18 by five okay so 3.6 okay is the point which is the point of inflection so now what are you going to do we are manually going to test at these points remember what did i say when we were talking about global maxima and minima so in an interval when you're trying to find out what's the max and the min value we put the value of the function in the end points of the interval which is your domain interval here and also at the stationary point so when you put it to you will end up getting zero here and this will give you a root two so this gives you a root two when you put an 18 by five you end up getting two under root of uh 18 by five minus two which is eight by five plus under root of four minus 18 by five which is again two by five okay how much does it come out to be root eight is four root two so four root two by root five so it's five root two by root five so that's actually root 10 oh this comes out to be root 10 so it has to be a maxima point actually yeah and four will again give us a four will again give us a zero and this is two root two two root two is actually root eight okay so out of all these things this is the maximum point and this is the minimum point now i would not have done all these things because since this is a closed interval since this is a closed interval your maximum and minimum value will be achievable at some point so there is no point choosing a b or a c in that case right but if i were given different different value figures over here then probably i would have done all these things but if you take my call i would have just marked this question in one second only i would not have even bothered to look at a b and c isn't it because your domain is closed domain is closed so we are not worried about any any point we not or any value of the function not been achievable is it fine so i saw this question in a complete way just to give you an idea about how to go about with these kind of questions as well let's take another one f of x is this expression this is a decreasing function of x in r find the set of possible values of a a is an indi a is a constant it's independent of x super easy question two minutes gone only four people have responded five four three one go okay this was actually a very easy question but many of you did not choose to respond most of you have gone by c which is actually the right option because if you differentiate this and you want this to be negative the derivative of this will become this okay so this is just a coefficient of x cube if you want this to be negative this should be a parabola which should be opening downwards because you're saying it is true for all x so this should be a downward opening parabola okay for downward opening parabola the coefficient of x square should be negative that means this should be negative which means which means your a should be between minus one to one okay now equal to one equal to minus one that is also permissible okay because in that case it will become a negative three and for that for that matter it is negative only correct so equal to one is also permissible equal to minus one is also permissible so in light of all this thing option c is the right option option c is the right option is it fine any questions any questions any concerns if this was an infected quadratic then you know that your discriminant should be negative okay so just to recall if you want to solve this particular you know inequality and let's say this is true for all x belonging to the domain of the function which is already a number then a should be negative number one and your discriminant should be less than zero by the way equal to zero if they want then this should be less than equal to zero a plus three oh okay okay yeah then in that case then in that case no then in that case what will happen b square minus four ac then you have to check this this part minus because b is not there no so minus four ac that should be negative that part you have to check right no aditya so it would be a downward opening parabola but yeah but zero it would be shifted up no correct correct in that case it will not be possible yes right because let's say even if it is a downward opening parabola at zero it would be at three that means it would be positive for some values of x yes in that case it will not be possible next one remember the domain is only x greater than zero so in this interval is this function increasing decreasing both increasing decreasing none of these simple less than a minute question okay five four three two one two okay so somebody had to mark so most of you have gone with b only one person has said a okay so if you differentiate this it'll become minus one by one plus x square derivative of this is minus one plus x if x is positive this will definitely be a negative quantity so it's a decreasing function next I think this would be the last question for the day it's actually a very easy question I mean one minute has gone but I can see just four of you having responded to this question it's quite easy actually not zero one everywhere okay should we end the poll five three two one go okay and a poll most of you have given option number a that's actually the right option see it's just a I watched this whole thing is going to be zero because zero to five by two this guy is going to give you a zero so it's a constant function so constant function is neither increasing nor decreasing so both of these options are gone global maxima is one so only thing you can say it is continuous in zero to five okay all right we'll stop here I think most of you would be now exhausted three hours of three hours almost 20 minutes of problem solving tomorrow's session will continue with the same exercise but in tomorrow's session I will have few questions on section b as well sorry on vectors in 3d as well especially in the last half an hour especially for people who are going to be writing their isc 12 isc students so cbc people can opt to leave during that time so seven o'clock to 7 30 I will have some sessions on vectors and 3d as well if you want to be a part cbc people you're most welcome but that will not be there for your board exam so there is a slight difference in the chapters for isc people and cbc people so especially for our isc friends I'll be keeping the last half an hour for vectors and a 3d cbc doesn't have vectors right in the board exam yeah they don't have vector and 3d they don't have only isc people have okay so don't worry prakul and raghav you can you know last 15 20 minutes maybe we can keep it for that in fact vectors everybody can participate because you have already done vectors in our regular session also okay 3d probably you will not be able to do many questions okay so thank you so much I hope you learned many things today whatever you had forgotten or missed out slipped out of your mind all those things got recalled what happened