All the topics that I am going to teach you Today his most important question is The chances of getting something like that are very high. If we revise from the questions then brother At least give the final touch to your preparation. There are subjective questions and there is haste in them. I don't want to mess it up by writing each step down. If you want to show this, then this is an important question. is to solve linear equations using Years of a Matrix In this we ask questions Through this you will learn how to make thorets and What type of questions do you expect? Is there any shortcut way to do this? Is there any short way to solve this? Its short form explains such questions when If anyone comes then catch one and integrate first Hello kids, how are you all? Brother, see you guys in the Hacker series Well, our series is over. We have given you one shot of each chapter. It has been done but now the board exams are already over. Your boards have arrived this Saturday, haven't they? I have my 12th paper on 8th. CBSE Boards has Maths paper and I hope all the children are preparing well. If you are doing it then as we promised I did that to you in one shot If you get it done then it is fine but at the last moment We will also get your revision done through this. Each thing gets revised by improving it one by one. No, and the effort is to cover all those topics. Whatever I am going to make you do today The most important question that comes Chances are very high that such questions will be asked. We will revise it so that you get that kind of Things suddenly go wrong and the exam I too wish you could do that thing well. Okay brother, I have hope from this hacker series. All the children had a lot of fun and it was very You guys might have enjoyed some of the most You must have learnt it son, it is more important and now it is your turn See if you are prepared to revise. I have done it and have just opened this video. So don't think that there is little for us. So you are ready, you must watch this. And try on your own whatever things I I am getting it done myself by forwarding it. Attempt the questions if you feel like it If your preparation is complete then brother it is final Just give it a touch and check your preparation. Check that your answers match correctly This is a very good idea before the final exams. There may be opportunities for all those children. For those who either want to prepare for the last moment they have to brush up and revise They have to do things, this is kind of the last The practice of the moment is called revision. They say that you have to do that or else those children Those who did their last minute work revision but now they want to see means a kind of like a mock understand that you You are not getting marks just because you try are doing to myself and questions and Let's see how many questions you get right. That's right, let's start brother. Let's start our complete revision in one shot. I will teach you those topics which I have this for you, what do you say hacker? The chapters I taught in the series I will revise only those chapters which are left. Sachin sir taught that and you should do that. will be given but these chapters which I We had them done, let's take a quick look at them I am going to do revision today, okay numbering This numbering is just six chapters. The first determinants are messed up. Application of Evers' Trigonometry Derivative Application of Integrals Then Vectors and Linear Programming These Six I had made you revise the chapters. We will do it one by one theory through important questions where I'll let you know if you need it, but Our focus will be on asking questions. Let's go from the board's point of view. Let's start with the first of the determinants The first question in Question Determine I have to do it again, friend, this is that time It is not that you sit and watch videos. Yes, I understand, I understand, I don't have to do that. You will understand it, right friend? If you have completed your preparation for the boards You studied once, twice, three times. You must have studied for school exams too. Now twice in the final exam, once or twice If you have done the revision then now you have to ask yourself I have to do it son, I have to do it myself after pausing. And the objective question is that Meaning, brother, the board has to be settled in a minute. In this you will get multiple correct questions. Multiple option questions will come. If you want to finish it in a minute Do them quickly and whatever is subjective The question is, don't rush them. I will write down each step and show you Okay, just like you write on the board. You have to write like that, not in a rough notebook I am doing it and I finished it quickly. Pause calmly like you would on the boards Think like this in your sheet as you are writing You will start writing the answer. okay in the subjective one so let's go Let's start with the first question The first question is this, now look at it. Possible values ​​of x are now asked. If there is no board paper, if this If there was an objective exam paper then x Why should we give the value of forest first? Because as we have two determinant matrices Sometimes some children do it Okay, this and this have been made equal to this and this Such determinants are made equal There are no such matrices that are equal. Yes, brother, both of these are equal i.e. 2x = 2 and These two are equal, i.e. x = 1, so x1 from both If it came then there would definitely be an option x = 1 Gives to those who have this habit This is his answer to equating matrices. If it comes and goes wrong then determine equal Brother, we have to make the determinants equal. How to make the determinants equal Their value should be equal brother too Determinants are equal if their values Are equal there correspondingly These two are not necessarily related to each other Correspondingly, the value should be equal. If you want then how is the value 2 * 1 2 * 1 -5 * 4 This is done, the value of LHS is given 2x * x - 6 * 4 so this becomes 2 - 20 = 2x s - 24 24 take it So 2 - 20 + 24 = 2x so this is it brother 20 is subtracted from 4 and 2. 6 is left. 6 = 2 × s 2 If we subtract then x s = 3 i.e. the value of x + - √3 has become brother + - √3 when the value of x s Also 3 comes x must have a plus minus sign You have to put two values. The square of √3 is also a quadratic equation. The square of minus √3 is 3. So there are four correct options, what can be done will most appropriate There is also some discussion about matrices within the main chapter. The part is given in NCERT so that The parts that are added are joints and everts. I can get the parts for you quickly. I will get it done quickly, so first This is a question which was asked in 2015. is this matrix 3/3 right the co factor of What does the element a21 mean? He also gave a hint about the meaning of second row. Two second row is the first column, this is the row number this is the column number so a21 is this element second row first column this is done second row and this is the first column so The element in the second row, first column is You know, his son has to find the factor. is the factor which is minus times minor hota hai if a pj is Odd and P is odd, right, here there are 2 P + The value of j is odd, so that means The factor will be minus times the minor in this Your minus sign will come and you will have to find the minor. What is used for minor extraction? for son, you removed this, you removed this, your which is deleting rows and columns of elements Yes, the one who survives is minor, isn't it? What is the minor of the element to 1 m21 m21 will be the determinate value of the Four Elements 6 -3 What's Left Make its determinant 6 -3 -7 3 this Their determinant value is minor. What is their determinate value? Minor is going to give, so look at this 6 - 3 - 7 3 and now its determinant means 6 * 3 18 And minus minus minus is subtracted and it becomes 21. So 18 - 21 becomes -3, so what is the minor of 21? It has come -3 It has come son but we have to factor is required so factor of 21 will be Minus times minor because i + j is odd i + What is the odds of h is 3 is 2 + 1 is 3 so minus of -3 i.e. what is the value of the factor It will take three steps, son. You have shown me more than this story. There is no need to sit and write this You are sitting and writing each and every thing, brother, number four It depends on whether the question is of one or two marks. It does, did you understand? Yes, it is a small objective. Then the steps are not important at all, but If the number two is subjective, then a little If you have written a lot of two or four then brother, tell me the whole story. Write it down brother, I have to earn four I will give you the number for free in the boards. So if there are long questions where there are four numbers what is it write proper things there it seems You know that friend, this is a small question. You should explain more in it but if No, it seems it is number four, but lengthy. So, it is yours, relax and do something better. Write down the steps etc., this is too much. No, you are writing each theorem and that's it. You wrote it so simply. You will get many numbers in the boards The children who are there are very many We would have taken tension about this thing that friend, we I don't know how to write on boards, sir, I will write. How come brother, you just write what you are thinking? Know what happens in the objective mind I think and write a lot of things. There is very little here, just one thing to focus on Put whatever you are thinking in the board papers No, keep writing what we were thinking. If J is odd then this happens so I wrote this Neither did we sit and he is also satisfied that brother He has not memorized it, he is not cheating. Lee, they just had to check this from somewhere. They have your art story. English paper should not be checked for writing No, they just want to check that. Do you know these depth things? It is not like that you have memorized it before me. I knew this question from some book somewhere. I have seen it and now I can directly write the answers by rote. Have you given it or copied someone else, then which one? It is very unlike boards I don't know whether you should worry or not. What can I ask, what steps should I write? They happen and this is not written by NCERT. If you change even a single word here and there, Marks are deducted, this happens many times in school He goes, right, the school teachers are there They deduct marks in NCERT, friend. I had written the next step, how come you did not do this? You did not write this, son, now it is yours Number C, now you are gone son, will it happen or not? Many teachers do this in school exams. It is a bit too much that one gets emotional. Yes, but this does not happen in the board, trust me. If you take the shortcut exactly right I gave it in two lines, just knock knock knock, I did it. I have not written anything here, I did this and that like this and put a minus sign and write 18 - 21 gave it and wrote this, it became three, write it like this If you give, I will give you zero or F like this Don't do it properly by writing I've shown it, okay, let's move on. Let's go ahead Matrix this ad joint and Now I have to remove the ad joint of 2 Batu. It comes through a trick, brother, do you know this trick? would be that if this were A B CD if a If the matrix is ​​AB CD then add joint This happens, it comes in the trick to remove it. Interchange the diagonal elements And put minus sign in B and C This is what the diagonal elements are called. Kara interchange who was Eddie who was son Eddie Get it done and reverse the signs of both of them, then For example, if a is this then add joint a Diagonal will become 27, change it to 72 and this If it is -5 then do it. If it is -1 then do it. two 11 must have learned that way yes and now this Do you understand the meaning of dash dash brother, add joint a D means dash I want to take a transposer, which is the first row, make it the first column Make the second row the second column. Two to 71 my 52 71 - 52 C option ours The answer has come in small pieces like this. The question has become ours, okay, objective. Look at the question, look at it brother, next a is a square matrix of order such that that its determinate value is given Determinate of a is of any one square matrix and any matrix of type 3/3 It will be or or understand its determinate value Five is given, that is, determinate of that is Five, but I asked what you have taken for adjoining. Determine now look here many times we I guess what will you do sir, I have given the matrix If not only this, if not even determined then one relation hota hai determinate of ad joint a is determine a to the power n - 1 very very This important formula is also in NCERT The property is Ed Joint's property Determine add joint a determine a key Power n - 1 Look at the added joint of two only There is a property in NCERT what is this n n is order n is order like what is n here is 3/3 because order three So what would you put in place of n? Let's say three Let's do one question first, then the other. I will also remind you of the property, so add joint. The determinant of a became the determinant of a power 3 - 1 means that 2 determines the whole of a If the son becomes square then it means determinate One is a fave, so it has a hole of 5. Square means 25 so determinant in 3/3 The value will come 25 B option will come son Option B will come, okay now I will tell you. I was wondering what the other property was, that too. Let's revise this ad joint. One is a * add joint a to if multiply Do matrix multiplication a a There is a matrix, its ad joint also comes multiplying that or adding a to a multiplied by ed combined a by a Did both determine a times Id titi of order n aa mean identity matrix of order n okay now all these I am not knowing the identity in things What is happening sir, am I wondering this? I have taught you the basics A B CD. You have it, right, because now these A B CDs also I will explain what the Bha matrix is. What is determinate and what is eye Not that much son, we cannot do all that in one shot. Can we revise one short Isn't it a chapter where a one-shot has been done? Each and every thing has been taught to you there, so you First, if you are reading this for the first time, then If the shot is not for you then this revision Revision lectures are not for you. This is a one shot of each chapter, please read it. So let's look at this property as well. Will see somewhere and use this property It will still be used somewhere So you remember this, A ad joint A Something like this happens, determine a into Identity Next Question Value of k for each which determine this matrix a is singular matrix is ​​now singular What is a matrix? Singular matrix It means singular matrix now if you don't know this then very A very important definition is employs Its determinate value should be zero. Okay, if the value of whatever determinant A matrix whose determinate value is zero then it is called a singular matrix Okay, so what is its determinate value? It will become determinant by doing a straight stick We'll give you 8 4 2 so this into this minus this into this k into k minus 8 32 And this value has to be kept zero because it is singular. If yes, then the determinant value is zero. This definition is also given in NCERT. 2k s = 32 to cut two so k s = 16 square will be removed again and plus minus will come Sk Is Plus - 4C Option Our Most It is apropos that we do not write plus or minus. In a hurry, I ticked this option and turned my head. Diya, this is a very common mistake, friend, am I right? Let me tell you, I made the same mistake in my 10th board exams. I had done my maths in 10th board. Maybe 98 came in or 99 or something like that, 98 came in. There I think in a six mark question I had to go somewhere and do this plus or minus. And I didn't do that as a solution. The entire solution ahead went wrong, two numbers If I had been cut, I would never have made this mistake. I have been speaking since that day in the 10th board exams. I made only one mistake, that is this sir, I mean I know Because when they go home and check So you understand it, right, or all the other answers? One was right, marks were deducted in this Maybe because I made a mistake here. So don't make this mistake, very much. This is an important thing, let's move on. Son, such a long answer question came. Can solve linear equations Linear Equations in Three Variables Using Matrix Method Gautam Bai's Five Pence Three Bags and One Instrument Box and Page A sum of 160 degrees five pen so let cost Off Late Cost Off one pen B write this brother Rupee X Late Cost of One Bag B rupees or late cost of one box which instrument box is b root okay so one bag A bag, a pen and a box cost If I accept this then he is saying that there are five pens. 5x will be one x of one pen is five There is one and five pens for children. The thing is, there is no need to write that plus Meaning of Three Bags One more box means only the cost He gave me brother and gave me money from the same shop Vikram Ba was given to Vikram by the same shopkeeper Brother, you are surprised that the shopkeeper cheated you. Did not invest, gave it to Vikram at the same cost Meaning the cost of one item is the same. If it remains then Vikram gets two pens i.e. 2 X got a bag plus va and three boxes I got 3z, I got this for ₹10, also Ankur By one pen means x + 2y + 4z = 250 This becomes our three linear equations. So, there are three equations in x and h. She has given 50 convert the given above situation into a matrix equation of Now you know three such We can represent linear equations as a single matrix. Son can convert it into equation ok You have to write all these steps like this Don't think that sir has not written the steps. If you write what I am writing, I will get full marks in the board. I am writing this much, this is enough, many People are too much, too much story They make it up and write it down, then they run out of time. I go to the exam and find out that only half of it is over. The maths paper is still left and I am writing it like this. This cost of each step sitting in the paper One pen is x so cost of five pen is fi All this is not so basic, not so basic He wants to know from you that you understand. Why did you write this, please write it next How to convert equation into matrix now It happens that brother, this is a matrix Hai Ko Fish Ka 531 21 3 531 213 more 124 in xva Judd equals to right side of 160 190 and what is 250 correct this is our matrix equation I got converted into this, we call it Matrix a call this x call this B Sir, please write it like this, yes write it, is there any problem? If not then a is equal to b a is equal to b Our matrix equation becomes where a is This and Be is This is the first part done find determinant a now they're saying a So a is nothing but 531 213 1 24 so its determinate when we We will go to take it out, son, come on, determinant. I will write it on the next page so determine This means that you will have to sit along the first row. Expands along with anyone Do it friend, let's do 5 along with the first one. * 4-6 then 5* 4-6 Correct 5 * 4-6 -3 * 8-3 then -3* 8-3 ok brother -3 * 8-3 + 1 no 4-1 + 1 not 4-1 so we have 5* -2 -3 * 5 + 3 then -10 - 15 + 3 then -10 became -1, that means it became -22, son. What is the value of determinate A -22? If it is done then this part is done, it is done gone find a Ivers an Ivers is also to extract an ivers of formula now we know add joint a up determinate a so a Evers' part we are doing is added joint a upon determine a ad joint a upon Determine a correct ad joint a upon Determine what we should do about it First you have to remove the ad joint brother, so rest here. Let's write it down from A is our matrix F3 1 and then there is 213 213 and then there's to 4 t 4. This gives us if ed If you want to remove the joint, son, then add the joint. what is the formula ad joint of The matrix is ​​that the cos of each element Write down the factors of each element. And transpose two right so add joint a be I will be the co factor of each element I will write and I will make you write like this in the exam too. You will make a mistake, you will forget the transposer first Write it above. Now we will calculate the value of each element. Son, we will find the factor, okay, like five We have just calculated the factor of Ka, in fact Five's Ko factor came -2 see five's factor came -2 ok ra came 4 the 8 8 minus the pa but - pa will come because This is odd, i + h is odd here. After this one will come 4 - 1 3, understood? I understood the game by sitting with everyone's love. Factor out the co of each element Factor out like I did in this question. I had taken out this one, where did it go in this one In the question, we factored each one. I had taken out 21 rupees for this, so I am for everyone. Co factor Let us find the factor of each one. Let's come to this element. See, the best technique for this is pencil. I will take help of this in the exam. I am taking the pencil, okay? removed the row and column of and what 4 * 3 12 4 * 3 12 - 2 can be done orally There is no need to show this much 4 * 3 12 - 2 what came 10 but -10 will come why -10 Why will it come sir because son -10 so It will come because we have 21 elements here. Now it is odd, if it is 2 2 then plus will come now. Look, why did you say pencil, now what are you doing? You will rub one end with a pencil. You will remove the column, you will remove the row and column of one. So 5 * 4 20 - 1 so this becomes 19 And there is no problem at all, the next one 23 do it brother blow this one then it becomes 10 to 3 has become but minus will come Got it because the odds are next 31st If you hurry, it will be a mistake There will be rest and relaxation from the 9 my 8 isn't it what it was 3 was Son, this is the only thing in the T one now. A minus will come and we will remove it. So we remove the row column of 3 15 - two 15 - two It happens to be 13, isn't it? 5th, 15th, my 13th, and one. You have to put a minus as well and the last one is 33 If the son is removed, it becomes 5-6 and then it becomes 1-1. Isn't it -1, we have got this too To remove the transposer low edge joint What to do finally add joint a ho Take the transpose of this whole thing which First Row is coming -53 My 10 19 from mine and 8 from mine and mine which is here It's come it's done its ad joint a and now Divide this by the determinant a. If we divide by determinate a, we get add Joint and determine each and every The formula will become ad joint this was Upon determine determine a means be Gaya - 22 is divided by 22 now, isn't it? Sir, why should I divide this by 2? No, leave it out like this, take the minus inside so that the negative is reduced a little bit, so 1 Up 22 Outside It's Done 25 - 3 10 - 197 my 81 This became a year, a year, son, this Gaya Correct Now only a year has come, this was the only thing to be taken out. Look, I didn't ask the same thing in the question. The value of the root but if you also If asked, they take it out just in case. Because such a question comes up, so a one The equation of bar b is called Let's write a is the inverse of x multiplied by Let a be the inverse * b or x be the matrix goes to a ivar * b so what we do is x will calculate the matrix if asked. So a ivars * b a ivars we just saw 1 Ba 22 21 - 8 5 - 19 13 - 371 This is an evers What should we multiply this by? b son b was yours 160 190 250 then 160 190 250 first row first column first row Let's continue with the second column and see what happens. If it comes, let's take it out and solve it completely. If we take it then 1/22 Outside first row first column is na so 2 * 160 320 + 1900 is 10 * 190, right? And 250 * 8 is 250. * 4 is 1000, so it becomes 2000 - 2000 is done - 2000 is done is this correct I have written that I checked the first element Take it easy, no hurry Write second row first column 16 160 * 5 16 5 80 so 800 and -1 * 190 is 19 19 361 so -36 became 10 and we have to do 13 * 250 brother 13 * I want to do 250, let's do 250 * 13 If it is there then you could have done * 10 * 3 So this is 750, this is 2500 2500 And 750 7 5 12 It has become 3250. It has become 3250. Correct. And let's take out the last one, son. Mine 160 no 3 16 3 48 so - 480 and 19 * 7 Look brother, I don't have the table of 19. Now it becomes 7 6 13 133 so p 1330 and in the last third row it is 1 * 250 so P 250 This is done, please check it. Check it out, okay, now we have x The matrix will become 1/22 out 1 by 22 Simplify every term outside, son. 1900 Gaya 2000 Out of My 100 120 Bach Sorry 220 is left, 100 is gone, 320 220 out of which remained, it became very cute Brother, look, it is also being cut from 22 800 okay, let's see what happens, friend. 3200 And 800 is 2000 2000 sorry 4000 4000 50 3610 out of 4050 36 out of 4050 10 goes then 36 goes then 400 390 and 50 390 and 50 is 440, am I right, 390 and 50 is 440 looks perfectly fine I need to remove it orally. I did not need to do it with such patience brother You seem to be running out of time, sir. It won't go away if you do it like this. You should finish your paper in one and a half to two hours. Take it easy, not in a hurry like a JE In this you have to play the board slowly Do it, there is a lot of time, see the next one. Brother, brother, what is this? Let's do this. I think it will cost 1330 rupees or 250 rupees. Gaya 50 60 70 80 1580 isn't it 3 two 5 and 8 okay and-- 480-- 480 became 1100, 1100 is left here son But what if 1100 rupees are saved then it is good. It's over because all our numbers It means that it is divisible by 22 Our nice matrix will come to everyone Divide by 22 and this becomes 10. Went 20 correct and it becomes 1100 diva ba 22 So 11 becomes 100 by two 50, this is 10 20. 50 our matrix x is 10 20 50 And remember who Matrix X was supposed to be? Let the matrix X be the root of x and y. Ko to yeh yaa your ex wa jade ho gaya From here you can write the value of x. 10 is or ki The value is 20 and the value of root is 50, son Correct, this is our answer, every one who Is The cost of the pen, bag and box is ours. She came out even though she asked in the question I didn't have this, but I did it just in Ask in case exam to solve Now you can do this part and the one You can definitely take out Sc-5a, brother. Square its hole square kar lena 5 times Taking a minus becomes a matrix question. He will go, this is an important question. Solving Linear Equations Using With the help of Ears of a Matrix Ears The linear equations are solved So, such a question comes in the board. There is always a long answer type let's move on to the next one To find the inverse, we are given a matrix given a ivar and b given a ivar If I have to find it out then why should I do it now? I am getting a very important one done We know that you are the first to tell the property Step what will we write we know that we know that's a property we know a They all have property, son. You must have read in NCERT b years a years is b evers a evers is good I said a lovely thing in the previous question. I forgot to tell you, look many times like here So good everything got cut from 22 so chances are Very high that our answer must have come correct. No, but sometimes what happens is that we get so much Obviously it is not visible whether the answer is correct or not. So we can also check, you know, these things. When I solve the linear equation and get whatever The answer is coming, what should you do to him? If you put it in the equation and check, then this much If you have done any long calculations somewhere If there is a mistake, it will be caught because We've done a lot, so what am I saying? Like this is an equation x + 2y + 4z = 250 so check it in the rough what x + 2y + 4z is 250 x means 10 of 2y Meaning 40 4z means 200 240 and 10 250 It is coming at all, it means our x y satisfies the root equations You can check the rest of the equations as well. equations, that is, we should do it in a way I got to know that this is the answer that came. If that is correct then I always checked it at the end. Do whatever XY roots are coming put Try it and see if it is wrong somewhere. True if these equations are satisfied All the meanings are 100% correct, let's come back. to this question a b evers b evers a Ivers, you know what you'll do now. Brother, I have a year, here is the B one. You will take out the evers brother The evers of p were just found out. How We Evers Found Out Who He Was We sat down and calculated the factor divided by the transpose determinant So what you had was Ivers came in and a I'm not doing the first step anymore. You will get B Evers B Evers Nikal Loge Kaise Ad Joint Add it by doing B upon determinant B Finding the joint Finding its determinant B Ivers will come B Ivers when Ivers comes a Ivers It is already given, look at it, invert it out. What would you take for A. Evers? First, B. Evers. Then a eiver so which b Inworld Do It Matrix Apply multiplication to whatever you have Yes, the answer will come, okay, so this is yours. Your homework is done, please complete it. Such a question may come on the board. Let's take another look today. I told you about a property, brother. The last question is about determinants. We will then move on to the next chapter today. I told you about a property, Ed. Finding the joint of a matrix a * ed Verify the joint that if it comes like this Now we have to remove the adducted joint of our left hand. The game is a joint of a is nothing But taking the factors of A Transposers see the elements of A Its factor is its factor see What is 1 nav mi 4 so it becomes -4 -3 Then you will delete its row and column Next will go 2 in 1 p 4 2 p 4 6 done But I will put a minus on my side of the odds. Because of this you If we calculate its odds then son then -4 -2 -6 This is also done -4 mite -6, now you should come. on the second row, first column Might minus 4 becomes Might minus -6 so 6s comes will go because why will six come because Minus minus we have odds here 2 + 1 The next 22nd element, which is 2 for 2 elements 1 -1 + 4 becomes 3 -1 + 4 becomes 3 and so on one Element 2 + 4 is right, 2 + 4 is 6 but -6 It will come because these are 23 elements, son, correct. To factor out the minor Minus applied next 31 so this becomes 4 p 2 4 p 2 6 32 32 let's see son so 2 plus 2 plus Look, if you had given a minus, it would not have been much. Confusion ensues I'll feel like no one will hurry here. It doesn't take all the time to factor out Minor has to be saved if factor question comes Evers' question came to Ed Joint. If any question comes, do it very carefully. Slowly do the rest of the things and speed up Show it in your exam, it is okay No, you will make a mistake if you show speed then -1 P 4 -1 Was this one written correctly or the previous one? I had applied minus when 2 ps 6 came. Okay - 1 becomes 4 and remains the same. Because 33 is even, just transpole now So our add joint A will become -3 -6 -6 63 -6 6 -63 What has become of us, son Ed Joint Now what do we have to do with A and Ed? Multiplying the joint is a add joint a Multiply and find the LHS We have to get out this was our -1 -2 -2 2 1-2 2-2 1 and add joint is - 366 -63 -6 -6 ee 63 correct now let's open multiplication First Row First If we do column first row first column then We're going to have 3 12 12 right 12 12 24 and not 25 has come, 25 would have been neither 24 nor 25 Isn't it yes, 25 has come and first row second When you do the column, it should come out to be zero, right? Because we have to bring identity -6 mine - 12 happened and it's coming exactly 12 zeros Great, very good, our zero is coming. Okay, in these questions, brother, minus Marks mean you can't be wrong. It should be because it will automatically detect If there is any mistake then first row third column If we look, it should come back to zero. -6 + 12 -6 Zero has become ours again Let's see what happens next to the second and first We will see -6 -6 + 12 will become zero again. Look at the second second, it should come to 25 then from Lets C second second 12 p 3 15 and 12 yes Exactly 25 is 15 and 12 a second 12 what did I say 12 6 to 12 P 3 P 12 Hey friend, it will be 27 in the first one too. I probably wrote it wrong, tee plus 12 plus 12 12 is 24 and 3 is 27, why 25? I have written that 27 is coming here only. 27 is coming here too, see, catch it The mistake has not been made, some mistake has been caught. You will make a mistake in this question, I am I was saying that yours might go wrong. This question is not possible for you. The mistake will be caught and that one will come as zero. It must have been there, see second third second third The next zero must be coming, 12 - 6 - 6 0 has come Okay brother, come to third first now friend. I watch it directly on third third, that too 27 must be coming 12 p 12 24 and ti 27 son 27 If it comes then these zeros will be coming, you check that. Taxation became this type of 27 common then 27 times aa3 is formed what will the LHS become Gone 27 Times i3 But Wanna You Determine a tom i3 do then determine a ki find out the value Determine A will be -1 not -3 or -2 How did you write these ee 6 and -2 to the minus 6 Sir, you should look at their factors. The elements in the first row are now removed. Take them and multiply them by the factor Look, I changed the -1 to -3 to -2 -6 -2 to -6 is the way it is I did the direct determinate expansion. Given then 3 p 12 p 12 27 done Determine a so RHS become will go to 27 times a3 i3 so LHS and RHS Same has come, property has been verified Okay, it's a matter of patience, don't be hasty. Let's move on brother Ivers Trigonometry functions now we come to ITF on ITF coming up on ITF small The chapter is very small, a question comes Okay, we'll quickly do this as well. Let's see your domain in ITF Learning the range table very well without you having to question it If you face any problem then the first question is this one 2024 Question number two has come. from Square of this plus Cossack Square So what do you have to say about such questions? I have to do it late or even half b theta and cut e vers 1/3 which is the angle If you take the fee, then our value which has to be calculated and is to the square theta plus cosec We have to calculate the value of square pa Well, theta squared is 1 p Square Theta coke square hota hai 1 plus cot square F what a great question and not an ever half If theta is the time constant then when You know it converts into So if n is evers half theta then this Write employees here, this employees na theta equal half and this one what ilai will cut phi to 1/3 so don't put theta in half Put two coats of water in one bath, son, then put one water in one bath. by 4 p pv by 9 then 2 p 1/4 p 1/9 Taking the LCM 36 Calculation by Taking the LCM It's okay to do this, this is very very This kind of important question will come again. next cos evers cos my 7 pi ba 3 now some What do people do out of habit, Sir Cos I have cut off the cos and the answer is -7 pi ba 3 Rang 7 pa ba 3 if you are saying both cut Gaye -7 Pi Ba 3 So this answer is your color Why is it coloured brother because this is the answer principle You can't even imagine what kind of ewers are in the range. Whatever answer you are giving for the value, that range It should come in, that is why I said domain range K quadrant learn karo beta so kas ivers How should the answer of cos come in the range I will bring it and try some tricks Range of cos iverses Meaning the answer should come from zero. The answer type should be zero in the middle of pi It should come in the middle of the pie so let's C first of all blow away this minus K K There is no use of minus inside Okay, and now this is a big angle. 7 Pa / 3 is an angle greater than 360, remove 2 Pa You can view it from any angle 2 pi minus two and write it down, basically the idea is small to him If we make it then we will get the LCM of 7 pi - 6 pi. 3rd-ranked Chris Evers 7-6, 3rd-ranked That means this thing is actually equal to k pa ba 3 and you can remove the evers cos Son, because the angle left is the principal If the angle of the range is then pa ba 3 is the correct Final Answer to This Question One Marker Like This It comes a lot Let's take a look at another one like this And let's see brother, I have to prove it, 9 pa Ba 8 - 9/4 Sa Ivers 1/3 is 9/4 Sa Ivers We have to prove that 22/3 is equal to this. How will we start with LHS, son LHS If I look at 9/4 then common will come and here Nice little pie batu will come pie batu minus sine times 1 by 3 over pi minus Sign Evers 1 Ba 3 Done Okay Good One You people know the matter, you will write it here. No that ी No T Sine Inverse V byth plus cos yv 1/3 i pi btu a The property is sine inverse x times cos inverse x pa batu then if I sine inverse take 1/3 over there so cos inverse is 1/3 will go to Pi over Minus Sign over Take 1/3 of its value and put it here. 9/4 In cos iverse 1/3 I can write this cos inverse 1/3 I want to change the sign inverse, see this 9/4 So here comes Cass Evers to just sign Evers If we want to change it then for that we can use triangle Let us take the help of if we take it as theta So what you do is let cos iverse 1/3 theta From here we will write the value of cos theta as 1/3 That is, the sine of theta value rv minus cos square theta these use If I do it then why bring up san theta sine Because san ivers have to create If we take the root of 8 by 9, we have sine theta So, from here the value of theta Done sign evers rat ba 3 so theta we proved Our theta was actually a sine. Ivers rat ba 3 hai so der for this value ho given 9/4 in which theta was theta value instead of branch iverse 2 √2 ba 3 and that's what our RHS is equal to. RHS equals son it has come got It Proved, let's move on to the next question, the next problem. Let's also look at 2tan iverse 1/2 p t Ivers 1/7 to proof t Ivers 31/1 Although this property has now been removed. inverse x + t inverse y but okay a little bit We must know the inverse of x + tan1 y It is the property of Evers X Pwa upon 1 my xwa if xwa ka Product one is smaller than aquar positive so This is a property, but the problem is this That's the first ton Evers has out one to fit in too Two outsiders don't fit We put this out just when we need to connect. Some have fits, so the first ten years So we put x, then we put in evers 2 x up my x s for -1 two x Belong to -1 to yeh hoti hai property to We write ton over half, right? Ewers 2x your 1 my ex S and let this not be written as 1/7 Let's give LHS, this has become son, right? Evers 1 My 3/4 And now let's do this, isn't it Evers? 4/3 plus no evers 1/7 Now we are ready to install this one, right? Evers Exp N Evers Va Lagai Jaaye N Ivers X Pwa Upv My Xva Xva Ka The product is also smaller than n, clear then x Pwa upon 1 mine xva xva 21 lcm Take it 7 * 4 28 If 3 is 21 then subtract 21 - 4 LCM If the 21 is cut tan1 It's 31/1, son. 31/1 This is what we needed to prove. How great. It's done, look how nice it is. See, an easy question has become an easy problem. You have such simple questions, right? in Evers' Trigonometry Let us move ahead to applications of derivatives. Now let's do a little bit of this too, right? We are revising those chapters which I told you in the one shot inside the 12th Hackers We had done all the chapters by him. I am revising it first, son. Two or three things in the application of derivatives There is a rate change derivative judge The rate change is a monotonicity in which There is talk of increasing and decreasing And there is a maximum minimum right let's go So let's start with the ladder 13 meters long is leaning against a vertical Wall rate change question seems to be 13 There is a ladder of one meter, CD, that is such a wall Bottom of the ladder is Dragged Away from the Wall Along the Ground at the rate of so this one is yours if this length x is then d is a dt2 centimetre per second is given by How fast is the height of the wall decreasing? Because when you pull it here, it naturally This CD will slide down brother If you pull the CD to this side, it will come down. So the value of y gives us d or ba dt1 will be negative how much is the height of the wall is decreasing when the foot of the ladder is 5 Meter away okay so the first relation is here x प्वा स = 13 Correct because it seems to be Pythagoras theorem Gai this is 13 meter another y please note I have considered it in meters so this The expression is slightly differentiable. Above Differentiable two hundred x dx2 + y and d va ba dt-2425 This wall is from the wall, this is your wall What I am saying correctly is that the one who has land and That is 5 meters, that is, what is the value of x? 5 So we have to find d by d when the value of x is 5 So we will put 5 for x. We will put 5 in place of x and d will become dt543 5 meters Employs y s = 13 s – 5 s i.e. y s The value will become 12 meters, so y here is 12 If we put it then - 5/6 centimeter per second That's it, now keep one thing in mind, Marcus. Don't give me any scope to cut, he has said How Fast Is the Height on the Wall Decreasing He himself has asked how to decrease the speed If it is doing so, then you will have to remove the minus. Because minus itself means decreasing. So it is decreasing at the rate of so you Write that height is decreasing Height is Decreasing at the rate off 5/6 centimetres per second because In decreasing, the meaning came automatically, write T. I should have given a height of C Decreasing at the rate of 5 by 6 centimeters per second The minus automatically turned into decreasing, right? I asked you at what rate the change is taking place then - Keep 5/6 but if you use the word 'decrease' then Remove the minus at 5/6 cm seconds will be let's see this one What happens in this is case study based Let's also see the question in the application brother There are many case studies in derivatives. The Traffic Police has installed over 24 This is the question of 2024, I have given you one shot. I also got it done, you see it okay This camera can capture speeding vehicles at Distance of 300 meters and even function in the Dark a camera is installed on a pole at the Height of 5 meter it detects a fine wave The height of the pole is 5 meters and it detects a car. Traveling away from the pole at a speed of 20 meters per second at any point x meters away From the base of the pole, you could have made a little figure Yes, this is the car, let's assume that brother Your car is somewhere here, okay? Yes, and he is saying that this pole that you have Its height is 5 meters, it detects cars Traveling away from the pole at a speed of 20 meters per second so it's going that way if If I take this to be x dx2 is 20 metres per second and the pole is of height The height of the pole is 5 meters, this is also given to you. It's done alright is the angle of elevation of the speed camera From the car at sea is theta, this angle You've also been called theta, this angle Theta is said to be on the basis of the ab Information Answer The Following Express Theta In terms of height of the camera so just here It has become very easy because of theta. theta becomes perpendicular upon base 5 ba x 5 ba x done right so express theta in Terms of height of the camera to end x In the term, no theta has come, he has said Express theta will become theta, right? Ivers 5 ba x Don't give it any chance brother Theta Had to express so a part we Theta is expressed in terms of height and x Next find d in terms of th / dx3 / dx8 is the derivative of tan1 differentiable of 5 / x due to the screw into chain rule Derivatives will also come from Son So-5 Ba S You will have to multiply and the chain rule will apply. differentiable If you forget, then d theta body is one Gaya x Span x Span 25 x step up x step 25 and that's it -5 times x s x s is cut off by x s so d theta b d one came -5 up x s p Our 25th part has also been expressed. Is there any other part brother, it is a C part Find the Rate of Change of Angle of Elevation With Respect to Time at the Instant When the car is 50 meters away from the pole Well now comes the main question that he said d theta / dx8 sum instant this is d theta / dx3 / dx3 by what to find x we ​​need to find is the rate of change of angle of elevation With respect to time means d th / take dt2 to so d th / dt-2425 dx1 Remember this is given at 20 meters Second at the instant now question again reading Find the rate of change of angle of Elevation with respect to time at the Instant when the car is 50 meters away From the pole, the value of x is 50, son. If you want to withdraw when x is 50 then you what to do is put x 50 x If you want to put it then -5 up 50 s + 20 35 in d a ba dt2 correct something like this happened so 20 p 100 ho Gaya Numeration In - 100 Upon correct down to 2500 p 25 2500 Q is 25 Went Correct 50 squared becomes 2500 and now If you subtract 25 from the top and 25 from the bottom, then 25 4 100 down becomes 100 + 1 so -4 up 101 Our answer has come, it is correct. Something happened -4 / 101 and in this on radians Second will be the answer what will be on radian Why seconds per radian second because angle So radians come in things like this and ivers is not ivers nor angle in radians It would be and if it was in respect of time then Seconds per radian will be -4/101 Please check the calculation I asked another option in this part. was given if the rate of change of angle of Elevation with Respect to the Time of Another car are talking about another car this Not of the car but of another car whose rate is off change of angle of mean theta with Respect to time distance of 50 meters From the base is 3/11 then find the speed means dx1 dt3 ba dt3 / dt3 up 101 to b part bhi kar hi taking We are doing the B part of the third part of this. This is the B part of the third which was another one I'll use the relation again, d theta byty e equal to minus of 5 my x sp 25 in d eq b a dt3 body is given 3 up 101 and this is given -5 up was again given 50 instead of one Look what is given in place of x, 50 Only 50 meters are given if so 2500 plus 25 so it will remain d a body If you want to remove it, son, then cut 5 If the F is cut then 500 paise becomes 5 505 Right 505 has become a body and this 101 also cut will go then d once dt1 dt1 d a ba dt1 done son, correct, one more thing This was saying rate of change of angle of Elevation within a distance of 50 meters from the base of the pole is okay so We just have to tell the speed, speed is Magni D If there is speed magnet then this speed Whatever happened to the car, it should be 15 meters per second. has become 15 meters per second SI We have formed a unit, so let's go, this is good. The problem was that it was asked in 2024. Yours Joe is that Now there is one study in which this type of thing also comes. It goes like this, it comes in which what There is a combination of increasing and decreasing Strictly Increasing Many Children Confuse I was watching in the chat when I taught Applications of Derivatives. What would Sir Strictly Increasing be? What is the difference between is and increasing There is no difference, friend, the point of the board See where to open and where to close from the view Many times what happens is that we open and close You take it too seriously Don't worry, first of all it will open for you. You have to keep it open every time you keep your phone open. Increasing or not deducting the number Yes, many in both Strictly Increasing What happens when at a point The derivative is going to be zero but its Derivative change is not happening nearby Like x k x k ok y = x k then dydekop.org what happens to 3x s so now at x = 0 you get Now you have to say increasing only. I have to cry because x equals 0 The left side is also positive, the right side is also positive So if the derivative is zero at any point But I am not changing the sign in left right. It is positive on both sides, isn't it? I told you carefully there. Haven't you heard that if you assume zero at this point It is coming and there is a plus here too, there is a plus here too. So now the derivative is not changing sign So that means its instantaneous at that point It has become zero but to its left and right If both are positive then still zero But what would you say, would you say it's increasing? If you look at the graph, the graph of x is like this It happens okay and if in left right If change is happening then it is best to keep it open. We can keep it closed as well, there is nothing wrong in it. Yes, but there may be a chance of cutting marks. If the sign changes from left to right So strictly increasing ho increasing Yes, you should always keep it open, okay, when will you close it? What to do when the sign is same in both left and right It is staying, let's see this one. let's look at this one find the intervals on which the function is stickily Increasing to fx-100ms 1 2 * x - 1 isn't it, x - 2 is equal to x-1 s was added so one x - 1 was left 3 * x - 1 s There was an x ​​- 2 Okay brother, this much is correct, now just a little Subtract x - 1 from x - 2 and this becomes If we open the brackets, 2x becomes 3x, 5x -6 -2 If it becomes -8 then this f 'x has come, what is this gone son f ' a has come f ' x has come f ' One has arrived, and I will keep it positive now. For Strictly Increasing what will I keep I'll keep it positive, okay, so keep it positive. to keep Wavy curve to keep positive It is not loud to use but lightly. Keeping you in mind, you are 8/5 second points Gaya and the third two are all you can do. Aloud is yours and in the last bigger than two If I am putting anything then it is positive Yes plus yes isn't it 3 4 5 put anything 8/5 If I put two then this will be negative for you. If it goes just one then there will be an overall minus Function expression and put less than 8/5 If I give it then both of them will become negative then It will become plus and I will put less than one Then it will become plus because what is on one is the whole square is the repeated linear factor Yes, right? So this is how the sign change will work. Now listen carefully, brother. We will listen to it, so now the question is this. Should I open on one or close on this then it is 8/5 Keep it open, the sign is changing to The sign is also changing on this in fact Maxima minima will occur because derivatives turning from plus to minus, from plus to minus Meaning increase then decrease then maxima 8/5 pe aayega and to pe what will come minimum Pay decrees will come, then increase decrees Then increase to two pay minimum will come to one pay what is happening nai dar maxima nar minim Actually you would call it increasing. Because the plus is on the left as well as on the right It is also plus, its sign has not changed. About okay, so now I understand this confusion. You wanted positive anyway, positive The area was shaded but we were close on one too. We will do it because both the left and right of one I have your plus sign, one is another Maxima nor Minim okay so basically Maxima Minim points have to be opened We have to keep all the remaining points close. We have to keep it for a while, so it's ours. Answer: What will be the final answer or minus? 8/5 from Infiniti Union open to infinity is the entire domain I will increase the function, son increase Will increase the function inside this whole will do a jo answer hai strictly Increasing will come in this, okay, I understand. I hope you understand, so you can also Strictly increasing is what we would call it. Because in both the left and right of one Positive is strictly decreasing where Minus is coming, that is strictly decreasing. If there is a minus, then brother sir, this is what is coming, isn't it? Minus so this is what is coming plus minus plus If it is a plus then only this minus is coming Strictly Decreasing has opened 8/5 from 2 what do some books do now Let's close it here on 8/5 as well. I closed it and closed it too, so that too That interval's point of view is not wrong. This is correct but let us look at the overall function So 8/5 was the maximum and 2 was the minimum correct But look at the interval to the left of 8/5. If you don't want to talk, increase your rights. If so, we are 8/5 on one side of In inequality, we say increase. If he is doing it then that is also not wrong but you Whatever I explained, you did the same on the board. You keep the intervals open. Keep it open okay wherever derivatives The sign of is changing but where the derivative Same sign is being kept on left and right also. There is a plus in both, suppose there is a minus If it was minus minus then it would have decreased there too. I would have been doing it because there is no door maxima nor minim at that point I I explained it to you using the graph of y = x. From x in k you saw the derivative at x = 0 It is becoming zero but its sign does not change. As far as plus is concerned, there is only plus on both sides, so zero on This graph keeps increasing, you Look at the graph, it is increasing at zero. You will see zero is not A maxima and minima in this zero pay This graph is increasing, is it okay? Now I hope this is clear to you. Conceptually, what does y equal mean? Should the two derivatives be kept zero or not? Don't keep it simple, just assume derivatives Not keeping zero gives you increasing Decreasing whether strictly written or not It is written, let's move on Let's move ahead brother, let's do a case study. Let's also do one of the maxima and minima You could have done the maxima and minima as well. Yes, you will see, this is a very good question. Study Read the following passage and answer the Question below carefully in an elliptical Sports field, this is an elliptical field. Authority Wants to Design a Rectangle Soccer Field Soccer means football with maximum Possible area good sports field is given bye i got it yes ok ok ok ok so it's something like this Brother, we will take care of your sports field. Values ​​as in the equation of a standard ellipse x s / a s + y s b s = 1 Okay speaking of the length and width of the rectangle field r 2x and 2y ok so Rectangle with Inskrinox inside it becomes 2x and width is 2y then this I take x and y values ​​so that the width If it becomes tava, that means where is your xva point? is on the ellipse and we can assume The Length and Width of the Rectangle Field This This Find the Area Function in Terms of x area of ​​this rectangle we get this A rectangular field inside an elliptical If you want to make your own soccer field, this is it. The football field is so wide is football field okay this is your football Field So, what do we do now? We have to create an area function, so the area will be area of ​​rectangle is will go 2x in 2y but ax and y are related by this relation Tova ska ba b Sk is 1 my a sba a s ie a smy x sc up a sc sava = b ba A under rut a s - x squa be Put the value of y here, so 4x * y replace b / a a √ a s - x This is our function, this is what we had to do. We had to bring in terms of x, then y I removed and made the area a function in terms of x area in terms of x Find the critical point of the function Ye Jo Area is a function of x has come, let's write a on the next page who has come 4 B Bye a x under root a square minus x square This is how it came 4 b ba x under root a square minus x square area function of X has come, what are you saying about it, it is critical Remove the point, i.e., put Ax as zero. Critical point means the derivative is zero. If it happens then we will do different 4 B Ba constant outside is x constant its Derivatives Write its power half and Gave plus y The derivative of the constant x must be one. Went Right, if you cut and beat him a little, then 4 will give him zero. Also have to keep 4 b ba then went to zero then minus x score upn rat a squa minus X sk p a sk min x sk equals 0 Let's take the LCM, son. So we have a s - 2x stop upon haste I did not do it like this and I wrote it down. We will keep the derivative zero, so a s = 2x s x Let me say the value of s x is a by √2 plus I am not minus because the dimension Isn't it the side length of the field? The value of x will be negative a bit, a ba √2 We have reached a critical point. The critical point is x = a √2 Use the first derivative test to find The length and width of the soccer field Maximize this area, so now what do we have to do? To do is to find the point of maxima The derivative is coming to zero, that is what will be coming Maxima, if you do this much then your number will be cut. I assumed that I would go after checking Maxima. Show me, he has clearly written that first First Derivative from Derivative Test Check it with a test, brother, do something amazing. I don't even know what the minima point is. Yes, this is it, so what should you do, son? Factoring this out is the derivative of Tha Ye F'x Came a-x then a + 2 √2 a + √2 a a + √2 x a - √2 root of a s - x s these you I was crying, please remove this, this was also positive If you remove a - √2, one was placed positive 0. So a = √2 and you are greater than a / √2. If you put it then it will be negative because of the cuffit of x It is negative in this and if you put it here If it is positive then the sign changes from plus to minus. How to change the sign of a/√2 It's been from plus to minus, son first derivative test you guys have read a ba √2 is changing its sign from plus to minus. Meaning increase then decrease so you will write Since the sins a' a changez sign from positive to negative since x = a √2 is point of Maxima has to write the complete sin ad x Changes sign from positive to negative There for x = a ba √2 is point of Maxima, now the point of Maxima has come. Now we had to find 2x and 2y, so x If the value of a = √2 then the value of 2x becomes √2 We want to find the dimension of the field, 2x and 2y va what will happen va waj brother ours Remember what was there or what came before you, remember it by heart a square minus x squared = b By a rat and score mine x score came x The value of square is entered and it becomes one and a half a minus a Batu to be bye A B Ba A A Bye got rote A and A got cut So the chant of B Bye has come or what has come, B Bye I have become a habit, I want a friend, I have become a fool. Bye Rat Tova Ho Gaya Rat B To Dimension Of The What will become of the field with B dimension Son, the dimension will be √2 and rate will be The dimensions of the rectum should be that is what we have Proven, now see the next part, that's it. We are talking about the same thing with the second derivative. Proof it using the second derivative test. Find the lengths 2x and 2y of the soccer field that Maximize its area so what I mean You did it with the first derivative, now what do you We have to do that this first derivative came This came as the first derivative, in this u Find the second derivative using the / v rule and has maxima then the second derivative can be To do negative proof, now you do this a-x I removed it from the first derivative. We have achieved maximum identification maximum Identification This is how it's supposed to be, these guys move forward. Now we are talking about the application of derivatives. We have got one like this in it A question related to geometry may come. You will do this yourself by opening the tank with Square base and vertical sides to b Constructed from a metal sheet so as to hold a given quantity of water so that the Cost of material will be lessed when The depth of the tank is half of its width cost to become already set okay so you can put it there This is a question of maximum and minimum, you can do it. Let us move on to the next chapter. Application of integrals is fine brother. Next, I had made you apply. Application of derivatives after Integrals are our only remaining vectors. I will get you done and after that I will tell you guys I will get these three things done through linear programming. We have chapters left, let's start. Application of integrals in small quantities There are chapters left, so let's see in this Nothing is a small application of What to do in integral area Under the graph has to be drawn if it The function va e f is a s u t t y... to integrate is to do A to B if it So if you remove this area If so, integral wadi is one To be this is the area wa value f a So if we put it in, it would be a to b f ecd ec One thing has to be noted in this If the graph falls below the x-axis Applying mode in the area that is negative falls in the area with x Axis let's see area of ​​the region Bounded by the curve scale 4 x and line x e 4a so saying y s e 4x make what you Everyone knows what it looks like and x = 4a, the line area will look like this Bonded by this s 256 ba 3 units, so which area are you talking about? This whole area is being talked about, son. This is 256 by 3 then using integration Find the value of OK, got it? The first task So that's what you need for A and B. Finding the coordinates of x is 4a. So put x to 4 y put x to 4a So a becomes plus minus 4a. So a becomes 4a. 4a b becomes 4a - 4a so what do we do now Son, we could have done only the upper part. Yes, let us remove this from the one above. Double the vertical strip value in this li y d a key strip value li row to 4a Integrate it, what is its range, zero to 4 because if this is a 26/3 total then This area will be halved. integral wadi x two times integral wadi x from zero to 4a this is equal to 256 by 3 Correct 256 ba 3 will be 0 to 4a from zero. Cut down the value of 4a to 4a, brother. Put the parabola's var r 4 ex √4 one one 0 to 4a 128 times 3 √ 4a take it out x to the power 1/2 d one then the integral of half the power of x becomes son x to the power 3/2 3/2 correct x's power 3/2 up 3/2 limits Have to put from 0 4a is 4a^3/2 so I write it here I'll try to save the next page. This much is left, so 2 * √ 4a and this is 4a^ 3/2 / 3 bottom one = 128 / 3 to the power of 4a becomes 3/2 and 1/2 of 4a Pa It is 2, and subtract this 2 and you will get 64. Son, half of 128 is 64, so 4a is + - 64 + - √ 64 + - 8 but a positive bola If it happened then I took the positive one 4a also. I have taken it, though I take plus or minus. If we take plus here everywhere then the value of a Your turn has come sir, this is how we If you show the calculation then you will not get marks I will not cut it, I am telling you this brother This is a complete proper calculation, there is no flaw in it. I have not made any figure, you have shown half the area. Given vertical strip made integral Created and removed code I have solved it, you can write it here for a and B for A and b put x e a 4a inv s = 4ax and then Just write this here by implementing this. It is enough, it is enough if the question is number four. If there is so much to write, then it is too much Let's go now in this area there is a very Area is an important concept integral of which is the area of ​​the ellipse Its area is Pa A, which is total You will also get proof that the area exists. I have also got the proof done that it should come. One shot should get you nothing Let's integrate 0 to a, this Take out the 1 part and multiply it by four. We give because of Let's look at the proof of this in the cemetery. The question came in 2024 for five marks. See the question above for the exact same proof. Using integration, find the area of Ellipse included between the lines x = -2 n2 is not asking for the complete ellipse which we have I have taken it out, I have remembered it, I have found it completely Ellipsis's son is asking here that a ellipse okay there's an ellipse but in that Don't ask for the area of ​​the entire ellipse. I am just asking - 2 to 2 So look, when I'm putting in y0, it's 4 0. remained This is automatic -4 0, they are saying If x = is between -2 and 2, tell me whether it is x = This will be -2 or x = 2, so you will get this much area. to tell that So this is the area of ​​the ellipse included. Between the Lines -2 And So What to You Son, if you want to remove this then you have to do this much We will take it out and multiply it by four. By Going By the Standard Logic Proof so y d We will integrate one and we have to find out this much. This x was 0, this was x = 2, this is 0, this is x = 2 So we integrate from 0 to 1 and from 4 to We will multiply our area and create it. It would be nice to dx1 take the LCM If we take y = 16 - x = 4 then y = 16 - x = 4 Given √1 - x into half the value of wa is this here But put it son, then 4 in Half Zero to Rat 16 Min Acc Square correct now this one is big The formula is this, it is equal to Times the Same term ps a sc btu sine inverse x b 4 This is what its form is This is what happens brother, what do we know about its formula? I have to do it, son, I have to do it from zero to T. Limits have to be entered from zero to two Look brother, if you put zero then this will also give zero. This too, forget about zero zero, just two. If you want to put it then cut this into 2 also. Look, all 2 in √1 my 4 p 16 sign If you add 2/4 plus zero, it will still be zero. I want to meet √1 I 16 Sa Iv 1/2 2 √12 means write √12 as 2 √3 then 4 √3 becomes 16 * pa ba 6 Look at this, you can subtract 4 √3 p 16 Pa/6 written down 8 Pa/3 from top to bottom Two cut off okay square units forward Apply it, this area will become good The question is sir, five in so few steps You will definitely get the numbers, brother. You don't have any page full of marks It is our misunderstanding that when we do not meet brother Boards will not take two-three extra sheets till I don't have full board marks until then. This does not happen, you will get steps You have to write it step by step, right? If you write even if it's just one sheet, I myself For example, I did not take an extra sheet. I was feeling though that My whole paper got extra today. I did not take the sheet, it seems I did it in very short I gave it but later I checked my marks and it came to 98. The location question was made very short. I remember it was a question about the area. Two marks were deducted in that, it would have been worth six marks. It was our time and I didn't even make a figure. In just one line I integrated I wrote the answer very quickly Like we do in objective paper Don't make the mistake of writing it in one or two lines. If you are number six then you will definitely get a cut butt If you have written this much properly This method is very safe and more secure than most. You want to stay and write something. Write down the formula you used in the corner. You don't need anything more than this, right? We will get marks many times, we will not take any steps. We start overthinking about it and find out No, I didn't ask for the steps, what did I ask? I don't know what you will ask me, brother. He is asking Maths, first understand his M. I need to check your maths in English. Don't have to check where to put the full stop Where will the laugh take place, where will the delay take place, so much Nothing happens, I give marks but They have to understand that you don't remember. If I give NCERT question then They came by rote, they don't want to Right in the boards let's move on The next area is enclosed by the region y = 4x s y = 4x s and y e 8x l 12 Let's hurry Let's do it and then we will move ahead in the area. Questions like this come again and again, don't they? Will move on to the next chapter Area of ​​the Graph of the region y ÷ 4x s The graph looks like this, isn't it, y = 4x, right? Word opening is this one upward The opening is y = 8x ÷ 12 y = 8x ÷ 12 some This would be a line going from 0 to 12 x0 So we have to solve for both y1 and Because we have to remove this area, son. If we want to calculate this area then Brother A and B will have to solve it. Let's solve Zara to get an end B Solve Parabola and Line So 4x S Eva 8x = 12 x = 2x = 3 S – 2x – 3 times 0 x – 3x P 1 times 0 x = 3 and -1 so this is the point -1 and what will be its y 4 parabola I put in x to -1 and this x is 3 9 * 4 36 Okay, so these two points come out. Now we have to find out the area of ​​the region. The area to be removed is the area wound between Let's apply the concept of two curves Minus lower is not so area between two what is the formula of curves so area Bonded between two curves, 3 x key Domain Likhi Wa Off Upper minus va of lower d One okay upper minus lower d let's do ex Son, what about this function of upper 8 a p 12 Write the one above first Otherwise it will be negative or of low What is below, look, the line is above like this Below is the parabola. The y of the parabola is 4x. S So we have integral might to 3 so 8 of ex Break the units apart, 8 of one becomes 4 of one. Square its done 12 Square its done 4 One of our integrations is 3MW. All you have to do is enter the limits I will write it on the next page so 4 these 3 q minus my cube is not 4 a s ok sorry 4 was a squa not a cube Square 4 x got scored and 12 x was So I'm 12 minus 3 What was the last term? 4/3 x cube is 3 cubed minus My Cube So this is 4 in 9 My 12 in 4 My 4/3 27 Calculate it correctly 8 for 32 12 Pho 48 4/3 in 28 then 50 80 minus this take 16 common Take 5 minus 7/3 so 16 * 15 - 7/3 16 * 8/3 128 ba 3 coming brother answer 128 ba 3 Is the answer correct? If it is absolutely correct then rest easy. Take it easy, there is no hurry, board The paper will be good brother, you just I have to do it patiently, don't rush or create a ruckus, friend. It is not a question of hours, time may be short The questions will not decrease, time will run out. You will fall short, this is the actual thing that is if you do it with patience If you don't hurry, you won't be beaten. You will do it slowly, not back down. It will take a lot of time, especially for those children. For those who appeared in competitive exams Have you prepared for NEET or JE? You have prepared for 12th and after that you Giving maths paper in 12th boards If yes then you will find the paper easy. It will be done in time, just don't worry. By preparation I don't mean just Sir, yes I had bought the batch, I just You don't have to buy a batch, you worked hard. Yes, it has happened but not a very good percentile. I but you had put in efforts so the board I won't cause you any trouble, so let's go now. There is one on inequality, yes do this also. I give it because when it comes to inequality You will say that sir did not get it done, otherwise it is one shot. I have got all of them done but see here also taking are then y sk lesion ect The graph of x y s e x is The y scale is small, meaning the inside portion The portion inside the parabola is on this side Isn't x big, meaning x goes to infinity? and x le equals y so x eva make First, we created the y and x coordinates of the parabola. Inside and y greater than x means y being larger should mean above the line above the line What is the area under the parabola? son above the line below the parabola so dis is the required area okay this is the Required Area Because you want y If you grow up, you will go above the line and If its y square is small then this parabola If you go inside, x will become bigger. y scale equals x means x tends to infinity If the parabola is inside the butt This is the only common area left above the line. Again, you should remove this point first. put the y coordinate of e x into it Put two y e aqua s e ax so vv It will come and integrate it now. Two 0 to Vava Off Upper minus wa of lower dx1 will be its y what is the y of the parabola There is √ one from here so √ xxxx0 do 0 to 0 and 0 to 1 √ x became 3/2 to the power of x 3/2 x Scrap 2 So that's 2/3 of mine. Half 1/6 That should be your answer 1/6 square The units are done, okay friend, it was easy ahead. If we grow, the area will also become ours now. Let's come to vectors, son, vectors And linear programming, just two topics now. remained in our revision, then this one Our portion will become the largest among vectors The first question is from 2024, three markers. Question to find angle between Very important, a triangle was given whose position vectors are given by The position vector is not a vector Remember brother what is position vector The coordinates of a vertex are If A is 2 AA cap - JP then A is Coordinates 2 -1 1 become B's position Vector means joining from the origin This means that the coordinates It is 3-4-4 and all the angles are being said I will draw an angle and tell you I will show you all the same Let's say if we need to find an angle If A is to be removed then what A If we look at this vector and this vector Remove and create the dot product between them If you have to do it then Vector A points away from A towards vector B So Final minus initial 1 - 2 1 - 2 aa caps plus minus 3 psv h cap plus minus 5 Myve's cap so we'll have minus eye cap minus eye cap minus eye cap what happened son a b vector has become a Vector like this a take out vector ac Vector AC will be Vector 3 Might I Cap -4 p 3 min 3 h cap -4 -1 - 5k cap then a And any two vectors taken out from this To find the angle, subtract two of these. If I had to take it out, I would have taken this and this out. Correct so now I have to write K a if I have to write K a Will be done If abc3 comes then Magnitia is abc1 * 1 -1 + 6 Please check it once -1 + 6 and + 30 is how much it is, it is 35 son abacus.com 35 Mud A Mod A has come, take it out too So this is 35 divided by its root 1 square root 2 Sq 6 Squa A has become 1 Squa P 3 Squa PS 5 Squa to 35 apan 36 40 41 How much is it? 25 9 and a 10 35 so rote 35 by 41 becomes son cos theta R 35 ba 41 done okay like this one The first angle came, now you do the same thing You can take out C and C also, it is okay. Let's check the calculation once. My 1 1 mine 3-5 3 mine 4 mine 4 Mov I Cap -2 J Cap -6 K Cap I Cap -3 h Cap - 5k Cap ok -1+ 6 + 30 36 31 35 more Magnitia A Forest Marker Vectors Come in Many One markers see a mud a diya hai 3 The range of K is given -3 to 2 modes are asked Now see a very nice thing, the meaning of mud ke So the scalar number is Sir's take out take Come on no problem butt mode out of the bun I will come and if I have given mud then mud 2 sorry Now that Mud has done these 2, I have to tell you its range. I want to tell its range whose mode is K*2 Because I wanted to tell you its range. Now the important thing to note is that if the chem If it is from here to here, then what is the range of mode k? It happened if If your -3 is set then the mode is zero brother It will happen, right? Minimum because there will be zero in the middle. Magnitude is minimum and maximum of mode How much will it be when I minus 3 mod 3 If it becomes max then it will become minimum value 0 * 2 0 and max 3 n 2 6 then 0 to 6 d Option The answer to this is going to be a little bit of mud was embedded in the concept of Let's move on to the following. is the unit vector with ij, then we know If you come and let us know then you should tell us. Which of the following is correct? When there is a difference, there is a difference. h is zero because it is a 90 degree angle If it happens then it is odd then it is zero and cross. Jav scalar number cross product a scalar The number cannot be odd, it is 0, this is correct Because between the aa cap and the k cap If the angle is 90 degrees then the dot product is zero It will be C0I cross, this is also wrong. Cross I0 happens, I cross J ke happens Isn't it like that, it goes with the eyes crossed? If this is also not wrong then C is your option. It will be corrected Option C will be corrected Okay, so these forest markers are vectors. would come to us The next question is one about projection. Let me ask you a question, it is very important. Projection questions come if A and B Vectors are given then find the rash of Projection of vector A and B find the ratio Projection I think something is wrong Projection of a on b u ratio of b on a Maybe we'll see a projection of a on b projection of a vector onto b Vector is add be cap like this right What is the projection of a on b? B cap is a similar projection of B What is a vector on a vector Badat is a cap, son, remember it very well. The one on which the projection is made There is a cap in the dot, right, what was on here? B so B but dot cap came on what was A so If A P Cap comes then this ratio has to be calculated. Think there is a little miss print in the language. This rash if you want to remove it then add B cap means Add your mud B means beed apna mode hai na unit The definition of a vector is given with a cap. Vector Apna Mode Y used instead of B I did the ratio of these two If I take it out, I will divide it and be upon mud. B bead A upon mud A so adot B and bead A Cut off, we have got mud and our mud. B Mud A our Mud B has arrived Mud A Mud B Can you take out Magni, look at it? 2 squares 1 square 2 squares done Its Its Done 5 Square P 3 Square P 4 Squa 4 ch 8 one result became this and this went to 25 and 25 50 50 means 5 √2 something like that It will happen that this was Ajoom's question that Find the ratio of projection of A on B projection of b on so this we I also asked a question on projection. Our projection is very important The formula is a b cap a b Cap'n let a and given vectors a and b, find unit Vector along the vector a cross b or this We will say that both are perpendicular to one unit. Find the vector a and b given vectors The perpendicular unit vectors of both How will come a cross b up mud a cross b will be the unit vector which is perpendicular to both a cross b / mud a cross b You already know how to find the cross product. You have achieved a lot in Physics also. In maths also IJ's 4 - 1 1 2 - 2 1 Cross with the help of correct determinant Product Best Way i Cap * -1 + 2 - J Cap * 4 - 2 + k cap * minus 8 p 2 correct so this is eye cap minus u Cap minus 6 Cap here comes your A cross B Its Magnitia is mod a cross b ho Will be Sun Score 2 Score 6 Score 41 Gaya son this upon this be unit vector gone this upon this finished answer let's next Area of ​​parallelogram whose adjacent Sides are very important to determine What is the area of ​​a parallelogram? of parallelogram parallelogram if a vector this is a this is b so the area of ​​this Parallelogram is given by mud a cross b This area is not halved in the parallelogram If there was a triangle then half mud a cross b It happens son, okay, so just tell you something. Don't cross these two with mud. Take it out, the answer will come, I will come again I am not doing it yet, I have removed the cross product. take it out like this I will take it, you can see this one. Let's see this one, brother Chota. Chapters have a lot of meaning in these vectors. It is also very scoring in JEE. Very easy questions in the boards comes mud a cross b ho squ a b s so mud a cross b means mud a mud b sa Theta direction is not required, mud is enough Whole Square Ed B Means Mud A Mud b cos theta hole Square is given by 400 Mud A Mud B Whole Square Ko Common Le Let's take it inside, it will become a sine square theta plus cos squared Theta this is one, this is one, mode also 5 given so mode a squa in 5 sc e 400 So mud a whole square became 400 b 25 okay so how much did this come to? 16 came to Maybe so mud a hole square became 4 * 4 Do not write the value of mud a as plus minus four. Dena Bhai Magnitia Mud a Key Value For That's it, okay, mud a's value has become four. So this was our question, Izzy. Problem: Easy problem, let's move on to the next one. Let's come to the next one brother if a b c r unit vector such that a p b p c Sum zero of is given to you to create I have already Told it straight squaring both Make its sides zero, its mode value is also zero If you do squaring both sides then this Mud A Ps Mud B Ps Mud c sk p 2ab b s c means 2a b Bead Sea Road Aye its sum son became zero but because Unit vectors are written as unit vectors Vectors means their Magni is one, son. This also one This also one This also one so 3 p 2 ed B Bead Seep seed a its value zero th le there go to add be plus bead c plus sd a Value -3 will be divided, final answer is given - 3/2 If you assume this in this question I would have just asked you Ed B, then I would have asked Ed B tell me if you can do this question If A asked B, A would ask B first. Write C and then square both Both of them would take mud while doing sides The minus on the side would have been removed in the mode and now Squaring both sides gives a square b S p 2 ad b is mod c square unit vector If this was one, I would have created a new question. I have made a new one from here Adot b comes to minus half this is also a The question could have been made in this or asked like a bead Taking the seed, all three would have come to minus half of all three value minus half aa Let's go, vectors are ours now we come Let's go to linear programming Programming is very good and easy Chapter is again scoring chapter above it A fixed type of question comes The optimization problem comes first. Above is an objective question with single correct answer. Let's look at a linear programming The problem deals with the optimization of The world knows about linear function, brother. If the name itself says linear then it is a linear function. It will be optimizing the linear function It performs a function of the type a + b We optimize, we maximize or Minimize common regions determined by all the Constraints of a linear programming The problem is called any number of constraints Or their equation in equations comes when area is made into a common area which comes What is the thing that is shed called? It is called feasible region, son, feasible region. He says I will create a feasible region right now, see. Which area will be the feasible region? like doing a Let's do one question 2024 three markers The question is an easy problem, solve it. Following Linear Programming Problem Graphically, the Z is 500x and the P300 is Our objective function, which we Want to maximize linear function Look at the X and Y, in this we Why can't you use calculus because There are two variables in calculus, we use single Maximize a function with a variable minimizing are subject to constraint this Equality is a constraint, so we What is the first thing you always do? Graph Let's make it son, let's make graphs of all this Neat and clean constraints Represent x y 2y on a graph First we will graph equal to 12. will enter zero ek ji dalenge tova 12 and wa zero If you put sorry x, you will get 6 If you enter no or zero then x1 will come. 06 more 12 We will have a line like this. it And we have to stay less than this. So you will have to take the lower portion of this line. Let's see the lower portion and then 2x pwa Take this 12, in this eight will be reversed son 0 becomes 12 in this becomes 60 If this too has to be kept then this too will go We have to stay below this, we have to stay below both of them. to stay under end 4x p 5va gr e 20 So 4x i 5th Equal to 20 plots first If we put zero in the value of 1x then y becomes 4 If we put the value of f in one then y will be 0 will go if F and 04 04 0450 Okay so when we have these two When I joined, something like this happened. Son, this is what has become of him. If you want to be bigger, if you want to be above this, then this Above them both below so this is the feasible If you find the common area of ​​the region then I want to stay above them, I want to stay below them also x Positive y positive means first If we have to stay in the quadrant then this Has the feasible region come and closed? Feasible region has come so optimization What the problem says is that if the feasible region If it is closed then the maximum or minimum Value comes from these corner points. Corner point of the physical region It happens son, you can come there, you can come there Maybe you can solve this point. All the rest are corner points, this is solved Can you do it, I think it works, yes it works If it goes then it will be rooted at all these corner points Find the value of the function. What do you do next? You will do the root function of all these corner points will extract the value of How many corner points are there brother corner Points are our friends Pa okay Pa so this is our Xva key value and this root function was 500 x p 300th we will find its value xva first would have taken F 50 means putting zero to the 5th of x. If it is then 2500 will come and then the second corner The point is 60 again. 04 06 and 44 find the value of the root of all these Yes son, the one who gets the biggest will Maximum value is what you have Call linear programming is solved like this is 60 6 5 30 3000 0 x 0 व 4 1200 x to 0 va to 6 6 3 18 1800 and 44 It means that 5 chha sa aa 4 common took 800 * 4 3200 Which is the largest value? Big value I 4 4 pe 3200 so this is your z The maximum value of the function is 3200. The maximum value will be at which point it will be 4 4 When both a and y are four fur then it What does it mean that these things Satisfying all the xva The maximum value of this thing in them should be 3200 and what value can it take when x and y Will both be equal every time, sir? It is not necessary that it will come to this The question has been asked right here Done in question Come on brother, let's see one more, let's see one more There is one I will get you one in which Sometimes open-ended becomes closed if There is a feasible region in the LPP problem If it is closed then it comes only at the boundary point Maximum Minimum Value Not Closed If it happens then I did a one shot there. had also done Let's solve the following here. LPP is to minimize graphically This time the minimize is also at the same end point. But the other condition which is given is this given So let's plot again what we will do Son Plot will x pw greater equals 8 so 2 x pwa equal to 8 will make 40 0 8 Isn't it x 0 tova 8 and that's kind of like this It is done and what do we have to do son? If you want to live greater, you will have to live above this. Next is Exp E 10 then 10 0 10 0 what else to do 0f 0f so down These two lines will be plotted by us. What we both have to do have to live above because both have son is the sign of greater than equal to so we get What to do is to take this region or see Taking this region Now I have to do one more step in the open-ended process. What you have to do first is So you need to know the visible region of this area. It is okay to see the boundary points. What are the three boundary points? points is this times three boundary points Who will son Xva's three boundaries Points are one so came 10 08 And don't let anyone in between say this boundary point because of the physical region Boundary points are one, two, three in this area. let's solve this, right? Let's solve this. to x Solve π e 10 and 2 x π e 8 have to do it So let's do one thing: the value of x is 10 times 2y would have kept it You can do it anyway, linear equation 20 min Fava Pva e 8 so 12 = 3 Vava's value is 4 Done y4 Done x2 Done I think 2 4 aa Let's check the point 2 4 Doing both 4 times 8 and two times 10 8 Okay, so 2 for this point. If it is done then this is the first step for you son. All the roots had to be removed at the boundary point pulls out all the roots If yes, then see what will come out to 10p. value 5x i 7th It was 10p when x became 10.50p. 56 came 24 on 5 to 10 p 7 4 28 38 it's here son 38 it's here right 38 It has come, so now all three look the same Minimum it seems to be 38 but you get it every time This is not the answer, you have to check. Something closed area happens, some check Don't do it, closed area means your The minimum has arrived, that is the minimum, the max has arrived That is Max but open area should come and open. area, so what should you do if this If it is minimum then you should also see a proof of it. Is it getting any less than this because What is happening in the open area cannot happen. Going forward, the closed area may reduce. So there was a fix, we looked into it. Can't go beyond the boundary point But there is no boundary ahead in this, right? So what we have to do is you have to check that Can I keep z less than 38 so that j Lesson 38: Let's just check what to do. Would I have to check it by keeping lesson 38 What was the value of 5 exp 7th, let's just keep it and see what it is We will also make a graph in the graph we had. If its graph also has to be added then its x and y to plot the graph I want to put x0 in if then y becomes 38 or 7 They are very dirty, if you put y0 38/5 and at least one point is good for this. Because brother, how did 38 come, it came from 24 If he is 24 then he is satisfying him. He must be doing 24 It will definitely work, 7 4 28 and 10 38 If he does then these two-three points will tell us have got 38 ba 7 38 ba 7 means ho Gaya a little bigger than five somewhere here 38 Ba 7 And 38/5 means 5 7 3 to almost so something almost happened here That is this and this point, this is it We have drawn this dotted line. Which one is made that we are checking? dotted line at the end line dotted line this is the line with the letter 'v' which we Keeping and checking Son, this is correct, we are keeping it and checking it. What more do we need in this less than 38 Should I live less than this? If you want to do less than this, then stay less than this It means listening carefully, less than this means its Below then what is the feasible region of this region If something common came or did not come then something about this Common comes from the feasible region of not coming I mean, it can't be less than this, brother. This is our physical region and if it is separated from 38 If you keep it low then you are getting this region, meaning something. This is our only point that is not coming to common It is common, so 38 has already come on it, that means this 38 cannot go less than if it has some common Suppose this line goes in place of Hypothetically it would have been like this and if it had been like this then now This region would have become common as its feasible region common would have come together i.e. our function which The objective function is that it must be less than 38 could in that case because it has some common The area comes with the feasible region If some common thing about inequality had come, that is It could have been less than 38, what would we have said then? that the minimum value can be less than 38 We don't know what could happen, whatever we want It may continue but at least we do not say anything about it. You can, but if something common is not coming like this This is your line in the question, this one 38 The reason for the line that came after keeping it less than this is The yellow one is coming, its physical region Nothing else common came from this, that is, this function You can't keep it less than 38. You wanted to see, brother, she looks minimum 38. I tried keeping it below 38 but found nothing. That means 38 is the minimum, so since it is open-ended You have to do the work if you assume some common If it had come, you would have said no minimum value but Now 38 has come so what will you write since Sins 5 Exp P 7va Le 38 Judge No Area present in feasible region There Four Root Kant B Less than 38 less than 38 which you are trying to keep It was not possible that it could not be happening 38 is the minimum possible value. Four h minimum is 38 This will be our final answer. Okay, I understand brother, you have to do it like this, that's all. It is simple, only two types of questions can be made in it. can one when someone is closed ended You have no problem with that One has to check at the boundary point itself. Your area is a feasible region which is unbounceable. Infighting If you find something common with the region, you can say it. Dena no minimum value if this was the maximum So we find the largest one which is 56 and then the largest which is z If we check by keeping it greater than 56 then what would we do Let us check by substituting 5x + 7y > 56 and check that Brother, is any person coming in common, if you come If he had gone, he would have become Maxima, he will definitely come. In this case, think for yourself. Common will come in maximum because brother Sir, how do you know that x has 5000 crores? Keep 10000 crores, it is there in the region but He will continue to give this maximum value of 56 will give a value much larger than so maxima Its does not exist will minimum exist tax went to this function okay why because The common region did not come in the minimum and maxima The common reason would have been this only concept so let's go with this one The session is over, we discussed I have taken my son, whatever is of a hacker. Questions which chapters I revised for you I got it revised for you. You will get revision of more chapters Lets for the remaining chapters will also be released. Start with the part of the marathon so most First thing, what topics are covered in it? If I do it, there are seven chapters in it which I'm going to cover you in this I will also explain the theory to whoever, wherever If there is a requirement with any questions then Matrices Relation and Function Continuity and differentiable integral differential Equations 3D Geometry and Probability The remaining part is being done by Tarun Sir. Okay, so these seven chapters cover our matrices relation function Continuity and Differentiable Integral Neither integral means both become infinite Define and done Differential Equation 3D Geometry and Probability Questions You will also tell me whatever is with you. Arrange for a copy and pen together. Keep practicing so that you too can have one The flow is being maintained, it should not happen that just We are watching it like a movie, then that Then you don't have a utilization Make sure whenever we do it, I mean whenever Watch this video and try to copy it. Sitting with a pen and practicing together Are let's start with the part of the This is our number one question. What are the givens of these matrices? The question is given a key value is given to us There is 1 -1 and 7 then what is written a s - 8a + k * i = 0 what is the value of k No, first of all you should accept it as you like. If you prepare for JE etc. It can also be from characteristic equation etc. It goes but now like the character equation If you don't have any idea then how will it work? Let's solve this by first removing a will have to remove a s, so a s How to find out that a * a is equal to I will tell you the shortcut method but you will do that in school. You won't be able to guess, right? Yes, objective. If you come for something else then you can do it, right? He will also explain how characteristic evasion Let's figure out 10 equals -1 and 7 is okay. So 10 -1 was 7, multiply 1 -17 here. Let's do it while multiplying. If we multiply row and column wise then If we multiply it row wise and column wise then What has come, one has come row and column wise Keep multiplying this row and this column. Row and this column are just doing multiplication. Same here too, keep multiplying -1 -7 then -8 came and then again here If you multiply it, what will be 77, 49 How to multiply rows and columns with Multiply Rows holding Columns Multiply Row with Holding Column multiply answer aa gaya a ska ja a s aa So what should I do with a s-8a? If you want to find out the value, then take it out and So there's 1 - 8 0 and 49 okay - 8a so 8 Before multiplying, multiply everything by 8. If you multiply it, what will come out to be 8 aa What will this go, it will come, it will cry, okay. What will this come, 8-8 came and how much will this come? 56 has come, right, equal to + k * i, so This value has come, i ive 0 i means Identity Matrix So how much did 1 - 8 come -7 has come, how much has come, 0 has come, okay And how much did it become? 0 became -8 49 - This comes to -7. So k + k * i = 0. What should I do now if I write -7 etc. take him to that side when you You will take me to the children, see what will become of me. = will come here and is equal to 700 and 7 Neither will anything go in minus there Now Sen will become plus, see from here If it becomes common then k * i is equal to a will be equal to 7 * i so the value of k here If you compare it with this, what will come out? Sen aa What happens in the identity matrix? If it is 101 then what will be the value of k? It will go equal to Sen in this way which Okay, so the second thing What did I tell you, from here a - 8a is there any short cut for this Is there any short way to solve this? Let me tell you the short method, look at this Short cut, if you assume this objective If it comes in the pattern then this is it, don't get involved in everything Dena came for this objective like value Here you come for one single choice Is there any short cut method or characteristic? It happens through equation, okay this is my detail. I won't look at it, I'll just write it down. how to write with fatt like 2 cos 2 For example, if you want to write an equation, how do you write it? So 10 -17 is the answer to this in one go. You have to find out such answers quickly. If you want to remove it then see what you will do a s - is Na a S - then what do you write after that know the sum of the diagonal elements here What will you write as the sum of the diagonal elements? which is called its trace, the sum of the diagonals elements * a equals p m * i = 0 end This is the answer, the answer will come immediately, okay mode Meaning what is that, it is determinate of a So see how we will find the answer, see. Quickly look at the equation here, you have to I did not read it if it is objective like Suppose there are 20 questions in it. If you want to do it quickly then how will you do it? Minus, look, see if you are enjoying it or not. Look, if you make these two equal, then you will make them equal. How much will it take? 1 + 7 is 8. + a Now how to find the determinant a - bc7 - 0 then how much came out, if 7 came out then 7 * y = 0 See, nothing had to be done with the option If you match the value of k, what will be the value of seven? It's here, okay, okay, this type of thing Look, I'm going to give you another question. hmm a i equal to is written here 1 2 3 and 4 are equal to ok and value from you He said find the value of it, right? Find the What is the value of k, do you have to tell me? Find the value of k is okay, isn't it like this The question is expected, isn't it a - 5 * a + k * i = 0 what is the value of If you are able to understand then if it is single choice Suppose the one with number one had come, You can write it down if you agree with the short ones. Either the long ones came or the very short ones came. In whichever category someone else comes in, then there You will have to write it there in this way It's okay to solve it but if she is single If it is going to happen then you will only get the option of match. If you want to get it done then it is okay to get it matched. s - 5a + k * i then determine the value of k Find its determinant of a and answer it will go okay the shortest way to tell that No, you have it, right, even for singles I'll show you the pattern plus you If, for example, suppose it was very short, it came like Because you get five formats. There are sections A, B, and C. Section D Section D then you Attempting strategy will be explained that Now let's see how to attempt this. This is the turn, it is very easy that if half If it happens then accept it as yours, accept it as yours. I will come in either section A or section B. will fall into one of the categories what happens for this what happens for this Equality of the matrices is there, right? Equality means all who are, whatever they are Correspondingly, everyone is equal You have to keep moving while doing this, see how you will balance it x - y = -1 aren't they just equal Correspondingly, the mention is with f equal to 2x - y = 0 exactly So let's subtract this once x and The value of y will also come if you minus it. What will be the value of x = 1? This is the equation Number one, this is equation number two. Correspondingly ment was equalized If you minus this, then minus x The value came to one and what is the value of y? After placing one, the value of y will come to will be equal to 2 when x = 1 and y = 2 So two things have come out x = 1 and y = 2 Now we have to find the other values ​​of z and w. Make it equal for him too, how will you make it equal? 2x + z is equal to how much aa gaya f k is equal to, so from here the value of z is ours since the value of x was one So what will be the value of z? 5 - 2 cc e Equation 3 is equal to what is the next step? 3z + w is equal to what is 13 So put the value of z here equal to 3 * 3 + w is equal to how much is 13 so w's What will be the value 13 - 9 which is 4 x also came y also came z also came w also came Gone 1 2 3 4 Look great, get the pattern It's great 1 2 3 and four so 1 2 3 and Fore arrived x1 arrived y2 arrived z3 arrived ond Whatever has come, it has come, it is fine, it is comfortable. Equality of the matrices applied to it Okay, so this is going to tell you the equality of The Matrix comes next only The multiplication part is a * b If you want to remove it, do your homework. Let me tell you, this is very easy, whatever you do It will be a bit difficult type and a little different. If I get any type, I will let you know. Let's go, this is homework number one, solve it. Keep doing it with your hand and then tell me about it. What was everyone's answer? Whatever homework was done, it's fine. If you do, please tell me in the comments. What are your answers, what will you do? Whenever we want to multiply Like I told you a s so how You will multiply this with a * b okay then after this this You will multiply it again with this You will multiply this, multiply this by this If you do it, your value will increase and I want to do column wise multiplication. Now comes the next question, this is a very valid question. What is the written form of the matrix a and b square matrix so a and b which are two is a square matrix with determinant You know the value of a The value of b is -1 determinant of b know value is equal to 3 what is the The value of 3 * mud of a * b is right if Multiplication by three at any time here It is written, isn't it, and the order was written to you. order here order here written If it happens then this three is not outside. Many times people would just give him a three. If it is there or not, then always assume it is like that The determinant of k * a is written here. But the k that comes out is k to the power n. There is, isn't there, here n what would be called order No, keep in mind what the order will be called. Many times we forgot and then got it done. Once I have completed the entire answer, always pay attention. Putting n to the power of k will keep the order. It will come out, for example, suppose there are three here, then Why will the cube of three not come out? Because the value of the order is so given. Look here, k to the power n is written. If so, how do you get rid of the three lovely children? cube then how much will come mud a * b so mud a * b then he will come 3's Q * mud a * mud All the values ​​given are equal to b, how much is this 27 has come, this -1 has come and this 3 has come So what was the final answer minus 81? That's what this value is, isn't it always meditation? when exiting k then k to the power n I expected such a question to come out. It is in the single world, isn't it? I can only come in this is fine next Next comes a bit of integration If we look at the matter, then this is Integration questions are fine for every type. If we keep looking at the questions, then we will have three questions right now. Watch more Matrix videos now If we meet again in the future, what is written? e to the power of x from 0 to one here And what is written below 1 p e to the Power x It is typed in which the option is not written. means very short or in short so we People cover every type of section. We are moving along fine because we have There are five sections in which you can ask yourself questions. If we get to see it, we will let you know. A section can come in this, what is it written? e to the power of - x / 1+ from 0 to 1 e to the power x so I don't understand anything What will we do if we have e to the power of x So let's take e to the power x here. take pay t okay then Differentiable to You can change the limits etc. later also. You can change it beforehand as well, like Take the first value of x, the value here is one If you put it, how much will it be if e becomes the power of 1 There is a limit, what will it become? Power 0 will become the limit now after this e power x which is in t so dx1 what is will go to e to the power x dt1 + e power x dx1 has come okay 1 up so what has become 1+ t became a t in multiply as well Gone is fine and dx2 / t so convert this to dt1 t You will see, it has been made in DT2. From one to e and then came here dt1 t will also come because dx2 is up in t this And the remaining one is t * 1 + t, so look at this If this is done then final integration Dear children, see what is being made from e From 1 to dt1, we have arrived at t s * 1 + t It is equal to correct, please check it once first. What we did was e to the power - x / 1 + e To the power x was okay, was yours Question: What did we do? e to the power of x e to the power of x is equal to dx2 e to the power x was equal to t so t dx1 So dx1 / t will be equal to, right, where I will also write dx2 / t and wherever e is If I see power x, I will write it as t, so t * 1+ t what has come dt3 so finally it has come I want to integrate it like this Whenever integration comes, t is excluded. Do it, take the t out, so take the t out If you do it then there is a way out of it. Like you can do it with partials etc. There is a way to do this, partial etc. This can also be done with partials etc. If so, do it with partials etc. This is a This is the problem that I told you about. whether short answer type or long answer type If you have such a problem then this is very good. This is the problem, so how can we solve it? If we solve it through partials then How to solve for partials a / 1+ t This must be expressed as + b / t + c / t s is how this thing is done I am going to express it this way How a / 1 + t b / t + c / t s and what is integration from one to e dt1 t s * 1 + t is equal to this type We can expect a few questions. You can do that from this one and dtc1000 -1 so put it here so 1 is our What will happen if the square of -1 comes then this How much value has come, one has come, shortcut approach through other wise what you get in school If you are told, you can do that as well. Take the LCM etc. then the values ​​in ABC put the value of t, put zero, put one, like this Also what are the shortcut methods from here now First, whatever power is the highest, We'll catch it like from here so t to the power -1 I did not go with this, I took the highest power If they take it then what is the highest power t s If we subtract this, what will come out to be t = 0 If it comes then find the value of c from here You can put zero here here If you don't enter zero, it will become undefined. If we put it here, c is equal to what? will go to 1 + 0 meaning one has come, if correct then a also C is gone, now the only thing left to do is to remove it. The most important thing is that we have a heavy task now. Look, I've already used row, use -1. If I have done it then I will not use that value. I will do it, like look here, I took out one -1. I have used Ro, I have used him, so I Apart from that, whatever value is left is used I will do it like if 0 and -1 are used then I will use one I will use it, I will use it too - 2 uses If I do this, I will use one. So if I use one then how much will it cost 1/1 * 1+1 so this two has come on this side How much did it take to get a / 1+ 1 then a / 2 plus equals b/1 + c/1 so what from here The value of b will come, now keep it here It came to 1/2, okay, this came to 1/2. plus b plus the value here is c The value of b was equal to one, so The value is coming to me is coming to -1 equal to a c finally its evaluated Isn't it the meaning of expressing this thing like this? You can do this in partial fractions Partial fraction is called partial fraction What is in the fraction? First write the linear value. Take 1 + t then t then t then t Even if you do it with the LCM approach No problem, then the values ​​of t are three. You will have to enter the values ​​1, 0, 1 and -1. Three equations will come from A B C and that too You can do it by solving it but I think this There's a little shortcut. What's the short cut? Find the value of t -1 and put it in it. Take out zero here, put it and then two. The values ​​have at least gone out, now only the third If you want to extract the value, then some such value In addition to 0 and -1, we will put one I have put a forest, will it come to you? If a / 2 + b / 1 then what is the value of b? -1 is here, now just start the integration Two, are you having fun? Have some questions? Look, you will get length, right? Expect that means everything is easy for you. We are going to meet, I just have to go there. We have to keep doing Patt Patt, okay, so that's why I have also kept some tough ones. Easy is also kept in it, this is the value of a what was the value of a brother it is one Neither dt1 1 + t has arrived one has arrived dt1 + t has come okay then next one to e So how much has it reached till now, what is the value of b was a was brother 1 b what what was -1 okay So it was -1, meaning it would come up minus 1. dt-2425 t is there and what value will have to be entered once e will have to be added, one will have to be added to this also - l f t once e have to be inserted once insert What will come in this is 1/t of space Integration will come to -1 / t once inserted will be e and once it has to be put 1 to the power of t Now put the values ​​n + 1 / n + 1 That's your final answer here. If we put what will come out, ln1 + e came out. Okay, and the second one becomes ln2. with a plus sign here with a minus sign here But if l n e - ln1 has already come then it comes out to be 0. So let me write it here for now. with a minus sign so equal to 1 / e - 1 Look how many wonderful things have come now. Look at what is being solved and solve it. If I do this, I get ln1 + e ln2 ln2 ln2 If ln1 is equal to 0 then it If we remove it, then -1 / e comes to + 1. So this is gone, this is also gone, so finally see the answer. It turned out so nicely ln1 + e - ln2 2 -1 /e and you can also access the log properties via If you solve then ln1 + e / 2 -1 by e dis Is the answer is no, isn't it a strong question that Did you enjoy it or not, see it like this Questions may come in your long form. So I will tell you about every type. like how will you do it in short answer How to do it in single choice, finally you will Feel that you will feel after this lecture That yes, such questions can come up, right? So if I don't minus here If I had done a plus here, it would have been very That was an easy question, if there is a minus here Instead, only plus would have been written here e If it had been written to the power of x, then that question would have been very It's just rubbish like 1+ e to the power of x If you take this then you can differentiate this one It was a very correct question, wasn't it? can you take e to the power x t so what is this First, e to the power of x is t. what became 1 + t became this at the bottom If it comes then t was already there and dx1 / t means this So this is done, d dt1 means dt2 / 1 + t Now as you have read in Parshall If we want to keep it exactly like that then a / 1+ t aa went b up t came then came c up I have come to you now Shortcut method to find the value Told you if you do it with LCM approach So okay, there are three values ​​of A B C. Bring the equation and you will solve it here. So a -1 put here ki row put here In short, any value here Put whatever is there, don't put anything else. So if there is no benefit then put a forest what is coming to you as soon as you put b's The value is coming out to be -1, then integration is applied. This is also in the log, this is also in the log and this is 1 up t Integration of s -1 / t is fun like this The questions you're going to get are fun, aren't they? Now look at the next question, if this is such a question then Are you going to get this type of question? It is written as x - 1 * e to the power - x * Integrate dx-data-grid automatically What can I say if a guy goes away on his own? I mean, I'm saying that you shouldn't integrate this integrate this whom using using by parts then integrate it using by parts and As soon as you integrate this guy into your You will go flying on your own, man. I will go and see, apply it by part, by part So what happens to the left part of x? This islet rule is called inverse logarithm. algebra trigonometry equation so this The last part comes, so its If integration happens then x is empty here. and integration of e to the power of x what comes if I say integration of Power of e - x dx-data-grid e to the power of x with a plus sign and d one came b check Take a check once, the value will come here is x to the power of e right here integration over here minus x minus e to the power of my x will give a minus sign and Its integration here which comes here is already derivative one of x okay If the value is coming then how much has it come plus I'm here, okay, so look here, what's here? Look what's coming here Look, look, it has come into plus, look, Here it is written in P plus, isn't it? What was written there, it was written in minus Look, it is written in minus there. I told you that this cutting will happen, cutting will happen The cutting is done, otherwise it would have been done If you have to do it again after adding it, then see this And this part of yours will be cancelled. Look, this ends in plus and this ends in minus. This is the only answer left, meaning by part one You will put it on automatically, I am telling you it will be over. will go to - x * e to the power - x + c equal okay this part is here okay straight away Integration is direct integration No, as soon as I grabbed one and applied it, Immediately you will see that whatever comes to you goes straight Look at the cancellation coming out. integration so see it cancelled out If it is correct then what is the answer - x * e to The Power - x That's right, now it's a question I'm going to give you homework, this is it. Homework number two follows a similar pattern. ok so I'll tell you the pattern see what I have to ask this question, how will you do this such a thing The question is which is yours like section Is it included in section B or C? can be used for this question what There is a criteria, nothing to do, see Let me tell you the easy way, see how to start. I will keep giving you hints so that you Solve it and if you can do it then here What is it, look dear children, from 0 to 1 e to the power of x is what is x / x + 1 time is squa dx1 so what does the little trick call it + 1 And add -1 and see it gets smaller. Look at the cleverness of the small trick, see from 0 to one to the power of e x look here this little chala If we say that then add + 1 and - 1 here Look, this is a small trick. Look, this is a small trick that happens now. As soon as you integrate, look ahead. You see, you do, from crying to the forest now If you divide it separately, how much will it be 1 / x + 1 is - with the coefficient 1 / x + 1 The square came and you will integrate dx1 The man will go walking on his own, I mean I'm saying it like you You will start integrating it from 0 Taking e to the power x / 1 + x then in What to do is apply pie part as soon as I am saying if you apply by part This guy will not go walking, he will finish walking. will go like we just dealt with e to the power x This is going to be dealt with like it was just dealt with now I asked a question a while ago So this will be dealt with, so by part in this What should I say about applying it, applying the left part and tell me what the answer was okay so We have also told you what the method will be. Now what will you do here after this? Will you apply, use the by parts, what did you say? Use the by parts that islet rule right Evers' Logarithm Algebra Trigonometry and expo angel ok karna karna to If you get the answer to this, then this is your homework number. Two has come, okay, is this type of pattern? Now comes the turn of differential equations. What happens in a differential equation? So the unit in the variable separable will be either If we get into the linear differential equation then will come in a pattern of homogenous either order okay to ask about degree etc so now What happens to us in variable separable is like what happens in homogenous Let's replace our Lada with another Ladawa and the one with the linear differential equation We have a part in what we have integrating factors etc. had to be calculated This is the equation that has to be checked. This is not the part that is straight forward You have it written as variable separable See how to find out You will not get a little more in the format If you have to create it then the properties of the log etc. like l a + l b l a equals l a - l b ie l a / b so this It is going to be used a lot, isn't it, like l a + l b if it is written in plus then what This will multiply a lot You are going to be used a lot, right? l a - l b is equal to what will come to ln2 in a divide by b is equal to minus Take it there and see what happens. is equal to what came c is now equal to l x + what is l y l a means what will come l n x equals y = c then xy2 to the power c now e to the power c what is c It will come means xy3 has come, it means the answer that is coming is If option B is coming then B is the correct one Whenever we ask this question, you think the option is correct, isn't it? If I ask a question, then my grandfather should be such that he just looks at me. We are handling it like a movie. Come on, a movie is going on, not a movie Solve the question of what you have to do We have to keep doing it, okay, let's come to this now. This is a differential equation, this is also a variable. The unit of separable is neither variable nor separable So this is homework number three, okay? You will get a lot of homework. Keep trying, we will keep telling you. to give you a hint so you know That yes brother, how to solve this How will these variables be separable? Separable children see how the variables are separable I found out, look here, look at this What is being built dydekop.org e to the power x * e to the power y + x s * e Power y came on if I could isolate it like I'm able to separate the y ones and If I am able to separate the x one then it is called are variables separable is correct so what do I Look, I can separate dydekop.org e to the power x + x s * dx5 see integrate it now do it Will you be able to do it or not, tell me quickly, will you be able to do it? No, the variables are separable, not easy e's Power - integration of y will give e to the power x integration of will come and integration of x s This is non-age variable separable. Parts now come somewhat reflexively and Transitive par ye hai na which functions In the part that follows, there are some more questions than this. You will see this many times Mainly people like these types of questions Sometimes he asks for the total number of Reflexively number of semantic relation vo I will also tell you one one and oneu function How do these questions come? Isn't it true that you will see it sometimes The one that is last, not the paragraph one The problem is that it can be captured in You must have seen it in the sample paper. When we discussed it, this came up in it. The part is of this type, right? So this is can be of the type like total number of If relations come then know those results It should be okay, it will make it into a story type. This is a case study type thing, like here. It is written Consider the relations, this one If the relation is given then you just know should be reflexively to always be what Should be a pair of a and a, right? reflexively, it should be so If it happens in extra then we will not get any relief. It's okay if I say the same thing In identity relation, what happens in that That should be Emma A's pair only. It is a must, it won't work otherwise. what needs to happen reflexively to be a All the pairs should be there, see here are there and it is also visible 2 2 is also visible Yes, 33 is also visible, it is visible, isn't it? These three parts are visible, meaning this Reflexes Nader Cymatic Knot Distributors look after it well Reflexively children see it is written but 21 where is it written to, it is written but 2 and to It is not written now when 2 and is not written means All pairs should be a is b then b is a If there should be a pair then it is not symmetric So for now, we can say this that reflexively This option is available, isn't it? This is also an option. Getting one time for transitive also Let's check for transiting what does transitive mean pair of a b If b c is a pair then a c must also be a pair I want it if it is a pair of a b, dear. children and if there is a pair of b c then a c in it There has to be a pair of then this is transiting and if all three are It happened reflexively as well If it becomes transitive then it is called What is called equivalence? are reflexively and transitively What's the difference reflexively Emma A pair of It should be mandatory, apart from that get He has no problem going but the cemetery In the case of if a is a pair of b then bama a Must be paired or else it will be cancelled He will go there, so this Vanma is 2, so 2 1 If not, it's gone transiting like look at it there is 1 2 okay then what is the second one is 2 3 so then second what has come 1 3 has come how has it come to And when two is finished then the pair of a and b was b If there was a pair of c then it came, then it means this Transitive is coming so if best If you select the option then this is what is coming up, isn't it? reflexively if any of the best If you select the option It is not reflexive, anyway we get it This option was gone because in single choice It is going to come, isn't it? That's why you have to do all this. This part should come, it can come. Like maybe this is covered in singles. Or else it will come in passage type. I can't do all this in the long run. You will see in long etc. Do you know the parts of the integral? So the probability part will come long etc., let's talk about integration The questions came in a slightly longer pattern Okay, so this is how it can be done now. Dear child, as if you have no one I want to know on to find out many minds If you want to find out, what is the way to do it? See whenever you want to know one by one It is one thing, see how it goes for school. Vanav whenever you want to know Vanav function then Vanav What is a function, which is that function? occurs when the value of fx1 is less than the value of fx2 is equal and x1 = x2 then it is called one One way is called forest, so this is The second way is to take its derivative derivative if derivative proper proper is positive if f ' x proper is positive Is it proper increasing or proper Decreasing is proper strictly Increasing or strictly decreasing then In his case, it becomes one, isn't it? It is proper, isn't it? Proper means proper means like this. Not monotonically increasing, strictly It is definitely increasing so x1 = x2 The third case has arrived, the third case becomes like this. Draw a line near you now parallel to To the X axis, we asked this question I also got it done, will you check it when In your sample paper, there is discussion in it also called line parallel to the x axis Do you want to draw a line parallel to the The X axis has to be drawn if it is more than One point pe cut is more than one Then this one one will come at the point. Local maxima also track from local minima It happens but if we see this much till now then See cute kids to check one one out What will I do when I see you here? Whenever the question starts, the domain and Codomain Look, look, this part is called domain. So this is called domain, okay, this is domain because you don't remove the codomain You can read whatever is written on this side. What is called codomain? This domain is called codomain. what is this codomain is this fxxx.pro So what if 2 equals 3, 3 cancels out? What is the value of x1 when it goes to x2? If this result comes then this will If it is called that then it is so, this is another way also Differentiable so fx3 came if this three come What does it mean if you go, what will happen to you? This three has come, what does it mean, it will come There is always an increasing part, so This one one has come now sir how do you know this is on two Let's look at the on to function to find out What is the method for on to function address to do on to function if you know If you want to do it, what is the way to do it? Codomain is equal to range if The codomain is equal to the range if If it goes then it is called on to fashion scenes The domain of the given is a real number If the range is also real numbers then the domain You can't just remove the range. You can remove the domain given in the market You will have to write the domain on this side. If it is given then it is given, this is Dominion is given, now what do you do? You just have to find out the range, you have children, right? What will be the range of all those figures of y = 3x will go as if the domain is a real number in it You will keep putting in all the inputs, what would be the range? When the output comes, all the values ​​come. If I go then I will take out the range, dear child That will be a real number only, isn't it a linear function? Isn't it a real number, so the codomain What does it mean if it is equal to range? This is the function that will come on both This is a one and on to both function. The biggest thing is called the forest. There is another name too, dear children, what do they say? what is it called injective function Injective: Keep all these names clear because This happens, these come, what do you call this? One is what is called an injective The function is called a mapping function. Yes, isn't it, and if I talk on to What is this called, Serge, of function? What does Actin say, Sergey? What are the functions of actinides? which is injectable and surgical So he is called the best, who is he? bijective functions are bijective meaning what will happen should be wps on to The return of these comes, right, if I Tell me whose Evers is he? Only the inverse of the bias function does f inverse x which is called okay So that's the story, okay. Thorates relation that can be defined on the a a to a is then Reflexively related formulas etc. If single choice etc. comes then You should know okay Reflexively or semically, two things only. If you see a transitive figure, It is very tough and for equivalence it is different. This is the only method to solve it, so mainly Those who can come either reflexively Yours is for the cemetery, right? reflexively there are results then Take care of that and see these are the results Look here, he is writing and it is shown. Okay, so here it comes reflexively. for what is to be extracted reflexively So this happens reflexively for 2 equal to the power n s - n reflexively If I talk about your SIMATIC So this relation comes out to be 2 to the power n * n 2 to the power of n * n + 1 / 2 ba 2 so it comes to this if I talk Let me do this, for example, suppose there are three, then three For reflexively so Reflexes mean 2 to the power of 6 So this means that 64 will be the relation which is 64 which are reflexively if this If I talk about the cemetery, how big is the cemetery? If the SIM goes, it becomes reflexive. What will happen if we talk about cemetery? is 2 to the power n * n + 1 / 2 the value of n given, it will be four, it will be five, which okay so how much is coming n*n + 1/2 means it will come to the power of 2 3 * 4 / 2 will become how much 2 will be If we get to the power of 6, what is the answer? 64. 64 relation symmetry also has 64 reflexes If you know the formula then you will do it, right? I got these formulas written down which are very important. Some of the things we've been reading are very All the things you can expect Now, as this question is written, this question Like you who are single, who is single I can also expect and that which Meaning, what is expected even in multi-choice ones You can do it and those in section B You can expect short answers in that too. Type K Bola Number of the Equivalence Relation containing what is equivalent Relation has to be explained in which one becomes two Containing is many times such a question comes up Bar Bolta Hai Symmetry and Transitive Containing is many times it will say SIMATIC end transitive but not reflexively is reflexively but not a transitive so two elements it gives you sometimes For example, 1 will give 2, 2 will give 3, 1 3 He will give it, he will give it, he will give it two-three times like this Apart from this, it will ask how much minimum is required. There is a requirement so that this equivalence Once the relationship is established, he will always use it. Minimum word, you see, minimum word like this Expect in Type Key Problem Case Study This can be done in the sample paper If you want to see, it is written 1 2 3 in it. speaking of the number of the equivalence relation And I told you about the equivalence relation. What is called a relation that is reflexive? is transitive what are those three called Both of them together have this brotherhood The equivalence that these three brothers have is if these All three brothers got together, RST brothers, so this What if we find all three brothers? What will be called equivalence? Equivalence will be called R ST means reflexively and transitively okay Now he said that he definitely needs Vantu. This is mandatory for me now. Reflexively, this is what you want. I need 1 as well, I need 2 as well and 33 as well. Apart from this it will not work These three are needed reflexively if I If I talk about SIMATIC then this is the meaning of a b I definitely need a pair, it's inside me. It is packed in the box, besides asking What is needed for the cemetery? Dear children, I would say that if there is 1 2 2 1 will have to make this mandatory There is a requirement, see more have come like 1 If 3 comes then 31 will also have to be entered. So that's why whenever you use it, see The word minimum must have been written as minimum, right? Minimum here too, word miss here If it is there then it will come with minimum written because otherwise Then more equivalents can be made, right? So that's why the word will always come in writing if It is a containing question, meaning it contains Minimum word use ho ke aayega r ho gaya s ho Now let's talk about t if it happens in this If I am talking about t then it is okay what did I tell you a b b a c If there is a pair of a then a pair of c will have to come, okay If I had said a is b then b is c then a is c A pair of will have to be found if both of them are there. Neither if dear child I talk like 1 is 2 and 21 has also come, so 1 and 22 Both will be formed if 1 is 2 and 2 is 1 So combining these two will make 1 and 2 2 It will also be made, so both of them are already there. So, it means this is already proven. Meaning, who has such a minimum requirement? Who needs even 1? Requirement is 2, requirement is 2 33 are required and 21 are available. If there is a requirement for four elements then like this You can expect short questions. in either case studies or single ones I am somewhere in these three Expectations can be made, right? If you are able to find it then from what we are telling you Some things are going to get stuck with you. You understand that this is here only for that. It would be fun if you get exactly like this in the end Exactly the same type, okay, so this Now the number of equivalence relation is done When his turn comes, he asks many times asks for the number of one function etc. or Into function asks what do you say sorry? Many one asks one one asks or Subjective will ask, okay, so this type of Questions keep coming one by one, right? I ask these types of questions, you'll see It is also there in NCERT in RD Sharma etc. It is not there but it is there in RD Sharma. There are chances that he may come in single. I want at least the single ones who are If the type comes, you will tick it, right? At least you know the formula. You won't miss it, like let's say your number is off. The reflex relation of 2 to the power of n - 1 Sir Sa Symmetry Ka Aa Gaya 2 to the Power of n * n + 1/2 then you will know that you will do it immediately, no You know that we will be sitting here, friend, tell me something. It is there, the work that you tell me will be done there, right? No one will tell you beforehand, will they? You will have to go mentally prepared from there. So this is about the number of the surgeon Speaking of Sergey Ektin from A to B. Surjecting is to tell the surgeon that codomain equal to range ho jaye hai na surge If the action becomes equal to the range, then first I Like I will tell you for the sake of one, for the sake of one Isn't it true that one is for one, what happens to the forest? Isn't it one for one if here dear Children, if there is an r element, isn't it, if here For example, if there are r elements in a, then r r It's an element, okay, it's an element, look at everyone. and if n inside b If there is an element then tell me what is the result I am giving you that there is an r element inside a. And inside b there is r element isn't it and where n is greater than r is not good condition made We will take three conditions and make a way It is possible that the value of r of n is greater than n is the value of r is Greater than n I am telling this for one forest Okay, one way would be to find the value of r. becomes equal to n okay then the third method If the value of r is less than n, right? This is the only way any other way can be There cannot be a way that, like there are three element, it has four elements in it There are four elements, there are three elements in it. It has two elements It has one element So how do you do that if r is greater than n? If its value is more then the answer How will zero come like here? You've got five, imagine that here. But suppose five elements come and If three elements come in it then that mapping Like you will have to do it with two of them together. Here is 1 2 3 4 and here is a and b So the forest 2 We will have to make it like this, then Arrow Do You will have to catch it, otherwise it will not be made. If it is equal then there is no problem. But if the same means more on that side If the element is more then what will be the answer Zero will always come, so here is the direct answer. Write it equal to zero and it will become zero. Number of one one function asks so possible If he speaks in this then n factorial so this will come in n factorial If it says that if n is greater then In his case it will come to n s * r factorial right pick up r from n and then Get this mapped so n s * r Factorial, so this condition holds when r gr is n or r is n or n gr is r if If such a question comes up, then it's okay to assume In this question, if you assume it is written here hota surge ectin likha ke injective lika It would have been written like this, yes na find the Number of the injective function from a to b a has elements 1 2 3 4 and b has Inside there are elements a and b and said Tell me the injective function, what is the direct answer What does zero say? The direct answer is zero. Why is there zero here, r gr n, isn't it here? There are four elements here, there are two elements here It is not possible if b was written in reverse. And if a was written here then the answer would have been 4c 2 * 2 factorial is greater then this side If it is more then the matter will be settled on this side If it is more then answer directly, answer directly. Zero comes if I talk on to Is it a function key or not? Have to look on to function for many There are many ways to do this, right? can be eased how to eased You can go like think, imagine here It will come 1 2 3 and then okay come here will equal one and two so if I talk What is the total number of functions? of functions then the total number of functions How can we do something like the total number of functions? If you have to remove it, what will come with the forest? What could be clubbing of one and two There might be clubbing then clubbing with two Could there be clubbing of one and two? What could clubbing with three be like? What is the clubbing of one and two to four? It could be one and two, meaning if I say total No. of function what is coming total Number of function will come total number of function then pass option for the first one How many options does the other person have? How many options do the third one have? The fourth one has two options, see one. One and two's club with two and one Two with three and one with two four How much did it come out? 2 to the power of 4 came out. This is the total. There is a number of functions, if I say this Here we have n elements, here we have r elements. If so, tell me what is the total number of functions If the total number of functions comes, then the total number Off function those who will come will come you have n to the power r of the first one Pass r option to the other one then r option then the third one has r option then n that is, it comes to the power of r, you can write this, right? Yes, total number of functions is this Your one function three criteria now I'm talking like, let's say, on two. If you want to find the function, then find the total. Remove the into from it, which into will be functions whose codomain is equal to The range may not be coming into him If we call a function, look at our mapping It got damaged, mapping is happening in it but How did I mess up a mapping here? Look, this will not come into, look, this, look, this It will not come, it will not come, look, it My possibilities got spoiled because This into did not come here with pay to The arrow was not made, look another arrow has come. Two has come, three has come, four has come, come here One went and two came, look at this mapping mapping Mapping Mapping Mapping Look at this with One Here the club is done here with pay to If the club is done, then one each is left out, two mappings. It is such that it is into everything else. There is no problem in the rest, just accept it. Done with Oneav Done with 2 3 4 2 So then that code equals to range came to happened with 3 42 happened with all of them If we add 16 in combination then it becomes 2 Power 4 - 2 then what will be the answer of 14 You can expect a question like this Yes, you understand, we are asking the same question. Which is a bit of an expected run, you don't do this I think it will be a very silly question. You have seen it in Physics, right? If I see it, anything can happen in this game. So I think this, oh friend, it is very easy. The question is coming, it came in chemistry. If ours comes then the game can be changed for us. Keep all the things ready sir, all this is in JE. Yes, definitely all this is in JE but Your RD Sharma also has it, right? The game may change, so we try. I am trying to understand that whatever good delivery can be done You are doing this because if the paper is tough then you If you are mentally ready for easy then everyone will do the same. I am ready, the tough paper is here, I created a ruckus. I gave it to you friend, I gave this marathon. We know all this, if you can understand then that is it Looking at questions that are a little expected which can be rotated a little, meaning We are looking at it as per CBSE. Okay, I mean, play some games with yourself. I can't even imagine, now it comes to you. I had to remove the pass f f, I had to remove the fog f is a given to you - 3 - x to the power 1/3 f f there are some questions, look at them a little I want to do that part in it. Because see, these are easy questions. You are doing it from somewhere or the other. You must have got it from somewhere, right? You can go if you want to do something different. So you need to level up a little bit. If we want to score 80 out of 80 then we will have to You will have to think a little while moving up the level. CBSE will have to decide how far it can reach. So we are carrying only that much, not even this. That we made the level very hard. No, we have to level it only to the extent that Meaning of yourself as can be expected If the paper is hard in some of these then Expect a lot of questions from this. What can you do, what can be done How can I turn around so many times? I saw the sample paper and then I After analysis, I have come to the conclusion that yes This question can be asked to children Definitely all this can be given, okay So like fxxx.pro What is 1/3 raised to the power of 3 and 3? It got cancelled, see how much has come from here 3 - 3 - x to the power 1/3 and 3 3 so my If it gets cancelled then how much will it cost? If x is raised to the power of 1/3 then what is the answer? x to the power of 1 is here, isn't it? x to the power of one is here. If it goes then it can come in composite fxxx.pro 3 is equal to x is correct now Your cemetery etc. comes and what is this symmetry called See, symmetry is any matrix, any is any matrix of any type Keep in mind that any type of matrix is The Matrix is ​​a given if you Forms of Cymatic and Skew Cymatic If I have to write then how does anyone write it? matrix a and you have to express it as SIMATIC Plus Skew Cymatic so how is this Express Karoge a P A Transpose Batu a - a transpose ba 2 with a plus sign It is very important to keep this in mind Things can come anyway, so this is called What is this called a symmetry matrix? What do you say dear children, cemetery? This is the symmetry part, what is it called? Hey dear children, this is called screw Is it cymatic, isn't it fun or not? This is the cemetery, this is the Skew cemetery Funny Funny Funny Will Ask Many Times Symmetry will say many times Skew Symetic There's nothing to be afraid of, there's nothing to be afraid of There is no such thing because there is symmetry. is a + a transpose but means p The value of a + a transpose ba 2 q is The value is a - a transpose b a 2 so q I am asking q, if I am asking then take out the q first. transposer like a you know dear Children look what is given a what is given 20 5 and 4 transposer nil lo kaise Transposing rows and columns by reversing them By changing it to 20 5 and 4, what did I do? Tell me what is skew symmetry a - a transpose ba 2 so what will you do a - a Transpose, you will put minus, say, two minus Do it 2 - 2 0 0 - 5 -5 5 - 0 5 very Good 4 - 4 0 Very good what will you do now If you divide this by two, suppose I would say that symmetry gives a + a transpose. If I asked, I would have asked, otherwise why would I take it out? So a - a transpose ba 2 chch is equal to equal to q now divide this by 2 so 0 arrived - 5/2 arrived 5/2 arrived and 0 arrived This is our ultimate answer, see, it is yes. Ultimate This is our ultimate to zero Zero and this and anyway if we skews Talking about the symmetric matrix, in which The elements that come along the diagonal It always comes to zero, whether it is 2 cross two Be it 3 cross 3 or four cross four Whatever elements come along the diagonal Those who are going to get zero are going to get zero and the rest The addition that will come will be zero, like all these. If you sum it up, it will become zero, right? Matrix is ​​symmetric, symmetry means Then the same pattern will be written, this will also come This may also come, so check that now. You will have to do it, right? So keep these things in mind. Keep that if Simetic and its came then fine Now let's move on let's move on There are many problems related to geometry. An easy problem is written like this You can expect those two numbers. It is going to happen, right? You can expect it. I have to write the equation off line many times. It will say parallel to the root axis is parallel to This is the x-axis plane, let's say parallel to The x axis is parallel to the y axis Nothing to do, what equation do we have? Write the equation of any line What are equations in Cartesian form? x - x1 / a ok then what comes y - y1 / b Then this comes z - z1 / c = after fighting This is equivalent to the equation that is at this point From which point are you passing? From which point are you passing 111 meaning Find the value of x1 y1 z1 parallel to the z Axis means a0, b0 and c There must be some value like alpha or something. If the value is x then it becomes x - 0 / 1 Sorry it became 0 Sorry x - 1 / 0 it became y - 1 / 0 come z - 1 / ba take some a e equ I fought because the value is not given. The z axis which is parallel to 0 0 3 will be 0 0 5 would be 0 if x y was written then its In case it means a b 0, right, here it is just z axis is written z axis parallel to z axis parallel to z axis swung parallel to x axis was written so here x Okay, if you understand then write it down x - x1 times y - y1 this is the equation point You will find these further things in the form I am going to meet, now it comes, it is continuous, right? Continuous This question is very important It is very important, isn't it? This question can be expected We're asking the same question that a lot It is expected, it is very important This is it, so this question is a little expected. This is about continuity and differentiability. What is the continuation function whenever f a = f a comes from I and f a is equal to If it comes then it is called continuous function. goes whenever f a e f a+ comes isn't it and If e is equal to f a- then it is called is a continuous function, isn't it, and if the limit The right hand limit and the left are not equal. Hand limit like this limit itself is not equal If I miss the function then it gets discontinued. If any of these three is not equal to If it goes then it will be discontinuous, now say this is y equal to what is written mud x All the functions with mode PMD x - 1 They are always going to come continuously like Mode of x is x Mode of x - 1 or Mode of x * x - 1 do it okay all these continuous Coming soon, how do we do this? You are a child, I would have told you its graph also. How will you graph this right now? The answer is that row and one per continuous Hai means sorry row and one pe ye continuous Yes, this is continuous, meaning overall this Continuous is Modest Function Yes if If we talk about differentiable then this non Differential aa jata ro and one pe sari We will discuss these things on this question. Because these questions are also there in the sample paper. tha okay so let's come to this that y e two what was written mud of x + mod x - 1 so this How is it done? Many times it got a minus. He gives it many times and puts a plus sign to scare me. There is nothing to be afraid of No, because it is very important What about the one who is very important? You have to look very carefully at all Hey, what is this question? Is y equal to what? It is written mud x + mud o x - 1 so what is this How will we know where it will break? It will break at the point, so equate it with x = 0. If this breaks on one then it will cry and this on one It's going to break isn't it, mud x always what It breaks on the row and mud x - 1 If it breaks at 1 then it is broken How do you break this down first by x < 0 and then by one You will break the bar between 0 and then Once you break x > 1, it will be divided into three intervals. will be broken into x < 0 at x from 0 to between x > 1 and x < 0 Both will turn out negative, meaning like this By doing minus and minus, both these are negative will turn out to be negative and when x > 0 then this So it will come positive but it will come negative. This will come to x + 1 with the same result. What will happen in his case then? If both test positive, then first what happens two minus then plus minus and Plus plus it breaks like this okay so now If you solve this then see the final conclusion what will happen if -2x + 1 what will happen if x The value of less is 0 then else if else if one When will it come when it is between Ro to Va So from here to the middle of the forest, this came and if x > 1 then what will come to be 2x - 1 aa its graph which always comes is boat A graph of type boat boat hey boat comes You must have boarded the boat, you must have boarded the boat now. This is the same boat, now I have to make it a good one. Also, the graph of m u x + m o x - 1 is given by It will be made now, it will always be made, how can it be made now? It is made here, make zero here One make this is the Now this is the graph, this is the graph. Graph, this graph has come, this is proper You might be seeing the continuous function. I can't see it, look proper continuous If it is a function then what is the value of this -2x + This is the graph of 1. What is this? The graph of 2x -1. What is this? This is the graph of y = 1. Just finished. This continuous graph is visible somewhere It is not visible but if I talk If I talk about differentiable then yes here non There is a difference here and here It will be non-differentiable anyway you check it too Can you differentiate? What does g say f ' of 0 - then -2 will come differentiable If I take it out, this forest will cry. Differentiable so to will come and ro will come So if I talk about Ro and One, what It's going to happen if you assume to tick If it comes then do it immediately, okay, so if I'll talk to you and your lovely children on One. What is coming will always come What's going to happen to Continuous and 0 and one? Is going to be non-differentiable I understand If the thing is coming then if it comes to ticking Here comes this question, look, I'm telling you. Which question if you came back after writing it like this? Instead of doing this, accept this question. It has spread in the market and everyone knows about it. I have already written this question, suppose I have come here with this question. Dear children, I came to your attention after writing equal to See, write m x - 1 + m x - 2 like this K came and said find out this continuous whether it is there or not and whether it is differentiable or not Neither can I tell whether it is continuous or not. Is it Continuous or Not and Differentiable Is it not to do nothing, to do nothing I may write 5 or 10 in the story of the mode plus No problem, it will be continuous. Always Continuous and the graph that comes out of it I know how he is going to come. The graph that comes will be the same, the same boat graph. So this continuous is going to come if I If I talk about differentiable, then what is this It will be differentiable, it will be known where. Its non differentiable it will come from Ro Equate this to rho. Equate x - 1 = 0 x - 2 = 0 then the values ​​of x are one and two. what will come out non non non non non x = 1 And what will come to pay, non-differentiable It will go just fine, look immediately and accept it like this I have come to tick, do it immediately Continuous at every where one and two pay then Continuous is about to come, it is a matter of tonning If you don't understand, then accept it as it is. It is written like this, tell me you made x - 1 mod x - 3 and write it like this and ask Continuous Bow Continuous Always Continuous Differentiable Hey, I'm telling you Just make it equal to zero and tick immediately. It has to be done, once you have the easy information then It is easy, if you don't have the information then it is too much There is a problem, isn't it? Let's go d ba d one and so on. Let's take out some cheap questions too. What is written inside the log? It is written under root 10x dy4 lag what is written as square √ 10x so If you have any problem with the first property of the log, etc. If you are still having trouble then try to make it a little easier brother. It has natural properties which are What is inside the log in front of you? will go to t to the power of x what's 1/2 what's 1/2 So take 1/2 down, how much is 1/2 I went here and how much did it cost. I got 10x off. So, x to the power of n will come down to n. What is the derivative of 10x? So dy2 will come l what is the derivative of 10x There is a thing called the chain rule. The chain rule If we apply the derivative of t x then log t x what's the derivative of 1 up t x f t x How much will the derivative come from s x where We have to find how much 45 is 45 45 then x = pa Ba 4 pa how much will it cost 1/2 t pa ba 4 How much is 1 and how much will it cost, t to 2 If it gets cancelled then how much will the answer be? equal to ok what chain is installed in it Rule applied chain rule so ever f of g is one so what if we have differentiable Comes first, take it out Sorry differentiable first f off Whether it's gx3, four or five, it's something to be afraid of. No, we will do the outside first and then the inside. The more people do it, the better it will be for the people first. What is the derivative of log 1 / x? Then the derivative of x is equal to 1 / t x * s x is there right so all these if if questions It came in a very easy type of way, okay Now comes the next bit of x + y = l x + y Then the same chain rule will have to be applied, right? And this will be your homework or something. So you will also do your homework, right? So this is your homework okay this is your homework tell We are checking whether he is alive or not. Isn't it so? What is the sinx-cosx derivative of one y How much is the derivative of dydekop.org Who knows, Alpha Beta Gamma, maybe it Can be expected in singles Alpha Beta in single ones see big The expectation in the one Is there an expectation of probability or not? Then the expectation from your integration If so, then this type is either from AOD There is expectation, okay, so if I talk like I deal with chapter seven I am just telling you a line from it. Makes an angle of alpha beta gamma again Value is saying that alpha cos2 alpha + cos2 beta + cos2 ga see in this He keeps changing the question, sometimes he says that aa ps beta + k s gamma sa s aa + sa s beta + sa s gamma ever cos2 a + cos2 Beta, there is a relationship, we have it, right? Is the relationship a very important relationship? The relation comes K S F + K S B + K S Gama, this was seen anywhere in the market. This is always one, isn't it, always one, so this What happened brother cos1 alpha + k s beta + k s gamma = 1 var alpha beta gamma What makes the angle with the x-axis? axis and z axis if this is the thing sir If I want it, what will come, just this Replace it with what if you want it. What will you do with this also? Replace this, what will come 1 - Sa S B Very nice what will come 1 - Sa S gamma = e equals 1 1 1 add take the sum Take it, add it brother, add it, it has been cancelled. So 2 is equal to what will it be, 1 + 1 is 2 goes sa saf + sa s beta + sa s Gama, if this comes then make it two in it Only two questions come, one is this and the third is this. This is it, now what will you do with 1 + k alpha There are formulas for multiple angles, you will apply them. See from the strip what formula it is. Do you know what alpha is? It is is 1 + cos2 a diva ba 2 very good then Similarly, if I do k square beta If I talk about what comes out to be 1 + cos2 b / 2 Come, if I talk about Gama, he How much will come 1 + cos2 ga / ba 2 come na is equal to 1 now solve this will do 2 will come in LCM, will say to two Come on, go, go there, that new and one will go away How much will you get after getting this, it will be a lot. Well look, the 3 Plus is here. cos2 alpha ps cos2 beta p cos2 gamma send this thing called λ equal to equ2 over there If you give it quickly then cos2 alpha ps cos2 beta ps cos2 gamma i equal to kitna aa will go -1 -1 -1 -1 -1 cos2 af + cos2 beta if this comes then to Do it, if this comes then do all three questions. Variety has been told that variety is important. Bhaiya important variety, variety is very It is important that this is an edible variety. Variety is very important, isn't it? That's why stars are made, aren't they? Can be expected in singles Isn't it any of these three? Any one can be expected now Every one gets a little probability You must be looking at the question which is a little Expected, do you understand those seven chapters? I told you about starting intake from me, intake differential equations starting at I told you that I am doing it with those four people. Yes, the other six are there and they are doing it sir. Now it will be like A and B are the two events. It is written that this marking raises many questions. Especially these questions are not expected. Sometimes the question of independence will arise and sometimes Conditional questions will arise a lot. more and this is the question that you say I was probability and I should talk This is your question of integration. I can expect big from the big one The big number in which it is there, in which A little bit of calculation may be required here. What all are given, dear children, look at it. What is the probability of a person being given Given is 2 Probability of b is given The probability of a is then What is the probability of a / b? Tell me something Do not do nothing Probability of If you want to know a/b then how does it happen? See whenever finding the probability of a / b is conditional here it means is b already done what is the arrival of a There will be probability, this is probability. What comes in off an intersection b upon The probability of b is Probability of a difference from b upon what is the probability of so if my Probability of producing cute babies of A intersection B so how to Dear children, how will you remove this? The soul or small soul is placed, it is has a probability of a plus a probability of b minus the probability of a intersection B toy will take out from the soul an intersection Look what a mess B has got, it's all settled. ok poof has come pattu has come very good This is coming out to be zero, isn't it going to be zero? So the probability of a intersection b e equal to 0 means a and b are the Disjoint set up mutual exclusive To whom it will be said that this cry has come, it is over now. Brother, the above row is done, a intersection b then a and b r the jo tumhare ho gaye ye a Intersection b 0 came, meaning this mutual exclusive event done okay right So it's very important to write this down. This relation has to be derived between a and b, right? The probability of a / b is whenever written So what is the probability of a Intersection b your probe this top questions I am expecting it, can I ask anything? If this is okay then next comes Baby, whatever comes next is your order. Let's also ask a question about the degree of order. over the degree so the order over the degree I have expected this question to come up, right? Questions of this type of question order degree Take it some other time and it will come in single version if If you have to come then either those from section B I am in section B, I mean all these are short which came in the category of long one Let us tell you that it can come in the long version. Because there are only five sections here and there. Turning into a case study that is expected That can be done with a little relation It can be expected that the relation is No, who told you that the total number of what functions can come either pick on that Will he take it or will he take it with someone related to you? Relation means how many order pairs It is not necessary that it will be made etc. If you want to make equivalence then keep Vanma to The one after that is a little bit It can now be expected that Dear children, what to take out, order and I have to tell you my degree, see what order will come. It always comes, the degree may or may not come. It could come if he gets Paulino's form. It is good if you can write, you will not be able to write So don't come The order will always come, whatever the order will be Highest order will be highest degree Highest Differential Equation to Order Always comes, always there is order If it is coming then what should I order for this thing? It's like look here it's written in the forest If this is written in two then whatever order comes will be in two Order is fixed now if If I had said the order, I would have come to talk. Degree of Degree Will Be the Highest Power Of the given order highest power if him It is not necessary that you can write degree If this Paulinot then great like this Paulino is visible in the form of d ba DX6 is coming, hey, we can't see that. This is the confusion that a person gets into. I am getting a six, don't watch it. We have to look at this market whose order comes If only his power is seen, then If we talk about order, then two will come degree If we talk about it, then two will come, this 3/2, so anyway will also not come because in degree fraction They do not come in orders, they come in fractions. Your natural number will come one, two will come three will come okay so this is same as b The market is completely spoiled. Okay, let's do the question that I am making. Those who make them, it seems a little that Some things like this may come up, okay? Now, let's take a little bit of our probability base. Let's look at the question a little bit more, okay? The question can be slightly expected. Sometimes it says contract, sometimes it says Is Stipe's question whether the same person will speak? It may come a little bit, that's okay, questions about this What if he comes or he will say the same thing I will do it brother, I will either tell the truth or will say truth and false will say contra Meaning it's the opposite of what I'm saying. He will say that he does not listen to me. I say that will not happen at all, so that A little expert can ask this type of question. Whatever has been said, speak the truth. that its probability is 25 Meaning the probability of a that is going to come is How much will it take to equal 3/4 of its a Compleños The probability of speaking is very good very nice very nice now this way is saying the probability of B telling the truth what is the method for 8/8 ba 100 means 4/5 If he comes and does not tell the truth then there is a probability of B Compliant 4/5 means how much did you get, 1/5 Now he has said probability. State which contradicts the statement My O contractor means both brothers Opposite working means one true said So one false said means it means If we say the probability of a, it will say False only then the statement will be complete Then the probability of b telling the truth And he said, "False is over, this is the answer." Find out how much it is, this is 3/4 probability of a Ye Hai Compliant The probability of b has been found, meaning how much 4/5 is the probability of a Compliant If you do it then how much will it come, 7 / ba 20 so 7 / ba Do you know what 20 is? 7 divided by 20. is the probability that they are You must be doing contrast, meaning he said the truth. So that falls, if it is said to be true then it falls, then this It is said many times that it is contrary to the dictum. That saying the same statement means it is true. If it goes then falls would have formed two categories in it Yes, this question is very important for everyone. I expected this type of thing, right? Now this is going to be some homework for you. Some homework will also be done on continuity and So this is your homework over differentiable. Very important question, very important Question one, you have mud x + mud x - 1 Tell me, I had mentioned two-three varieties out of that. This is also very important mode anywhere. Show me the mud, sir, I am scared. The question of mud is going to come up, isn't it? It seems that where one feels scared, that is where one will come. It is in the question, that is what is going to come, okay? There is nothing to be afraid of if you see this. The soul inside breaks down as soon as I see the twist. should x one x is positive - one one x is Negative oh ho oh ho it always breaks If someone doesn't know, he can make and break stars. If it looks like stars then it will have mode x mud x n x is positive my a n x is negative then Breaking it means once it is m x like this Can't break, can't define, define will have to be x positive times x positive with which will come x s will come - x s a will go when x is negative will come to row plus You will check and then once again on the row minus Differentiable by the way these continuous and Differential will come, check it ok I want to tell you this won't work, oh yes, you're watching Yes, isn't it? Let's look at the second question. The second question is about ad joint. Property base can come, right, add joint The ad joint is very important. very very very heavy level Isn't it important to ask the ad joint? It can be expected this time, right? So see what happens, see what the property is. It says there is a property a*ed It is a joint of A, isn't it, tell me a very important thing These are the questions, aren't they? Can be expected added joint of a * a i equal to what comes mud * aa very Important is what you are writing among these Some questions can be expected first Very heavy level important second Determine of add joint of a are very these What happens when we raise to the power n -1 Here comes Made Ray to the power n-1 okay This is very important, very important It's a game changer brother, game changer from here Some can be expected, let's see These two properties are very much If it is important then have some expectations If it can go then what is a given in it The square matrix is ​​given by Whose order is it, three is enough Determine of a know-2 is asked The determinant of ed is equal to the joint of 2a Na ad joint of kya bata hai mode re to the power n - 1 so dear children look here what is written ad joint of 2a so this What will happen if m is raised to the power 2a * n - 1 Maharaj, here it is written 2a, so here 2 It will come here, a is written here, so a will come. If 3a is written then 3 will come, if 10 lakh is written then 10 If lakhs will come then it is something to be afraid of. What is the determinant of 2a to the power n - 1 n Order O order how much is three amazing so Determine how much to the power of 2a Amazing, now from here I told you to go out. If he comes, the poor guy won't It will happen, this is the main mistake, isn't it? Whatever happens will happen if you just 2a The game will be over when you get fired. Also, keep in mind that k * a is the determinant of If this is seen anywhere in the market Find out what k to the power n * mod of a Keep in mind that if you take it out, it will be completely finished. This is the game changer here. How much will the cube of 2 be squared and What will come, do you understand the square of mud? If you draw 2 squares, the game is over. Why does the cube of 2 become a cube? Brother, I have an order, when his three orders are over then this will come. 2 to the power of 6 is how much given It is very good, so how much did it cost? To the power 8 has come, it is written somewhere 2 re to the Power 8 Very nice question Very nice question that can be expected Property base is definitely a problem. You are going to get it or not It is there in the market because you saw it It might be that we have seen two or three such things in the sample paper also. The meaning of this type of question was discussed This is important, this is one of the matrices Whatever we are saying is important It's important, it's okay, even more so than Hannah Evers. A little can be expected then Ivers Let's do a problem of the type with matrices Look, you can expect two or three questions. Minimum is not the matrix too much It's important, okay, now here comes the next one. and b r the Invertebrate force means its everse ageist doing of the same order than chuch of the The following statements are correct and incomplete. This will not be ever this a in be ever this a P B Inners, look, I will tell you what happens. Look dear child, whenever you take it out like this What is an A-Evers to you? Well, an A-Evers. If you don't know, then write down whatever comes to mind. Dear child, he comes with a joint of a Upon mode of a isn't it just for them comes to mind when A is non-singular The matrix is ​​non-singular whenever matrix would be Whenever this is a non-singular matrix is A which is a non-singular matrix non-singular so whenever this non-singular The matrix will be non-singular only when the inverse There will be existence otherwise this ever will not come. No, yes no, it will not come, it is a non-singular not in the singular matrix This happens only in the case of non-singular words. The value comes in the case of non-singular I'm okay so now non single will come one thing I told you this, what did I tell you the second thing now? Like if I want to check this then it is like this in this If you turn it on, dear children, see what It will come mud a * mud of b evers and Mud of b evers what comes 1 up mud of b It's over, it's written correctly, it's very good. look at this look at this what will come mud of a Evers mud of b evers so hang him upside down / mud a * 1 up mode of b b mode of evers If you take it out, it will come upside down, it will be very nice. also written this would be a property of A evers The reversal law looks like the added joint of a * There is also b, like you have ad joint To find out the value of a * b, this also seems counterintuitive. Add joint of be in add joint of a same In this way, the reversal law applies here too. This property does not exist in plus. So where did you impose a different level of law? is being b evers p a is not so different level law is going on so I think D is the Correct option is ok, not correct is asked If I had asked correctly then all three would have been fine. This is to be expected a little bit in Evers. Whatever everyone is saying, it could be Expectation is let's see which of these What becomes heavy spreads in the market Now let's look at the second question, which is that your independent probability base Let's also see that there are many different questions. We're wondering which type is best. I have a question for you, right? It can be expected that you Homework will be fine for is e to the power x is going to be the mapping Done, yes this is homework, it will be fun. You are an injective function which I was telling you that the mapping function So this will be your homework, okay, look. Three elements are given, four elements are given I want to tell you injective mapping, see this is the The one that you had written was the number of one function. Number of functions r > n becomes r = n n = r is his question here three There are three elements, dear children. element and there are four elements here It will come, it will come if there were four and three here. So then it doesn't come here there are three and here If there is four on it, will it come, will it come, I got it written Has gone You have told me the solution, sir, what else can I say? I am not able to tell you, then tell me how will it work? Even after telling the solution, you do the wrong thing. Tell me what you will do, how will you do it? I asked the questions and I am telling you those. I will do your homework for you. yours Yes, that's why we are of the type that you are telling us. Questions are also being given, speak the truth In what cases do they like to agree with Each other agrees with each other in starting the Same fact is agreeing there is contra doing He was agreeing, what did you tell him? Did you ask these questions just now or not? He will either agree or disagree I got the disagreement done. Now this agreement question regarding contract Yes, I agree, are you able to understand this question? We have done it, so keep giving the homework. I am wondering where to practice brother. How will you know how the market is doing? The question will come, right? This is done, what else brother, what else Yes, the question is to practice differently. different type okay now here comes my love Children, this question is a very lovely question. Is this property base? We are asking the same question. Which is Gadar, Gadar means the one who creates trouble. The question is, will it create a ruckus, okay, now this is written Look dear, from zero to pi ba f a ofv P 10xx We are asking the same question that Market means what is expected in the market System, your what is this market, right? Market by the way raita first so what here It is written zero se le pa ba f a 1 p 10x d one is written see nothing If you understand then there is only one thing in the market Should King God King God worry? I did not understand what to put How to plant it, only King God can do this can formulate the question and it's a little bit Expected question, good question, good The question is, can the number three come? Your section is number three. P can be expected if this is Equal to dear children what is this from 0 Laker Pa Ba 4 and what is this inside Elan has come in 1 p 10x and Up 1 p t t b this is what happens listen to this story If you haven't heard it yet, then listen to it today, my t b Upon this, it will come 1 p n b this formula It happens that you must have learnt it in childhood, na pa ba 4 What happens and how much will it cost dear Kids 1 - 10x up 1 p 10x then King God This value is coming after making it alive Now let's pick this up and keep it aside, it is too, isn't it? Keep it here so that I am equal to you dear Will the child come? This is a very strange question. Expected dear children from zero to pa / What will come to 4 ln2 plus 1? 1 in Gaya Market What will come to 10x upon 1 + 10x and dx4 after taking LCM will come to ln2/1 + 10x more You must have heard about dx9, listen to log a - log b Yes, you got a chance to listen to it, let's listen to it today people. / b which is log a - l b from 0 Taking pa / 4 gives ln2 * dx2 minus from 0 to pa / 4 ln1 + t x with sine And dx8 are you able to understand now you will see after this Try to look at this carefully. Picture this when you take the picture very carefully you will find this i equal to i i which we considered as i i aa i also You can accept it, you can also accept it by your name. Believe it, whatever you want, bring it here, come here. If 2i is equal to what will come out to be ln2 How much will come here, from Ro to Pa Ba 4 1dx and then p a b a 4 will come out because x Integration will give us ln2 * pa = 4 Just what is the value of aa coming pa ba 8 * ln2 see answer see answer don't trap me like this Meaning he will not give me the option, I don't know. I don't like it, give the same answer, Pa Ba 8 * ln2 so the answer is coming out to be pa ba 8 * ln2 written in any So this is the same as / 16 * ln4 because of ln2 If you can write then take this down If you go then pa / 8 ln2 see option match Getting it done is also a huge task here. If you are able to understand how to match options then this is It is expected in section 2 3 It is expected, isn't it, if not two, then three I expected this to be the one This great section becomes too short We are going to tell you in three, that's why In the section What You Can Expect Maybe this is this type of question which is this Aaya can be the best in the world Important Questions on 101 Expected this question to be the shortest Distance Shortest Distance or Question Aana It is the shortest distance, whether it is JE's Be it an exam, whether it is an exam here or not, it is very It is very important, it has to come like this If the question is correct then the question is about short distance Practice as much as possible very very very Very, it's enough, time is up, very, very Very very very important question by the way Whatever we are doing is very very way This but it has 10 times veri, right, v to the power of v. Infinity is important, right? Let's go to this. What will be the first thing you will do in this question? There is direction This is ratio, this is also the equation of a line. Which is the shortest distance to be found? This is a line, how did you know that these are queue lines? Its direction ratio indicates There is a direction ratio, right, because here Look at the direction ratio, it is 3 -1 1, -3 2 and 4 If it is correct then the direction ratio is are not collinear or parallel to you If so, then it means that parallel wala It seems the condition is of these lines only. So first let's take out the point a1 a1 who will be satisfying this line So how much will come to 38 and what will be 3 b1 Direction ratio so what is that 3 -1 and 1 take out a2 what is a2 that is in this line He is satisfying what is coming -3 7 and 6 p2 what is the directivity ratio -32 and 4 Okay, there's nothing to do because For this, the shortest distance formula comes a2 my a1 dot b1 cross b2 in mode b1 cross b bus ts it a2 my a1 to pat It will go away, just have to do minus Do only minus and whatever value comes, mode Put it in because you have to find out the distance No, if you give birth to cute a2-a1 children What Just keep doing minus with I I J J K So how much -3 -3 has come along -6 has come -1 It's visible here and 6 - 3 3 very Excellent, very nice, intricate work done It's already here -6 -1 and -3 3 by the way If you take out the common also then it will become like this, right? Take out the mango here too, it will look beautiful. Picture-2 -5 and now find the equivalent of V are b1 cross b2 very nice b1 cross b2 Find b1 cross b2 and find j and Awesome and here's how much it is 3 -1 more So this comes out to be 3 -1 and 1 -3 and 2 and 4 equal to check check check Yes, very good, what should I do now? Children, we multiply with a cap Let's try it and see how much you can get in a cap -4 -2 How much is coming with a cap equal to -6 -3 and -1 so -1 came, very good cap How much is coming with 6 -3 so 3 has come very good very good Okay, now we will take out its mud, it is very A beautiful picture comes when we multiply it. What will you do with this, dear children? Let's know this, multiply it with this If you do this, J will be multiplied with I. will multiply with j with k So, how much will the square of -6 yield? plus - the square of 15 plus the square of And what will come in the mode inside the under root 6 S S 15 S S 3 S Very good so that The remaining bus is left inside the under route only, meaning This is the value that is left, which means its answer It's coming something like this, it's coming under 6 s 15 s + 3 s inside the root is If you take out the common answer then how much will it be 2s + 5 s + 1 s 25 and 4 29 29 and 1 30 then 3 √3 then 3 √3 This question remains, this result is expected to put a2 - a1 b1 a b2 up mod b1 cross b2 b1 b2 what is direction ratio what is the direction a2 a1 the point that He is satisfying the line. This is the passing point, this is also the line to pass The point is to do it, mud will come, suppose There is no problem in getting negative answer. Because the shortest distance is first. Check if there are any skew lines or not. because it turned out there were no skew lines and You applied this formula parallel line There is a separate question for that a2 - a1 cross b up mud of b so there b b you same We will get the same there b like 3 -1 1 Here you will get 3 - 1 and 1 then then This is what happens if the direction ratio becomes parallel. Then it will be called a parallel vector if The answer is zero, which will not happen. Meaning it's intersecting again because Look at its short distance of only three are of type either will be intersecting Then the answer will be zero, he will definitely ask. No, there is another one parallel to that Direction ratio which is in LDA times Should the third one come skew la so this The one with the skew line is the one Mainly asks for. Very important heavy skew lines Level is important brother, that's why I did it 50 times I put infinity, I put something, meaning something There will be a motive, the motive is sir, the motive is okay Everything is clear to you now, isn't it? The next question is coming, I am seeing which one. The question that can be expected from you is So dear children, this is a little question It's a bit of an Evers type, expect it a little bit can be done inversely to a type Ivers has come, this has to be taken out, you have written a You have this eve to draw What did I do if I calculated the value of 14a evers? Told you the short way to remove evers Let me tell you what an ever is. See Short method: If this type of thing comes What is that in objective base ad joint of a by mud of a okay what would Had a joint of a by mud of a so dear How will children determine the mode of a of a then mud a how much will come 3 * 2 pv equal to how much it has come from it has come mode has come Sir, is there anything to remove the ad joint? See the method of ad joint matrix formed by factoring and its Transposer days will pass if So what to do in the objective bus, see these flip both He is the main diagonal, what should I do with him? If you want to turn it over, turn it over, then T and Th have come and those what to do in both is to put a minus So this minus minus has become a joint. This mode is off, now just write the answer. What will be the value of inners 1/7 aa went and how much did it come 2 - 1 of 1 and 3 Equal If you do that then how much will 14a -1 come to Times have appeared in 2 - 1 1 and 3 Solve it properly and see how much you get. 4 - 2 equals 2 and 6 okay so I Well, let me tell you about it here in a shortcut way. If it comes in objective, suppose If it comes to subjective then remove it completely I will have to show you one by one. I am saying that if the one mark one comes then then you can remove it directly okay so if Any matrix is ​​​​making it a star If you will write that you have made the children stars, right? is any matrix ABCD and you get what to do with the septum what to do with the ad joint off a has to be taken out, right, to cross to Because a little bit can be expected That is why if you want to remove it without any problem Think about what to say without thinking and use less brain Flip it over, flip it over, put a minus in it. Is it very easy or not, someone is in your mind Not stress, otherwise it will take you out. It will appear as if the reason for one factor is removed. If you take d then the reason will come to be factor If you take it out, a will appear, hence take the photo. So mine will come to c because that's the co factor. Whatever happens, minus minor is also included in it. It happens that -1 to the power of a p j a a j bus It will take time there, then its transposer Everything is readymade here, everything is readymade in the market You just need to flip it over and put a minus sign in it. If you want to apply it then it is there So what about transposers, at least here. It will come immediately, okay, so I have to do this. You should be able to solve at least 20 questions. Those who are single should go at least Look less because they will come formula based Those who come, if the formulas come immediately, then at least What less is there, just accept it If you are giving question paper out of 80 If I get 17-18 out of those 20 questions If questions arise there, then our for instant plus becomes a whole different Energy of that level comes but it does not come there. Now you will attempt 18 questions and get 18 marks. Once you are sure, you can move ahead. If you do it, it will be of a different level. Confidence comes because you are able to understand You don't have to worry about single choice. Because if it gets stuck then it will be completely My paper will get spoiled, it means The first question is, hey, it's not happening, friend. I didn't ask the second question and they tried. Let's do whatever strategy is there. what happens psychologically is that that paper What will happen in starting like starting The three or four questions that arise It becomes a little tough so that the child The poor guy gets scared, he's there for three hours. waste half an hour there and you Then later you will feel like, oh man, this is two and a half hours. It is left, so we should try whenever we get the paper. If you give then try that if the first one If it is not happening then move on move on move on Go ahead, let's say the first question. You didn't see the second one, you didn't see the third one. I didn't see it, come on the fourth one, I am not able to make the fifth one. But come on, somewhere or the other, one will be made Confidence is very important, okay now like this It may happen that you already have more marks, my friend. We will bring our lung ones and It turned out that there was an error in some calculations. Come, then take a different level of confidence. Our target is that whenever we paper Let our target be that our confidence should be high. This is very important, especially in the initial phase. If our paper gets spoiled in the initial phase or did we misplace something in the marking I gave it or wrote something wrong, I am very confident Let it happen and very much a D Motivational Do that friend, at least for half an hour My part should be half 45 minutes, questions in it I want to get maximum and one I should have confidence that I can score 15, 10, 15, 20 marks. I have already made it, now whatever I make it It will only go plus, if you are able to understand then one Confidence will come, that's what I am saying. Only through single choice will you get it and single What are choices sometimes something The questions will be so heavy that you You can do it only if you know the formula. Because there is a tick case there. I also have to show something, solve it there. I am not getting the number, if it is ticked then it is not correct. If it is over then that is what I am saying I want to show you how to do it, so I am trying to make the single choice He can ask what types of tough people are there. What can you ask, like, suppose I told you the total number of Reflex relation, now you have read it or not. Reflex relation has been read, total number of reflex relation has not fallen then There number of tot number of one one function I asked, do you know the one-one function, hey friend. Was there a total in the market? We didn't do it, that's why we did all this Are you adding a joint of a you take out there I told you the trick. You will take out many from the butt, right? things that were explained in a short way can be approached to determine the value of k* a has come, you had read determinate of a that Hey friend, I made a mistake in taking it out. So wherever the child can make a mistake like Determine the edge of the same adjoining edge Add joint of two properties read No, the question came from him. The question came from him. If you go, you will go with those things which you have not studied. This will raise the main question and then I will think, oh friend, this So the paper is very tough friend, this is how we do heavy The level has arrived, so you have to do a little for it. I have to go to Physics mentally prepared I had seen that chaos, so that it does not happen again. We should go mentally prepared for it. So don't think about this paper, friend. Chemistry is done, Maths will also be done, right? So don't think like that, you should move ahead with your It is full preparation with full palm method going to prepare in a way with Okay, let's start next. Having fun with the question of things that don't See, the question should be raised now because the theory So we cannot study theory, right? There are many children who have some time to become Even on the first day, they keep searching for the market. I am getting something, they keep searching for it. Yes, isn't it? Look at the second question which is your homework will be done now homework okay So I'm going to mark homework too. homework so this is your homework okay so This is homework, it is very easy, how to add If you want to do it then this order is the basic part. How to add an order, this is a little This is a basic part, explain something in it. Not worth it, how do you add orders? Add only if you have the same order You will find that it is the same order, I just had to check it. That's fine in that, now comes yours. Part Yes, this is a little bit random. There should be a random question on the variable also. On variables etc., we see random variables and also look at the question of independent Okay, this is on random etc., not on this. Sometimes Pisces will ask you questions. It will ask you to find the variance and then it will say find the value of k so these are random variable questions I have slightly expected this type of question. Sometimes it will say that probability one x is less than 1. Probability when x is Lesson 2 Sometimes will give relation px1 aa key fa the value of k bola hai okay Value bola hai determine the meaning of Sometimes I am talking about distribution Sometimes he will ask for the mean and sometimes he will ask for the standard deviation. There is nothing to be afraid of, okay? See how to remove it whenever you need k If you want to find the value then the probability is Also find the sum of whatever you have given this x aa whatever You took out the variable x and it came out equal to everyone Whatever sum comes will be equal to one, right? Probability of x1 Probability of x2 For example, suppose you put the value of x, then Probability is 5 I am putting one So the probability of one is 1.5 Whatever I get from them will always be for the forest. equal means I said what does it mean that k Plus will come here 2k plus here will come out to be equal to k s + k = 1 so from here What value has come, 3k has come, so 3k has come An equation always comes out equal to + 2k - 1 = 0 will give two values ​​of k and will return one value There will be things that you will have to ignore. It will come in minus, it will come in fraction Meaning it will come in minus or from k to value two More than this will come, one such type will come The value you always have to make sense of Meaning there will always be a negative and a positive. It will come, see, I am solving it now, 3k s - k - 1 = 0 so the value of k is one 1/3 is coming and the value of one is coming -1 Look, if the minus value comes then the minus value So if you can't take the probability then minus It will not come, what does it mean it will come 1/3 come If it goes then I have found the value of k. Value dear children how much did I get 1/3 lot Well, what is the value of k, dear children? It has become equal to 1/3, now we have to find the mean. how to extract x aa p a aa It is done by multiplying, isn't it? Denote it by M. how to extract x aa p a aa By multiplying it becomes mu and If standard deviation has to be calculated then it What comes to e times the square of x - e The hole is square, doesn't it mean that This is how it has to be calculated x a s * p a a with a minus sign x a a * p a a of The square of the hole has to be calculated like this square of the hole of x s * p xi-xii This comes out to be equal to sigma, isn't it? So then that standard deviation part Sorry, this variance part is over. Done, this is equal to sigma square. sigma square, which is the variance, isn't it? Is sigma equal to variance? It happens that the standard deviation is okay here. So this happened, I haven't asked you yet, I asked you like this. As for me, how will I come out? Will I come out? 5 * k okay this will come first plus k key You will then put 1/3 of the value here. Then what will come next, it is 1 *. Then what will come to 1.5 * 2 * k so how much The 3k square has arrived, let's multiply. Go ahead and solve for 2 * k. Look, once the answer comes, I will come to Times. Now you have 4k sc + 2.5 * k k Now you had the value of k equal to Keep this, put 1/3 here and Do the calculation and this is how much it comes out to be 2.5 / 3 is here, so 9 is the LCM for me. It is visible and how much it came to 4 + 2.5 * 3 equal to means this is 7.5 + 4 / 9 and that's it Answer has come 11.5 / 9 ok It is 11.5/9 so it is coming to MK It is equal, right? I mean, what do you have? The mean is gone and the variance is gone, if you ask then This result has not been asked to be calculated in this manner So this is a little expected of this type There is a problem, Pisces has come, various have come, right? Okay, one problem and you have to do what you have to. and then you type this I'll give you some homework, okay, a little bit. is expected over the random variable which What we expected is what we are seeing What is seen in the market is Expected, okay, then independent Let's see some type of questions, right? This question is a little bit like this I was saying that, for example, suppose this type of It can also come in my mean variation Or else tell me this type of question I am here, look, this is written, look at it what is it look what is written The probability distribution is given by variable in that x is a and p is a Look at the values, it is written 0 1 2 3. According to probabilities given, he said find the value of k Determine the value of probability one x is equal to 2 then asking or x gr 2 What will come on pay or find the value of This plus this, what will be its value? These types of questions are a bit expected. What I was telling you is that there are three There are four types of varieties available either Standard Deviation Above the Mean On top of that, there is little variety of this type. It is important that this question is not captured. Maybe this question is a little bit about the big one. Can be expected on the bigger one Or your section that you Section three became section F in them This can be expected of you This one is a side question, okay? What is important is the base etc. Whatever you expected has happened This can be done on section number four which A little long answer type has an integral We will definitely do that and this is also fine. So they are going to tell you what Expected a little more from you We have to focus on our relation function. etc. are you watching this I am not expecting long ones It is expected that either you or one If it comes in either two then last So I am telling you this also. Continuity is differentiable, his question Can't expect much in the long run In short or short type single then look at all Each one is going to get something from this. okay now comes the cute baby like its What is the value of k to be calculated? Do the sum of all and find out if equal to one then k + k / 2 plus here comes k / 4 + k / 8 = equal to 1 over here 8 come in LCM How much will come if it goes 8 + 4 + 2 + 1 Divide by 8 = equal to 1 so k from here The value will be 8 divided by 12 + 2 is 14 and 11 so this is 15 so 8 divided by a It is equal to 15, okay, please check it. + 4 + 2 2 + 1 Okay, so here it is. Your 15 is a diva 15 k ki The value came, so it came, now he was saying Probability one x is let = 2 so look at this. See, I'll tell you how to remove it. Now you try and see the probability. of x = 0 plus the probability of x = 1 ps The probability of x = 2 is equal to Multiply from here, sorry, plus it. Just look at k + k / 2 and k / 4. 2 p k / 4 its value has to be found then k Find out the LCM of 4+2+1 / 4 means equal to 7k / 4 and k The value of the last is 8/1 so you have to put Okay, it's 8/1 so you have to putt 7* If 8 / 15 * 4 is equal to this value 14/1 so this is 14/1 okay now if this If x is the probability of 2 then If I say इग्र 2 then what would have come 2 means this come Then k/8 came and what was this 8/1 So this is 1/1, okay, so now these two If you add 'G' then only one is visible 14/1 If you take the sum of all of them then they will come to one place. If it goes then this is the method then such questions If you can expect a little Pisces people, whatever I told you, okay, these two The variety of types is enormous It is important that you understand, so we I will tell you about all the varieties. What to expect now If it comes on independent then this independent But this question can be expected a little. Maybe this type of question is okay. Problem in Mathematics is given to have So there's a problem that's in mathematics. is it not given to a student whose chance of Solving R then chance of solving his Given is the probability of solving it The givens are 1/2, 1/3 and 1/4, right? The probability of 1/2 is 33 out of 50 and on 25 if event of their of Solving are independent means of solving It is independent, it doesn't matter to anyone. This is the probability that this problem will B Solved, I have to solve this now, solve it. What if even one problem is solved? Nana will solve it, just accept it. Probability of a given to us Dear children, look at the probability of a The probability of solving it is given by And it is 1/2, isn't it, and the probability of b is If I talk about its probability What is given 1/3 is given, isn't it given 1/3 is given and the probability of c which is given If it is 1/4 then there is no need to solve it. Probability will also come if even one If the problem is solved then it will be called salt No, I have a lot of such cases. I'll go and pick it up and solve it. I did not solve both of these, I solved this one. Both were not solved, these were solved If you don't do it then it is possible that brother's I solved both, but I didn't solve this. Did you solve both or not solve this or all three? If solved, there will be many such cases. What should we do when there are more cases? If you want to be, think in reverse then move in reverse Minus Probability of a Complainant If he corrects it then something will be solved. I don't want it solved, I don't want it solved I have to get it done because if I get it solved Given as if solved by a solved I got the problem solved by one or two people. It will also be solved by three, even then we have That case was removed by minus one. Because I don't want this case. minus the probability of a Compliant Sometimes there are four, sometimes If there are five, start by minus one. Do you understand if he says that At least the case comes and then his case They separate, but mostly yours This type of thing comes in boards like Che etc. I come in at least two cases with one case. But there are not that many expectations here. So 1/2 comes out, 2/3 comes out and 3/4 comes out, so 2 2 is cancelled 33 is cancelled so How much did it come to? 1 - 1/4 which is equal to 3/4 So 3/4 is the Answer: You had asked such questions. It is a good thing if it was done then it is not It comes in four types and five types. who is independent like the one with the gun Here comes the example of salt. These are the questions that will mainly be asked to you. Do you understand that you expect such questions? You can do it in short answers or multiple answers. Anything can be put in it, okay? It is a very valid question, it is quite correct. the question is look what a question it is this is very This is a lovely question from Vector, isn't it? The question is can this be done in 3D? Now like what is written here, one is the line m that is passing So -4 2 and -3 is parallel to the Line this is parallel to this -4 2 more -3 is parallel to the line this Parallel is okay and speaking vector Equation of the line is this again speaking p k aa tell me what do you mean by this here This happens whenever we solve the vector equation If you write, how do you write, dear children? r vector is equal to a vector ps lada times b vector, we write whether we write what a vector is or not Let this be the point on the curve Satisfying means passing the line This is the passing point. Isn't it, this is the point that happens He is passing, this is the passing point What is called directivity ratio? great very nice it's called direction ratio and this is called passing the point through which it is passing If yes then the passing point is given to us. This is the passing point, see it already. Given -4, 2 and -3, see, same here. Look it's written -4 2 and -3 now we have Just what needs to be taken out, needs to be taken out. He told something, he said what The direction ratio will be parallel to this. So it will be parallel, meaning first of this If we find the direction ratio, then first x Make a quota of one, how much will fit here x + 2 / ba -4 Find the divisor of x and y + 3 Up-2 Very good u Divide from here So how much will it come to, z - 3 / 3/2, this is it. Direction ratio direction ratio that will come What will they come -4 -2 and 3/2 see this will come This Maharaj will come, this Maharaj will come and this Maharaj will come -4 -2 or 3/2 you can do it like this be -4 -2 and 3/2 now this is like you said to two Multiplied by? I gave it and multiplied it by 10 lakhs. Its direction ratio will be because its It will be parallel, if it is parallel then fight Times is involved in this, so can anyone Keep multiplying, first see this, this is it or not -4 -2 3/2 neither this will come nor this It will come or can it come or can it not come This is what will come, how can this come if I Let me multiply it by 2, see, it becomes 8, 4. and -3 if I multiply this by -2 If I do this then it will come because this is because of this If parallel comes then see that in the option. It will be necessary because there are many possibilities. Meaning, its parallel vectors are many If everything is possible then it will come along with it. Multiply that by Lada and this is what happens. Its R means multiply it by two, 3 multiply by -5 Multiply by 8 to get 10 lakh Multiply as much as you can Karo B will come, they multiply by -2 It has been given that these three options are not available. These are very strange questions, do you understand what they mean? The level can be increased to a little expert Section B or Section Two or Level Three Okay, then one more question, let's see. Let's do it, I hope it's okay You must have understood things, right? What did you do to create these types of problems? Making This Is Your Homework Direct This is the question, we have asked it somewhat directly. The question is that we did not do that much. I stressed that yes, it is a little difficult. Understand the types of questions that may be asked You are the difficult type of person you are A few game changer questions Because the rest who will be easier So you get this question now. Like this is a standard question for you. d a up x s + a s these formula based This is a question that the whole world has. If the whole world knows it, then where are you? You will be able to move forward and speak to the world The question paper was very tough, friend, all this I have not read it yet, so all this Everything is completely formula based, isn't it? then the whole world will come because you I have studied in school, brother, I have worked hard. If it is going to be different then we should You have to get ready for this, here it comes, look 1/2a equal to ln8 / x + a + c so the program You should know everything and consider yourself as one went under rat x s + a s all this has come You just need to know the formulas or the game is all About the formula, whoever knows the formula I don't know the formula for victory, the game is over. Look, you must know these types of formulas. a s - x s, isn't it? All these are formula based. There is a problem with these formulas, isn't it? If the question is based then this is expected It can go some more than this, so this You will definitely be able to make it, right? It's already daytime, tomorrow is the exam, so when? Now this will be done, the day after tomorrow everything will be done. Up under rat x s - a s so how much did it come will come inside l x p under root x s - a is equal to s + c Look at the method formulas, what is there here? It will look like this one with x spack Sometimes it will feel like x s my a sk, everything has been written. If you cannot see then formula based questions If you got it then it is good, it did not come. So say it brother, Jai Shri Ram, right? This is formula based so maybe things will work out. Maybe it didn't come, I can't say anything. If you have to make it tough then this will come, right? Now let me also tell you this in the manner How to bitwise higher order derivatives and if whatever is Your variable is on x if theta k in respect to y in respect to theta so How to remove dydekop.org 2 theta now also its derivative So, take out dx3 and 2c theta. what will come to the derivative of k the derivative of theta How much will it come to - cos theta, okay cos2 of theta What is the derivative of 2 with a plus sign? * sin2 theta That's it, yes minus in it keep changing the sign of the minus because k theta the derivative of minus sa comes to sa theta derivative of k comes in theta, it's over Keep this here, it has become too common. So look what came out is k theta - cos2 theta Look down at the denominator in two times How much has sine theta minus sin come theta, look, this is amazing, isn't it amazing, cos theta Minus bitter theta is coming sign thi minus I am coming along, okay, I had asked you this question. Hey how How will it work, how will it not work, how will it work Will work Maharaj will be shocked by this. The law is not in the steps of the ok this is homework Keep writing your homework while getting it done okay to go this is your homework okay If they see something different then this Is your homework right a s = 3i like this We have also solved the problem of See what they say in the sample paper. can a make a I already told you, okay, so this is Your homework and whatever it is, let's just check this out. Let's see this and then do it, our done. Let's do it Maharaj, right? And then one more question. who will give you homework and Plus I'll tell you what type a little bit In a long answer, I will tell you how to do this. Okay, so look, this was the question. It asks for d2 y / dx2 to do nothing. Children dy2 becomes y / dx2 Property of log First solve the log and then Solving is okay so log key first Solve the property l a / b written If so, what will come, l a - l b o ho