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Ar all 9rgs % Heforn Z, OZr Ctdrc J ? Elaborak....

Question

Ar all 9rgs % Heforn Z, OZr Ctdrc J ? Elaborak.

Ar all 9rgs % Heforn Z, OZr Ctdrc J ? Elaborak.



Answers

$e^{\text {log } z}$ equals (a) $z$ (b) $|z|$ (c) $z+2 k \pi i, k \in \mathbf{I}$ (d) none of these

So in the given question we have an expression and we are asked that if I z cube plus z squared i is a two plus that square Miner said plus I is equal to zero, what is the model is offset? So this is being asked in the question. So let's take Z square as a common factor from here. So we will have that square times. I said plus one and we can write the second part. This us minus zed minus I right. And from this part let's take I as a common factor. So that squared times I said plus one minus we took I as a common factor so that by that by I minus one is equal to zero and we know that I is equal to the square root of minus one. So water is the value of one minus I. Right, so let's multiplying both the numerator and the denominator with I. So we will have I divided by I square if I is equal to the square root of minus one, I squared is equal to minus one. So the denominator would over here would be equal to minus one which means I divided by minus one is minus I. So one by I is minus I right? One by I is equal to minus I. So we can make that substitution over here. So what we will have over here is that square times ISAT plus one minus I minus ISAT minus one. Right. And from here we can take minus one outside and then we will have the expression that square times Isaac plus one. Unless I times I said plus one equal to zero. So now we can take Isaac plus one as a common tractor and then we will have I zero plus one times zero square plus I that squared plus I is equal to zero. So this is what we have by simplifying the expression. Right? So now from this we can Right? I did. I said plus one is equal to zero. Are that squared plus I is equal to zero? Right? We can. Right? Since I said plus one times square plus I is equal to zero. We can reach these two conclusions from the from the about quick question and from this weekend, right that either zed as equal to minus one divided by I or then square is equal to minus I. Right? So when we take the model is offset in both of these cases we will have if we take, you know, one by I. S minus I. So we can write this as minus of minus I. Which means that is equal to I. And if you take the models of said in this case we will have said as equal to models of I. So how we take models as if we have a complex number of the form X plus I. Y. The model is is taken as the model is is taken as the square root of X squared plus Y squared. Right? So over here we have the models of the complex number zero plus I. Right? So the models would be the square root of zero square plus one square, which is equal to one. So in the first case the models of Z is equal to one. And now let's check the second case. Right? That is when that square is equal to minus pi. So if this is the case, when we take the when we take the models over here, we will still have the model list of that square. When we take the models over here, we will still get the modelers as one. right. So in both cases, when we take the models of that, we are getting the models as one, which means the required answer is one right. And the option in the question, which gives is the value of models of Zelda's. One is option it. So this is the required and set that is uh, the option a give 60 models of that as one. So I hope that all the steps of this problem are clear. I hope you understood the method. Thank you.

I've been given here this relation more of their negative two equals to more than 81 find out this value here. So what we can do, can we have model the square. So we got him from here that we should square here. The both side of the equation will square the boat saturday question here. So we get more than negative two. We're equals true. Where more than negative one script. So for the formula the formula is given us if we have to contact numbers A negative big almost product given us Mordecai square plus Mark V square negative A. B. Don't you get negative A. Can you get me this moment? Now going to apply and we get more of these were glass before negative V. To conjugate. There will be to the and negative. We have we're going to get to there'll be to the congregation equals, we get four times more to the square plus one negative the negative Z. Conjugated. Stop. We're going to simplify this. We get here morty sphere plus four negative to the negatives to the conjugated equals before. Not this group plus four negative policy negative. Or they go on Juliet. So now I'm gonna find out here. Three more the square. So we get here three more the script. Next we have negative four real deep. What does four candles with this? And next we get here negative told me negative two. The conjugate very close to zero. So we have three more the square negative too. The plus the Contrave particular zero. So we know now the plus the conjugate that gives two times real deep. So three more of these were negative too, times two times real with the equal Tito. So from here to get three north, these were negative four times real deep. Here's the question little Both that is coming out to be a deal. Thank you

In the given question we are told to find the imaginary part of the complex number, zinc. Right? So we have a few options which are given us one by two I times Zd plus Z bar. The second option is one by two times Z minus Z bar. The third option is one by two times that plus Z. Bar. And the fourth option is one by two I times Z minus Z. Bar. So we need to choose and choose an answer from any of these four options. So in the options we can see that we have Z and Z bar. Right? So if we consider these at the complex numbers, letters of the form expressed by Y, then then that burger is denoting the conjugate of the complex numbers said, which is given by X minus I will. Right? So the imaginary part of zen is white, right? It is why? So what we need to find is to find which relation given these options, give us the answer as white. So what we're going to do is to take the first option and we can see that in each options we have that plus that bar. Or that is that minors that bar. So let's find what is that? Plus that bar. Right? So the place that bar is expressed by white plus X minus iy. Which is equal to two X. So here when we add that injured bar we have we are eliminating the uh imaginary part of the complex numbers. Right? So what about that minors at bar? We will have expressed I write minus express minus X plus I. Y. And this would give us plus two. I. Right to I to a Y would be the result of that bitterness. That bad. So the imaginary part of that is given by Y. So in order to take wife from this, what we need to do is to multiply one by two, right? So when we multiply one by two i on both sides of the abo equation, we will have one by two I times that minus that bar is equal to Y, which is the required imaginary part of the complex numbers. It. So this is given as the option one by two, I times that minus the bar is the option D. So D is the correct answer. So I hope you understood the method. Thank you.

I have given you this relation or even placido Holtzberg. It was more demons were blessed. And then the script project last year, the one they do congregate there purely imaginary to work on here. Everyone need to so we're able to walk on that they want you to pointed it what we can do here you just have to explain this. So I'm here. We get more vivant square plus more The two square blocks 31 need to conjugate the minute you know, plus the one conjugate V two equals more V one square plus more the to understand filled out and we get the one the to contribute plus Even congregate the two equal deal. Now president of basics here people let's just take a complex number that has given us explosive hurdle. Why? And if I do it conjugated act towards its conjugate that is next negative right away. So I will if I just for all of this I will get two weeks a real value. Right? That's real value. But if that complex number is purely imaginary, everything's just only carried away and the negative Erdogan it is coming out to the video and this condition is that is why that means it says it means that that one get to conjugate plus its conjugate conjugal visit to equal zero. But that means this complex number should be purely imaginary. So we can go with demon little conjugate is purely imaginary option. A and to be correct. That's correct. And he says that is even over the two is be ugly emmanuel. So you can work on it. We have the one need to congregate not even over there too. What was he to live it? Easy to get that's coming up with the even or was he too? Times more details spread. Let's see here there's judas here. Well you just divide by three to nothing will change. Now we know that Z one Z. To conjugate this purely in january and that's a real value. More zero square the real value. So if you get a real value care and here we have a purely imagining number. So that will give us is even over the two is coming out to be a purely imaginary number. Well, we can say option B is also correct. We have it's even over the tube. Is your lead demanding Auckland is that he said that They wanted to conjugate plus they even conjugate the two as equal of europe. That is only the impact. The only to conjugate plus even conjugate the two equal to zero reaction. T. Is also correct. This is that all Z one, Z two are the waters is or right triangle. So I just draw here a right triangle behaving and this is all. This is the one does this needed. No, we have learned that it's even what they do was even over there too is the only in mandarin. So you can get like a iota or they've given us kate in the bar Toyota by or two. So that means if you just rotate here the two here we just start from the two. We rotate the two by angle of 90. My angle of 90 it could be plus or minus we have here. So we can take it as okay is plus or minus. We have so the angle would be plus or minus. Nothing will matter. So we just read by 90° here being here either the one or you too. Your angle is 90 years. So that satisfies this. Okay Into the bar iota by over june because we have the university is purely imaginary so we can see that those even the two are the voices of our right angle triangle. To the option D. It's also courage. Thank you


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