Transcript for:
Coordinate Systems & Complex Numbers

Bismillahramaheem A Nasteen. So students do exercise 1.5 A little chit-chat before we begin. Then we will do exercise 1.5 question number Let's start one. so question number one Two important points before understanding this yes son. One is the rectangular coordinate system and another polar coordinate system Which you have to do in question number one. So First of all I will tell you about rectangular Let me explain the coordinate system. Its Later kids I will tell you about polar coordinates I will tell you the system. So first of all you all People should put headings on their registers. Rectangular Coordinate Rectangular coordinate system. so most First, kids, let us understand that this is your What do you have nearby? Is it okay son? So the most First of all, you all know that we are a We have to take a Cartesian plane. this plane We have four parts. Here But children, we first coordinate this They keep on saying this. Kids, you have these in seconds It happens. Apart from this you also have third and This is its fourth coordinate. Ok Son? On this side you have positive x There are axes. negative x on this side Axis happens. negative this side up What happens to positive y and below kids? y is negative. all these things you They would have already completed their studies in 9th or 10th. Is it okay son? What's in the First Quadrant It happens? x is positive and so is y What happens? Positive. Second Quadrant in which x is negative, y is positive Is. In the third quadrant, x is also negative, y It is also negative. in the fourth quadrant x is positive and y is what children Is? y is negative. The angle here would be It is zero in the beginning. here at The angle is 90°. There are 180 children here It happens. Here you have children 270 And here when the circle is complete What does this angle become? It becomes 360. what do you guys think about this mid point Do you say it? They keep talking about origin. And what is the point at the origin? Does it happen kids? Zero is zero. So That's why I have written O here. So what does it mean here son? This is the origin. well in this system you guys draw the points what is that thing called? G Plot They say. In this system you can What are people saying? It's called plot. For For example, you may get some point. he said Tell me brother please draw this. Its you Make a plot. so in rectangular you guys How to make it? for example is It should come to you in this way, you should get two and three. It may come into question. He should tell us that brother You give us a plot draw of it or 3 2 may come or 2 or 3 may come, anything may come, son So how do we do it then? See the most First of all you guys made a plane like this It has to be made. Is it okay son? like this You guys could have drawn a plane here There are. Now look at this positive x axis Is. The first point you have is on the x axis Is. The second one is of the y axis. so you Suppose there is one here and pay two here. Ok? Similarly, this is positive y. The forest here Is. Here's to. And here you can make three suggestions Let's do it. So positive two. so this positive two you point upwards like this Let's bring it up to three. because the second point If you have three then you can mix these two together which we mix in and in this way it Your plot gets drawn, son. If two, three or any point comes then this the way you draw but what we're going to do today He is studying the topic in Exercise 1.5 Son, we have polar coordinates System. Ok? So now I'm going to raise it I am here and heading towards Polar of the coordinate system. now let's raise it Are. Yes. yes students, now we will study We have the rectangular coordinate system. Now we have the polar coordinate system. Is it okay son? So the polar coordinate system I use R and theta. We do. Is it okay son? So you all know that here you should have distance also It can be and radius also. For For example, you have this circle and this is the midpoint of the circle. So you are anybody take the point it will be fixed and what will that Will you call me son? will be called the radius. and this There is a fixed point. from this point here The distance from this point to the circle I am fixed that. and here is what you have The angle that is being formed, what is that angle called What would have happened son? what is that angle called It happens? is called theta. Ok? So It is something like this. but here at we have a little statement Change is also going to happen. So that statement now The way I am speaking, speak in Urdu I have been. You people too are in the same way Write it down in the register. First of all I say when you are with me at a fixed distance Let's rotate our point in this. What do you want to write first, son? when you your point with a fixed distance What do you make me do in this, son? getting it rotated if yes then rotate it within the fixed distance and the angle it will make in the rotation What is the system called, son? Polar It is called the coordinate system. Ok Son? Saying the statement once again Am. Write it in Urdu. the way I I told you to write it like this. Then I'll give you Let me tell you what it means. What did I get written? When you are with a fixed distance your What do you make the point do in this, son? You get it rotated? When you fix any of your Rotate this point with the point in it If you get it done then it rotates through a fixed distance and the angle it will make in the rotation What is that system called? Polar It is called the coordinate system. So I to you Draw a car caisson plane here I will do it. Son, you understand everything now will go. Look, this is the point here. This is the mid point here, this point here What is pay called? First of all, we have to The point must be known. This is the point which is This point is called the beta pole. Pole. Correct Is? In the rectangular coordinate system It is called the origin. Polar Coordinate System This point which is in mid, what is it called Here? We call this a pole, son. Ok? And this is what will become of you Will you go? The radius will be formed. and here what This is the angle, this will become theta. Ok? So We have understood this till here. means this what will these points make together The son will make you have r theta. Now we I understood the statement I just mentioned when You can move your point within a fixed distance Let us rotate it in this. When you're someone your point with fixed distance What do you make me do in this, son? getting it rotated Are. Is it okay son? and from a fixed distance The amount of rotation it will do is fixed distance the more rotation he does from it the better son and The angle formed in the rotation and the rotation whatever angle is formed in it what will it be called It's called the polar coordinate system. Son, this is the pole that we have, right? This pole, son, we have this pole what is it six point okay son this What points do we have? is a fixed point. this is the axis we have here Axis what is it called? Polar Axis It is called. what is this called kids? It is called the polar axis. and it is the same thing positive x axis, this negative, this positive y, this is negative y, the angle here is zero. It's 90 here. It's 180 here. here at It is 270. And here when the circle is completed So there is 360 here, son. Ok? So What happens in the circle? Fixed Point Joe What happens is that it is the center of the circle. Circle what happens in son? In the circle The fixed point in the circle is fixed What is the point? center of There is a circle. So you know about this thing. I am telling you these MCQs. These Things can come to you in MCQs Son. Ok? and the fixed distance would have been Isn't there a fixed distance in the circle? The fixed distance in the circle what is that son? There is a radius. Is it okay? So these are some things that you Have to remember this. Now on to question number one Son, let's go. inshallah you You will understand everything. so question Number one, now I raise this, son. di students question number one key We have the statement. It is saying that Plot the following points. So you have given them I have to plot it, son. very handy There is a question. If you stay with me First part is given G question Of number one. So let's start quickly Are. 2 75 degree. Is it okay son? So you have this 75° And two lamps are given along with it. Now this two What does it mean? This two means that you How many circles do people have to make? Two Have to apply. So its radius is two Is. So here whatever value you have You have to make that many circles. and here The angle given to you is 75°. Is it okay son? So you guys first of all You have to make two circles approximately like this. Is it okay son? and its here like this You have to make a Cartesian plane. Are you ok Are you gone son? That's it, your figure is ready. Son. So first of all this is just an estimate. Now what does the paper guy have to check? keep it like this or keep it d from here this is the mid point that we have You have a polar point, right? from here to The distance till here is one. The distance from here to here What happened to you? It's two. If If there were three here then how many circles would we have Would you make it son? Make three circles. Ok? So First of all the two became a circle and the angle which Isn't it? The angle is very important here. 75° son 75° first you should know Needed See, I just told you that The angle from here to here is 0 to 90 so between 0 to 90 you have 75° will come. So 75° will come between 0 to 90. If you make your figure like this, it will be proper So it will be fun. 0 90 180 270 And what will come here, children? 360 will come. So here you can guess Make an angle. Ok? So you guys scale I will make it with help. So here you have it G 75 You will make a completely straight line and here You will draw a point to the end at 75°. Is it okay son? So your G plot is ready. it will be question number one part number one Of. Here you have made a small statement You have to write it down for every question. I I am writing just for this part. Balance You guys will write the parts yourself. You What will people write? from Figure the Polar What coordinates will you write, son? you did this After making it, you have to write this statement. From Figure by Figure the Polar Coordinate from Figure You must write this down yourself. The Polar From Figure the Polar Coordinate Two 75°. Is it okay son? Write this for every question Is. The coordinates given here will be the same will come. Represent a Point to Represent represent a point two unit Represent a point two units away from the Poll Away from the Poll away from the pole at the angle of 75° at the angle of at the angle of 75° so son for every question that you You must have given the pass value, you just changed the value It has to be done here as well as here This will be completed for you. Ok? So now I will get the second part done and give it to you. I will give it to you in second part. So you Inshallah son Majeed will understand. So Write the second part which is -3 and 120°. Second part son, second part of all of you pass -3 120° so we could have made a G space here yes son. A space was created here. so second part -3 120° Son. so first of all the three which are your has a radius. So what does this three mean? Is? How many circles will you make? Three circles Will apply. Here pay two meant two circles. Here three means that three circles are the most You will apply it first. You guys at a guess I will apply it, son. your circles are good I will be able to do it. So you should try it that we have to make cute circles Son. This is how you can guess the circle And from the middle you can create the y axis. Are. Son, with the help of scale. and here in What do you want to make? the x axis. Let's go It's that simple. Three Circle, Three Distance means now from the fixed point here But what is the distance till here? Forest. What has happened? Two. and what is this Gone are the kids? Three. now it is -3 so here What happened to you? -1 -2 And what happened here, son? -3 ho Went. Now it is 120°. So take a look from here The angle from here to here is 0 to 90 Went. Even 90 became 180. 182 270 Done. What happened to 272 here? be 360 Went. Is it okay son? Now look at the minus here is in. 120° you have it. So 120° It will be made somewhere here. Minus to you now I will tell you how it will be made. Look, it is 120 It will be built somewhere near you. This which I have put dots. Ok? you can see It must have been so. hopefully. I name it here I am giving you P's. You have it original There is no point. So this P prime becomes 3 And 120. 3 and 120. you will say sir this is -3 Is. Why did you write three? these three Above it is positive 120 right? Actually you are minus for you have to be exactly in direct line of it You will have to draw a point here. You have seen its absolutely straight line here I have to draw a point, son, a straight line exactly which will be its axis as if it were a mirror If its axis falls on the negative side then You need to drag this point downwards p will come here to -3 and 120° negative For this you will have to make him his son. Ok? So here we have the second part. And you can use this statement for the second part also. I will write it down, children. So now we are in the third Let's move on to the part. third part key On the side. This is my third part. if these You will write the statement which I have given this first The part statement is written. you guys I will write it exactly like this. The third part is 2 π / 6 First of all you guys did it this way First of all there is two, so you draw two circles. Do it, son. Draw two circles. you so will you draw a cute circle Son. Your drawing is better than mine. So I'm just drawing on the board. I drew two circles roughly. Now its You have to pass the line in between Son. y axis And what happened, children? the x axis. Make a complete figure. 0 to 90 90 to 180 180 to 270 and what happened here? 360 ah Went. Now it is π/6. The value of π is 180. 180 divided by six is 30. 30 There is an angle, right? So the angle of 30 is equal to the angle between 0 and 90 It will be formed in the middle and it is also positive. So If you are positive then there is no tension. so here But it will be built somewhere around this place, son. We will roughly calculate two and π / 6 is fine Son? Yes it is obvious that this has happened and this What happened sir? It's two. This is what it is You must understand what this is telling us. Is? The circle is telling. telling the radius What will be the radius, son? Ok? so this We had second part and third part. Now Let's do part number four, son. Yes Students now we have the part number You all have four, so its radius That is five, so how many circles do we have now? I will have to make five circles, son If you need it then make it, Bismillah First of all this is the first circle this kids second five right so radius If there are five then make a five circle, this is done g third 1 2 3 five to make what happened ji four okay son these four You can make a circle here by guessing People also and these five are fine son five If yes, then make a five circle and use the scale to you guys passed this line in between If you want to take it, then you have a scale, son So suppose my figure is very good If it is being made then take this and complete it I told you all to complete it quickly Son, here it is zero, here it is 90 there is 180 here 270 70 and here 360 son 0 90 180 is now five for you, see now Yes, you can also write here that this is forest. this is two this is three this is four and this what is the point it's five okay Son 5π / 6 multiply 5 by 180 Divide 150 by six and you get 150 by 5 Multiply 180 by six and divide by 180 Where will it come? 180 will come from 90 and 180 in between 150 so this point is this point What are you making, son? 5 and 150 are made Actually. Ok? So 150 points which is 90 And you can use it anywhere between 180 You can draw, son. so this is positive too If it is there then there is no issue. So this is your The point is fixed. and here is your It's 150, son. Ok? so here you are We will write down point P 5 / 6 So yours is ready. Part number four. Now let's go son Let's move on to part number five. Yes On to part number five. very handy There is a question. By enjoying your Solve the questions son. Part Number We have five. part number 5 -5/2 -5/2 and π/3 so if you divide five by two you get If you do it then this 2 something comes. 5 / 2 If you do this on your calculator, Comes for children 2.5 comes. what is this coming Is? 2.5 I am writing this to explain it to you I have been. 2.5 The value of pi is 180 divided by 6 30 comes. This way you have this point It is being made. 2.5 for naa beta you will get three You will have to draw circles. Is it okay? Then This will cost you 2.5 dr. so live here First of all the First Circle, then seconds And then the third. So you have to make three circles. It is natural to fall. Is it okay son? So this is g y Axis came over it. and this is what you have from the mid to the x axis I I drew it, son. So you guys are doing the scale Making it with help. I know you guys So, these are good kids. So here comes the G angle Gone 0, here comes 90, Here comes 180, 270, 360. Is it okay son? So now we need -2.5 and 30° firstly 30° between 0 to 90 It will reach 30°. 30° will come between 0 to 90. Meaning it will become 30° somewhere around here. Correct Is? Now why did I put this point here? Have you stopped it? here's why I did this Have you stopped? Because you have -2.5. Right now we are making it 30° on the positive. So here you would write positive 2.5. Or you can write it down, of course, if you are not able to see it late If so, you can of course write it here, what will you write we will take it son not minus not minus because We are on the positive right now, we are its reflection make g positive 5/2 and π/6 which It is written in the question, you guys write the same We will take it and here it is done g p prime Gaya prime means that we have just now I have made it on positive side. Now if we If we draw it on negative then its axis it will come downwards. he is ours It will become a question. So son this is in mid pay Why did I apply it? Because its The answer is it comes to you 2 something. So 2 Something, it won't go on three, right? Are you getting the point? 2 something is three I will not go to pay. So this is 2.2 and It will come within three. So that's why this We will stop in the middle. Ok Son? So, me and you too This has to be stopped in the middle. Three Till then you should not mix it because it is 2.5 It is coming. right now it is on positive side so I have written this positive 5/ 2. Now you Make its axis downwards son or oh You should not mix this also. You have also Don't mix it, son. Ok? So remember three This shouldn't go till then. not this till three Should go. We will stop it right here We give it. Is it okay son? So here it We will name our original point We will give P's and P will come -5/2 And π / 6 so its axis is downward It will be made on the side. its axis is downward Because it is in negative. So this is our son I have it, part number five. I hope G You all have understood it too. Did you understand it, son? Would you not have understood it? So someone please tell me. I am explaining in a very simple way. What I stated in part number one You had it written below each part, right? I have to write it. This statement. Now I'm going I am moving towards the next part, son. You guys should also write the next part, son. Write down part number six. Well done. Part Number six is -3. Part number six. -3 - 3 - 2 / 3 if you If we solve 2π / 3 then we get the answer I think it comes to 120. 2 * 180ed Bye 3G 120 is coming. which means that It will be made something like this. -3 more - 120 is coming. Something is coming this way. Ok? So now both of these are negative. Now we have to make it non clockwise. Let's make it clockwise, son. So Here ji is first three, right? so three You guys make a circle. This First Circle This is second radius three, right? So you have three The radius will have to be made. So these three Radius is formed. Now, kids, from amongst this You have taken the line. like this and like this From. Is it okay son? so here I am now I will get it moved clockwise. So here it is What will the angle be? 0 It is -90 here. Here - 180 - 270 , 360. Is it okay son? So first of all, if we Let's talk about -10 so -10 is coming to -90 And -80 or -10 should go somewhere here will form. somewhere here it will be formed which is x and this Right now we are on the positive side, so we are giving the name of p prime. 3 - 120 is okay? now we have its axis which is You will obviously have to draw it upwards, son. so this is its x o you can use its scale If you make it like this then it will become absolutely straight Yours. Is it okay son? So here is your PM will come. P will come to -3 and - 120. Three Joe This is representing the three circles Is. 1 2 and three. Is it okay son? so like this This will become your plot draw. you guys what should I do? this is a statement below I had it written. You made that statement Son, you have to write it down for every part. Correct Is? So let's go to the next one. This is what ours was The pass was part number six. Now let's go Let's move on to part number seven. Yes Students now we have part number seven Is. Yes, I have written it. If I If I divide nine by two then son It comes to me at 4.5. Ok? and if I multiply 19 by 12 by 19 by pi If I multiply it and divide it by 12 then Coming to me right here - 285. Is it okay son? So the value can be created in this way It has been happening. First of all it's 4.5. so it means We will have to make five circles of it. So Son, first of all you guys make five circles Approximately. Is it okay son? this is first Two Three. Ok? Four And here comes son five. Ok? Now you People have to create the y axis. from the circle You have to pass the line. Mid to Near You guys do it. and the x axis you guys mid Please pass it through y axis son. Ok? So yes five here now we have point have been done. First, seconds, This is third, this is four and this is five. Ok? Now So here we have 4.5 - 285 - 285 yes son this is 0° This 90 means -90 - 180 - 270 - 360 Okay son, first of all G -285 Whatever it is, it falls between -270 and -360 if you go then we will be here Now let's create our own original angle Son. So this will be created here on G mid. You People will make dots with the help of the scale. Now Why was this made mid pay? Because it's -4.5. Is it okay son? So now we're going to call this P prime. Let me give you the name. and here g positive 4.5 And - 285. Is it okay son? Now you are here Its axis has to be made like this. and here You will stop at the middle and this will become a P point. will go -9/2 and -19 /1π so for minus you have to calculate that You will have to make it necessary, son. Ok? So Here also if you look at the value of fraction Write it down and don't give the checker a chance So it is the best. -9 /1π so the angle is very important here son Is. So this was part number seven. Part Number I am coming towards eight, son. If I convert -5 to If I divide the pay then I get something I get -2.5. Ok? 5π /1 If I Here I multiply 5 by pi and divide it by 12 then I get 75° for the kids. Ok? So Now g is 2.5 so I have three circles here Will have to make it. This is first, this second And this is the third. Is it okay son? then you guys g this way the positive y axis and the children like this the x axis. Are you alright, son? Now first of all Let's make it 75°. This is 0° 90° 180° 270° 360° son. So it is 75 degrees son. one then The angle is negative. Axis for Negative It is always done, son. so for that First let us create positive 75 here. And we will stop it at mid. This I will not mix it. Ok? because these are 2.5 Is. So let me name this P prime. This. Positive 5/2 And 5π /1. Is it okay son? Now if I get this x I want to make it, I will make it till here because this is 5/2 and this should be called p I will give it -5/2 5π /1 In the beginning I had written a line Here in the first part you must have heard that line every time You have to make it mandatory below the graph, son If you want to build it below the plot then this is our Pass is question number one. Now we ask the question Let's move on to number two. g students Now we have question number two and In question number two we have beta polar Have to create a form. So the polar form is Nor do we have anything like this. = r cos theta + iot sin theta son this is our It has no polar form. First of all Before we start question number two, we Let us understand how this polar form is formed Is? Then we will do question number two. So This is a very easy question, son. So the most First of all, what happened to you? Yes The Cartesian plane is done. This is how we people form a line parallel to y axis here and this mid point is divided by this point What do we kids do here? yes let's join this point with. Ok? So here we have parallel to x Axis. This is done, G.Y. This became x. And What happened, son? y is done. And this We give it the name of r. Is it okay son? And What will be our angle here? will be? will be theta. So first of all I'll tell you Let me tell you that this polar form is for kids How is it made? Ok? so polar form Before making it, you should draw this thing completely. I have to take it. After this children you found Have to do it. Ok? So how do we find r Are? You have a small formula pythagoras theorem by Pythagoras Pythagoras Theorem. So what is this, kids? it What happens? Square of Hypotenuse square plus of equal to base Square of Perpendicular. Is it okay son? So The hypotenuse we have r. So this is r's The square will come. This is the base. Base we have x so this is the square of x and Perpendicular. What do we have? Yes Perpendicular. so what is perpendicular Will you go? g y so if I look at both sides of this If I take the square root then the value of r is I have the square root of x² + y². So What is the value of r coming out? of x² + y² Square root. so these kids we have The value of r is coming. So first of all we You had to find r here. now we are here But you will find two values yourself. One So we will find sin and secondly we will find will find cos. Is it okay son? So the most First if I talk about sin then sin theta which has the formula for Perpendicular over hypotension occurs son You guys son, you have come after studying in 9th 10th have been done. so sin theta equal to according to What is two figure perpendicular equal to here? y of. And what is the hypotenuse equal to? r of. So y will come over r. so sin theta Perpendicular over hypotenuse. Ok Son? So y over r r come here and multiply with sin then y = r sin theta Will go. So this is our value of y. Ok? Now if I find cos from here I do it. There is a formula for cos, kids Base Over Hypotension. ok sir? so cos theta equals To base is x and hypotenuse we have r. If we multiply r here by x equal to will come r cos theta okay son So we have the general definition of complex numbers there is a form what is it it is z equiv = x + iotty ok x here is a real number And this y is an imagery number, you all know it. if son then z = x x is equal to what is x equal to r cos theta is r cos theta plus iot y y is equal to It turns out that r sin theta is r so instead of y I put put kids r sin theta if I I will take r common among the two if I buy both If I take r common from this then we have this polar form will come son so i have put it in common If we take it then what is the value of z? r cosθ +iot sin theta theta so here we have it son Which form has arrived? the polar form that we Had to prove it. Ok? now here is a There is one more small thing. If we take this in short If we want to write it in the form then what else can we do can you write? We can also write this as are z = r cis theta. z = r cis theta. so this is also its short has the form beta cos theta and sin theta. So what do we have for this, son? Short In the form. Now if we take an angle from here If you want to find out, if you want to find out theta then the angle which It always comes out of the tan. So What is the formula for tan theta? Perpendicular over base. so these kids tan There is a formula for theta. tan theta equals to perpendicular here is y and the base here what is it son? is x. Perpendicular y is And what base do we have? is x because We also have to find out theta here Is. So in polar form we also get theta Will have to take it out. So theta equal to what a Will you go? tan inverse y over x so here we have theta cube The value will come. One more and I will give you a good news Let me tell you here. this y which is it There is an imaginary part of z. Iminary The part is y whose son? of z. and x Whatever happens, whose real part is it? of z. So the value of theta can be something like this It can. θ = tan inverse imaginary part of z over Real part of z. Is it okay son? So you guys The value of theta can also be calculated in this way Are. And what can you write to this? theta can also be called the argument of Z. Ok Son? So here the argument of z is, Its value is fixed with you Is. From -π to π. argument of z which Its value is - 180 to +80 This value is fixed for children. In this There is never any changing. did you remember this Have to keep it. This is very important for you These are Mcqs son. Ok? Now this is Our heart has been proven. this is our theta also Done. Now a couple of other small things Are. I will tell you that also. Then We come to our question number two. Then inshallah you will get question number two It will seem very easy, son. Ok? Now I I am erasing this. This side is now yours The pass is present in the recording. So this Son, I will erase it. Is it okay? Now There are just two or three more small things. Then you will understand everything about this What do we have to do in the question, son? Correct Is? So first of all you should have a plane I'm gone son. You have this in the first quadrant Done. It's been seconds. This became third. And what happened? G Fourth Quadrant Done. Is it okay? It is like that, right? First Both are positive in the quadrant. Any two values. one in a second negative, if there is a positive value then it In seconds. in the third both would be negative Are. One positive and one negative in the fourth It has value, son. Ok? Other than this Let me tell you one more thing. if you have this is positive positive value and that If it is the first quadrant, then you are theta there. I don't change it, son. Ok? over there Whatever the answer is, you would write the same. Are. Is it okay son? In the First Quadrant If you are finding theta and have questions Both the values are positive. Both the values of the question are positive Then you do not change the angle. Ok Son? if the second quadrant if the second If the quadrant is being formed then you can find it there What do you do out of it? would put a minus Are. Meaning for example question comes like this let's go. First it is negative, then it is positive what do you do to theta π - theta Whatever your angle, you can find it what do you do from it son minus it and if you are in the third quadrant And if your child is third coordinate so what would you do there is there my son pie what do you guys know we are doing it plus we are doing it and What do you do in the Fourth Quadrant? you guys are minusing 2π fourth If the sign in the quadrant is positive of a negative value if the negative is negative So what do you do with pie, son? Plus We keep doing it. So in my opinion you You have understood the basics, son. Ok? So these were some basics that we had. Now we People go to their question number two On the side. this is my son i am going to raise him I am trying. You guys question number two See now. Now you don't have to do a lot of things It should be clear what is by Allah's order Have been done. Is it okay? So now we are You can do question number two, son. Correct Is? So write down question number two. Take it. part number one is saying that You have to change it into polar form. question number two part number one 4 + 3iota son question number two part number one This is what we have: 4 + 3iota. So the most First we have to find its r. The first value is x. The second value is y. So the formula to find r is x² + under root of y². Is it okay son? So we its We are bringing out the polar form. then the value of x Four has arrived. The value of y came out to be three. Child. Ok? So r is equal to 4. Square 16 3 square 9 r = 16 + 9 25 so r equal to what we have here will he come? g 5 square root of 25 is mine what happens son? There would be 5. First you r I have to take it out. Then you also find theta Is. So I just told you the formula for theta. I had explained it to you. Proof has also been done. tan inver y over x this is g I told you theta value here I even got the formula proof done here, son Pe if you remember. Then, θ = θ = tan inverse The value of y is three and the value of x is 4 Is. So it becomes 3/4. Yes. value of theta What did we get? 3/4 so here now What do we make? make a space Are. Now you all should use your calculator Write on it. tan inverse 3/4 calculator for your Is it nearby? then press shift press shift. press the tan button and Will write 3/4. Is it okay son? then the answer comes will go 38.86 What will theta come to? 386 So this son Angle came to us. yes yes Sure, what do you put on it Is? You can also write the degree sign in this way You have to apply it on it, son. Ok? So Now what did you have with this polar form that I just got it proved. z = r cos θ + iotsinθ Are you okay son? We have this polar You had the form, right? Then, z = what happened in place of r 5 cos θ ji 38.86 38.86 Plus + iot sin Theta was replaced by 38.86 Is it okay son? So here we have our The values have been put. So instead of r, 5 This value came in place of cos theta theta and Here also we have put value in place of theta Gave. So yes, our answer has come first Of the part. Now let's go to the second part Towards. Students now we have the next There is a part and believe me the next part is very It's easy so I divided it into two parts So that your time gets saved, son. Ok? So the second part is 1 + iot. So If we look at this part here We have the value of x as one and so is the value of y The value is one. x is also a forest y is also a forest. So Let us first find r. So x² + y² 1² + 1² So the square of 1 is also one. 1 is square one. 1 + 1 equals 2. r The value of square will be 2, son. Correct Is? Then we have to find theta. tan inverse y over x value of y one of x Value one. I divided the forest into one and one came Went. What is tan inverse one equal to? 45 K. and 45 we can write as π / 4 The value of pi is 180. 180 for four pay If we divide it, we will get 45. Is it okay son? So after this we have its polar form I took it out. The formula for polar form is z = r cos theta +iotsin theta. The value of r put the square 2 cos θ in place of π/4 Iotsin was π/4 instead of θ see these two Positive is Positive then First Quadrant Isn't it true that x is also positive and y is also positive? If it is the first quadrant then the first We do not change the angle in a quadrant okay son so now we have a third part is third part is 1/2 + square 3/2 iot so firstly we find r We'll take the square of x² 1/2 + the square of y Square 3/2 will become square 1/4. square root of square cut three two The square son of will become four. Ok? Now I am taking LCM here. LCM has the same value in divide by . So 1 + 3 What happened? Values with Four and Divide I came once. kids what are four to four Will you go? Cancelled. So the forest will be saved here. And The under route of the forest is a forest. The value of r will become one. Then we have theta I have to take it out. tan inverse y over x of y The value of put x is put which We wrote the values of y and x above here Is. Son, the iota value is who has y and who does not have iota That is the value of x. So two to two What happened children? It got cancelled. Correct Is? If square 3 is divided by one then Square 3 has arrived. tan inverse square 3 Is equal to 60. and 60 we multiply by π / 3 Can write. you write it on your calculator Take it. tan inverse square 3 so that's your I will only get 60. and 60 to π / 3 degrees Convert to radians by π / 180 By multiplying. So the value of π is one 180 If we divide this by three we will get 60. Now this is also the first quadrant because Both values are positive. so therefore The angle will not change. So now we have its Children, write down the polar form that we found. Have to do it. r cos theta +iot sin theta of r Value one has arrived. The value of theta is π / 3 sin The value of theta is π/3 which we have considered here. I have found the value of theta, son. to the forest If we multiply this then this is the value It will come again and this will be our answer. Coming to part number four. Part Number Four is -5/2 -5 3/2 iot. First of all, we r has to be found. So first of all, son We have to find r. Ok? so this Whose values do you have as children? r is equal to. Is it okay? This is the formula for r It happens. So r = x² + y² so I have this I quickly wrote theta here. so this r is it son r is it okay? sorry i wrote z I had r so the value of x² x is -5/2 plus -5 / 3/2 This is the value of y you have. 5 square of 25 square of 2 4 of 5 Square cut under root of 25 square So Three was saved. 25 3R I made it 75 And what did I do with the square of two? Four. Where did this minus go? this became a minus Beta plus because power is even above. Its Later I took its LCM. Divide If the value is same then there are 25 + 75 children I made it 100 and the divider was four It has come just like that. then 100 I divided it by four and got 25. 25 What is the under root of? It is five. This is the value of r. Now we have theta Have to take it out. So it is easy to find theta This is the only way. Formula tan inverse y / x θ = tan inverse y put the value of -5 3/2 x value of put is -5/2 so here I I am doing the cutting. Just look at the minus The minus was cut off. Two was cut into two. From five Five was cut. Is it okay son? So just and Just what is left? tan inverse 3 which is equal to 60 and we write 60 as We can write π/3. Now I wrote here Pe edge z lie in coordinate three. this which You actually have this function for z is equal to the complex number. Ok Son? So this in coordinate three gives us How did you know? I just gave you a I showed you the coordinates. I said was that if with both the values If it is negative then what would be its coordinate is it son? It becomes three. and the third What's in the pie to go in the quadrant We do? Plus do it. If you remember back the video a bit and you will find that To go to the third quadrant, Sir what did you say? Sir had said that the third What's in the pie to go in the quadrant We do? Plus do it. Whichever angle comes Is. so three now we can solve this I will take it. Three multiplied by π is 3π. What will happen to 3π + π? 4π / 3 Here I have I have taken its LCM, son. once again Let me tell you. There is a forest in its divide, right? So Multiply the remaining values below cross multiplying the values Are. Multiply the below values Cross multiply the remaining values to get 3π 3 to pi with 3π 1 to π with π 3π + π 4π / 3 So this is the value of g theta. Now We have to find its polar form. So Polar To get the form, I can raise it I will give it here. this is all you have you have the recording of it Children. So take out the polar form. I think you all know the formula. Then, z = = r cos theta r cos θ +iot sin θ z = r turns out to be 5 cos θ What has come out? 4π / 3 4π / 3 θ arrived iotsin theta is replaced by theta children 4π / 3 So we will get the answer to this part Yes, I hope you all understand number four. Yes, the students have come to us now part number five is what I have on the board I have already done it right son, so first of all Don't rationalize what you want to do We have to take it, see, there is 1 + iot in the divider So multiply by 1 - iot and minus 1 Divide by iot. Is it okay son? after this The sign above is the same. if the sign is same then this hole will become square of. down a + b a - b a The square of 1 is the square of one. Let's put the square in place. - b² then It will become iot². up square formula opened a² + b² - 2ab becomes what? 1 of The square of one iot is -1 So minus and minus became plus. of iot Square -1 - 2iot 1 + 1 2 so what happened to one from one Son? It got cancelled. Similarly two to two It got cancelled. So the empty -iot was saved. Now We can easily find out x and y from here. You can, son. So how will you do it? See. Now there is no real part here. then what will be x? What will be 0 and y? -1 is forest with iot and if minus is -1 then firstly you r Take it out. The formula to find r is Is. The formula to find r is x² + under root of y². then the value of x is 0 and the value of y is 0 The value of -1 is then the square of 0 is 0 - 1² + 1. r equal to two will become one. Are you alright, son? Now we find theta. theta equation The formula is tan inverse y over x okay is it son? Then, θ = tan inver y over x x will be replaced by 0. We this What do you do with the minus? would take it out Are. Is it okay son? So we can exclude this minus. Let's take it out. So here it will come to θ = - tan inverse - tan inverse infinity something Over 0 equals infinity. θ = -π / 2 will come. θ = -π / 2 then it is at 90 There is value. So now we are not aware of its polarity Let's take out the form. ok here's one It is a small concept. a small concept I will also tell you the page number. this which There is page number 15 of a note in your book, right? The box is made. So you guys have Have to do it well. Is it okay son? this is a note A box of is made. You guys see this I will take it. and besides this there is another one That was the point I just read. Ambulatory I will be with you in the upcoming videos as well I will discuss it. Ok? So now we take this Let us take out the polar form. question is over We do. z = r cos theta + iot sin theta. The value of R is one, the value of R is Forest. and here cos theta will replace -π/2 Is it okay son? +iot sin theta is replaced by -π/2 So son, this will be our answer. these are ours You will get its polar form, son. So I hope you have understood it. Now we live Let's move on to part number six. Book Now My son has come to you in the market It is available. you guys as soon as possible Purchase the book because in the book There are important points. important things Are. So you guys did not discuss these things I have to do it, son. Ok? You people have seen is also there. So let's move on to part number six. Are. Part number six. Now I'm gonna raise it I will do it. yes I will give it to you quickly son I will raise it and my part number six I will come here. Okay, I'm done. This has happened, the anger has increased, son, don't panic It is first of all. you would have come to part number six Are. that which I have 3 + iot Part Number Six 3 +iot Over 1 Plus 3iot So what's the first thing you want to do? Its conjugate has to be taken. First of all its What do you want to do? have to take the conjugate Son. take it because it comes in divide If so, we have to finish it Is. 3 + iot 1 + 3iot 1 - 3iot over 1 - 3iot fixed bugs? Now multiply this by 3 to get one From 3 Look at 3, minus has come. 3 will come from under root three and This iota will also come with it. Are you ok Went? This is under root. This is also under route. If this is done twice then only three will remain. Square Ho to Square Under Root Cut Will go. Now it's the turn of this iot. Iota to Forest From Iota it will come with this - 3iot² over 1 square a² - b² So here I open the formula Gave a² - b² closed. Ok? 3 - 3iot +iot - 2iot equals iot² -1 It happens. So minus and minus are plus Will go. Over 1 square Under 1 square If the root is cut then it will become 3iot². 3 + 3 2 3 will come - 2iot Over 1 + 3 will come. If iot² is -1 then Minus and minus will become plus. the one above From Value to Common 3 - Iot will come down to 1 + 3 will become 4. So 1 + 3 What have I done to the child? made it 4 Is. So let's move on to the next step. Let's make space here, son. here Now our space has been created. So 2 * 1 = 2 2 * 2 4 so what will come? 3 - Iot Over 2 is fine sir. Now this which There is two in the divide, this is with both of us we will separate it with this also and with this also I will write about both of them separately. Correct Is? So what is x in this? 3/2 and what is y in this? -1 / most Let us first find r. x² + y² under root r = Write the value of x. Write the value of y in the square of 3/2 -1/2 The square of the hole and the square root of the top Will go. r = 3/4 square to under root cut If he goes, three will remain. Square Four of 2 Will be done. The square of -1 is 1. of 2 The square will become 4. = 3 + 1 / 4 here I took its LCM I took it, my son. ok sir? then next Let's go. Next you would have made a space here Are. Yes. ok sir. Done. 3 + 1 is 4 4 / 4 Will go. 4 / 4 Four cut from four One and under one There is only route one. So the value of r is Forest. Ok? Now we need theta. ta = tan inver y x well one thing keeps coming to my mind again and again It has been happening. I had a small concept I should have told you about this thing also. just Will you give me a second? If you have a If you don't have seconds then you can skip the video If you move a little ahead then next you will see this You will get the continuation, son. then 1 second Please give it to me. Just give me just a second. That point I feel I have not made somewhere You must have noted it somewhere. ok next Let's go. So, I will tell you further. It is very hot right now. so ta = ok do you want So do not write minus here. At the moment the tan The value of inverse y is 1/2 the value of x 3/2. Ok? So in the previous question also Do not write the minus and then find theta Take it through the coordinates. what we just now I have done part number five, right? You are here in that. But of course don't write minus here and theta Take it out. through whom you son her through coordinates through to to to cut θ = tan inverse 1 / 3 tan inverse is equal to 1 / 3 60. tan inverse 1 / 3 is equal to 30. If you write it on the calculator, you will get 30. on the 30th We can write π / 6 30, what do we write Can you? π / 6 Now observe that x is positive, y It is negative. if x is positive then y is negative Is. So here these are our coordinates Which coordinate will be made, son? Fourth coordinate will be formed. Ok? So Forth What will you write in the quadrant? Theta Ly In Fourth Quadrant. Theta Lie In Fourth Quadrant. So I'll direct this I will do it just fine. I'll plot this theta on y I will make it negative, son. Ok? -π/6 It is alright sir. So see the next one. Now you Find its polar form. Polar Form what happens? = r cos theta + iot sin theta z = r instead of one cos in place of θ -π / 6 +iot sin We replace θ with -π/6 so this is our remaining answer. sorry part number of gaya ji second part Six Now we go to Part Seven. Now We have the last question number two It's a part, son. This is a very easy question. 3 + 4iot 4 + 3iot conjugate taken g. Multiply and divide by minus. Now up multiplying the value by Am. three to 4 to 12 three to 3 to 9 iot 4 to 4 16 4 to 3 12 iot² 12 will come anyway. -9 + 16 + 7 iot² If it is -1 then this minus changes to plus Will go. 12 + 12 24 7iot came just like that. 16 + 9 will come. 16 + 9 is 25. on this 25th I will write about both of them separately. Son. Ok? then the first value of x is and name the second value y I will give it to you. After this now I can find r I will take it. what is the formula for r? x² + under root of y². So son this is the value of x put of y. Took its square. Now Divide by same value is 625 and above If we plus the value then this is 625 625 625 were cancelled and one was saved. under the root of the forest If there is one then r comes. Now we can find theta I will take it. tan inverse y over x tan inverse y The value of is 7 / 25 the value of x It is 24/ 25. If 25 is subtracted from 25 then it is tan Inverse 7/24 is here. If you guys want it then Of course don't find the tan inverse. this tan Even if we put the inverse instead of theta Your ans will be correct. How son? Look at it this way. Instead of z = r we have 1 cos r It's theta, right? What is this theta equal to? tan inverse 7 / 24 then in place of theta tan inverse 7 / 24 plus iot sign Replacing theta tan inverse 7 / 24 so instead of theta, tan is inverse 7 / 24 You can also do these puts. its inverse You can still place a put instead of theta. So both these answers are fine for you son. So This is question number two, son. Now we Let's move on to the next question. Four is given by cos 5π/ 3 + iot sin 5π / 3 So son, we have the first part Is. So, let’s start our calculator. By picking it up. ok sir? so all game calculator That this is going to happen now, children. So the most First g z = 4 cos cos 5π / 3 + iots sin 5π / 3 is it right The question is first of all yes we have to answer this Let us set z equal to 4 so z = 4 Type cos on the calculator 5 π / 3 5 * 180 Write it down of course or the first one work out 5*180 divided by 3 so get 300 Here is its answer. Is it okay son? So cos Withdraw 300. cos 300 so answer is coming G 1/2 + Iot Find sin 300, son. So sin 300 is its Answer is coming -² 3/2 ok sir. So Next z = 4 then 2 LCM comes out to be the remaining went 1 plus plus don't write plus minus minus square 3iot to i took the lcm of this 1 + minus minus square 3iot left 2 * 1 2 * 2 = 4 z = 2 1 - square 3 So what will happen to us, son? The answer of part number one will come. if you If you want to take it a little further then Multiply this two inside. 2 - 2 Square 3iot. Ok? so like this too Can do it. So kids this is the first part We have it. Now we go on our On the second side, it is given that z = 3/2. Write down the second part G. Well done. second part = 3/2 And what is given along with it sir? Also given is cos 7π / 6 cos 7π/ 6 + iot sin 7π / 6 is correct? So first of all, you We have to take it late and it is equal to z. After that, the son You on your calculator We have to find out 7π/6. 7 * 180 Bye 6 so 210 is coming. then find cos 210 Take it kids. Minus square 3/2 is coming Find out plus incomeot sin 210. sin 210 then Its answer is coming -1/2 so this is for all of you Children, this is the answer you will get on the calculator. Correct Is? Then, z = 3/2 - Square 3/2 -1/2 iot plus minus minus 1/2 and this Iot has arrived. z equal to now this here There will be two ways to solve it. you this three multiply it by 3/2 so this You can also take minus common here You can take it and even if you don't take it, it's not an issue Is. So this minus came to three just like that What will happen if 3 becomes 2 and 2 becomes 2? Four. Then this minus came. Three to One three two with two with four and together Iot. So this is our answer, it will come in seconds Of the part. Now we will watch the third part which is given something like this. z m = z m = r there is no issue if these r and r = 7 is r = 7 and you have the argument of z it is given son then you write something like this take the third part na z mode and z mode Whose number is equal to r and whose number is equal to r 7 and the argument of z, this is what you have is equal to theta so theta here 23 is equal to pi / 12. this argument which of z is this is equal to theta. if you If you look at the beginning of the lecture, I saw there But I told you this is the argument of z Son. Ok? So if you look at this in polar form if you want to write it in , then z = r cos theta theta The value of is 23π /1 + iodin theta theta value is 23π /1 okay children, then our answer is based on the value of z = r You have 23π /1 and all of you have your own Calculate 23 * 180 on the calculator Divided by 12 then the answer is 345 z = 7 cos 345 you guys find out son cos 345 so the answer is coming 6 + 2 / 4 +iot sin 345 sin 345 so the answer coming out is is - 6 + 2 / 4 So son, what did I do to answer this? I have given it here, I have put it here, children. Ok? So I have posted its answers here It has been replaced. cos 345 this is the answer Will go. plus iot sin 345 you have this The answer will come. Now you have done a small job Have to do it. multiply this se in I have to get it done. say okay I'll put it there I will multiply it on the other side. So Let's create space on this side as well. Any It is not an issue. Let's create space here, son. Ok? So z = 7 was multiplied. 7 6 + 2 / 4 ok sir. then plus 7 - 6 + 2 / 4 and what came out with it? Ayot also arrived. so what did I do here What did the kids do? to this value as well to this value also with both I got it multiplied separately. So, this is our answer. Students This is the next part. part number four is And these are the easy parts. now three parts I have got it done for you. now the rest of the parts are also like this. So I thought yes time Let us save you. beta part number four See. z mode is given 11 means this is r The value is 11 and theta is given as -1π/12. Is it okay son? So you have this polar form it occurs. First, we will write it in polar form Let's change it. the value of r is 11 theta The value of -1π /12 + iot sin theta -1π /1 Now you can calculate 11 on your calculator Multiply it with 180. Ok Son? and divide it by 12 and you will get the answer It has been 165. This answer of cos 165 is coming. - square 6 - square 2/4 + iot sin Son of 165, this answer is coming. Square 2 - Square 6/4 okay? then let's go to next yes children. Now on this 11 I am in I am multiplying it. See when I'm I will multiply it with both of them If there is a minus then I took out the common minus Took it. So this is square 6 and this is minus It's a plus. Square 2 and divide by 4 I have arrived. I have also shared this 11 with him If multiplied, then 11 square 2 - Square 6 divided by 4 and equal to I came just like that. And this is our answer. Yes. This is a very easy question. Part Number Five. The mode of Z is given as 10/3 which means this The value of r is and the argument of z is given is -17π /1 so the argument of z is This is equal to theta. so I wrote given theta = -17π /12 so I firstly I have changed it to polar form, son. So the polar form z = r instead of 10/3 cos r theta theta instead of -17π/12 + iot sin theta so instead of theta we get -17π /12 Now you need to pick up your calculator. pick up g write 17 * 180 and divide in If you write 12 then the answer coming is 255, son If we take out cos 255 then we get this answer Square 2 - Square 6 / 4 and then +iot If you calculate sin 255 then you get the answer square 6 + square 2 / 4 so this is the answer Used to be. Ok? So this is the one below the divide I have also tried the 10/3 that is outside I have multiplied it and I have calculated 10/3 for it I have also done this with both of them separately. what did you do son? I multiplied it Is. So you guys have to understand the next step did. See 2 * 2 4 2 5 10 so this 5 comes been 3 2 r I've put six here 5 / 6 The above value remained the same Ok son, I will do the cutting here as well. I am getting 2 * 5 10 2 * 2 = 4 so this is five It came like that 3 * 2 = 6 became square 6 + square 2 came as it was and This is our answer for part number five. So let's move on to part number six which that is 2 cos - 33 +iotsin -33 its Type cos - 33 on the calculator and you will get the answer 0.8387 then type sin -33 on your calculator The answer will be - 0.546 So if we multiply this two with this, then The answer is 1.673 I multiplied two with this as well So the answer is 1.08 893 plus minus what Done? It got minus, son. Ok? So The last part is part number seven. r is given. z mode is given 12. The argument of z given is pi. to all of you do you know that the value of z mode is this We have children equal to r. And The value of argument z is equal to it occurs? Equal to theta. Ok? So I Made its polar form. z = r cos θ +iotsin theta. What is the value of r given Is? 12 The value of cos θ is π then +iot sin θ Write the value of π cos 180, sin will come to be -1 Write 180, 0 will now come inside this 12 I am multiplying 12 with -1 -12 12 with 0 so 0 will come out to be 0 This is question number three, so first of all this You have to note down the questions, do it orally You have to get it on your register son, after that Then you have to come to question number four You have to come, come now we will talk about question number four Let's go to the side. yes students are with us now This is question number four. We have our Z1 and z2 both exist. of question In the statement it is telling us that brother z1+ Find z2 also. Z1 - Z2 also, Z1 Z2 also and Z1 Z Z1 over Z2 these four parts are given Are. If we first look at the simplest model of Z1, then take out form nine 5π / 4 if we calculate it son here 5π / 4 5 * 180 / 4 225 So if I take out cos 225 here If I take it then our work will become easier. - 2 / 2 minus square 2/2 + iot sine 225 If I calculate it then - square 2/2 so it means Both the answers are in which of our son Are you? Coming into minus. Ok? So In which of our answers are these two coming? In the minus. cos 225 and sin 225 too. so this what will happen to plus or minus? Be minus Will go. So we're going to create another step right here. instead we would have minus it right here Are. -iot Square 2 / 2 okay son? cos 225 - 2 / 2 sin 225 also - square 2 / 2 coming Is. In the same way you could have taken out the son z2 below as well Are. 5 the value of π is 180 180 over three divided by 60 cos 60 gives the value 1/2 + iot sin 60 sin 60 has a value Square 3/2 Okay son so first of all This is how you calculated its values I have to take it. Now we have the first part z1 + z2 So what do we have from z1? Yes It turns out z1 is 9 - square 2 / 2 - iot square 2 / 2 this is the value of z1 plus z2 5 1/2 + iot square 3/2 so son now you What are people supposed to do? don't solve it Have to give. Have to solve it. no to inside Multiply it. -9 square 2 / 2 - 9 square 2 / 2 and this one is like that Similarly multiply nine with this also And don't multiply it with this also. Ok? Now it's the turn of this five. multiply 5 by 1 on 5/2 and 5 3 to 5 3/ 2 and this iot will also come. So I think you all have understood by now It has been done. Now what did you guys do? See with which there is no iota with this value and if there is no iota with this value then You guys write this down together. -9² 2/2 + 5/2 You wrote these two together And with whomever it comes, you guys take the common If we take iota common then this value is plus First write down 5 square 3/2 and this minus Write down the value later, 9² 2 / 2 then I took the common one with whom I came Okay son, so what do you guys do from here now? Do this, children should take LCM If we take LCM then what will come here? -9 square 2 +5 ok son? Just like this Here also you have to take its LCM. So what will be left on top? 5 3 - 9 2 / 2 so Here also what have we taken from this Children? Have taken LCM. above and also Below as well. So let's look at the LCM. Divide has the same value. So the above value I will write it as it is. Just like this Divide has the same value. the one above We will write the value as it is. So This is part number one we have, which is I have solved it and shown it to you. Now Insha Allah, Allah will do good to you in this, son. second part is z1 - z2 i am in this I have started changing the second part. So What do you guys want to do? you people have I have to note it down again, son. Ok? So again you guys second part again do. The minus will come here. here at what will also come? The minus will come. Now minus If he goes inside what will he sign? Change. -5/2 - 5 3/2 You have to solve it again, son. So this -5/2 which is the real value of I have to put the complex together. So To put the complex together we have See, there is a minus with this too. also with this There is a minus. So now we have to take the -iot common It will have to be So the plus will come here. 5 3/2 This is also a plus and this is -9 2/2 this is also a plus Because now children what common thing did we take outside Took minus because what was with both We had a minus and its LCM came out So this will come here -5 will come and Here the iot of G minus will come and here So this is our second number. My part is also done, children, it is Allah's order So now we move on to our third part On the side. I am starting to raise this, son. Any If you have any issue then you can tell us in the comment section You have to ask me. So now we live Let's do the third part of this which is my Pass in the form of multiplication. Ok? so look at this z1 z2 I have put it here Write the formula for multiplication Is. for multiplication and for division For. So you have to remember this formula. So I have written the formula for z1 z2. It's the third part, son. It's the third part. I have written the formula for Z1 Z2 here Is. What is the formula sir? R1 R2 then what is it son? to multiply cos θ1 + θ2 + iocin θ1 - θ2 So here we have the children's formula is of z1z2. Ok? Then θ1 + θ2 + θ1 + θ2 Is it okay son? So this is the plus we have. So R1 and R2 now this is R1 in Z1 and R2 in Z2 What is? R1 is 9. R2 is 5 then 9 * 5 then cos cos θ1 + θ2 now θ1 whether you want it or not Write down the value. 5π / 4 and π / 3 are If you write down these values of θ1 and θ2, then this It will be the best. So this doesn't matter in the calculation It will become easy. 5π/4 + this θ1 5π/4 θ2π / 3 5/4 + π / 3 then Look, you are also feeling that this too This is cos, this is also cos, so this will be θ1, this will be θ2 will happen then +iot Then add the angles related to sin Take 5π / 4 , π / 3 So son, now I have calculated both the sin ones also. What have you done to the angles? have done plus Son. So let's go to the next one. Yes. Next Let's walk towards the steps. And what ours is What will be the next step? 9 * 5 45 cos now I am taking its LCM. 12 LCM will come. 4 * 3 = 12 3 * 5 15 3 * 4 is 12, son. 3 * 4 is 12. then plus iot sin I am taking 12 LCM. 4 3 12 3 5 15 and 3 4 is 12, son. It is alright sir. So What did we take from him here, son? Have taken LCM. So let's go to the next one. Yes. What will come to us next? 45 45 cos 15 + 4π 15 + 4π will become 19π /1 and 15 + 4 there too 19 π / 12 So son I hope you all reach here Now you understand how we performed this step have done it. Is it okay son? so next step What will you guys do now? zee next I will ask questions. This is our answer son Part number three. Now you guys will go let's move on to part number four and I'll I am starting to feel jealous, son. part number four Let's move towards. I have started raising it. take this sir. I have done it. Part Number Four. Yes. You have it now in recording is present. So inshallah you will have no problem Not everyone will experience this, son. No problem whatsoever It will be. So see part number four. which I have z1 /z2 so z1 /z2 is Look at the formula, I have written r1 there / r2 then what is written cos θ1 - θ2 + Iotsin θ1 - θ2 so son I have this formula Which you all must have remembered by now. Correct Is? So let's move on to the next one. here now Let me add a little bit of ink. So all of you People, son, put your values. Well done Put the value. Look at this, it's r19. is r25. So what happened? 9 / 5 cos θ1 is 5π / 4 and kids θ2 is I have θ2 is π / 3 cos of θ1 and the cos of θ2 then +iot sin y is this formula when the value In the form of divide we have to find θ1 is g 5π/4 θ2 is g we have π/3 So it is clear to all of you that no one has even It is not a matter of concern. then next View 9 / 5 cos Let's take the LCM. 12 LCM will come 15π 3 * 4 will become 12. Plus Iot sin 12 LCM will arrive. 3 5 15 3 4 12 ho Will go. Are you alright, son? so here you go We have also taken LCM. 9/5 cos 15 - 4 becomes 11π and there 15 - 4 will become 11π only. Is it okay son? so this We had question number four. Now We move on to question number five. Students now we have the question number five which is same as question number four We have to do it, son. Then z1 is given by near you. This value is z1. z2 is given Is. You have z1 + z2 z1 - then z1z2 and z1 / I have to find z2, son. Ok? So the most First, I'm going to solve z1 a little bit. Am. I have calculated cos 23/12π I have taken it out. So this value has come, son. Correct Is? And in this way sin 23/12 pi is calculated. So this value has come. You all know that How did you take it out? Multiply 23 by π and divided it by 12. And what your If the pass value comes then what will you do with it Do you want it, son? I have to take the cos. 23 * 180 By 12 it is 345, son. Ok? So You divided this by 345 and 11, i.e. 11 * 180 / 12 This is 165 coming up. So Z2 which is 165 to It has been solved and you have Z1 I told you, son. Ok? Now You have to solve the first part first First Z1 + Z2. So I wrote it here g Z1+Z2 first part. so this is for z1 The answer is 7 Square 6 + Square 2 / 4 then plus + 7 Square 2 - Square 7 / 4iot So here it is Gone are the days when Z1 has arrived, son! what do we have come to the z1 now with this We have to add z2, I have z2 -1 Square 6 + Square 2 / 4 and + 11 Square 6 - Square 2 Square 6 - square 2 / 4 iot okay son so this The full value z1 is obtained and this full value z2 is obtained If it has come then we have understood it, so now let's see We z1 square 6 + square 2 / 4 square 6 + 2/4 what do you need to do Children? Solving Rail with Rail Is. So what does 7 - 11 become 7 - 11? -4 and the value in brackets will be square 6 + square 2 / 4 okay? Similarly, this is The value you have is 7 squares with iot 2 - square 6/4 ios and so on It is Iota + 7 + 11 + 18 it will come son. Correct Is? And no more, apart from that we are short here You will have to do some work. By solving this First, we have to do one thing here. Son. Ok? See this square 2 - Square is 6. These Square 6 - Square It is 2. So we have to do the same with this. So Then what do we do? like these two It's beans. these two are the same so I I solved it. 7 - 11 - 4 now these two There are no beans. so we have to solve this first have to do it. -7 Square 6 + Square 2/4iot So minus when I took the common from here this So by the value of minus squared 6 plus The square becomes 6 and the square of plus 2 The square of minus becomes 2. in a similar way which This is the last value that you have The children will come back as they are. this one which value is +1 Now I have taken the minus common here, this The value is benign. the one inside the bracket I took the value from here minus the common value and got the same I did it, son. So here we will see from four Four what happened to you? yes its cancelled here Done. Yes. I cancelled four to four has given it. So the remaining value is I will write it here. Minuses square 6 + square 2 so this value remains Went. This value was saved -7 +1 +4 +4 And what was left with it? Square 6 - Square 2 Iot. Is it okay son? Now this The values in brackets are the same, right? so therefore We can solve this. then next g We have four in its division also. Now! What happened to four from four? Cancelled. Look If its divider has four then that four will definitely come Once. So what is left here? Minuses Square 6 + Square 2 + Square 6 - Square 2 Iot so this is ours The answer will come. yes we have just done this with plus I have done the first part. Now we are going to take seconds Let's participate. Yes. The second part is ours Near. here the minus is z1 - z2 so here We can put a minus in the big bracket like this Let's apply it. Now son, I am in this I have started changing. this is the second part no sir? Z1 - Z2 exactly z second part This is what we have now Z1 - Z2 ok son So, for that, we have to do the same thing Rather I do one thing, I I will raise it, okay now what do I do about it If I change it, son, I won't be angry and I'll get you back here again It's not an issue, just go ahead I have left the above value as it is z1 is here minus this is here z2. Is it okay son? so next 7 I am writing this first value. Square 6 + square 2 / 4 this value came as is. then this plus 7 Square 2 - Square 6 / 4 This is the second The value also came the same way. Minus inside Will go. It will change its sign. these minuses And what will this minus become? + 11 Square 6 + Square 2 / 4 ok son And this is minus and this is plus minus 11 Square 6 - Square 2 / 4 Iot Ye Jo The outside minus is this minus of z1 - z2 What will he sign when he goes inside? He will change it, son. Ok? then next Let's walk towards the steps. View Square 6 + Square 2/4 Square 6 + Square 2/4 So what do I write 7 and 11 7 and 11 Give? 18 Both 7 and 11 have a plus. so my Brother, what should I write to him? 18 Write Give. 7 and 11 becomes 18. and come along with will Square 6 + Square 2 / 4 ok son So I solved for real values. I have taken it with which I have not iota This one was real and this one is real, now this I have to do this to do this What will have to be taken minus from this common you will have to take minus 7 7 squared of minus 6 minus squared two by 4 iots so what did i do here from this value I minus these two If you have taken common from outside then 7 these Square will become 6 plus and this square 2 minus will become over 4iot and The last value remained the same - 1 -1 Square 6 - Square 2/4th is what it is The last value, son, we have used this Don't say anything. So now here the cutting Let's do it. 2 * 2 = 4 2 * 9 18 ok Yes. So what is left here? yes here escaped g9/ 2 square 6 + square 2 / 4 here g-7 -1 - 18 Is it okay son? And what came along with it? Square 6 - Square 2 / 4 Iot So Here But we will still cut it. 2 * 2 4 2 * 9 18 So what is our answer? 9/2 Square 6 + Square 2 / 4 - 9/2 Square 6 - Square 2 / 2 and together what will come? Iota will come. So this is ours g z1 - z2 will arrive. Now we are going to its third Walk towards the part. we have third part Is. We have to find z1 z2. So I Its formula was written as r1 r2 cosθ + 1 + cos θ1 + θ2 +iotsinθ1 + θ2 is it okay son? So R1 R2 R1 is 7 R2 is 11 7 * 11 then this cos 2π g θ1 + θ2 23/12π θ1π /1 θ2 is then plus iot sin sin of g First angle is θ1 and second angle is θ2 you have it right son cos it came just like that now this is the same value 23 + 11 divided by 34 12 came in and pi also came along with it Like 23 + 11 34 /1 and what came with it Son? The pie has also arrived. After that I what have you done? multiply 34 by 34 on the calculator Multiplied by 180 and divided I got it done on 12th and the answer came as 510 I took COS 510 and it came to you. cos 510 - square 3/2 then plus iotsin 510 so 510 has come 1/2 is ok son So multiply 77 by - 77 3/2 77 multiplied by one is 77 7/2 This was its third part. Now our I have the fourth part. towards the fourth part Let's go son. part number four i I will do the changing myself. Quickly you People should also write it down. Part number four. Now we have Have to find Z1 / Z2. Is it okay son? So what would be its formula Is? g R1 / R2 What happens to the R1 / R2 marker Went? Without a marker you never know which hand is which If ever any of these comes then its yes the formula is r1 / r2 this is that The formula is written in the previous question I have also told you cos θ1 - θ2 sinθ1 - θ2 so what will come here G minus will come and what will happen here too If it goes minus sir you will make it 23 - 11 People calculate 23 - 11 23 - 11 on their calculator So it is 12. 23 - 11 equals 12. 23 - 11 12 will come. 23 - 11 what will come Children? 12 will come. Similarly here also 23 - 11 12 will come son. ok sir. so here But yes I made the change because divide What is the value in? G is in minus Not in the middle. Here comes the minus. So what has come here? R17 is here. R2 11 I have arrived. It's 7/1 sir. what happened here But? 7/1 so 12 will be subtracted from 12 so cos pi He will come, right? If 12 is subtracted from 12 then cos π and Children, what is the value of cos π? cos Calculate the value of 180 as -1 on a calculator Of course take it. 12 has been cut from 12. so here Here it is 7 / 11 cos 180 - 1 sin 180 What is it equal to? is equal to zero It happens. ok sir. sin 180 is equal to 0 cos 180 - 1 7 /1 multiply by -1 We did -7 /1 and multiplied it by 0. If you do it then you will get zero only. So this is ours Yes the answer will come brother. we have this part number is four and question number is five Already happened. Now we will come to question number six. pass is question number six which is first and It was and is now. So if you have the first If you have prepared questions with videos then He is also right. He is also one of the best on You can do that because 20% of the paper is SLO Based can also come son. Ok? So I have already written question number six here. Am. argument z1 z2 argument z1 + argument z2 is it ok son? So look at me I am lying down. you have z equals na r cosθ , iot sin theta is equal to this right now I'm also taking another z r cos theta now why did i buy two z because i have z1 and z2. There are two z. So I made one Named z1's and the other z2's. if here If it's R1 then this is R2. If this is θ1 then this is θ2 Is. If this is θ2 then this is θ2. Ok Son? So absolutely yes, this θ1 will come. the forest is So all these will become ones and all here will become twos If it is there then there will be two here. Is it okay son? So Now I am taking its left hand side. The argument Z1 is the left hand side Arguments Z1 Z2. Is it okay son? So I would recommend Z1 Z2 to everyone Let me solve it first. What is my Z1 Near? R1 cos θ1 +iot sin θ1 Firstly I calculate Z1 Z2 together I am getting it multiplied. here comes Z1 And with this I'll write Z2. R2 cos θ2 +iotsin θ2 ok son so let's see the next one actually We have to find out the value of the argument from here. Is. Ok? so what about multiplication It happens? In previous question also you have have done it. So this is R1. This is R2 with this one. Similarly, this one R2 will come out. This one R2, you have to write it like this Is. Ok? So this is R1. This is R2. so this will come to R1 R2 this is θ1. This is θ2. So How do we do this in multiplication? By adding θ1 + θ2 similarly here also +iotsin θ1 + θ2 is that okay son? so next if we Check its magnitude from here Magnitude If we look at the magnitude of Z1 Z2 So that's what we have for R1 R2 and if we look at its argument Z1 Z2 from this The magnitude is R1 R2 and if we calculate it look at the argument of Z1 Z2 so what is that will be? argument if we look at what will that be for z1 z2? That would be θ1 + θ2 this angle is actually We have an argument for this, son. Ok? This was previous when I introduced I had given you these things in that too I had defined it at that time, son. Ok? So See next g. These θ1 and θ2 are the real ones These are the arguments of z1 and z2. Ok Son? So θ1 and θ2 are the arguments z1 z2 θ1 and θ2 are what we actually have The argument of z1 and the argument of z2 are Son. Ok? So this is our proof Will go. This is the first part. Now we live Let's move towards the second part. second part We will see, son. I will write the second part I am thinking that you all will listen to the second part If you note down the questions along with me Believe me, it will be very beneficial, very short There is a question and there is an easy question and there is a difficult question no son there is nothing difficult in this okay so let's go I have raised it and I am now moving on to the second part Take the second part, I have done it son I will write the second part from my book. What is the second part telling us by seeing it? argument z1 z2 Arguments z1 over z2 , Arguments z1 - Argument z2 so son first of all you have read z1 and z2 I have to take it lying down. Ok? After this, you What is the plan? Z1 over Z2 means left hand I have to take a side. You have to take the left hand side which is Arguments Z1 over Z2 argument What value of Z1 do I have? Z1 Keys The value is R1 cos θ1 +iot sinθ1 This is the Z1 I have, it is fixed sir It's all fine over R2 and below Z2 I am writing the value of or you can simply write it like this Write it down right now, right now, like this Let's write down that we are finding Z1 Z2. Ok? So it will be easy for you will go. up z1 down z2 cos θ2 +iotsin θ2 so if you guys write it this way then it will be It will be much better. Is it okay son? So Let's go to the next one. z1 over z2 = This is r1 r2. It will come just like that. r1 r2 I wrote it like that. What am I writing here? cos divide Isn't it? The angle in the divide is called It is written in minus. θ1 - θ2 cos θ1 - θ2 plus iot this na at the center Actually we have its properties also. So what do we write divide in? to the angle It is written in minus. Is it okay son? So this is how you guys have to do it. These Your next step will be done. Now we go Son, we are moving towards the next step. Let's create a G space here, son. Here But let's create space. So from here, if you If we find the value of magnitude then The value of magnitude that we have is is the value of R1 over R2 magnitude. So You write it here Magnitude. Is it okay son? What will be the magnitude? Z1Z2 , R1 R2 so what do we get from this Son? The magnitude has arrived. Now if you of this Remove arguments. The argument that is This is the angle that we form, not the bracket I have this argument. Ok? if you Now let us extract its arguments. Z1 over Z2 This is the angle which is actually There is an argument. like previous part I was there. So θ1 is the argument of z1 z1 is it ok? I have also given you this formula It was told in the introduction lecture when we were doing the polar form and so on and θ2 which is That's the argument of z2. so the argument of Z1 over Z2 θ1 is the argument of Z1 and θ2 Argument of Z2. then it will be proved Son. The means has been proven. So, this is our I have question number six. Now we Move on to the next question. We have Question number six and its statement Is. is telling us that the divide statement it is written divide z1 = 6 cos 150 + iotsin 150 z2 = 3 cos 30 + iotsin 30 and express in the term of x + iotty then In which term did you guys write this, son? Is? Meaning in terms of x + iota In terms of complex form you have seen this I have to write it down, son. Ok? In this term. So I will tell you all the formula for divide. I have done it. z1 / z2 = r1 / r2 cos θ1 – θ2 +iotsinθ1 - θ2 So here we have the formula for r1 The value of g above r1 below r2 is 6/3 cos This is θ1 with cos this is θ2 150 - 30 Then + Iotsin g above θ1 and below θ2 So 150 - 30 so 150 - 30 becomes 120 150 - 30 became 120. cos 120 - 1/2 sin 120 Calculate 3/2 on your calculator. To Ko multiplied by one 2 / 2 minus by the way I have just arrived. then this to When I multiplied it by 3, I wrote 2 3/ 2 Son, what happened from two to two? Canceled Done. So the -1 is saved. here also from two Two got cancelled. So Iot Under 3 Survived Son. So this is our form x + ioty of. x + ioty ok son? this question It was number six. Question number seven key Let's read the statement. What is it telling us? This is a very easy question. so I have already done it. Seven is saying That yes multiply. now you are here Have to multiply it. Z1 is given by 2 cos 60 + iotsin 60 z2 Given 5 cos 90 + Iotsin 90 and you have to express it is in the form x + iotty. Is it okay son? So the formula for z1 z2 is r1r2 cos θ1 + θ2 + iotsin θ1 + θ2 so we have If there is a formula for multiply then son r1 is above r2 is down 2*5 is up θ1 is down θ2 is so 60 + 90 60 + 90 then plus iot sign now If you come to sin then θ1 is above θ2 is below 60 + 90 is ok son so 60 + 90 is 150 here also it became 150 cos it became 150 - 3/2 sin 150 becomes 1/2 of your We will calculate it using the calculator. now this 10 I am multiplying inside. On the 10th So -10 = 3 divided by two. Went. Then multiply this 10 by this one When I did it, I got 10/2. and this is like that I came just like that, son. Now here we have Did the cutting. 2 * 1 = 2 2 * 5 2 * 1 = 2 2 * 5 then -5 3 + 5iota, so this is our answer. This is question number seven. Now let's go Are towards question number eight. Students Now we have question number eight key There is a statement. You guys from your own books Do you have a PDF or a book, I will read it. telling us to find the modulus and Argument of z. Finding the modulus too and the arguments also have to be found. Meaning We also have to find theta. just understand the matter Is. Question number is eight. you have this The given equation is -2 – 2iot. So first we Let us find its modulus. so you Everybody knows that the first value is means x² is always plus in the center and y² so -2² + -2² up under root of r formula is x² + y² under root so this is the same thing We have. So the square of -2 is 4 of -2. square 4 + 4 8 taken under root 2 2 so Here is your module. Now what have we Do you want to find it, son? its argument of z also have to be found. means find theta also Have to do it. so the argument of z or theta is There is a formula. tan inverse y over x y The value of is -2. The value of x is -2. of y What happened to -2 in place of -2 x instead of -2 -2? It got cancelled. So the forest arrived. tan inverse 1 It is equal to 45. and you will take 45 degrees we can write son tan inverse one of 45° And 45° can be written as π / 4. Means pie Divide the value of 180 180 by four So 45 now look you have both of these negative Is. when both of them are negative then where are they Will you bring it? In the third quadrant. So Theta will lie in the third quadrant. So we plus our answer in pi Have to give. So multiply this answer by pi This was done because both are negative and These two are lying in the third quadrant I am son. So pi plus π + π / 4 So what will be the answer? 5π / 4 is correct, right? How is that? multiply four by pi Give. 4π + π / 4 took the LCM, right? the forest beneath Is. cross multiply the bottom one Multiply the values by the rest of the values Cross multiply. So 4π + π 5π / 4 comes Will go. So son, this is the argument we have z also came and modulus also came. So Question number is eight. Now let's go Towards question number nine. Question Number It's nine. telling us that the right Equation arguments. you found the equation The argument has to be entered z times - 2 +iot = 2π / 3 ok? You have to find such an equation which will be found out from here and which will be = it can also be like x + iotty, your son Near. Ok? So you have calculated the equation from here We have to find it and its form is x +ioty Now we have to find the equation. In the equation Then we would not have the concept of IOT. Ok? So let's start now. To all First z is equal to what? If we have the sum of x + iotty If we take it z times then it is with iota which Whatever is signed will be changed. this is plus If it is iota then it will become iot of minus. So this will give us a bar of z. Now I Substitute the value of this z bar into our equation I will put it. So this argument came z times z times what is equal to x-ioty Is. so x-y comes to this z bar And this value remained the same. Ok? And 2π/3 came out the same way. Next See I put the real thing together. x- 2 And of these two I have taken ayot + ayot Took the common. this is real na x - 2 I have put this together. and both of these Of the lines I'm making Iot I took the common one. So +iot became common. Iot came common. In its place, the forest survived and The minus of y is left over here. Now what have we want to do it? See, this is in the form of x + iotty Whatever he has said in the statement has been done. What should we do now, son? of the argument formula means that formula with theta we have Have to apply. So what was the formula? The value of tan inverse y over x y is 1 - y The value of x is x - 2 tan inver y over x and This 2π / 3 has come as it is. These argument means theta formula we have I have applied here, son. So here is tan is the inverse. Go to this side and become tan positive Will be done. Now you have calculated it on the calculator The value has to be found out. of tan 2π/3. To Ko Multiplied by pi means by 180 If we divide it by three then we get 120. So tan 120 I solved it on my calculator So this is what I got under 3 tan of G minus 120 minus under 3 and this value It will come back as it is. Is it okay son? So I wrote 1 - y as is. Gave. x - 2 is being divided here. Here What will happen when you come, son? be multiplied Will go. So -² 3 multiplied by x - Square 3 is multiplied by -2 so here But I made a small mistake. Minus and these What will become minus? It will be a plus. So I think I have understood this. Now we have to understand something. this forest is like that Went. - 3x becomes plus on this side and this is +2 3 came on this side and became minus and -y on this What happened by going to the side, son? Plus now I twisted it a bit and wrote it down This way. This is coming to minus 2 3. Ok Son? So what is this minus and this minus Went? It has become a plus. Ok? So you to this This equation can also be written like this Are. This +y should go to the other side and change it to -y. So square 3x this +y coming on this side -y Done. This is plus 1 and this is minus 2 Square 3 will remain as is. And What will be left in is equal to two? Zero. So This is your required equation question Number nine. If you do not understand anything then You should ask as per your wish. Now let's go Moving on to question number 10. g students Now we have question number 10 and this This is your son's question. Statement Read it quickly. He is saying that if z = x + ioty he is saying that if z x + The argument z times - 1 + is equal to iotty 2iot over z bar + 1 - 2iot = 9π / 4 k It is equal. So you have to do this show The statement says show that x² + y² - 4x + 2y - 5 By solving this we get The equation should come to you, son. Ok? So let's start. Yes. Bismillah We do. There's a bit more work. right now we I am doing question number 10. So son See, you have z given to you. First of all x + iotty if you take out the z Who will this be equal to? x - iot k. This way right? So after this, now a small A statement is written about your exercise On the back side. argument z1 over z2 this is Who is this equal to, son? Meaning of argument z1 - argument z2 You guys minus the divided value You can write in it. so the purpose of saying this What is? This is the divided value, right? I started writing this value in minus Am. The argument should come with the top value first will go. z times - 1 + 2iot then minus then The below value will appear. so here you go I have broken the divide. z times + 1 - 2iot So this is your question children It is also important. This could also come on the board It is a long question. Is it okay son? The question number is 10. so let's go to the next Yes. Arguments z times z times is equal to what? x - ioty Of. x-ioty instead of z bar -1 + 2iot Argument g x - ioty instead of z times + 1 - 2iot = 9π / 4 okay son so here we go now what is the next step going to be argument now here I'm separating the real hmm x - 1 this is real with you and iot I am separating the ones +iot 2 - y will come. 2 - y so I took the common number and made it 2 - y Is. Similarly, children, I have also said something here This is what I am going to do. The real ones were separated. x + Taking 1 and -iot common to 2 + y 9π / 4 Yes. ok sir. So till here you all are now I have understood it. well here if you See, there is a plus here, then you can see it here also If you take the common plus then go ahead and then You might have a little issue too, son. Ok? because here plus +iot common taken If it has happened then here also we will take +iot as common Let's take it. so with this g -y and is -2 ok with that? so as not to go further We don't have to change the sign again and again Son. Ok? So this is this kind of G form You have it ready. now what do we do here want to do it? g tan inverse of y over x The formula argument is theta itself No. So its formula is tan inverse y over x so the formula which is applying this so this is tan inverse y y value of 2 - the value of y x is x - 1 tan inverse y over x Similarly here also tan inverse The value of y is -y - 2 is the value of x g x + 1 = 9π/4 so till here g now you all If you have understood then move on to the next one I am here. Yes, now we will create space Son. So look, there is a small formula which you guys have further studied in trigonometry I have to go and study also. So this formula from now on It is being used in this question. So now It's a compulsion, son. It is our compulsion. Now this We have to get it done, son. Ok? and yours too it is a compulsion. You will also have to do this now. So This is the formula. There is a small formula Son. tan inver a - tan inver b then this is the formula children tan inverse a - b over 1 + ab tan Inverse a - b over 1 + ab this is the formula of ours, son The pass is present in trigonometry. So now This formula is being used. tan inverse a is - tan inverse of this value is b. These The first value is a. The second value is b. So I'm putting here. See a whom Is it equal? a is equal to 2 - y over x - 1 And what is b equal to? - y - 2 over of x + 1. so this is a and what is this son near you? is b. Is it okay? tan inverse a - tan inverse b so I would put that here hmm its on the right hand side left hand No need to put on the side because the left hand side is already Has been made. So I put it on the right hand side I am doing it. tan inverse a - b then what is the value of a? 2-yx- 1 - What is the value of b b ? -y – 2x + 1 a - b a - b over 1 +ab 1 + The value of ab a is 2 - y over x - 1 b The value is y - 2 over x + 1 so children this Now we put our values in the formula 2 - y over x - 1 - 2 - y over x + 1 is then divided by g in the same way. Let's move ahead. Time doesn't go by own. Let's go to the next one. yes the one above We will have to take the LCM of the values. So There is only one way to take LCM. the bottom one multiply the values Multiply the bottom values by x - 1 The easiest way to find the LCM of x + 1 Is. and cross multiply the remaining values Son. Ok? So this is x + 1 with this It will be multiplied. 2 - y x + 1 then this center minus and - y - 2 x - 1 so son now we have solved this It has been cross multiplied. so in the same way Next also we can multiply this I will give. Ok? so in divide in divide What is happening? yes look at whose divide Nothing happens in his divide What happens? There is a forest. Lady Divide Nothing happens in its divide in the forest It happens. Now this is a single value. This is a is a single value. So I will also give the LCM of this I am taking it. So son here now I am in space I am making it so that you people don't get mixed up go here. So I have a little something here now You guys are under tension. ok sir. Ok? So This is how we create space. So son Now the LCM of the bottom value is x - 1 x + 1 LCM will come. x – 1, x + 1 what will it come to? LCM will come. Is it okay son? yes exactly equal to This value will come. 9π / 4 equals to this The value we get is 9π/4. Ok Son? This value will be 9π/4. Ok? So x - 1 x + 1 multiplied by one. cross multiply so what am I writing Am? g x − 1 x + 1 x - 1 x + 1 is multiplied by one got cross multiply and this one is above If multiplied by the value of 2 - y - y - 2 is here. x - 1 x + 1 Look at this one I'll draw an arrow here. Cross Multiply like this one above this one If multiplied by the value then this is the upper one The value has arrived. I'm going to move it a little higher now I will do it and write it down. This LCM has arrived. x - 1 and x + 1 so here comes the LCM Also of the value. So till here we go now I have understood it. when we apply equals here Otherwise it would not have been planted here. Ok? So now we have to understand something. 0 = 1 What does x - 1 become x + 1? yes it is cancelled Went. We made it disappear here Son. So next day we have to discuss this with each other what needs to be done? have to multiply No. So how to multiply? See G just. 2 with x 2x 2 with 1 2 then -y with x - xy -y with +1 so -y so son I These two were multiplied together Here. Ok? So I will tell you this Where did the inverse go? He will tell you right now Are. Write the tan inverse here. Is. Ok? So you have calculated this tan inverse You have to write it near the middle line. Is it okay son? So now this minus is yours It will come just like that. students these minus joe It has come just like that. -y to x Multiply - multiply xy -y by -1 + multiply y -2 by x from -2x -2 to -1 to +2 over what happened Yes? Over this, the LCM is cut right? it was a bean We have. x - 1 x + 1 what will this become? will become x² - 1. What will happen to this? x²- 1 The formula a² - b². and here g I will multiply this. 2 to -y So I'll write -2y with it. these brackets Of course if you don't apply it, it is not an issue. Because Now the sign outside is not of minus. So 2 with -y -2y 2 then 2 with this -2 -4 then -y to -y with + y² then -y with -2 then plus 2y will come to = tan will come to 9π / 4. So tan here is inverse Was. What did I do by going to this side? has been made positive. so up to here brother now you Everyone has understood it now. Now this side is yours is also present in the pass recording and you There is understanding too. Now I'm done with it Am. Yes. Is it okay? so i asked you it is not there and it has been raised also and you I understand this, son. Ok? then start We do. Yes. Now a little further from here Our journey remains to be done. a little bit of question There is also length. I've also done Speedly. If you slow down the video a bit and watch it If you take it, it will be beneficial for you. Ok? So Let's see what will happen next? Its us Let us open the bracket. 2x + 2 2x + 2 - xy - y This first value will come as is Went. The minus will go in and its sign will change will give. So minus minus + xy - y plus 2x - 2 Are you okay, son? So what did I do? This I got the minus multiplied inside Gave. look down below g x² - 1 - 2y ok sir here we are here with you Let's solve this together. -2y to plus what do we do with it here? Yes, I cancel it out. Ok? So What do you have left? rest x² -1 and -4 Here, the remainder is -1 and -4 minus 5. I will write it down. and plus y² of the remainder. x² -5 plus y² left over g below. Are you ok Went? what came in equal to two? The forest has arrived. Multiply n by pi. Four Pay Divide it and take 10 out of it, then this will be yours One will come near you. Ok? Now a little higher Let's see what remains in the value. Yes We have. 2x and this 2x 4x is saved. Yes. plus 2 plus 2 minus 2 cut Went. minus of xy plus of xy to our g If the pass is cut then minus y and minus y left over minus y minus y minus 2y This value is saved and is getting divided here then this value comes with it when If we multiply then what will be left here is x² - 5 + y² So this value is left with you, son will go. What do we do now? A little bit of this Let me write it down in some order. In what way? First wrote x² + y². Ok? + 4x coming on this side minus 4x Done. minus 2y comes on this side plus It's 2y now. And this minus 5 remains as well Went. equal to 0 which is required Write down the equation or what is it proved note down. So son, this is your question. It has been solved. which is required Equation. So this is the question we have, son What was the number? Yes, the question number was 10. Now We move on to question number 11. Now We have question number 11 and its answer is There is a statement. telling us that if z = x + It is written in the income statement. End argument z - 2 - 3iot minus argument z + 2 + 3iot = 2π So you have to show that 2y = 3x son what should come out of this? near you The answer must be obvious. Ok? then start We do. Yes, I do Bismillah. To all What do you want to do first? replace z with x + I have to put it in Ioty. Ok? So I what did you write? put z = x + ioty in this question. so what is this Will you go? g argument x + iot in place of z this z in place of -2 - 3iot It came just like that. Minuses Argument instead of z came children x+ioty 2 + 3iot will be equal to each other to 2π will come together okay so next Let's go towards the step, what do you have? will go here your argument will come g added the real values together x - Meaning of 2 and iota values together you took iot common ok son this is it like here also the argument Here also son, real values should be brought together You have to pay tax and you have to take IOT common So what will come in y + 3 equals two? 2π aa Will go. So I think you all here I have understood this till now. Now we apply the formula It has to be done. The argument is basically theta itself Is. tan inverse y over x then y - 3 over x - 2 Similarly here also tan inverse y over x then y + 3 x + 2 = 2π. Is it okay son? so here Pe sir we have applied the formula. Now! We read the back pay formula. tan inver a - tan inverse b tan in question number 10 Inverse a - tan inverse b is equal to Does it happen kids? tan inverse a - b over 1 + ab so this is the kids formula we have Is. What do you guys want to do in this? Own The values have to be put in. Ok? tan The inverse of this is a-- tan inverse of this is b. So Who did this equal? yes you have tan The inverse is written first. tan inverse a - b then what is the value of a - b a ? y- 3 x + y - 3 over x - 2 This gives the value of a - what about b is there any value? y + 3 over x + 2 so kids I put this here Gave. Here tan inverse remains the same I have arrived. replace this a with y - 3 x - 2 y - 3 x - 2 then this minus remains the same I have arrived. This is b, y + 3x + 2, I put it and the value of 1 + a b a over y - 3 Over x - 2 y - 3 x - 2 This is the value of a and The value of b is obtained y + 3 over x + 2 okay son so here now we have What do we do with a and b in the formula? Gave? I put my ji down. So in my opinion Yes, there is no problem even here. What did you guys do here, son? Here on the right hand side a and b of a and b of What has happened to the value? put it Is. So, I have understood till here. Even I have understood it. tan inverse a - b a - b over 1 + ab i repeat this again and again so I am staying here so that there is no problem. Ye Ji Absolutely sir. I write g tan inverse here I have been. I am writing in the meantime Inverse. It is alright sir. so live here now I will create space, son. Now I'll create a space here so that I will get maximum space, right? So tan inverse this g tan inverse this comes as it is Went. Now I am taking LCM here. Rather, let us do one thing. Let's do one thing. You can leave out the tan inverse. First of this Take the LCM. x – 2 x + 2 so what is the result g lcm The lower value of this upper value multiply it, son, here we get 2π along with this 2π will also come naturally which is 2π it will come with this okay so below Multiply the values containing Cross multiply the remaining values x + 2 to y - 3 y-3 and x + 2 will come out. Is it okay son? And What will happen if x - 2 is divided by y + 3? x – 2 y + 3 then this value will be available to you. And What will come in the over? G over Over here also I am taking LCM. x – 2 x + 2 multiplied by 1. x – 2 x + 2 is plus y-3 And y + 3 and what is the LCM of g x - 2 x + 2 So here I have given this value what have you done to it, have you solved it The upper value in case of LCM The LCM is taken apart from the bottom x - I multiplied 2x + 2 by one Gave. I have written this one. Then, x – 2 If x + 2 is multiplied by one then x - 2x + 2 will come out. Ok? and here y - 3 and y + 3 y - 3 and y + 3 comes out like that. And this LCM has come in the divide, son. Correct Is? So where did the tan inverse go? tan inverse Coming here tan will become positive. be tan 2π He will go, son. Ok? So tan here is the inverse Was. Coming on this side 10 has become positive. So, till here you all have understood now. So here, now let me create a space Something like this so that we get maximum space May I get it. So this son is LCM. This from this What will happen? this gets cancelled out Will go. So I'm multiplying y by this. I am among ourselves. y with x is divided by yx y with 2 with 2y y with both of them i It has been multiplied. Now it's the turn of -3 -3 with x -3 with 2 -3 with 2 When I did it, it was 6 to the minus. Went. Is it okay son? So this minus here I came just like that. and what is happening here Is? Yes I am multiplying both of them Am. xy + 3x - 2y - 6 okay son over this LCM so These two values of LCM got cancelled out what remains is x² - 4 a - b a + b x² - 4 then a + a - b a + b y² - 9 okay son then Here the G formula is closed tan 2π tan 360 what is the equivalent to you? Yes tan 360 is equal to zero. then tan 360 ji what happened? It became zero. Let's go to the next one. Yes. This xy has arrived. These Plus 2y comes out. This comes to 3x of minus. this plus minus 6 of 20 also came. If the minus goes in then its He will change the sign. - xy what will come next? g minus of 3x minus of 2y and plus 6 = 0 okay son so this is xy plus of 2y minus of 3x minus of 6 minus of xy minus of 3x minus and minus plus 2y minus and minus plus 6 this is divide Where did the value go? The value to be divided is It came from zero and got multiplied. so what is the answer Done? It became zero. so let's go to the next Yes. xy is cut from xy and after this something is cut Is? G-6 is cut off from +6x. so the rest What was saved? 2y + 2y 4y - 3x - 3x - 6x = 0 Two of these If we take the common factor then 3x is the remainder of 2y - will become equal to 0. Ok Son? Then, - 3x = 0 / 2 will come. to be multiplied here Used to be. It got divided below zero. Ok? so next i'm doing this here Am. I don't have space. So this is this This will happen step after step. 2y 2y – 3x = 0 - 3x Here the plus will become 3x. so this Our proof will be proved, son. So this is what we Had to prove it. What did you want to prove? 2y = 3x 2y = 3x then Hans will write the prove. So Son, we have question number 11. Now let's move on to question number 12. yes students now we have question number It is 12 and its statement is. he is saying That solves the equation. solve equations Is. z - 2iot = z bar + 2 given mode Is. Is it okay son? So first of all I lay down We have done z = x + iotayy and z times x - iotayy Will be done. Now I put z here. Gave. I put x + iotty here yes son x + ioty has arrived and - 2 ioty is like that Similarly, what is z times z times equal to? x- if ioty is given then put x-ioty in its place I gave it to you, son. Now x remained as it was. Out of these two, I took Iot Common. x + 2 remains as it is. and this -ioty I have written it here, son. Ok? Now what do we take from this? I will take the mode. So how to take the mode? Yes? square of x will go up and the square of y - 2 will come down. Meaning The square of real and imaginary will arrive. Similarly here also the square of real and Always in the center plus imaginary The square will come. Ok? Now taking Square on both sides. when we are on both sides If you take pay square then from square the square root cancels out Son. Ok? and besides what have I Have you done it? x² arrived just like that. On this I The formula of square has been opened. y² a² + b² - 2ab Here also a² 2² 4 2ab 2 * 2 = 4x and this is the square of y He will go, son. So our next step What will come near? x² + y² + 4 - 4y - x² + x² here is minus Went. +4 becomes minus here. + 4x here - It became 4x. + y² comes here -y² equals to what did I get into? Zero has arrived. So up to this Yes I think it is clear to you all now Already happened. So see the next step. from x² x² got cancelled. from y2 to y2 and then to corresponding Like son +4 to -4 I cancelled it. So what's left is minus 4y minus 4x So out of these two I took the minus 4 If we take the common value then what remains is y + x = 0 -4 from this and -4 from this if you take common y + x will be saved. So in my opinion you Everyone has understood this till now. -4 here It is being multiplied. so -4 here It is getting multiplied, right? So -4 over here is 0. What will happen when you come down? It will be divided. So what will happen to this? It will become 0 / -4. So, is everything clear till here? I have done this x + y It is written. If you can't see it. I'll write it up a bit Children. So this comes out to be x + y, okay? Now If I divide 0 by -4. on 0 If I divide by -4 the answer will be 0 Will go. then x + y = 0 so this is your son's required equation Is. You can also look at this equation like this Can write. The children who wrote like this There is no issue in the paper, son. Ok? So this will also be your answer. so this We had question number 11. Question Let's go to number 12. He is saying that Solve the equation. solve this equation Two brothers. Let's do it right now. what is given has he? z is equal to question number 13 12 is done. I am running on GZ 13 Yes. 5z + 4 5z + 4 +iot this is in your book The question is there, son. and 5z times 5z times - 3 + 2iota so this is the question This is its form available now from here You have to find the equation. Is it okay son? So That work has to be done first. z which is Who is this equal to? x + iotty k. Then z times will be equal to what? x - iot k. ok sir. so now we're here let's put 5 as is z put x + ioty and + 4 in place of . +iot and this module will come as is. This Similarly, here also the five came as it was Went. x - ioty instead of z bar - 3 + 2iot This is the same as before kids will come. So in my opinion, up to now Clearly, there is no difficulty till here It isn't there at all. There is no difficulty right now. g5 multiplied by 5x it would have been mine You don't have to panic. Is it okay son? Five was multiplied by both. then g+4+iot will come then this value It will come to you just like that. Ok? This Similarly, here also we multiplied 5 5x - 5ioty - 3 + 2iot This value came as is. Now we Let's get the real thing together. 5x + 4 is I did it together. The Real One values that do not have iota and The one who has iota took the common one. iota common 5y+1 will come and together what will come? Yes, this mode will also come. This On the side we have separated rare and imaginary I gave it to you, son. same way here too 5x - 3 real different and iot different plus iot common 2 - 5y so this value will come out something like this iota is common and 2 - 5y so this modulus is also I will come here. Is it okay son? so this Iot if you took common then this is with this plus of is 2 and plus it is 5y to the minus. Correct Is? So what do we do now? Now we have We have to take its modulus. modulus g now Knows how to take care of children. Is it okay? So how do you take it now? Yes you all know it Let's take the square of the rail. 5x + 4 holes Square and iminary of square 5y + 1 What do you put square and above it, son? Square root. ok sir. same here too 5x - Square Plus of 3 2 - 5y square of and what will come above it? The square root will come. now what did you do Is? I have to write G taking on both sides. Taking Square On Both sides. So kids, now on both sides If when we take square then from square What will be the square root of this? It should be cancelled. You guys should write this line, son. So This is the best line. to check You will understand where the square root went Has gone. Is it okay? So if you don't write If we cut it then there is no issue then. Then he gives her some excuse and Might as well cross. if he doesn't understand So do it in such a way that the person checking it can understand it. let's go. Every step also needs to be directed It wouldn't have happened, son. Ok? So this is our life Which question is going on, son? here it goes This is our question number 13. Ok Yes. So now I have created a space here Am. Now I have started doing it here. See g the square root will cancel out here. The square of a + b will open. 5x² 25x² Will be done. The square of 4 will become 16. a² 2ab 2 * 5 10 will become 40. here comes the g 2ab 40x its also opening square Am. 5y² It will become 25y². 1 squared becomes 1 And 2 * 5 will become 10. 2 * 5 10 2ab so in my opinion till here So now this is this square You guys have been opening it since childhood. Childhood I am opening it from here only. now on this side You guys should open it too. so what came here Will you go? 25x² 3² will become 9. 2 * 5 will become 10 3. , 30x will come. The square of 2 on that side is 4 square of 5 25y² And 2ab 2 * 2 = 4 4 * 5 20 2ab comes out. Correct is it son? So here you can have some fun you will see Just look a little bit at what is here now It is going to happen. See this is + 25x² equal On this side of + 25x² this cancels out. See, when this plus comes here it will become a minus then It will also be cancelled. So we will cancel this Let's do it. 25y² it's here and it's is 25y². so what did I do too Children? It was cancelled. Ok? so the rest Let's see what's left. 16 + 1 17 40x 40x and this 10y fixed bugs? Where did this forest go? 16 + 1 is 17 Went. Ok? Now look, 9 + 4 is 13 Is. 13 will become minus here. Its After -30x Here the plus will become 30x. and this minus 20 Here plus 20 equals 0. After this, son, I will make it 40 + 30 70. 10y+ I will do 20y 30y. I will do +7 -3 + 4 Equation 0 will come. 10y + 20y 30 40 + 30 70 taken two common from the whole equation 35x 2 15 r 2 * 2 ok son the whole equation Took two common from 35x + 15y + 2 = 0 / 2 then 35x + 15y + 2 = 0 V is the required equation. So Son, this was our question number 13. Where I am present, son, but you don't understand. Comment in the section you guys will ask and you guys It is natural for people to ask and I try That I should also reply to every child, son. Ok? So this was the question we had. What was the number? It was 13. Now let's go Let's move on to question number 14. Write it down Question number 14. Now I'm going to raise it I am trying. The recording is present in Mr. The recording also exists. If someone notes If you don’t do that, he will pause the video. Son Saathsa ji, make notes along with me, son Create your notes. Is it okay? So the notes If I make you then Inshallah at the end Everything you need will be ready. again There will be no need to take notes. just if If you forget any step then you will come back quickly Will see more on YouTube. Is it okay, son? So don't go to academy tuition. when a thing You are getting everything. Academy Tuition Even if you are studying from YouTube Then you should leave the academy tuition, son. Leave it. Then you can judge yourself Let's say that brother, even after going to the academy, we If you are studying from YouTube then go to Academy There is no benefit of it. Is it okay son? So write it, question number 14 is saying that g is z x + ioty you have the equation Need to find 3z times - 2 + iot , 3z + Iot so son we have a question Question number is 14 which I am going to write now. I am done, Bismillah, let's start Let's do it, Allah will do good to you son, if it's ok First let us take z = x + y So what will happen to the children z times? will become x-ioty. So you can put it here now instead of 3 z times x -ioty - 2 +iot This value came as is. Similarly here also instead of 3z we get x + ioty Ok? and the plus of iot is same as it is I have arrived. multiplying three inside Am. 3x - 3ioty And this -2 +iot will come like that. This Similarly here also multiply the three inside I am doing it. 3x + 3ioty And this plus item will come as is. 3x And -2 we have done this together. 3x- 2 We did this together. In both of these We have taken the common of plus. So There is a forest along this iota. with this iota of minus 3y is. So this is 1 - 3y This way we will apply the modulus. fixed bugs Son? So next here also 3x is different common 3y + 1 so son here I am now this I have made this kind of form and I am hoping Yes, you will not understand step two. you are mine We will ask the right person also. Is it okay son? Question Number 14 is in progress. What do we do now? the square root of this is the square of x Square and iot value of 3x - 2 square of 1 - square of 3y hole. This like here also x² the square of 3y + 1 and what comes up? g square root is here. so live till now It is not a matter of any issue, son. Ok? What do you want to do now? You have written taking Write square on both sides son. Taking Come on, the marker is about to burst. Why? Because this has been such a long lecture, son. So there must be a marker, son. the man is tired So why won't the marker work? Ok? So Taking Square On both Sides. so now we square both sides when then what is the square root of the square Will be done? It will be cancelled. so live here Now, I create space. enough for me Need all the space. So I am living it I am opening the square. the square root is then It's over, right? So what will be 3x? 9x² 2² 4 fixed? a2 + b2 - 2ab 2 * 3 6 6 * 2 = 12 so this is 2ab of minus Went. 2 * 3 6 6 * 2 12 wherever 1 square 1 1 3y becomes 9y² - 2ab 2 * 3 6y = 3x what will happen ji it will become 9x² ji here 3x squared is 9x² here 3y a + b² so what will happen in 3y the square of 9y plus one 2ab 2 * 3 6 6y okay son so 9y square plus one 2 * 3 will become 6y so we can understand till here It has arrived. Now, next time we have some great fun We are going to step. + 9x² to + 9x² missing I did it. + 9y² becomes + 9y². Yes. He has been given leave. Now here we have What was saved? 4 + 1 5 minus 12x minus 6y this is left over g plus 1 here aa to the minus of 1 plus of 6y here aa to 6y = 0 of minus is clear till here Sir it is clear to us, 5 - 1 became 4 minus 12x - 6y - 6y so what is this Yes - 6y - 6y becomes -1y. Is it okay son? Now let's take out the four common parts from the whole equation Are. -Let's take 4 common ones. So this is a minus 1 has arrived. 4 * 3 becomes 12. and here also 4 3 R what happened? It's 12. 3y = 0 Went. everyone took the sign of minus 4 common Changes will happen. -4 Go there and get divided Will go. then 3x + 3y - 1 = 0 / -4 3x + 3y -1 = 0 Students, now we have the equation but he said find the g points Do it. So how do we find the points? So first of all kids you have to convert x to zero If you want to put then the point y will appear. Then you want to put y as zero so x The point of will come. Is it okay son? So I am a I work. change this equation a little bit I will write it down on top. Ok? so that we Let me get some space. 3x + 3y – 1 = 0 So first of all you have to put Whom? x has to be put to 0. this x then 0 It will happen, right? then 3y - 1 = 0 3y = 1 then y What will come equal to? y = 1 / 3 then one This point has arrived. See, he said points Find it. So first we put x as zero If we do this, we get the point of y. This answer has come. Now we have set y to zero as well. Have to putt. So here I will create a space I will take it son. make a space here Am. So once you put x as zero and then once again put y here in this In the equation. So now I'm putting Put y = 0 so this y will become 0, right? So what was saved? 3x - 1 = 0 3x - 1 = 0 3x = 1 and what will be the point of x? 1/3x= 1 / 3 So son, we have both the points We have found them. This is your life now it is final There is an answer. So this is the question we have Number G students are next for us pass question number 16 in which is given z = x + iot w = 1 - iotz over z - iot w which is the mode is equal to one and you I have to prove that what z is is what aana is The real one should come, meaning the real part should come There should not be any imaginary part in this The answer is okay son, so we have to prove it If it is there then look at it and you will understand the matter, w is ours Pass 1 - Ayotz Over z - iot w = 1 - z we have x + ioty which is is present in the question. z we have x + iot ty - iot w = iot in multiplied xiot and iot²y over this x will come and what will happen here Sir, out of these two, the plus iot is common y will come to - 1. = 1 - xiot iot² will be equal to -1 then minus minus becomes plus and y becomes The over divide value remains the same for now Write down x + iot y - 1 so I think its clear till here Is. ok sir. moving on to the next step Are. Let's move on to the next step. Correct is it son? Then, w = = 1 - xiot Rather do one thing. a shorty Let's do some work. 1 + y we would have split this up Are. Let's isolate the x iot. Meaning Separate the real part and the imaginary part. Similarly this divides in the divide so first It is separated from. x +iot y - 1 Now we take its mode. Both mode of w on side 1 + y - xiot x + iot So y - 1 is correct sir. Ok? What is the mode of next w equal to? of the forest then w will be replaced by one 1 + y x xiot Over x + iot y - 1 g exactly g with both modes even If it is there then we would have written these modes separately Are. Now this value is being divided here. What will happen if you come here? Multiply x + iot y - 1 mod equal to mod 1 + y - xiot then If it is beta mode then you all know we are in mode what do you do by taking the square root of x² And The square of y - 1 will come out. ok sir. Next here also square like this root 1 + y Hole in square center with plus and - x The square of the hole is so rare and imaginary We took the square of the part x² Here I'm taking the square of this y² + 1 - 2y means the square of a - b equal equal to a² + b² + 2ab so here also I opened the square - x² will become + x². Where did the under route go? If there is under route on both the sides then from under route The under route was cancelled. or understand it like this Let's see what we took on both sides? Took the square. So make the next step You can take it from both sides. from under root What will happen? It will be cancelled. So Let's move on to the next step. Next Step Let's see son what will be the next step? x² + y2 + 1 x² + y² + 1 - 2y this +1 + y² + 2y all these The value will come here. It will become minus. come here -2y And -x² = 0 this is all plus. on this side what will happen? It will become minus further. So Let's go to the next one. went from +x² to +x². , y² went from -y² + 1 to -1. fixed bugs? So -2y - 2y - 4y 4 will be divided here. If y is divided by 0 and 4, the result will be 0. fixed bugs? So z is equal to what in our Near? Given in the question is x + ioty So y is equal to 0. then z = x + iot0 If z = 0 then z = x. Iot from 0 When multiplied, it came out to be 0. Now this is the x This x we have is a real number. Ok Son? So this is the real part we have. Correct Is? So z is a real number or a real part Proven. Is it okay son? So we have this Question number 16 is 16. Now we are going to question number 17 17 which is z2 mode = 1 you found to do z2 - z1 over 1 - z1 times z2 and this This is the hole mode, son. Ok? So I have considered this value. I multiplying and dividing z2 mode From. Multiply by what? From Divide z2 mode. Now what is the way to remember this? z1 The mode already exists. then we have z2 We will multiply and divide by mode. If there was a z2 mode. Ok? then we z1 mode z1 times mode multiply and Would have divided it. so z2 times z2 mode times Multiplied and divided. the rest of these Both values came as they were. Above In the value given below you have entered the above value You didn't say anything till the end. Correct is it son? So the next z2 from one Multiplied by z2, it will appear 2 times. on top of it Is there a bar as well? Multiply Z2 with so do it z1 times z2 z2 times this is z2 times No? So what happened to this one as well? It got multiplied. the upper value You didn't say anything. you have seen it like this I have written it. Z2 times the same comes Went. Z1's Bar This son has come just as he was. Z2 Z2 You have to remember this again. we have this There is property. Z2 Z2 times these kids we have equal of Z2 mode. You guys have to remember this. Ok? Ok? This is also asked in MCQs Can. So z2 times z2 this is equal to of z2 mode. Remember this. So the top value It came back as it was. z2 times now this z2 times Where did it go? Now the thing to understand here is What is that? Look, Z2 has arrived. These z1 came once also. Now this z2 bar is The z2 mode that we have is equal to Is? of the forest. so what is this z2 mode Will you go? It will become a forest. This Z2 mode is over Will go. This Z2 mode will also end. What will happen to this Z2 and this Z2, son? Forest It will be done. Ok? So after this G Z2 - The Z1 will arrive. Z2 - Z1 whichever bar is equal to It is of Z2 - Z1 mode. So this is complete What will be the value of this? be cancelled out will go. And its answer will be one. so this Son, question number is 17. now we are next Let's move towards the question. Students now Next we have question number 15 Is. Yes I am saying that N AC source supply A voltage V equiv 120 Question number is 15, son. one such The source is what is applying the voltage. V = 120 cos π/4 + iot sin π/4 correct Son? After this it is written to a Circuit with Impedance. The impedance that he has It is written z = 1 + 3iot / 2k after this he says calculate the current so current you mean i what is it Have to calculate. Ok? so we I have understood the statement. this z given is 1 + 3iot / 2 so I have this two with both I have written it separately. fixed bugs Son? Now the V which is there in Ohm's law We have to find I, find the current Is. This is equal to ZI. in fact it would have been It is RI. So we are representing R here. are from Z because in the question he said Z is given. So we have to find I i = V OverZ will be found by us. Meaning of Z Multiplication is happening here. coming here It will be divided. So V over Z so give us I need an equation. We need one in V. One We need the equation beta Z in. So the one with v We have the equation. watch na v over The formula for i is being made. Now we find z Have to do it. So to find z we need to First, let us find r. So we How do you find it, son? from here x² + y² so x² + y² so this becomes 1/4. The square root of the square was subtracted. be 3/4 Went. Its four LCM will come. 1 + 3 is 4 Will go. So what will happen here son? From Four Four will be cancelled. then the forest will be saved and The under route of the forest is equal to the forest It happens. r = 1 will arrive. Now we have this There is a formula to find the angle because Now we have to find z, so for z we need I also need r, I also need theta, right? So theta The formula is tan inverse y over x to y what is the value of g iot 3/2 ours we have y and over x what is x g we have 1/2 3/2 over 1/2 ok son so what here It's done, if G2 to 2 is cancelled then tan Inverse 3 is tan inverse 3 is equal to 60. 60 we get π / 3 Can write. degrees to radians Have you converted it or the value of pi is 180, 180 divided by three is 60 then find z are doing it. Here you will find the polar form of z Do you know what happens? r cos theta +iotsin theta. So what does r turn out to be? Forest. These What we found out. All this work that did we do that to find r and θz have done it. Because in the current formula z It is coming to us. Is it okay son? Then the number of The value came out to be one. The value of theta is π / 3 The value of theta comes to π/3. Is it okay son? I repeated the same step here by mistake Is done. I have written it exactly the same. So Of course if you write it only once then it is a good thing Is. So these steps are written same to same. Son. So it came like this. fixed bugs? So now let's go to the next one. I….to you I told you the formula. Ohm's law in terms of Impedance. We have to find V = ZI I Current. Then V will be over Z. so here Divided, right? so v overz you last I studied the formula for divide in the exercise Is. r1 r2 cos θ1 - θ2 + isosin θ1 - θ2 So I explained to you that in the divide What happens here at the center? Minuses It happens. So remember this V over, we Represented by r1 over r2. Divide If it is in then apply the divide formula Gave. And what is v over equal to? of i. well what i am putting here yes i am putting r120. Look, V R1 is written above so this is R1 120 ok So θ1 is π /4, the angle it has I represent θ1 by θ1 as π / 4 And son, I have seen the angle of this I represented it by θ2, so θ2 is mine what is π / 3 and r2 one I have calculated this as r2 and I represented it by θ2 Given the values of r1 and θ1, the value of r1 θ1 And these are the values of r2 and θ2. So here I I have written all the values on it. Now I have The values have been replaced here. So What is v over equal to? is equal to i. r1 120 r21 r1 r2 cos θ1 - θ2 π / 4 - π / 3 + iotsinθ1 - θ2 So the term in the formula We have put the values in place. Then, i = 120 divided by one will give 120. cos it came as it was. 12 LCM aa Went. 4 * 3 is 12, so 3π is left. 3 4 It is 12. So 4π comes here. So I think you understand what I am saying Has been. We have taken LCM here, son. in the same way in the same way plus iot in the same way will come. The sign will come as is. and here But we have to take 12 LCM. 4 * 3 12 If it happens then it will reach 3π. then 3 4 is 12 4 pie will come and here I said The bracket has been closed. then let's go to next Yes, till here, Insha Allah's orders There is no problem with it. remember that What are we finding here? i I am finding it, son. Ok? So Next G, now let's create a space here So that you people do not face any problem. Take G Marker would be in trouble. Now what What should I do, son? So the marker got upset No. Now even the markers are saying Ustaad ji t Stop it. ok i = 120 cos 3π - 4π -π will come to -π /1 + iot sin 3π - 4π -π will come over 12. Ok Son? so i equal to 10th class you People have studied trigonometry In the chapter. If you haven't read it then understand it. If the angle of cos has a minus sign then it will be It becomes a plus. If at an angle of cos If a minus comes then it becomes a plus. his reason you read there that the circle is Would have been completed in the fourth quadrant Is. Because of which cos remains positive. The fourth coordinate is the house of cos Son. Ok? so that's why cos there remains positive. and if in the angle of cos If a minus comes then it becomes a plus. But If minus comes along with sin then it does not happen. plus minus minus iotsin π / 12 so this will be our answer kids yes question number 15 is about allah Our work has been completed by order, so this Our G is the required current which we what did i have to do son i had to find out ok son so this is question number 15 Now we move on to 16, number 16 tells us It is said that an AC circuit has an impedance z = 3 - 6iota so z is g we have 3 - 6iota is there son after this he is saying that The voltage is 90 + 30 ios. it is saying Find the current in both rectangles. and its Later he is saying that you have called this current as polar You have to write it in the form also, son. Ok? To all First I told you about back question I current I have told you the formula. So whose I Is it equal, son? V is equal to over Z Is. What is I equal to? V overZ K. So Children V is equal to whom? 90 + 30 Iot Of. And what is Z equal to? 3 - 6ot's. Above If we take three commons from the value given then it is 30 + 10 will come. Three common from the bottom value If you take it then it will get 1 - 2 ios. Three to Three What happened children? It got cancelled. Three Three got cancelled. So the rest of this value We have some left. Now I rationalized it have done it. So here it is 1 - 2iot. So 1 + Multiply by 2iot and also divide by it. Now I I am multiplying this. 30th in the forest with 30 to two with 60 10iota to one 10iot with 10 to 20iot² one with one one minus plus minus 2iota added to 2iot would give 4iot². So The matter is over. Let's go to the next one. 30 Just like that 60 + 10 becomes 70. iot² - is equal to 1. So what is this plus in will it change? in minus and 20 like that By the way, 1 - iot² is equal to 1. So It is clear brother. Let's go to the next one. 30 – 20 will become 10. 70 yoot as it is I came just like that. 1 + 4 1 + 4 divides 5 Will be done. Now these five, these two It is with. So 5 * 2 = 10. 5 14 70 It happens. So 14 iot has come, son. so this Five is with both. you can cut it I will give. So this means that we got current. It has arrived. So now you have called this current as polar Have to take it in the form. so polar form This polar bear is to take you all to The formula of the form comes. r cos theta +iotsin theta. So first we will find r Then we will find theta. Ok Son? So first I'm finding r. So 2 square of and the square of 14 is 2 Square of 4 14 Square of 196 4 + 196 200 will come. If you take the under route of 200 then you will get 10 2 This will give us the value of r. Correct is it son? So we will find theta. tan The value of inverse y over x y is 14. of x If the value of two is 14 / 2 2 7 14 then tan Inverse 7 is here. write tan on calculator Inverse 7 so this is the answer 81.86 So son, now we have to find its polar form It is about current. Then i = r cos theta + iotsin theta this is the formula we have. of r The value of put is tan 2, this is what we have here I have found the value of r. value of theta The value of theta is 81.86 81.86 this is our answer son question Number 16. Now let's move on to 17. Then there is 17. telling us to encrypt the World code. You need to encrypt. Any I have to send a message to World Code. So his The one given is 2 - see iot At the end he has written that he has done it. Then It is saying that you can then decrypt it as well. Give. You also have to decrypt it back Is. First of all I am encrypting it Am. Look, you have the letter C O D E If it is a code then the letter C in the alphabet is It comes at number three. o letter comes on 15 Is. In the alphabet, the letter D comes at number four. Is. The letter E comes on five. Now we have To complete the complex number. so baby we did it with this 0 came 0 came 0 Iota and 0 iota have been added. Ok Son? Now you have to encrypt it. This What does the code have to do with it, son? Encrypt to do. See, when you somehow got the message If you want to get a job done then you must have someone to do it There will be a will as well. So some way or the other It will also happen. so here is what it's for That's 2 - iot. So 3 + 0 iots to 2 - What do you have to pay through IOT? Multiply. Is it okay son? So this is C's Encryption will be done. This is what it is You must have got it naturally. So 3 * 2 = 6 If three is multiplied by iota then it becomes 6 - 3 iot has arrived. Then whatever it is What will be the encrypted form, son? See Oh, that one comes at number 15. So 15+ 0 What is the given value of iot and its key? 2 - iota then 15 * 2 is 30 and 15 is divided by iota Multiplied by 15. Thereafter, the son If it is D then D comes at number four. Of the alphabet. So 4 + 0 iot 4 + 0 iot and Its key is 2 - iot. So this is us encryption Son, this is how you can extract the encrypted form Are. 4 * 2 = 8 four to iota multiplied by 4 iot now g e is e which It is the fifth number in the alphabet so 5+ 0 iot and its key is g 2 - iot 5 * 2 = 10 5 multiplied by iota 5 iota So this is our encrypted form. Now We also need to find out its decrypted form. Is. So how to get the decrypted form? Are? The encrypted form that has come, son You have to divide it among your keys. Who An encrypted form has arrived, you have to pay on it What is the plan? Have to divide. If you see then The answer for C that came is 6 - 3iot. The answer for C has come, son. What came to you? 6 - 3iot. So its What will be the decrypted form? I'll give this a 2 - I will divide it on iot. Answer to C 2-What will I do on iOS, son? Divide By Give. Ok? So, from the above value Took three commons. 2 - Iot has arrived. 2 -iot So 2 -iot got cancelled and three came. So who is C? what was the answer? Three was the answer. So decrypted In this we go inverse. So C is what That was the third number in the alphabet. So Only the third number has appeared in the decrypted. So With this I complete the complex number. Give. After this son D is sorry O is O O The answer that came was 30 - 15. Now we have If you want to extract its decrypted form then You will divide this into 2 -iot If we divide it on whatever it is then I divided it into 2 -iot above I bought 15 common cards for the kids from the given value If you have taken it then what is left 2 - This is what If it is cut and it comes to 15 then it is absolutely ours If the correct form has come then what is this 15 son with this complex number number I will complete it. Which letter is this? Oh it is. Ok? Now it's G-d. D's What was the encrypted form? 8 - 4iotas. So I divided 8 - 4iota by 2 -Whatever happened on iot would have happened. The upper value Took four commons from. 2 -iot survived. 2 -iota cancels out to 2 -iot. so four aa Went. So what is four? g is d. So we its Complex numbers are also complete here. will that do. So what do we have, son? is d. Now last time I don't have any space Was. I put the last one on this side Children. So what about the last encrypted Form I think? 10 - 5iots. So we have this We will divide it. 10 - 5iot over 2 - iot. Is it okay son? So from here on top, five common Took it. This is cut. So this comes to us Son. So you can write five like this. 5 + 0iot. So here we have 17. Now we Let's move on to 18. Students Question Question number 18, 19 and question number 17 are same These are some questions, son. I need to understand the matter. The letter given in 18 is Q You I Z means quiz has been given son. So Q which is the 17th letter of the alphabet Is. It is U21. It doesn't happen. Z 26 It happens. With this we have complex numbers It has to be completed. By applying zero iota. Now that we have the key to it The answer given for question no. 18 is 3 - 3iota so we divided 17 + 0iot by 3 - 3iota If you want to multiply then this answer will come 21 + 0iota to be multiplied by this then it becomes 63 - 63 will come 9 + 0 has to be divided by 3 - 3 So the answer will be 26 + 0iota = 3 - 3 If you want to do it then it will come out to be 78 - 78, this is zero So 26 is multiplied by both of them If it goes then this is our encrypted form Went. Now we have described the form also in this way It has to be taken out in this manner only. the answer which has come You have to divide it among your keys. Correct is it son? So 17 is the common number from the above value. Went. What happened to him? Cancelled. So 17 I was saved. Then we have complex numbers It has to be completed. So this is the number We have q. what is this number of ours Near? It is Q. Now the second answer is 63 - 63iot We have to divide this by key. So we will take 21 commons from the above value. When this is subtracted from this, it becomes 21. complex I completed the number by adding 0 iot and 21 + 0 iot son is equal to u. in the same way Now the third number is 27 - 27iot now this one We have to divide it into parts. of we have 3 - 3iot so the upper value Nine Common will come from. This value is deducted from then 9 + 0 iot and 9 + 0iot for i It is equal, son. Ok? same as last This is the number that we have described Sorry encrypted form has come on 78 - 78 We will divide this and this answer will be 26 + 0 iot and this is equal to z so this question The number is 18. Let's go to question number 19 Son. So the letter 19 is class. So, C is the third number in the alphabet. It comes. L 12 12th number A first number S comes in 19th place and S comes in 19th place Is. After this I have the complex number I have completed it. That is how I did it I have taken out the encrypted form, son. his The given thing is -3 + 4iota. So if I multiply these two then the answer will be It will come to -9 + 12iota. I will do both of these So this answer, this answer, this answer is for you now, son. I know about this thing. Ok? So here it comes The encrypted form arrived at last do you know the described form We would divide the encrypted data into keys Are. So we divided the encrypted data into Did. I took three common ones from both. this cut If it goes then the answer remains three. and after this Complex number 0 Iot completed and We have this number, son. Ok? Similarly, the second number is -36 + 48 Iot divided by this pay t. It is from this When it was cut, the answer came 12 and after that complete the complex number and 12 which this is we have this is equal to l so this The letters will reach us together, son Okay, like this, this whole value is cut from this It's okay, it's the last one, right? -3 + 4iot when If you divide it on key then it will be cut So empty one will remain 1 + 0iot so this is our pass is equal to a. Similarly -57 + 76 is the number which is from here I am from either of the two I will take 19 common ones. If it is cut by this then s will come Will go. Is it okay son? because 19 will come Na 19 + 0iot 19 which is on s. Just like this The one with s below is also the same. So this too It will be cut. So this will come to 19 + 0iot so Will you get the answer to this, children? s come Will go. So these question numbers 18, 19, 17 are very It is an easy question and the same question. The exercise has been completed. very lengthy It was exercise. allah pak ji is with you all Are. You just have to work hard. see you In the next lecture. Allah Hafiz.