Class 10 Maths Chapter 1 Exercise 1.3 New Book 2026 - Class 10 Math Ex 1.3 New Book - Punjab Board

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[00:00:02][bell ringing] Assalamu alaikum everyone. Welcome to my YouTube channel The Problem Solver. In today's lecture, we will solve Exercise 1.3 Complete of Class 10 Maths Chapter One.

[00:00:12]Chapter number one is Complex Numbers. And the new book for class 10 maths is Punjab Board 2026. So before starting the exercise, we will discuss some topics. These are the basic concepts. After that we will solve the question. So the first topic is the complex plane and the argon plane.

[00:00:29]What is called a complex plane or an argon plane? You must be aware of this xy plane.

[00:00:34]Ok? This is the x axis with positive numbers on the right side. There are negative numbers on the left side. Similarly, this is the y axis. There are positive numbers here. There are negative numbers here.

[00:00:44]And the point where the x axis and y axis intersect is called the origin.

[00:00:49]Here x is also zero, y is also zero.

[00:00:51]Right? So if you have to represent any complex number in this xy plane then this plane will be called complex plane or argon plane. Ok? So for example, if we have any complex number z = -1 + 2iot and we have to represent it in this complex plane.

[00:01:13]So what will be his method? You will write this in the order payer form.

[00:01:17]Now see what is its real part? -1 What is comma and imaginary part?

[00:01:22]Two. So you will represent this order pair here.

[00:01:25]Right? Now you know that whenever an order pair is written, this first value is x and this is y.

[00:01:33]Ok? So in the same way you might have drawn graphs etc. So you must have got this idea.

[00:01:38]Ok? Now what does this tell us that this real part is okay? This will show the x axis. Ok?

[00:01:47]Whenever we represent a complex number in the complex plane, its real part will be shown on the x-axis.

[00:01:53]So now here you have to write the real axis instead of the x-axis and the y axis will show the imaginary part.

[00:02:01]Ok? Do not write the y axis. Now we will write the primary axis here.

[00:02:05]Ok? Now let us see which is the real part in this? -1. So -1 here we will see this -1 and from here you have to draw a dotted line like this. How far?

[00:02:18]Its imaginary part is to go there and to go here. Ok? So, to whom will this point be shown? to -1 + 2iot. Ok? This point will represent this complex number.

[00:02:32]I hope it's clear. So the simple thing is that in the complex plane the x axis will show the real axis. The y axis will show the imaginary axis.

[00:02:42]Ok? So whatever complex number it is, you can represent it very easily here.

[00:02:47]Now the next topic is conjugate of a complex number which we have also discussed in the previous exercise.

[00:02:53]But in this exercise there are many questions related to this which we will solve. So what is the conjugate of a complex number?

[00:02:59]And how do you find it? If you take this complex number z = -1 + 2iot, okay, so if you want to find its conjugate, then what will be the method, that you will take it z times, okay, this is showing to whom that we are finding the conjugate of this complex number, the conjugate of -1 + 2iot, so when you take the conjugate, you have to change the sign of the minor part, simply if it is positive then it will become negative, if there is a negative sign here then it will become positive.

[00:03:31]Now see -1 this real part will remain like this. If we change this positive sign to negative then this will be its conjugate. Ok?

[00:03:39]This is the conjugate of a complex number. Now if you want to represent this in a complex plane, this is the simple way.

[00:03:45]What is its real part? -1 And what is the imaginary part of it? -2. Ok? So here you see -1 is the real part on the x-axis. Ok? -1 and -2 these are points so -1 and -2 okay? This show will run on -1-2 days. Ok? So I hope you understand how to represent a complex number as a point. The next topic is modulus of a complex number. So here we will discuss what is modulus? How to find the modulus of any complex number? What is its formula? And how is the formula made? Ok? We will discuss all these things in detail.

[00:04:25]So if you just want to remember the formula then you can note down the formula and move to the next questions. But I would suggest that you should understand these concepts. Ok? So if you have any complex number for example z = x + yiot. Ok?

[00:04:43]is any complex number. So you know that now you can write this in the order pair form.

[00:04:47]Its real part and its inationary part. Ok? So if you represent it here. Let's suppose this is the point. Ok? xy Now if you have any complex number.

[00:04:59]I have just told you. We have also discussed it. Let's suppose the complex number is 2 + 2iot. Ok? So here if you have one and two. Ok? Similarly, one and two. So its real part is also two, its imaginary part is also two.

[00:05:14]Ok? So real part two and imaginary part two also. So this will be the point. Ok? Now what is the modulus? This point, this complex number, its distance from the origin, is called modulus. Ok? If you want to find this distance, the distance of this point from the origin, then this distance is called modulus.

[00:05:37]Now how to do this find? From here you have to drop a perpendicular line.

[00:05:41]Ok? So this will form a right angle triangle for us. Ok?

[00:05:46]Now this is a 90° angle.

[00:05:49]What is the side opposite to it? Hypotension. Ok? This means that the distance we want to find is equal to the hypotenuse. Ok? So using the Pythagorean theorem, we're going to find the modulus of this complex number here.

[00:06:02]Now see what is this distance?

[00:06:04]is equal to x. Ok? The real part is x. Which is the imminent part. Ok?

[00:06:10]How much is this distance? This is equal to y. If we talk about an example, what is this distance equal to? To K and this distance is equal to what? This is also equal to two.

[00:06:18]Ok? So we take at random a general complex symbol if x and y are ok?

[00:06:24]So this is equal to x and this is equal to y.

[00:06:27]So if you want to find this distance then you know its real part also. It also seems to be an imaginary part.

[00:06:33]So what is the hypotenuse squared equal to? of the base square plus the perpendicular square. What is Hypotenuse Square Is Equal to Base and Perpendicular? Real and imaginary parts which we know. x² + y² Now what do you have to do on both sides? Square root is to take ah root.

[00:06:50]Ok? Because we have to find the value of hypotenuse. Here we will take the square root. The square will cancel out with the square root. Here also we have to apply square root. Now here when we apply square root, we use both plus and minus signs.

[00:07:02]But here you will use only plus. Will not use the minus sign.

[00:07:07]What is its reason? That you are finding the distance.

[00:07:09]Ok? This modulus that you are finding is basically distance and distance is always positive. Ok?

[00:07:17]Distance is always positive. It is never negative, hence it will always have a positive sign. So this hypotenuse, which is basically the distance we're looking for, what is it equal to? Square root of x² + y². Ok? This is the formula to find modulus. If you want to find the modulus of any complex number, what is it equal to? Modulus of z = square x² + y² ok?

[00:07:44]What is this distance equal to? Square x² + y². So I hope you have understood how this formula is made. Now any complex number is the same if you take the example 2 + 2iota. Ok? So two is its real part and two is also its imaginary part. So this is the point.

[00:08:02]Ok? Now see, we know that this real part is also equal to two. The Iminary Part is also equal to Two. Meaning that you wrote the value of the real part.

[00:08:11]Take the square of the minor part, plus it and apply the whole square root. If you solve this then you will get the answer.

[00:08:17]So I hope it's clear what is modulus?

[00:08:21]When you represent any complex number in the complex plane, the distance of that point from the origin is called the modulus. And what is its formula? That is, take the square of the real part of the complex number plus the square of the imaginary part and apply the whole square root. So now you will use this formula in questions.

[00:08:41]Now let us solve the question.

[00:08:44]So this is question number one of Class and Maths Chapter One Exercise 1.3. Find the modulus of the following complex numbers. You have to find the modulus of these complex numbers.

[00:08:52]We have discussed Phula in detail. What is the formula? Modulus of z = square x² + y² That means you have to take the square of the real part.

[00:09:03]Plus, take the square of the imaginary part and apply the whole square root.

[00:09:06]Ok? So the first part is 4 + 3iota. First of all, you should let z = 4 + 3iota, this complex number we have made equal to z. It has been named Z. Ok? Now you have to find its modulus. So how will you find it? The real part is four.

[00:09:24]Take its square. Imaginary part is three and you have to remember this thing that do not write iota along with it. Ok? We will just write the number and apply the whole square root.

[00:09:33]So what will the square of 4 become? 16 + 9 becomes the square root of 25. If you take its square root, the answer will be five. This will be its modulus.

[00:09:45]Next part number two is -5 - 4iota. So here also, first of all you let z = -5 - 4iota. Now after that you have to find its modulus. So what is the modulus of z equal to? Its real part is -5 and we will take its square. Ok? Plus what is the imminent part? -4. Ok?

[00:10:06]You have to take care of the sign. If there is a minus sign, you have to write that also. Now if you take the square of -5, then take the square of 25 + -4, add 16 and we will get the value 41. Now this cannot be simplified further. Ok? 41 is not a perfect square of any number.

[00:10:25]So this will be the answer.

[00:10:29]Next part number three is 3/5 - 4 / 5iota first of all let's lay it down here again.

[00:10:35]Let z = 3/5 - 4/5 iot. Ok? Now you have to find its modulus. What is the formula whose square root you have to take? The square of the real part plus the square of the imaginary part.

[00:10:53]Solve this. What will happen if we take the square of 3/5? 3² 9 5 squared is 25 plus -4 squared is 16 5 squared is 25. Now here you have to solve it by taking LCM. So look, what's the denominator of both here? 25 will be the LCM. Ok? When the denominator is same then it is LCM and we convert the numerators as it is plus minus.

[00:11:18]Ok? We add 9 and 16, and what is the square root of 9 + 16 equal to? That will be equal to 25 so 25 / 25 will cancel out as the square root of one and the answer will be one.

[00:11:36]Next part number four is -2 - 3iot.

[00:11:39]Here also, first of all we let z = - 2 - 3iot.

[00:11:47]Now you have to find its modulus. So what will you do? We will take the square of the real part – square 2. Plus the imaginary part - square 3, we will take its square and apply the whole square root.

[00:12:00]Now look here, minus minus will become plus.

[00:12:02]Ok? Isn't it square? Minus minus plus and the square root will cancel with it. So what value will we get? Two. Similarly, here too, minus minus becomes plus, and the square root cancels out with the square. So what will come? 3 and then the whole square root.

[00:12:17]I hope you know what the square root of two means? That's 2 to the power of 1/2. Ok? And then its squared, so this two and this two cancel out.

[00:12:27]Ok? Now if we add this then we will get the answer square 5. This is question number two. If z1 = 2 + 7iota and z2 = 4 - 3iota then verify that. Two complex numbers are given and we have to verify, that is, prove that the left side is equal to the right side in all these parts. Ok? And this is a very easy question. All you have to do here is understand the question carefully, what conditions are given on the left side and right side? Now look what is there on the left side here? z1 + z2 is × So what this means is that we have to take its conjugate.

[00:13:03]Ok? First we will add z1 and z2. Then we will take its conjugate. What is on the right side? is a conjugate of Z1. This means that the conjugate of Z1 has to be taken first. Then we have to take the conjugate of Z2.

[00:13:13]Whatever answer they give, they have to add it.

[00:13:16]Ok? And the answer of left side and right side should be equal. First of all we write the left hand side. Ok?

[00:13:22]What is the left hand side equal to? Z1 + Z2 times.

[00:13:27]So now here first of all you have to add Z1 and Z2.

[00:13:30]Ok? Both z1 and z2 are complex numbers given. We add these.

[00:13:34]2 + 7iota is. The second complex number is 4 - 3iota. If you add them, their real parts will be added.

[00:13:42]What will become 2 and 4 6 7 - 3iota?

[00:13:47]4iota. So this is the answer to z1 + z2. Now you have to take its conjugate. Ok? Now z1 + z2 conjugate = 6 + 4iota, its answer was, if we take its conjugate then what will the conjugate be equal to. Whenever we take the conjugate, the sign of the primary part changes.

[00:14:07]6 - 4iota, so what is the left hand side equal to? 6 - 4iota, okay now what is the right hand side, right hand side z1 bar + z2 bar, so first we will take the conjugate of both the complex numbers. Then we will add them. So let's take the conjugate of Z1.

[00:14:28]What is Z1? We will take its conjugate 2 + 7iota.

[00:14:31]What will it be equal to? 2 - 7iota K.

[00:14:34]Similarly, take the conjugate of Z2. 4 - 3iota. Ok?

[00:14:40]Take its conjugate. So the sign of the primary part will change.

[00:14:43]What will happen if there is a minus here? Plus. Now you have to add both of these. Ok? Z1 times + Z2 times. Now z1 times + z2 times is equal to what? 2 - 7iota K. Plus what is the value of z2 times? 4 + 3iot If we add these, we will get the answer. What will 2 + 4 6 - 7iota + 3iota be equal to? -4iot K. So this is the answer on the right side. Now what is both the left side and the right side? Are equal.

[00:15:15]So you write it here. Ok? First we found z1 + z2 here. Then we took its conjugate, so this was the left hand side and this was equal to the right hand side.

[00:15:25]So at the end you should mention that the left hand side is equal to the right hand side.

[00:15:34]Next is part number two. Now what is there here? On the left side, z1 and z2 are being multiplied and then conjugated.

[00:15:41]Meaning that first we will multiply both of them.

[00:15:43]Whatever the answer is, its conjugate has to be taken. And what's on the right side is the conjugate of z1.

[00:15:47]is the conjugate of z2 and we have to multiply them. Ok? So first of all we consider the left hand side.

[00:15:53]What is the left hand side equal to? z1 * z2 and then its conjugate. So first we have to multiply these two. We multiply z1 and z2.

[00:16:03]What is z1? 2 + 7iota is z2 4 - 3iota has to be multiplied, 2 will be multiplied first by this complete bracket + 7iota this will be multiplied by this complete bracket we have to simplify it, okay so multiply 2 and 4, 8 2 and - 3iota have to be multiplied, there is no sign with this, meaning it is plus, so multiply plus minus minus 2 and 3iota 6iota now 7iota and 4 have to be multiplied, so 28 7iota and -3 iota will be multiplied plus minus minus 73 times 21 and iot² will come. Wherever iot² is, you have to write its value -1 so 8 now with these two there is iot. We can do plus or minus these. So what is 28 - 6iota equal to? 22 Iot K. - The value of 21 iot² is -1 so multiply 8 + 22iot - 21 by -1 + 21 so this is equal to 8 and 21 add 29 + 22iot what is the value of this? of z1 * z2. Now what was on the left side was that we multiply these and take their conjugate. So what will the conjugate of z1 * z2 be equal to? We will take the conjugate of 29 + 22 iot. Ok?

[00:17:26]So this is a positive sign, with the immediate part it will become negative.

[00:17:32]29 - 22iot. Ok? This is the left hand side.

[00:17:38]Now we solve the right hand side.

[00:17:41]So what is the right hand side? That we have to multiply the conjugate of z1 and the conjugate of z2.

[00:17:46]So first we find the conjugate of z1 and z2. z1 and z2 are givens. What will be the conjugate of z1? The sign of the imminent part will change.

[00:17:55]Ok? So what will happen to this? 2 - 7iota. Similarly, when we take the conjugate of z2, what will happen when the sign of the primary part changes? 4 + 3iota. Ok? Now we have to multiply both of these.

[00:18:07]z1 bar * z2 bar 2 - 7iota and this is equal to what? 4 + 3iota k. Now you know how to do multiplication. We will first multiply 2 by the full bracket and then multiply -7iot by this full bracket.

[00:18:26]Multiply 2 and 4 8 Multiply 2 and 3iota 6iota - 7iota and 4.

[00:18:31]Ok? It has a plus sign with it. Minus plus minus 7 4 times 28 minus plus minus 7iot and 3iota will multiply. So 73 times 21 and the value of iot² iot² will be written as -1 8. Both of these are imaginary parts. Both have iota with them. We can do plus or minus these.

[00:18:53]So what is 6iot - 28 equal to? - 22iot K. What is the value of -21 iot²?

[00:19:02]-1 so multiply 8 - 22iot.

[00:19:08]What will -21 * -1 be equal to? So 8 + 21 = 29 - 22iot So this is our answer to the right hand side which is equal to the left hand side. Ok? So what is both the left hand side and the right hand side? Are equal.

[00:19:28]Part number three is z1 / z2 is its conjugate. This means that first both will divide the complex numbers. We will take the conjugate of the answer. What's on the right side is that we have to divide the conjugate of z1 and the conjugate of z2, and both sides should be equal. So what is the left hand side? The left hand side is equal to z1 / z2 right? And then there's its conjugate. So first we divide both the complex numbers.

[00:19:53]z1 / z2 Also, you should definitely mention here that first we will find the value of z1 / z2. So what is z1? 2 + 7iota and z2 is 4 - 3iota Now how to divide two complex numbers is that we will take the conjugate of the complex number in the denominator, what will be the conjugate of 4 - 3iota, the sign of the minor part will change, it will become 4 + 3iota, it has to be multiplied by the numerator as well as the denominator, right, so now here we will multiply them, both of them will be multiplied, okay, so first multiply by this complete bracket + 7iota will be multiplied by this complete bracket.

[00:20:39]Now the denominator has the same pattern. It is four. It is 3iota. Once there is a minus sign, once there is a plus sign. So which formula will we use? a - b * a + b and what is this equal to? of a² - b². Ok?

[00:20:56]We will use this formula. Meaning that the square of the first number has to be taken. Find out the square of the minus second number also.

[00:21:02]Ok? Whenever you have complex numbers like this or any of these, you have multiplication in brackets. Ok? That the first number is also the same. The second number is also the same. Once there is a minus sign in the middle. There is a plus sign once. So take the square of the first number.

[00:21:18]Take the square of the minus second number. Now we have to simplify this.

[00:21:20]Ok? Multiplying two and four is 8. Multiplying 3iota by 2 gives 6 + 7iota multiplied by 4 equals 28. Multiplying 7iota by 3iota will give 21iot² / 16 - 9iot² Now see here the @ will come as it is.

[00:21:39]Both these imaginary parts will be added. So what is 6 + 28 equal to? 34iot K. + 21 will write the value of iot² as -1. Ok?

[00:21:51]What is iot² equal to? -1 K. The value of 16 - 9 iot² is -1. Now you have to simplify it further. So multiply 8 + 34iot + 21 and - by -1, it becomes -2 divided by 16 and this -9 - 1 multiplied by +9, so what will 8 - 21 be equal to? This will be equal to 13 - 30 okay? + 34iot and divided by 25 Now you have to separate the real and imaginary parts. Ok? In the last step, whose value is -13 / 25 + 34 / 25iota?

[00:22:37]Now you have to take the conjugate of z1 / z2. Now look, we cannot simplify this further here.

[00:22:42]If we could simplify it here then we had to simplify these also.

[00:22:46]So we have found the value of z1 / z2.

[00:22:52]Now we have to take its conjugate. So the conjugate when taken is -13/25 + 34/25.

[00:23:01]When we take its conjugate, what will it be equal to?

[00:23:03]Here the sign is positive with the minor part, so it will become negative. -13/25 - 34/25 Iot. Ok? This is the answer on the left side.

[00:23:13]Now we will solve the right side and its answer should also be the same. Let's solve the right hand side. Ok?

[00:23:21]In this we have to divide z1 conjugate and z2 conjugate.

[00:23:24]So z1 and z2 are given to us in the question.

[00:23:28]We have already found their conjugates in the first parts.

[00:23:29]When finding a conjugate, the sign of the immediate part is changed.

[00:23:33]What will the conjugate of z1 be equal to? 2 - 7iota K. And the conjugate of z2 is equal to what? 4 + 3iota k. This minus has become plus and here it was plus so it became minus. Now we have to divide these two. Ok?

[00:23:45]So z1 conjugate is equal to what? 2 - What is the conjugate of 7iota and z2 equal to? 4 + 3iota k. So again, dividing by the complex number in the denominator, what's the conjugate? The sign of the imminent part will change. 4 - 3iota This has to be multiplied here also and it has to be multiplied by the denominator also. And now we will simplify them. Ok? This complex number will be multiplied by this. This complex number will be multiplied by this.

[00:24:14]So this two will first be multiplied by this entire bracket.

[00:24:19]Plus sorry -7iot. Ok? This will be multiplied by this entire bracket. And which formula will be used in the denominator?

[00:24:28]a + b * a - b. So the square of the first number a² - b² means the square of the second number.

[00:24:35]Now you must have understood this formula.

[00:24:38]Ok? So now we have to simplify them further.

[00:24:41]Multiplying two and 4 will give 8 2 and -3iota multiplying -6iot. -7iota will be multiplied by 4.

[00:24:49]Minus plus minus 74 times 28 iot minus minus plus 73 times 21 iot and iota will multiply to become iot², okay 7iot and 3iota, so 7 3 times 21 along with iot² will take the square of 4, 16 - 9iot², write down the value of iota², where it is -1, now solve these quickly. Ok? So the at real part will come as it is. Both of these are accompanied by iota. We can do plus or minus these.

[00:25:19]So what is -28 - 6iota equal to?

[00:25:22]-34iota K. The value of + 21 iot² is -1 16 - 9iot² The value of -1 8 - 34iot. Now + 21 and -1 multiply to -21 16 -9 and -1 multiply to +9 then what will 8 - 21 be equal to? -13 k.

[00:25:46]Both are real parts and -34iot by 25. What needs to be done in the last step is that this number in the denominator should be separated from the real part and the imaginary part, so -13 / 25 - 34 / 25iot. So this is the answer for the right hand side that we had z1 bar / z2 bar and this was the answer for the left hand side as well, so the left hand side is equal to the right hand side, this is what we had to prove. Question number three is if z = 5 - 2iota then verify that in this question a complex number is given and we have to verify. That means we have to prove in all these parts that the left side is equal to the right side. And basically these are the properties of complex numbers.

[00:26:33]Ok? And these properties are true for all complex numbers.

[00:26:37]be any complex number. Ok? So, it's 5 - 2iota over here.

[00:26:41]These properties are true no matter what complex number is in its place.

[00:26:43]These properties will hold.

[00:26:45]Meaning that the left side will be equal to the right side. Ok? So the first part is, first of all, whatever questions you have, right? The way it is here. So you have to see what is said on the left side and the right side. Ok?

[00:26:57]Once you understand this, these questions are very easy. Now see what is its left side? z times and then again times.

[00:27:05]What does it mean? is that we will take the conjugate of the given complex number z. Whatever conjugate answer we get, we will take its conjugate again. So the answer should be equal to z. Ok? So z is a complex number given 5 - 2iota.

[00:27:23]What do you do first? Its conjugate is to be taken. So when we take conjugate, what happens is that the sign of the immediate part changes. So z times is equal to what? 5 + 2iot K. Now we will again take the conjugate of this z bar.

[00:27:39]Ok? So again in the next step we take the conjugate. Ok? We will take the conjugate of this z bar. Meaning that the conjugate of this 5 + 2iota has to be taken. So what answer will we get? The sign of the imminent part will change. So this is equal to 5 - 2iot.

[00:27:56]So z conjugate and then again when we have conjugate, what is this equal to?

[00:28:02]is equal to z. And this is what we wanted to show, that it should be equal to z. Ok?

[00:28:10]Next part number two is the modulus of z is equal to the modulus of the conjugate of Z if you find the conjugate modulus of z or find the modulus of its conjugate.

[00:28:23]The answer should be the same. The left side should be equal to the right side. then so be it. We will prove it. Ok? z is a complex number given by 5 - 2iot. We have to find its modulus.

[00:28:35]What is the formula to find modulus? We have learned.

[00:28:37]Modulus of z = square x² + y² ok?

[00:28:42]Whose square root do you want to take? The square of the real part plus the square of the imaginary part.

[00:28:48]Ok? So what will this be equal to?

[00:28:52]Take the square of 5 25 - Take the square of 2 4 Modulus of z = s 29 Okay? This is the answer on the left side.

[00:29:02]Now we solve the right side. Ok? So what's on the right side is that we have to take the conjugate of z times. z times is equal to what? π - 2iota, right? So what will its conjugate be equal to? of π + 2iota. Now you have to find its modulus. So again we will take the square of the real part. Plus we will take the square of the minor part. Now see whenever you find the modulus of any complex number or find the modulus of its conjugate.

[00:29:28]Look here, five were positive and two were negative. Ok? Too sorry was negative. There was a negative sign with two. So when we took its square, it became positive. So whether the real and imaginary part here is positive or negative. When we square them, they become positive.

[00:29:44]So find the modulus of any complex number or find the modulus of its conjugate. The answer will be the same.

[00:29:51]Ok? Because here also, if there was a minus sign, it has become a plus. There is already a plus sign here. So we will have the same answer. Let's find the modulus of 25 + 4 and this is equal to the conjugate of z.

[00:30:05]Ok? Square root of 29. So the left side is equal to the right side. Meaning that the modulus of Z is equal to the modulus of the conjugate of Z. What is part number three?

[00:30:17]That is to find the modulus of z. And we need to find the modulus of negative z. The answer of both should be equal. Ok?

[00:30:24]So we have already found the modulus of z here.

[00:30:26]Still, let us solve this question also. Ok? What is z equal to? 5 - 2iot K. So we solve the left hand side. To find the modulus of z. We will take the square of the real part. Plus we will take the square of the minor part.

[00:30:40]Ok? And we will apply the whole square root. So it will become 25 + 4.

[00:30:47]And this is equal to square 29.

[00:30:51]Now you have to take the conjugate of minus. Ok? So what would -z mean? That the complex number given should be multiplied by the minus sign. Ok? The complex number given is to be multiplied by the minus sign. Ok?

[00:31:05]minus sign * by z 5 - 2iota. So when you multiply by the minus sign, minus plus becomes -5 and minus minus + becomes 200. Ok? So that means when you find the negative of any complex number, what it means is that the positive sign is made negative and the negative sign will become positive.

[00:31:24]See What is a Complex Number? 5 - 2iot What negative z means is that what was positive will become negative. If it is negative then it will become positive. Now you have to find its modulus.

[00:31:33]So again, now see, here it is -5, so when you take its square, it will also become positive. This is the answer you will get. Take the square of 2.

[00:31:45]So 25 + 4 and this is equal to 29 squared so the left hand side is equal to the right hand side which means that the modulus of z is equal to the modulus [clears throat] of negative z and this is what we had to prove.

[00:31:59]Next is part number four.

[00:32:02]What is the left side? So you have to multiply z and z conjugate and what is the right side is you will find the modulus of z and take its square. So the answer on both sides should be equal. First of all we write down the complex number.

[00:32:15]What is z equal to which is given in the question 5 - 2iot and what will its conjugate be equal to? The sign of the imminent part will change.

[00:32:24]5 + 2iota So first we solve the left hand side. The left hand side is what we have to multiply by z and z times.

[00:32:31]What is z equal to? 5 - 2iota K.

[00:32:33]What is its conjugate equal to? 5 + 2iota. Now see which formula will be used here?

[00:32:39]This number is also the same. 2iot is also a bean. There is a minus sign once. There is a plus sign once.

[00:32:43]is a complex number. And when we take its conjugate and multiply them both, only the sign in between changes. So which formula will we use here? a - b * a + b and this is equal to a² - b² okay? There is also a here. b is. Ok? Once it is minus, once it is plus. So take the square of the first number. Take the square of the minus second number.

[00:33:08]What is the square of 5² - 2iot² 5 equal to? We have to take the square of 25 - 2 iot. Ok? So take the square of 2 into 4 iot². Now, wherever iot² is, we will write down its value -1 25 - 4 iot² is -1. Now these will be multiplied by -4 and -1.

[00:33:24]So what will happen?

[00:33:26]+ 4 its value will come to 29 which is on the left hand side. Now we have to solve the right hand side, which is to take the square of the modulus of z. So first we will find the modulus and then take its square. Ok? The right hand side is the modulus of z². So first we have to find the modulus of z. z is equal to 5 - 2iota.

[00:33:49]We have already found its modulus in parts.

[00:33:51]Ok? But let us solve it step by step here also.

[00:33:54]What will its modulus be equal to? We will take the square of the real part. Plus, take the square of the minor part and write it in brackets so that if there is a negative sign here too, then it remains in the brackets and there is no mistake. So 25 + 4 this is equal to square 29 this is the value of the modulus of z.

[00:34:12]What do we do now? We have to find its square.

[00:34:14]Ok? We will take its square.

[00:34:16]When we take the square of modulus of z, the value of modulus of z is square 29. If we take its square, then the square will cancel out with the square root. So, the square of modulus of z is equal to what? The left hand side of 29 was also 29 and the right hand side is also 29. So, this is the thing that you had to prove.

[00:34:36]What is z z bar z * z bar equal to?

[00:34:41]Part number five of modulus of z² is modulus of z equal to modulus of -z bar. Ok? So we have found the modulus of z many times.

[00:34:52]But here also we solve it again.

[00:34:55]Ok? What is z equal to? 5 - 2iot K. Its modulus has to be found.

[00:35:00]So we will take the square root. Whose? The square of the real part plus the square of the imaginary part. So 25 + 4 and this is equal to square 29. This is the value we have on the left hand side.

[00:35:17]Now we have to solve the right hand side. So what's on the right hand side is find the modulus of -z times. Ok? Meaning that first we will find z bar. Then we will multiply it with the minus sign. Then we will take the modulus. Ok? So z times is equal to what? Z5 - 2iot. z times is going to be equal to 5 + 2iota. Now you have to multiply this with the minus sign. Ok? If you solve it step by step then there will be no mistake. So - z times, okay? We will multiply z times by the minus sign.

[00:35:47]Minus multiplied by z times is 5 + 2iota. So what is this equal to? Minus plus -5 - + - 2iot Now you have to take its conjugate modulus.

[00:35:59]Ok? We will find its modulus. So what is the formula to find modulus?

[00:36:04]Take the square of the real part. Ok? That's why I told you to write it in square brackets and then take the square. Because if there is a minus sign then there will be no mistake in it. -5 is its square plus the real part is -2 is its square so take the square of -5 which is equal to 25 take the square of -2 4 and this is equal to the square of 29 so this is the value of the right hand side we have, okay here we have found the right hand side.

[00:36:33]So what is the answer for both the left hand side and the right hand side? Is equal. Ok?

[00:36:40]Ok? What was on the right hand side is find the modulus of -z times. So first we found z times. Then it was multiplied by the minus sign.

[00:36:46]Then its modulus is found. So what is the answer for both sides? That is equal. This is question number four. If z = 4 - 3iota.

[00:37:01]A complex number is given and we have to verify, prove that all four of them are equal. Ok? So we'll solve this first.

[00:37:08]Then like this, then like this and then like this.

[00:37:10]Ok? We will solve all four step by step.

[00:37:12]And then we will show that the answers to all four are equal. Ok? It should be equal. So what's first?

[00:37:20]Modulus of z. Ok? So the z number is 4 - 3iota. First we find its modulus.

[00:37:27]What is the formula to find modulus?

[00:37:30]Finding the modulus of any complex number. So what is the square root of x² + y² x? Real part. By this is the minor part, so the real part is four plus the minor part is -3 and then the whole square root, so what will it be equal to, here we just solve it, 16 + 9 square will become 25 and this is equal to 5, okay, we have found its value.

[00:37:54]So we will name this the first equation.

[00:37:56]Ok? Now we have to find its value. - Find the modulus of z.

[00:38:01]So first we'll do -z find.

[00:38:03]z is equal to 4 - 3iota. So we have to multiply this with the minus sign.

[00:38:09]Ok? When you multiply minus * by z with minus sign, it becomes minus plus -4 minus minus + 3iota. So whenever you want to find the negative of any complex number, what will happen is that you have to change the sign. Ok? Here it is positive 4. It has become negative. Was -3iota is +3iota. Now you have to find its modulus.

[00:38:31]Ok?

[00:38:33]So the real part is -4, we will take its square plus the imaginary part is 3, we will take its square, we will solve it further, 16 + 9 which is equal to square 25 and what will this be equal to 5, let us name it equation of two. Ok? We have found its value. Now let us find its value.

[00:38:51]What does it mean? We have already solved it earlier, not z times, which means we will take the conjugate of z. Whatever is conjugated, we will take its conjugated again.

[00:39:03]So first we write down the number z. 4 - 3iot is correct? Now we have to take its conjugate. So the conjugate of z is equal to what? The sign of the imminent part will change. Now we will take its conjugate back. Ok? We'll take the conjugate of z times.

[00:39:23]You can write the conjugate of z bar again on the next lines.

[00:39:25]Ok? 4 - will become 3iota.

[00:39:30]Now we have to find the modulus of this number that we have.

[00:39:34]Modulus of z times okay? We will find its modulus.

[00:39:38]So what is its real part? What is 4 plus imaginary part? -3 and take its square root. Solve 16 + 9 = square 25 and its answer will also be equal to 5. Ok?

[00:39:55]What is the last part? That's -z times we want to find the modulus. Ok? So - z times okay z times what do we have? z is 4 - 3iota okay? What do you have to do now? Its conjugate has to be found. Ok? - is z times. Meaning that we will find it z times. Then we will multiply it with a minus sign. z times is equal to what? 4 + 3iota k. And then multiply it with the minus sign. So when we multiply this by the minus sign, it will become -4 and this will become -3iot. Now you have to find its modulus. -z will find the modulus of the bar. We will take the square of the real part plus the square of the imaginary part.

[00:40:39]So 16 + 9 = square 25, so what will this also be equal to? What is minus z times equal to?

[00:40:46]5 k. Let's name it the fourth equation.

[00:40:49]So now let us see what is the answer of the four parts that we have solved, second, third and what is the answer equal to? Five K. So at the end you write that from equation number one to three and five these are their final answers, right, what are these four? Are equal.

[00:41:05]You can write this property at the end that since they all are equal. Ok? The last question is number five. If z1 = 2 + 3iota and z2 = -1 +iot then evaluate and you have to find their values. Now what's in part number one? What does RE mean? Real part. Ok? At the beginning of this lecture, I had explained these things to you in the first lecture of chapter number one.

[00:41:31]Ok? That if you are asked to find the real part of any complex number, it will be written like this. Now how to find the real part? z1 * z2 means that these two complex numbers will be multiplied. We will write the answer in the form a + b iot which is the standard form of complex numbers.

[00:41:50]Right? Then you have to tell the real part of it.

[00:41:54]So which will be the real part? This one does n't have iota with it. And we have to explain the imaginary part of this also. Ok?

[00:42:00]Imaginary part of whom? If z1 * z2 then what will be its imaginary part? With which there is iota. So what do you have to do first?

[00:42:07]Both of these have to be multiplied. We find z1 * z2 z1 * z2. 2 + 3iota is the first complex number and -1 +iot is z2. We'll multiply two by this entire bracket.

[00:42:20]+ 3iota will multiply this by the entire bracket. Multiply two by -1 to get -2. Multiply two by iot, 2iot.

[00:42:30]3iota is to be multiplied by -1. Ok? So plus minus will become minus.

[00:42:35]You have to take care of the sign - 3iot and if you multiply 3iot by +iot then it becomes + 3iot², right? 3iot * iot becomes 3iot² -2. Now look at these two, there's iota. We can do plus or minus these. Both of these are imminent parts. What is 2iot - 3iota equal to? The value of -1iot + 3iot² is -1.

[00:43:03]So -2 Now here whether we write 1 with iot or not, even if we write -iot, it still means that it is -1iot only. Ok? So there is no need to write forest. Multiplying +3 and -1 will become -3. Both of these are real parts. -2 and -3 will equal -5 -iot. Whose answer is this? of z1 * z2.

[00:43:23]What do you have to do now?

[00:43:25]What is said in part number one? Whose real part should I tell? Ok? of z1 and z2. So part number one is the real part of z1 * z2, what is the real part of this? -5 and what is the second part? The minor part of z1 * z2 is what we have multiplied both the complex numbers. Its imaginary part. What is the imaginary part?

[00:43:51]Which is the number with iota.

[00:43:52]What number goes with iota? There is no number written here. That means it's a forest, right?

[00:43:56]Look here, when we solved it, you must also write the sign with it.

[00:44:02]Ok? What number is with iota? -1 This is the imaginary part of it and this will be their answer. So exercise 1.3 is complete here. I hope you have understood all these questions.

[00:44:14]Inshallah we will solve the next exercise in the next lecture.

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