This comprehensive live session covers two essential Class 9 mathematics chapters: Introduction to Linear Polynomials and The World of Numbers. The session demonstrates how to identify polynomial degree, coefficients, and constant terms; evaluate polynomials at specific points; solve age-related word problems using algebraic equations; calculate areas of rectangles with fixed and variable dimensions; express linear growth and decay patterns; determine slope and y-intercept of lines; find equations of lines passing through two points; classify numbers as rational or irrational; prove irrationality using contradiction method; and understand the historical base-12 counting system. The instructor emphasizes starting from basic concepts and building systematically to develop strong mathematical foundations for Class 10 preparation.
Class 9 Maths PT-1 Marathon LIVE | Ch-2 Introduction to Linear Polynomials & Ch-3 World of Numbers
Added:Thank you.
A very lovely hello to all my class 9th ninjas champions. How are you all everyone? So I I I really apologize for yesterday's live. There was little technical glitch. So that's why I I was not able to go further on live and the live was suddenly stopped. So I really really apologize. I know you guys have missed one session, one important session. So no problem. We'll be covering everything whatever we have missed yesterday. I'm going to cover up everything today live session in this class everyone. So do let me know everything is fixed. Am I visible to you all? Screen is visible and I am I audible to you all. Everyone please do let me know in chat section you all are there and join fast. Come on this session is going to be gamechanging for you all because we will be covering two chapters today. One is introduction to linear polinomials and other one is the world of numbers from basics to advanc.
I will be starting from very basic questions to advanced questions as per the requirement.
Hi hi every hi Sura. Yeah. Hi. Hi Shaki.
Hello everyone.
Very good.
How are you all champions? So let's let's let's wait a little while so that other can join and so that all can get advantage of this live session. Okay.
Do some good work, noble work for others also. So let's wait two to three minutes so that all can join. Hi everyone. The student who is new here for the first time on the channel in this live class.
I welcome you everyone.
If you are I mean watching for the first time this live class on this channel.
Good. Very good Shaki.
Okay.
Join everyone quickly. Come on fast.
Hi. Hi Sep. Hello Sepia. Hello to all.
Hi everyone. So let's begin today's session without wasting a moment without wasting a second. So now I have already told you guys I'm going to cover up two chapters today. two chapter entire introduction to linear polomials and the world of numbers. So we are starting with introduction to linear polinomials which is chapter number two of your book kit man. Okay. Hi very good. So moving to the question number one presenting you champions. Look at the question number one.
Look at the question number one. For the polomial px = 6 x ^ 4 - xq + 5 x² - 9 write down its degree very easy question the cofficient of x² and the cofficient of x and the constant term. So I'm starting from very basic question. Don't think the questions are so easy. Oh my god, the questions are quality questions ahead. So have patience. Wait a little while. Okay. So what will be the answer of the question? So my dear champion, we have this linear polomial.
Not linear. Highest power is four. So this is not linear polomial. This is by quadratic polomial. By the way, this is polomial. But the degree decides the identity of polomial. degree is four.
Here the degree of the polomial degree is equal to four. The highest power is called degree highest and better to say highest power. By the way in mathematics term this is exponent. Better to say exponent. The highest exponent highest exponent is four. So that will be the degree. Hi everyone.
Is it clear everyone? The highest power will be four. And now cofficient of x².
Where is x²? This is x².
And the constant value which is connected with x² that is called cofficient. The constant number which is attached with variable is called coefficient. So we have coefficient of x² and that will be equal to what?
Five. Simple. Now min - 9 there is no any variable there is no any x x² xq x nothing is there only number is there only numeric value is there and only numeric value is called constant term so here here the constant term will be equal to minus 9 if the number is in negative form write it down as it is negative if the number is positive write it down for example if I ask What is the cofficient of x cq? So the cofficient of x cq will become minus one. Why? Because this one is in negative form. So if the number is in negative write down in negative form as it is. I hope this is clear to you all. Degree is four. Coefficient of x² will be five and the constant term will be 9 minus 9. Clear everyone write it down in chat section. If this question is clear to you all please write it down. So degree is clear coefficient is clear and constant term is clear. Moving ahead to the next question. Okay.
Find the value of polomial px = 4x - 7 at x =0 at x= to -2 at x x = to 3. Now my dear champion do you know we have this polomial px is equal to 4x - 7. P means polomial and the polomial is in the terms of x in the term of x. That's why here is x. Now what will be the value of polomial at x= to0. So put at put at the place of x0.
So this will be 4 into 0 - 7 4 into 0 4 into 0 - 7. So the answer will be - 7.
Now if I ask you find out find out p minus2 because we have to find out at x= to minus2. So now put x will be minus2. So this is equal to this is equal to 4 into -2 4 into -2 - 7 - 7. Now solve this. This will be -8 - 7. So the result will be -15.
The result will be -15. So the value of the polomial will be -5 at x = -2. Now find at x = to 3. So p at 3. So now x will become 3. So 4 into 3. 4 into 3 - 7. So this is 4 12 - 7 which is equal to 5. So these are the answers. Yes.
Hello sir. Please welcome me. Of course.
Yeah. Welcome wholeheartedly. Welcome.
Welcome everyone. Who is this by the way?
I am Dave. Welcome Dave. Who is this?
Arun.
Arun, I have one. Yeah, please ask now.
Whatever you want to ask, please. I'm here to solve your queries. I'm here to solve your questions. Come on, ask me.
Whatever you have asked. Come on, champions. Ninth ninjas. I hope this is clear to you all. So, we have one polomial and we have to find out the value of this polomial at x= to0, at x= to -2, at x= to 3. So put now put x= to0 x= to -3 x= 3. So we have this answer - 7 - 15 and 7. Now also find the value of qt. Now the polomial is q in the terms of t. So this is the second part. So we have one more polomial q is polomial and in the terms of t and the polomial look like 3 t² - 5t + 2. Now we have to find out the value at t = minus1. t is equal to minus1. Now put here minus1. So every t will be replaced by minus1 whole square and here t will be min -1 and this is + 2. Now solve this -1² will be 1 positive 1 because -1 into -1 will be 1. So 3 1 are 3 and this is - plus 5 1 are 5. So this is + 5 + 2. Now solve this 5 3 8 10 this is the answer. 10 is the answer. Hi sir. Hi Shbhar Ani. Hi welcome to Digsha Vidantu's platform. I hope this question is clear to you. So let me explain again. There was polomial given to you and we have to find out the value of this polomial at x= to0 at x= to -2 at x= to 3 and the another part of this question is again we have one polomial given to you and we have to find out the value of this polomial at x= to minus1 and I have solved there.
These are the three parts, three solutions of part one and this is the solution of part two. Clear everyone?
Hi. Hello to all. Hello to all the students who is new here. The student who is watching this live session for the first time. I really welcome you everyone from my bottom of my heart.
Moving ahead question number. Now look at the question. This question is little logical question but very easy and such questions will be coming in your exam at your school. Yes my dear question these are the conceptual questions and every maths teachers loves these type of questions clear everyone so let's do what is the question here a father's present age is four times his son's presentage very good very good after 6 years okay okay after 6 years the sum of their ages will be 62 years find their present ages so listen champion all class 9th student Listen, in every age type question, this is called age concept. Listen everyone, please. In every age type questions, always always find the number of persons present in the question. What you have to do first? First find out how many persons are there. So if I talk about this question, father is there and son is there. How many persons? Two persons.
How many individuals are there? Two individuals are there. Father and son.
Okay. And there there is some relation given in their ages.
Four times age after 6 years. So there is some story given. And we have to find out percentages. So in every age type questions always let the percentages first. Always do this. always let the percentage because see for example if I ask you champions listen let me shift my system here okay if I ask you what was my age 3 years back what was my age 3 years back you can't give me answer of this question because you don't know now what was my age 3 years back why you don't know because you don't know my percentage until and unless Yes, the present age is not clear to you how you will be able to find out the age of any individual 3 years 4 years back or 3 years 4 years after you can't. So in every age concept question always let the present ages first always. So in this question we are talking about father and son and there is some story given some conditions are given and what we have to find their percentages we don't know. So let this first let this first. So let I'm solving the question. Let the father's father's what champion present age. Let the father's present age be x years.
Don't forget to write these units x years. For example, if you are letting the speed kilometer per hour, yep or m/s should be there. So whatever you are letting that should be there x years.
Okay. And and let the son's percentage percentage be y years. So suppose first do this work. This is the proper way champion.
This is the proper method to solve age type question. Always suppose the present ages first. Always let the present ages first. Now read the question carefully from the starting.
Now read the question carefully. What is the question? A father's present age is is is means we are talking about present time. This is present tense. Is after 6 years means here we are talking about future. Future tense.
You have to decide whether we are talking about past, whether we are talking about present, whether we are talking about future. This it should be clear, right? So if we are talking about a father's present age is is is it means this is present. And what is the father's present ages? X we have supposed. So according to question, let me write here AQ. AQ means according to question X is equal to four times. Times means what? Champion times means what?
Multiply. So we have to multiply four to which four times his son's percentage and what is the son's percentage? What is the son's percentage? Sun's present age is y we have supposed. So this is so we have first equation x = 4 y. This is the first equation. Now read this second information. Second information. And what is the second information? After 6 years, my dear champion, father and son both are human beings. And if both both are human beings, so they both will be 6 years older. If we talk about the age after 6 years, so the both both father and son both will be 6 years older.
After 6 years future, so both will be 6 years older. Now after 6 years, the sum of their ages. So after 6 years father will be x + 6 plus because this is sum of their ages. Sum means add and son's age will be y + 6 because son's present age is y. So after 6 years he will also be 6 years older. And if we do sum of these two the result is equal to 62. Can you see here? This is 62. Now let let let's simplify this. So this is x + y and 6 + 6 will be 12 and 12 will go there. So we can write here 62 - 12. So this is equal to x + y and 62 - 12 will be 50. Wow. We have second equation this first polomial second polomial. Now we can substitute equation 1 into 2. So if we substitute I'm writing here substituting substituting first in second. So let's substitute I'm doing here. Okay. So x is equal to 4 y. So x will come here at this place. x + y. So x will come here 4 y. So this will look like 4 y + y because we have replaced x is equal to 4 y. So 4 y + y will be 50. So 4 y + y will be 5 y 5 y = 50. So y = 10. Once you get y, you can find out x because x y = 50. Now you guys can put here. So x will become x= 4 y. So x will be equal to 4 into 10 which is equal to 40. My dear some student what they used to do after getting the value of x and y they leave the answer here. I I found lot of time in answer sheet don't do this because the question is age. The question is asking find the present age of father and son. So we have supposed x is the father's present age and y is the son's present age. So don't leave your answer only at don't leave your answer till here. You have to write the answer in words. So, so, so the father's so the father's present age father's present age be what? X we have 40 this is X 40 years right and what sons son's present age be what what we had supposed this is y and y we have is 10 so this is 10 years 10 years this is the answer when you follow a proper procedure in maths when you follow when you make your answer presentable you know it really impress the evaluator and the evaluator will become happy to see these things. So toppers never forget these things. So after getting X and Y now focus what was X and Y? X and Y was ages. So make sure to write in words at the end.
Yes. Clear everyone. Hi Summer. Hi Summer can you explain? I hope this is clear to you all. Chapter name this is this is chapter introduction to polomials. Chapter number two champion introduction to polomials linear polomials.
Moving to next question clear some are clear. This is introduction to linear polomials. Look at the question. Wow what a question. Question number four.
Now look at the question everyone.
Question number four. So the question is the length of a rectangle is fixed.
Okay. Okay, listen champion. Listen, listen listen listen.
There is a rectangle.
There is a rectangle and the length of rectangle is fixed. So let's let let let's let let look at the diagram here.
Let's assume this is a rectangle, right? And the length of the rectangle is fixed. This is 15 cm.
This is a rectangle and the length of the rectangle is fixed. Now read the question. Read the question carefully always. Don't be in such hurry. Oh my god, I want answer. I want answer. Don't do this. Have patience. Stay calm.
Follow the language. Read the comprehensive. I mean, read the English language of the question. Well, understand it. Yeah, sir.
Hi, Summer. Summer is asking how are you? Yeah, I'm absolutely fine. How you how are you guys? Okay. Now, so length of the rectangle is fixed at 15 cm. Wow.
Now read this statement. Find its area.
Okay. Area. Find its area. Can you see here this language? Find its area. And do you know champion area of any rectangle is equal to length the product of length?
Some technical problem guys. Just length into breadth. This is the formula to calculate the area of any rectangle. Now look, so to calculate the area of rectangle, we have length but we don't have breath. Have patience. We have breath also. And we have breath. Three types of breath. When the breath is 6 cm. So first we have to take breath as 6 cm. In second case we have to take breath as 9 cm. In third case we have to take breath as 12 cm. So how many areas will be there? Three areas. Three types of rectangles are there. So first case, first case here is when length is 15 cm and breadth is 66 m. And the second case in this question will be when length is as it is and the breadth has increased.
So this is length will remain as it is and the breadth is 9 cm. Everything is centimeter now. So make sure to write in centimeter as it is. This is not meter.
It will go in cm. Right? And the third case is this is the third case right champion.
So uh yeah this is the third case when length is 15 cm the breadth is 12 cm. So we have three types of rectangles this this. Now if you have three types of rectangles so the area will also be three types. So first area will be what?
15 length into breadth because the formula we have is length into breadth.
So this will be 19 cm square. Area will always go in a square. My dear unit plays important role in exam. If you are not writing unit this will this habit will be continue in higher classes. In 10th class and board exam your marks will be deducted. So make sure to follow the importance of unit vary from class 9th. Okay. Now here area will be length.
Length is 15 and this is 9. So this will be 135 cm²ared.
Similarly in third case if I talk about here area will be equal to length which is 15 into breadth which is 150.
So this is 180 cm²ared.
If you don't trust on my calculation you guys can verify.
By the way you should trust on me on my calculation. Okay. Who is writing? Hi sir. Uh summer summer summer many science class very soon champion 15 into 6 very good champion cozy high shr okay 191 135 180 perfect cozy shriy perfect clear everyone this is not the end this is not the end because few thing is also left hence write the linear pattern for the area wow we have to write the linear pattern so what will be the linear pattern listen champion so if I talk about here linear pattern will look like for example for example area I'm representing area by a and here in this question length is fixed as 15 cm right but breadth is not fixed sometime bread is 6 6 cm sometime the breadth is 9 cm sometime the breadth is 12 cm it means that the breath is variable and if the breath is is a variable we have to show the brid using any variable any alphabetic I mean x y whatever you can use as per your wish okay so the formula will look like linear linear pattern the linear pattern will become if I talk about linear pattern for this case the linear pattern will look like a is equal to a is equal to 15 is fixed 15 into let's say for breadth For bread which is variable of of course I'm taking a small B. To represent the breadth I'm taking a small B. So we can write here A is equal to 15 B. This will be the linear pattern and B is not fixed. First we have to take B as 6 cm.
In second case we have to take B as 9 cm. In third case we have to take B as 12 cm. So B is the variable. Breath is variable. So this will be the linear pattern. Clear everyone?
Uh hi everyone this is sir this or is this just hard one please? Ah wait wait wait wait wait shi we have hard one questions also ahead. Have patience. See my method to to teach anything is I always start from basics. Once your basics are clear once your basic concepts are clear you guys will be able to you know break any questions comfortably. I mean competency high level high standard question will be nothing for you. Always start from basics. Don't try to jump directly to the tough one. Don't try to jump directly to the you know high standard questions. Otherwise when you will not be able to get the correct answer your motivation will drastically down. So better to follow proper rhythm. Clear?
Hi is it clear?
Hello everyone. Hi Servesh. Please put one video sample question paper for maths. Yeah service don't worry we are have see my dear champions listen we all the teachers are dedicated for you to deliver quality content that will be relevant for you for your exam and all so we we got what we are doing here all the faculty members we all are ensuring to make strong foundation for your class 10th this is our target ultimately right so have patience clear everyone summer is it clear to you fine sir it's okay very good very good moving ahead look at this question a sapling is 40 cm tall when planted. Okay. Okay.
When planted and grows 3.5 cm every month. Sapling means a plant I mean the plant sample when when we plant uh when we you know when we take a plant buy sample from you know the person who sells the plant you know little small size of plant to plantation for the plantation procedure right. So basically that is called sapling small you know small teeny teeny lovely lovely plants right that saplings are there so initially there is a plant sapling listen let's say this is a plant sapling I don't know my drawing is good or not you guys are going to rate my drawing w yeah let's say this is a plant sapling and initially the height of plant sapling was 40 cm this was the height of the plant sapling before plantation I mean at the time of plantation we can say okay when planted after plantation it is growing by 3.5 cm every month getting my point 3.5 cm every month perfect write an expression for its height after t months after t months so listen champion listen so for height for height Right? I'm using a letter T.
Correct? And for time and that is in basically month, we are talking about month. So time in month I'm using here a small t right. So what how the expression will look like listen champion. So h will be equal to and what was the initial height? It was it was the initial height at the time of plantation. So 40 plus now in every coming month. Now in every coming month.
Just a second guys. I got a call. Little urgent call. Just a second guys. Give me a second.
Can you hear me everyone? Ah sorry.
Sorry I got a call. It was little urgent. Little urgent call. Uh my mom called me. So I can't deny mom's call.
You should not deny mom's call whenever your mom's call now. So that's why.
Okay. So who is this? Uh let me let me have a look to all the chats. Hi. Who is I'm Tanvi? Hi. Hi Tanvi. How are you?
Cool everyone. So let's am I audible to you now? Can you hear me everyone?
Please do let me know in chat section.
Yes sir, you're audible. Am I audible to you all? Am I audible?
Come on. Yeah. Say thumbs up. Send thumbs up if I'm audible. If I'm visible to you all, send me a thumbs up.
Perfect. Shriru, thank you. Shrui is always smiling. Superb. Keep smiling.
Shri.
Yeah. Perfect. Okay. So, let's move ahead. So, listen champion. The question is at the time of plantation the height of the sapling was 40 m 40 cm. Perfect.
And the height of the plant is gradually increasing 3.5 cm every month. Right?
And we have to write down the expression for this information. So 40 cm was initial height at the time of plantation. So that will be added right.
And every month, every coming month 3.5 cm increasing. So the equation will look like 3.5 into t. This will be the algebraic expression. 40 40 was the initial length, initial height of the plant at the time of plantation. And now 3.5 cm is increasing every month. every first month, second month. So this T is variable. So time in month this is variable.
Time in month this is variable. First month, second month, third month, fourth month, fifth month, whatever is there.
Is it clear sir? How to do graph? Please teach and give. Don't worry. Don't worry. Okay, listen champion. So this is variable. I hope this part is done.
Expression for its height h after t months. This is done. Now moving to the second information what we have to find here find its height after 8 months. So my dear champion t will be 8.
Now put t is the variable because t is changing first month second month third month fourth month t is changing. T is keep changing right. So t will be eight.
So put here. So height height I'm writing here after after 8 months. Okay champions after 8 months.
So the result will be uh 40 + 3.5 into 8.
Can anyone multiply this please? 3.5 into 8 what will be there? So 8 5 are 40 40 28 3 are 24 24 3 27 I think 27 will come 85 we can check. Yeah don't take risk. Mhm. 3.5 into 8 85 40 carry 4 8 3 24 Wow my calculation is too good. 27 will be there. So now add 27.
Okay. So this will be 67 and the unit is cm. So this will be 67 cm. So after 8 months the height of plant will become 67 cm.
Oh 8 3 24 20. Oh my god. This will be 28 champion.
How I made mistake. Oh my god. So I can't seem this is 28. So this will be 68 cm right champion 8 5 are 40 8 5 are 40 40 carry 4 8 24 24 4 28 sir how to do graph have patient survey sir don't worry is it clear to everyone now the one one more part here is why this is linear growth oh my god growth means what what do you mean by growth growth means increasing now something is increasing gradually increasing and here the height of the plant is increasing now. So that's why this is linear growth and because because this variable part is positive variable part is positive. If the variable part would have become negative if the v variable part would have become negative then it is called linear decay but this is linear growth because the height of the plant is gradually increasing increasing increasing with passing of month every month 3.5 cm increasing. Moving to next question.
Question number six. Wow, look at the question.
Yes, look at the question.
A laptop is bought for rupees 48,000 and its value falls by 6,000. Value falls.
You know this is normal. This is common sense my dear. When you buy any gadget with the passing of time with the passes of time the cost of the gadget will be decreasing will decreases. Right? And this is called what the cost is the value falls. So initially someone bought laptop at 48,000 rupees and after some time every year the rate of the cost of the value of the laptop is decreasing.
False means decreasing. False means decreasing. The value of the laptop is decreasing 6,000 every year. So this is the example of champion linear decay. This is the example of linear decay. This is called linear decay because the cost of the laptop is gradually decreasing. Now moving ahead write the value. Okay. So we have to represent value by v and uh time with t. Perfect. So what will be the equation? The equation will look like v is equal to what was the initial cost at the time of buying? So that was 48,000 minus. Why minus? Why minus? Because falls down 6,000 every year. Every year falls down decreasing. The cost of the laptop is decreasing. So we have to put here negative sign. Can you see here?
This is false decreasing. In previous question it was growing growing growing growing growing. And if something is growing means something is increasing.
So that's why positive sign is there.
But here false means decreasing that's why negative sign and here is 6,000 and that is with time t every year. So this will be the expression v is equal to 48,000 minus 6,000 into t. This will be the linear expression. Now moving ahead find its value after 3 years. Come on you have three years. So t is equal to now t is equal to 3 years. So let me write complete information after after t is equal to 3 year the value the value of laptop what will be the value of laptop after 3 years so this will be v is equal to 48,000 was the initial cost and minus 6,000 which is increasing every year and we are talking about after 3 years so 6,000 into 3 so this will become 48,000 000 - 6 3 are 18. So subtract 80,000 18,000 from 48,000. So that will be I think 30,000 rupees. So this will be the cost of the laptop value of the laptop after after 3 years. Clear everyone 30,000. Hi sir. Hi Akash. Hi Akash Shani. How are you champ? Clear to all everyone. Clear.
And this is the example of linear decay because the cost the value of laptop is decreasing with time with year with each coming year the value of the laptop is decreasing and constantly 6,000 is decreasing every year 6,000 first year after buying the laptop in first year 6,000 decrease in second year 6,000 third year 6,000 so this is called linear and something is decreasing so DK that's a linear decay moving ahead question number seven clear everyone Look at the question.
A cab charges A cab charges fixed booking fixed booking fee. Okay. So there is a fixed booking fee plus plus a fixed rate a fixed rate per kilometer.
So this will be the variable.
You know there is a cab person who is charging a fixed price. If you are sitting in a cab and deboarding, you have boarded and just deboarded from the car, you have to pay this much of amount that is fixed and he is charging some I mean some piece of amount for coming per kilometer. So this is the variable because for upcoming 1 km, 2 km, third km, fourth kilm he will be charging something there. So this is variable.
This is the fixed amount. This is the variable. Okay. Now now Okay. and and uh something is there some person a person 8 km ride cost rupees 190 and 40 14 km ride cost 301 310 rupees if the fair y so we have to represent total fair we have to represent total fair with the letter y and uh distance and distance that is in kilometer basically we I have to represent by x and the equation follow y = ax + b. Find a and b. What a question. This is good question I would say. So listen champion. So according to question I'm solving the question now. Look at the screen. Listen champion. So when a person took the cab for 8 kilometer when a person took the cab to travel a certain distance of 8 kilometer and he paid rupees 190 right champion he paid rupees 190 so how the equation will look like listen so total fair total fair is represented as a y so y will be equal to b is the total fair let's say this b is the fixed fair fixed booking charge is E B plus this let's say this is A a A a rupees he is charging or she is charging for each kilometer and for each kilometer we are here taking x so this will be ax this will the equation will look like now put this whole equation compare this from this information so x is equal to 8 kilometer so what will the equation look like so b is what fixed price so b plus a we don't know A but the person traveled 8 km. The person traveled 8 km. So this will be B + 8 A and the total bill Y and total B is 190.
This is the first equation. And what will be the second equation? The person traveled for uh the person took the cab for 14 km and he paid rupees 300 as fair. So this is the fixed charge and he traveled 14 kilometer 14 into and uh the person is charging a rupees for each kilometer but we don't know that. So 14 a and what was the total bill that was 310 we have two equations this equation and this equation. Now do one thing now subtract both the equations. So if we subtract negative sign will come here - cancel the b cancelled. Now 8 a minus 14 a this will look like - 6 a because we are subtracting 14 a from 8 a 8 a is a smaller and 14 a is greater so negative sign will carry forward this is 190 - 310 how much will be there? So if we subtract this so uh it will look like 0 will come you can imagine a 3 1 10 you can reverse it like this 310 you like this zero will come two will come and this is two so 120 but with negative sign because we are subtracting greater number from a smaller one. So this will be 120. Clear everyone? Now minus minus cancel. So 6 a is equal to 120. So a will be 20. Wow. A will be 20. And once you get a, you guys can put a either in equation one or equation two. You can put in any equation. So So uh b will be b + 8 a and a we have is 20. 20 is equal to this is 190. Now solve this. 8 this is 160 is equal to 190. So 160 will go there. So B is equal to 30. This is not the answer. Please write the answer in word. At the end after getting the value A and B, please make sure to write the answer in words. Because because A is what? A is what? A is fixed rate per kilometer. And B is what? B fixed booking price booking fee. So here so so fixed booking fee fixed booking fee is what that is B 30 rupees and fixed if you make your answer presentable now basically your evaluator become happy make your answer should very presentable and impressive fixed rate per kilometer and that is equal to a and which is 20. So this is 20 rupees. So this is the answer. Clear everyone?
Yeah. Hi sir. Hi Aman. Hi Bala MKkesh.
Clear to all everyone champions class 9th ninjas clear to all. This is a good question I would say. Very good question everyone. If this question is clear to all, if you guys are really enjoying there, give me thumbs up. Give me some hearts. Show me your presence.
If this is clear to you all, thank you.
Balam MKkesh. If this is clear to you all, give me thumbs up. Everyone send me some hearts. Come on. Moving ahead.
Question number eight.
Find the slope and y intercept of each line and state which of them are parallel. Wow, what a question. Literally crystal clear, sir. Bala MKkesh. Wow. Yeah.
Crystal clear. Superior. Hats up to you Bala MKkesh. So see my dear champion. If you concentrate well now, if you are if you don't know the solution, how to solve what should be the starting of the solution. You don't know. But at least you guys can look at the screen. How the way I'm solving the question? You guys can see the method. Mathematics is just observation. How sir is solving the question. Come on, focus. Focus will clear your doubts. Trust me because I used to follow very easy method to solve the question. I can make your mathematics very magical. Trust me everyone. Trust me champions. Okay.
Moving ahead.
Find the slope and y intercept. Very good. So we have to find out slope and y intercept here in this question of each line. And this is from your ganit mji. This question is from these questions are from your GitH. And this topic is also there. So my dear champion don't ignore the topic because this is very important. And tell me let me tell you one thing honestly the topics you guys are studying here in class 9th now these topics exactly same you guys will be getting in class 11th mathematics.
Yes my dear and that is why CBSC is expecting to build up your foundation so that you guys will be feeling very relaxed and comfortable with when you will be solving maths in class 11th and that is why uh CBSC has released a new textbook but after doing some modification changes in new textbook that is for your own benefits some student might be thinking CBSC is enemy CBSC is not care for student come on feel happy and feel very proud guys the books are literally I don't know what happening other thing in CBSC there result delay and lot of things bad things unexpected things disappointing things are going on I agree but if I talk about I mean uh topics point of view the quality contents point of view this is literally very good yeah so look at the question so here in this question we have to find out slope and y intercept so my dear let me give you a very magical method so here we are talking about line why we are we why we are same line because all the equation if you observe here all the variables has power one x power 1 y power 1 x power 1 y there is no power two there is no power three so it means they all are representing a line they all are representing a straight line right so my dear champion so try to convert try to modify these lines in the form of y is equal to m x + c.
Try to make in this format.
This format try to make try to convert these lines in this format. y is equal to mx + c.
Yes, champions. The coefficient of y must be 1. Here the cofficient of y is one. So this equation is already in this format. Here the cofficient of y is three. So this equation the given equation is not in the format of this.
So three will go there. Y should be the coefficient of Y should be one. Here the cofficient of Y is two but the cofficient of Y should be one. Here the cofficient of Y is five but the cofficient of Y should be one. This is the rule. This is the rule. The cofficient of Y must be one. Okay. So here in this equation once you converted once you have converted these equation in this term automatically m will be your slope whatever the number is there and this c can you see this c this c is basically y intercept this is called y intercept so let's solve the question let's do the first question everyone please focus on the solution so we have the we have first equation The first equation y = -4x + c. Wow.
If you compare this linear equation y is equal to m x + c because I have already told you every straight lines are found in this format. Every straight line are you know remain in this format. So make sure to write the linear equation in this format. The coefficient of y should be one. Now compare. So if you compare, if you compare m will become minus4 and c will become five. So my dear champion this is the answer. Slope is equal to -4 and y intercept.
Y intercept will be five. Wow. This is the answer. It's magic. It's magic. It's magic guys. Sorry ammon uh this is English channel so I would not be able to use Hindi at all champion sorry to say but see this is mathematics in mathematics if you are facing little language barrier now but you can see the solution the way I'm solving the question at least you'll be getting something don't worry in mathematics sometime language not matters right if you look at the solution you can solve now now second question we have second question 3 y is = 9x - 12. This is the equation. Now what we have to do? Try to make this equation like this.
Try to make this equation like this. How we can make here the cofficient of y is one but here the cofficient of y is three. So now do one thing divide this three is in multiplication. So send this three that side that side. So y will look like y is equal to this will become 9x 3x - 12x 3. Clear everyone? Now cancel if if if they are cancelling you can cancel. 3 4 are 12 and this is 3 3 are 9. Now write the final equation. y is = 3x. Oh my god. Now look at this equation. This equation is resembling like this.
Don't worry about negative and positive sign. Don't worry. The equation should look like this. Negative positive signs are just operation. You don't need to worry about that. Now compare this. If you compare this, If you compare this, y is equal to mx + c. My dear champion, what is m? m will become 3. And what is c? c will become four. And what is m? m is called slope. c is called y intercept. You got the answer. So here slope is three. And y intercept will be y intercept will be champion what? - 4. I hope this question clear to you all. Now moving to third question here. Third one.
So third question is 2y is equal to 6x + 7. Again there is problem. The problem is the cofficient of y is 2. But what we want? We want cofficient of y should be one. Here the cofficient of y is two. Here the cofficient of y is one. So how we can make cofficient of y as one. two will go that side. So if this two will go that side, the equation will look like y = 6 by 2x and 7 will also be divided by 2.
Right? Now if they are cancelling, please cancel. 7 is not canceling. So keep that as it is. So 3x + 7 by 2. My dear champion, now this equation is exactly like this. So m will be slope.
So slope is equal to 3 and y intercept will be y intercept will be 7 by2. I hope this question is clear to you all. Is there anything challenging? Tell me honestly champions class 9th champions tell me honestly. Is there anything challenging in this question? Similarly, fourth question what we have to do 5 y = -2x 20x sorry + 15. What is the problem here? The problem is the cofficient of y is five but we want cofficient of y should be one according to the general format of line. So how we can make cofficient of y as one? The five will go there right side. Someone is writing sir y 7 is divided upon two. Okay, listen.
Listen. Who is asking?
Nirmal. Hi Nurmal. Listen champion.
Listen.
2 Y is at the left hand side. 2 Y is at left hand side. And two is in multiplication with Y. Right? So if this 2 Y will go right hand side. So two will divided two will be divided to this entire function not only six. two will be divided to entire function to entire right hand side. So 6 x2 and 7x2 also for example if I'm writing here Romesh three Romesh is equal to four Rahul sorry four bharat four bharat plus dilli don't get confused by the way three raomesh is equal to four bharat plus dilli now I want to make r alone I want to make r alone so what we need to do three is in multiplication ation we have to send three to the right hand side. So my dear champion three will divide 4 B also and three will divide D also entire right hand side will be divided by three. Clear to you? Clear to you normal? Is it clear to you now? I hope this is clear to you now. I hope this is clear to you now by taking this example.
If this is clear to you, please do let me know in chat section. Yes sir. Clear.
Yes sir. Clear. Clear to you. Okay. Now solve this question.
5 y = - 20x + 15. So y will become again 5 will go there. So this is - 20x / 5 for 15 divided by 5. Now we can cancel if they are cancelling 5 3 are 15. So y = y = - 4x because that was in negative form already negative form plus 5 3 are 15. So wow. Now compare compare this equation. Compare this whole format from this. I have already told you you don't need to worry about sign. S can be anything. It can be positive. It can be negative. It can be anything. So when we compare this entire equation from this one. So the cofficient of x is four and the cofficient of x is m. m is called slope. So we have slope is equal to yeah slope can also be negative. Yeah don't worry. And y intercept y intercept here will be equal to three.
So this is the answer.
Slope is min -4 and y intercept is three. Clear? Clear to all.
Moving ahead. Next question.
Okay. Next question. The graph of a linear polom. Someone was asking about graph in previous chat. Who is there sir? How to do how to draw? How to do graph teach and give? Okay. Yeah. Try this question everyone.
Yes. Everyone.
Yeah. Look at the question. Question number nine for you.
Yes. Question number nine for you.
The question number nine is the graph of a linear polomial px passes through the points 2a 7 5a 16. Find px if the coordinates of the point where its graph cuts the x-axis and y axis.
Really this is very good question.
I'm making here five star for this question.
This is good question. Five star question.
Five star question. Look at the question champion. So basically the question is the graph of a linear polomial px.
There is a polomial linear. There is a polomial but linear. And do you know what is mean by linear polomial? Linear polomial means on graph it is always a straight line. Please make sure if there is a linear polomial linear polomial means a straight line. When you draw the graph of the linear polomial it will be always a straight line. So basically there is a straight line and a straight line is passing through this point. So we have to find out first equation. Who is this? Hi Gurudev. Gurv.
Okay. So we have to find out equation first. So listen champion my dear champion when a line when a line passes through two points what happens when a line passes through two points for example x1 y1 the first point is x1 y1 this is let's say a this is a point this is a Right?
And we have one more point B here.
Coordinate is X2, Y2. Right? So when a line passes through these two points, we can write equation. There is a format of equation. Equation will look like y - y1 is equal to y2 - y1 / x2 - x1 into x - x1. This is the formula to write the equation. This is the formula. And now my dear champion, if you write these two points x1 y1, this is x1 y1. This is x2 and this is y2. Now put the values here. Now so y - y1 we have it 7.
Okay. - 7 is equal to y2 - y1. 16 - 7 divided by x2 - x1 5 - 2 into x - x1 x1 is 2 x - 2. Now solve this. So y - 7 is equal to 16 - 7 will be 9 and 5 - 2 will be 3 and this is x - 2. So 3 3's are 9. So we have y - 7 is equal to 3 into x - 2. Now could you please solve this? So y - 7 is = 3x and 3 2's are 6 - 6 and - 7 will go there. So y = 3x + 1. This is the equation. This is the equation of straight line which is passing through these two points these two points. So my dear champion this formula is very important very very important. This formula is very important to write the equation of a straight line when two points are given to you from which the line is passing. Clear to you all? Hello sir.
Hi. Clear to you all. So we have this equation. Answer is not completed. We we we we haven't get answer. The question is here. Find px.
Okay. So y is called px. So px is equal to px is equal to what? 3x + 1. This will be px.
px is basically called y.
This is y.
Mhm. This is y. Now find the coordinates of the points where it it where its graph cuts the x-axis. So first point cuts x-axis. So my dear champion when it will be cutting x-axis so at this point ycoordinate will be zero. Clear everyone? When a line cutting the x-axis the y-coordinate will be zero. So we have this equation on y is there. So put polomial as zero. The value of polinomial it means yx is 0. So when we put yx =0. So this will be 3x + 1. So here uh 3x will be -1. So here x will be -1 / 3. So the coordinate will be coordinate will be -1 by 3, 0. This will be the coordinate when this line will be intersecting x-axis. Second, when it cut y-axis, so my dear champion, when it cut y-axis, any line when a line cuted y-axis, for example, look, this is we have x-axis and y-axis and a line is passing through this. So my dear champion, at this point, at this point here, ycoordinate will be zero and at this point yaxis, the x coordinate will be zero. This is the rule. At this point, y coordinate will be zero. At this point, x coordinate will be zero.
So, when a line cutting the y-axis, when a line when this line is cutting the y-axis, it indicates that it indicates that the x coordinate will become zero.
This is y-axis. So, x coordinate will become zero. Now, put x coordinate 0 here. So, y will be equal to 3 into 0 + 1. So, y we have is equal to 1. So, the coordinate will become what? 0 comma 1.
This is the answer.
Clear everyone? This is the answer. So listen champion let me explain this question again. So basically the question is there is a linear polomial and linear polomial means a straight line and this straight line is passing through this point 2a 7 and 5a 16.
Find px. So this is the first question.
Find px. I mean find the polomial. So when a line passes through two points we have this formula. We have this formula to find out the equation of line. This is this will give you what equation of what?
Linear line when when two points are given to you. Two points. This is called twintoint formula. Two points are given. So when two points are given, consider first point as x1 y1 and consider second point x2 y2 and use this formula. After doing the mathematical calculation, you get the polomial. And when this line a straight line is passing through x-axis, y coordinate will be zero. So on putting y is zero, we have x - 1x 3. So the coordinate will become this. When this line cuts y-axis x coordinate will be zero and on putting x0 we have y is equal to 1. So the coordinate will be 0a 1. This was the question very good question as per examination point of view. Very important. Let me write here very very very important very important question this is. So the student who is new here go and hit the subscribe button and don't forget to hit the like button of this live session. Go and press the like button also. Is this question clear to you all champion?
Clear to you all champion? Please do let me know. Yes sir, this is clear.
Clear to all. Moving ahead. So let's move. This was all about introduction to linear polomials. So I have discussed all good questions, highle questions.
Now moving to next chapter that we are going to cover up here in the live class and the chapter name is the world of numbers. So let's solve the question world of numbers. So moving ahead to the first question of the world of numbers.
Okay, look at the first question. The question is one column of ishango bone.
Ishango boon shows number 11, 13, 17 and 19. What do these numbers what do these numbers have in common?
List the next three numbers that fit the same pattern. My dear champion, we have 11, we have 13, we have 17 and we have 19. Do you know they all are prime numbers?
All are prime numbers. And I hope you all know the prime numbers. What numbers are called prime number? Which is called prime number? The number which are having two factors only two factors one and itself.
The last part can't understand where yes we vlog I don't know your name bro where where is this champion is this listen listen listen is this so you have this straight line and this straight line is passing through x and y axis listen this is x-axis each student is important for me if anyone is having any kind of confusion I can repeat again and again for you right so we have this straight And this straight line is passing just imagine if this straight line is passing through this. So here this straight line is cutting the x-axis. So if the straight line is cutting the x-axis at this point at this point. So my dear champion here ycoordinate will be zero.
And if the straight line is cutting here y-axis this is y axis. So at this point x coordinate will be zero. So on cutting x-axis we are taking y as zero. On cutting y ais we are taking x is zero.
And on putting the values we can get X and Y simultaneously. Clear? I am back now. Is it clear? Now is it clear to you?
Clear to you?
So it's all about now see champion.
Listen ch listen to me carefully. When a point is lying on X-axis. Listen this is very important and it will be helping you out in class 11th. So these are the foundation topics for class 11th.
Listen. So when a point is on x-axis just imagine if the point is on x-axis it indicates that the ycoordinate will be zero here. If a point is on y-axis if a point is lying on on y-axis I'm talking about on not in first quadrant not in third second quadrant neither in third quadrant nor in fourth quadrant I'm talking about on y-axis. So if a point is on y-axis it means that x coordinate will be zero. Simple. So when the line is cutting X axis, Y coordinate will be zero. And on putting the value, there is basic mathematical calculation.
When the line is cutting Y-axis, X coordinate will be zero. And doing the basic mathematical calculation, you can get the value. Is it clear to you now?
Someone is there. Blog G, I don't know your name, bro. Is it clear to you now?
Moving ahead. So let's do first question of the world of numbers. So the first question of the world of numbers is there are four numbers given to you 11 13 17 19. This is the pattern. Okay.
What do these numbers have in common?
And the common thing is all are prime number. This is the common thing because 11 can divide by 11 and 1 itself. 17 can be divided by 17 and 1 itself. 13 can be divided by 1 and 13 itself. 19 can be divided by only 1 and 19. Right? So the the these numbers are having only two factors and only two factors numbers are called only two factors. Such numbers are called prime numbers. Only two factors one and itself number itself. I hope this is clear to you all. These all these numbers are prime number. Now the second part of the question is list three next three prime number after 19. Listen champion. So after 19 after 19 after 19 the prime numbers oh my god what happened I did anything okay sorry by mistake I erased actually so this is 11 13 17 19 so let me write here all are all are prime numbers why prime numbers because only two factors only factors are only factors only factors are one and itself.
Okay. Now what will be the next three numbers in this pattern following this pattern. So after 19 after 19 20 21 22 23 So 23 will be the next prime number because 23 can be divided by only one and 23 itself. After 23 24 is not a prime number. 25 is a composite number.
26 is a composite number. 27 is a composite number. 28 is a composite number. But 29 is a prime number because 29 can be divided by 1 n itself.
Okay. After 29, 30 is a composite number. 31 is a prime number because 31 can be divided by only 1 and 31. So this is our answer.
Next three number after 19. Clear everyone.
Who is this? Should we use the formula only when the points lie on the axis?
Yeah. Yeah, bro. You can use clear everyone. Clear? This question is clear to you all. So after 19, after 19 in this pattern the consecutive prime numbers will be 23, 29 and 31. The next three prime numbers will be 23, 29, 31.
If this is clear to you all, give me thumbs up in chat section. Yes sir, this is clear. We all are really enjoying the session.
What are co-prime numbers? Someone is asking there what are co-prime numbers?
Listen champion. So basically co-prime is not a number. Co-prime is a relation.
Listen, someone is asking co-prime number. Listen champion. Co-prime is not a number. Co-prime is a relation.
Relation between two numbers, three numbers. Okay. So co-prime numbers are all those numbers which are having common factor only as one. For example, if I'm writing 7 and 23, I have two numbers here, 7 and 23. Now, if I ask you, tell me a number. Tell me a number which can divide 7 as well as 23. Is there any common factor which can divide 7 and 23? No, my dear. No, my dear champion, there is no any such number which can divide 7 as well as 23.
No, not possible.
They can be divided by only one. One can divide seven and one can divide 23 also.
So they have only one highest common factor will be one. Except one there is no any other number which can divide both. So such numbers are called co-prime numbers. You can write 3 17 they are also co-prime numbers. For example 23 and 29 they are also co-prime number. For example 13 13 and 11 they are also co-prime numbers co-prime pairs clear everyone is it clear to you all no no it is not necessary that they will be consecutive or not is it clear to you all we blog vlog so is it clear to you co-prime number is it clear to you so co-prime number is not a number it is a relation relation between two pair of the numbers basically I hope this is clear to you all.
Clear to you all. You all are enjoying the session. Tell me honestly, honestly, please tell me honestly you guys are enjoying the session. Yes sir. Shrui is there. How? Okay sir.
Clear. So if you all are enjoying the session do let me know. Send me thumbs up. Send me some hearts. At least I I I I deserve little you know your involvement your So show me your presence guys. Interact with me. Send me some hearts.
I want to see the first student today name who will be sending me some hearts or thumbs up there in chat section. Who is there? Who is there? Who is there?
Okay, moving ahead. No one is there. Yes sir. Shrui is there. Thank you so much.
Shri, thank you so much.
Vlog is there. Blogger here. Please tell me your name here. Correct name, bro.
Tell me your correct name.
Okay. Wow. Heart is there. Thank you so much. Okay. Moving ahead.
Okay. So the next question is at noon temperature in high altitude desert of Ladak is recorded 4° centigrade. Okay.
So the initial temperature was 4° centigrade. By midnight, so in night at around 12 and 1:00 a.m. in midnight it drops by 15° centiggrade. So initial temperature was 44° centigrade and the temperature has been dropped by 15° centigrade. So what is the midnight temperature? Everyone knows midnight means cool cool weather cool temperature. So uh temperature at midnight.
I have changed my font. I have changed my writing. Okay. So, temperature at midnight. So, yeah, listen. The initial temperature was 4° centigrade now. And the temperature is decreased. Decreased.
Come on. Yeah. Drop. Drop. Drop means what? Decreased.
Decreased. And decrease means minus sign.
Decrease mean minus sign. So, minus 15° centigrade. So, 4° centigrade. -5° centigrade will be -12. Oh my god, how the people are surviving in such temperature? -2° centiggrade. Oh my god.
So this is the answer.
Uh myself Sanjana. Okay. Thank you so much Sanjana. SR sir you are doing a lot of for us. Thank you. Thank you. This is the religion. This is the religion of a teacher Shriy. This is our responsibility and we the teachers on this channel are ensuring to build strong foundation strong base for you guys so that you guys will be feeling relaxed and comfortable in class 10th.
This is our intention to give you I mean some quality session you know so that we could deliver quality content quality live classes so that you will be getting good marks in your ninth as well as you guys can ensure strong foundation strong base for your class 10th. This is our intention and goal. Let's achieve all together. Okay, everyone. So, minus 12° centiggrade is correct answer. Next, this is very easy question by the way.
So, the question is prove that rational prove that the rational numbers 2x3 and 4x 6 are equal. Obviously, they are equal because this is 2x3 and this is 4x 6, right? So, this is obviously this is equal. Why? Because if we multiply in this we can cancel this also but let me follow this method. If we multiply two in numerator and denominator in the this fraction. So if we multiply same number this is possible. This operation is valid in mathematics. You guys can multiply same number in numerator as well as in denominator. So if we are multiplying and don't do anything to the right hand side fraction. So 2 2 4 and this is 3 3 2 6. Oh my god both are equal.
That's why both are equal to each other.
I am back. I am back myself.
Who is this?
Hats off. Thank you. Thank you so much.
Clear to you. 4x 6 4x 6 is equal to. If you want to follow, if you want to operate right hand side, you can operate right hand side. It will be cancelled by initially. So 2x3 2x3 equal. You can follow this method also. So you are free to apply any method.
I hope this is clear to you all. Moving ahead to the next question. Next question. Next question. Next question.
Okay. Look at the next question.
Classify each number as rational or irrational giving giving its fraction form wherever possible. Root 81 and this and this. Wow, what a question. Good question. I would say hi sir. Who is this? Atar. Hi Atar. How are you?
Sir sir don't understand sir can we cross multiply yeah you can cross multiply also this is the method which is given in class 9 solution who is there who could not understand oh shriy le everyone is important for me I'm doing this question again for you again for you everyone so the question is we have 2x3 fraction and we have 4x6 fraction And the question is prove that both fractions are equal to each other. Let me show you one important thing. If I ask you champion 1x 6 and 2x 7 are they equal to each other? Not at all. They are not equal. Because when you do cross multiplication the cross multiplication 7 into 1 will be 7. 7 1 are 7. And cross multiplication of these two numbers 6 2 are 12. Oh my god. they are not equal to each other and if they are not equal to each other it indicates that the given fractions are also not equal to each other. So we can follow this method cross multiplication. Now if we do cross multiplication here in this question. So my dear 2 into 6 this will multiply with this and this will be multiply with this 4 3 are 3 into 4. Oh my god 6 2's are 12 and four. Oh wow. RS is equal to ves. So here we can write our hs is equal to rhs it implies that 2x3 is equal to 4x 6.
Clear to you all? Last one please. Triny now. Is it clear to how are you sir? Hi Atar, thank you so much for asking. I'm definitely fine. I'm okay. How are you champion? Tell me at yes yes yes we can apply. Is it clear now? Shruy. Is it clear to you? Moving ahead. Look at the question. Classify each numbers as rational. Okay. So we have few numbers given. First number by the way this is root. Okay. This is root 81. This is root 12. So we have one number, two number, three number, four number and five numbers. Five numbers given to us and we have to identify whether the given numbers is rational or irrational. So look at the first first is roo<unk> 81. Oh my god, roo<unk> 81 is what? 81 is a perfect square. So 81 can be written as 9 into 9 and we have pair in under root. So if we have pair in under root. So my dear champion we can take one and root will be removed. And nine means what? 9 by one any number can be written by any number can be represented as one. If I'm writing here 10 you can write 10 by 1.
No problem is there. If I'm writing here Bangalore we can write Bangalore by one.
If I'm writing here lo by one no problem is there. Right? Right everyone. Okay. Yes. Is it clear? So 9 by 1 and this is the format of P by Q.
Hence this is a rational number. I know you guys know rational number. The numbers in the form of P by Q where Q should not be equal to zero. Oh second question roo<unk> 12 roo<unk>2 can be written as 2 into 2 into 3 2's are 4 4 3 are 12. Now we have pair pair of two inside the root. So if we have pair we can take 1 2 out 2<unk>3 and 2<unk>3 roo<unk>3 is a rational number irrational number and when we multiply irrational irrational so the number comes is always irrational.
So because of if root3 would not be there. So this would be I mean rational number but root3 is there and root3 is a multiply with two. So because of this it has become irrational number. It has become irrational number. Now third third question is 0.333 and so on. So this is basically a repeating decimal non-terminating repeating decimal. This is called such decimals are called non-terminating because remainder is not zero. So this is called non-terminating. The division is going on going on going on going on going on going on. So this is called non-terminating non-terminating but repeating or we can say recurring non-terminating repeating decimal and all the non-terminating repeating decimals are always they can be they can be expressed they can be expressed in the form of P by Q. So they all are such decimals are always rational number. So this is the format of rational number because the repetition is very systematic. 3 33 33 33 3 33 3 Okay.
Fourth is fourth number is 0.12 345 1 2 3 4 Oh wow 1 2 3 4 5 and so on. Now can you see here champion? Here the repetition is very very systematic. 1 2 3 4 5 1 2 3 4 5 1 2 3 It means that same group, same bunch of number is repeating continuously till infinity same number.
So whether two numbers are all together repeating, three numbers are all together repeating, four numbers are all together repeating. If repeating is there, if repeating is there, such numbers are always rational. So this is also what? This is also this is also as it is as this upper one we can I can mark this I hope you know it means non-terminating repeating and if non-terminating repeating is there so it can be expressed in the form of P by Q and hence this will be what a rational number clear everyone this will be rational number okay now fifth fifth question is one point after decimal the number is 01 1. Now 0 0 1.
Now 0 0 0 1. Now 0 0 0. And what kind of repetition is this? Very unsistatic repetition. Here after decimal look at the bunch of the number. This is 01 and this is now 0 01. Now 0 01. Now 0 0 1. So in every repetition one more zero is increasing one more zero is adding so this repetition is not systematic same bunch of number is not repeating so this is called non-terminating non-re repeating decimal and such numbers are always irrational because we cannot express such numbers in the form of p byq so what I had done mistakenly I have erased everything sorry guys so is it possible to so I have I I hope you guys have already understood. Let me solve the last question. So this is 1.01 0 01 0 0 1 0 01 and so on. So this is called non-terminating non-terminating non-re repeating because you know the repetition is not systematic. The repetition is very unsistatic. Someone something is 01 now 01 now 01 now 0. This repetition is not systematic. So non-terminating and this is called non-re repeating or we can also use non non-terminating non-recurring and such numbers such numbers can't such numbers can't be expressed in the form of pyq hence they are they are irrational number they are irrational number clear everyone I hope this is clear to you all if the numerator are uh terminating numbers are terminating Is it rational sir? Yes. Yes. If the decimal is terminating, look for example, such numbers are called terminating. If I divide 10 from five, the result will be two. So division is completely done. The remainder is zero. So if the remainder is zero, such divisions are called terminating because the division is terminating.
The the division has ended up. So this is called terminating. And if you are getting two here, so 2 by 1 this is rational. So terminating decimals are always rational. Moving ahead to the next question. Is someone asking any doubt?
Uh how are you? Hi. If the number is it clear to you vlog uh sorry Sanjana if the if the numbers are terminating?
Yeah obviously of course always if so can you have provide an example? I have already given you example 10x2 12x 3 12x 3 is equal to 4. So that is terminating.
Okay. Oh this is important question.
Express.
What is it?
Yeah. Anyone clear to everyone?
Just a second. Champions.
Okay. Express 0.99999 in the form of P by Q. Are you surprised by your answer with your teacher and classmates? I'm your teacher. Yeah.
discuss with me okay with your teacher and classmates discuss why the answer makes sense okay let's do so first part let's do the first part first part is convert this decimal in the form of pq and obviously we can convert this number in form of pq why because the decimal is non-terminating but repeating and if non-terminating repeating decimal is there it means we can express that decimal in the form of p byq so let's assume Assume this is x is equal to 0.99 9 a a and so on.
Now listen champion sir please explain again which one who is asking Raki where were you Raki okay listen okay okay Raki look at this question this is fresh question new question for you please listen here my dear champion how to express this decimal in the form of p byq it means we have to convert this decimal in the form of fraction p byq means what p byq means what this is fraction and here the number is given in decimal so basically We want to convert decimal into fraction form. This is the question. We have to convert decimal into fraction. How to convert decimal into fraction? Listen champion. I'm going to give you very smart trick.
Okay?
Always always try to shift the decimal behind the repetition.
Always try to shift the decimal behind the first repetition. Here this should be your intention. For example, if the repetition is look like x=0.27 27 27. So here what should be the target? The target should be shifting of decimal here behind first repetition first bunch of repetition. First this is this should be your target. So my dear champion if we are shifting decimal to the one digit one digit position it means that we have to multiply this equation this format multiply by 10 on both sides because we are shifting decimal by one position one position. If we deal with this condition we are shifting decimal to the two position behind two digits. So we have to multiply this equation by 100. So which number will be multiplied? It depends on the position of shifting of decimal. So if here we are shifting decimal behind the first repetition. So it will be multiplied multiply by 10 on both sides.
On both sides and this is your equation number one by the way. Okay. So if we multiply 10 on both sides the equation will look like 10 x and this decimal will be shifted there. So now this will look like 0.99999.
So we can put here bar same if same number is repeating again and again we can so now the equation we don't need to now we don't need to show the zero because now the identity of zero has vanished because we have shifted point here this side so zero will not be there now this is your second equation I hope this is clear to you all so which number will be multiply it depends how much position how many position of decimal is shifting ahead depends it decides right so up. Now subtract subtract equation one from two.
Why I have decided to subtract equation one? Why not equation two from first?
Because 10 x is greater. This is a smaller my dear. This is simple. Now this is common sense. So always smaller will be subtract subtracting from uh greater. So this is 10 x - x. And here we have 9.9 bar minus what is this? 0.9 bar 0.99999 can also be written as now 0.9 bar because same number is repeating. So this is 9.9 bar minus 0.9 bar. Right?
Now subtract. So this is 9x and when we subtract 9 0.9 uh 9.9 from 0.9. So this will look like can we please subtract this? Uh this is what here what is this? This is 9.9 and this is 0.9 0.9999.
So this will be here this will be here 9.
Oh my god. Now if we calculate x so this is 9.9 and 9.9 is equal to 1.
And that is the reason why 0.99999999999 is approximately x is 0.9999 is approximately equal to 1 because we have proved in fraction form when we converted this number in fraction form we are getting one. So 0.9999 is also equal to one. This is the answer and that the question has instructed you to discuss with your teacher by your answer because you are surprised. Why we are surprised? The given number was in decimal 0.9 and we are getting x is equal to 1. This was x equal to this and we are getting x is equal to 1. So this is the surprising thing here. Clear everyone?
Clear everyone sir please explain which one?
Clear everyone. So x is equal to one also. Wow what a question. Now moving ahead without performing long division decide decide which I mean we have fraction. Hi hi Saki how are you? Okay so without performing long division we have three numbers three fractions 7 by 20 4x 15 and 13 by 250. Check whether the given fractions have terminating decimal and which are repeating.
Repeating or terminating we have to decide. Then verify your answer by long division. Okay. Okay. So my dear champion listen champion I'm going to tell you very important thing. So first fraction is first fraction is 7 by 20.
The question here exam important questions. Yes Shaki these are very important questions. Hi An hi an if you guys are enjoying the session do let me know in chat section. Yes sir we are genuinely enjoying the session. Thank you so much. Please write in chat section everyone and you if you haven't hit the subscribe button go and hit the sub subscribe button of this channel and don't forget to hit the like button of this live session. Okay. So we have 7 by 20. Now we have to identify this fraction is having terminating decimal or non-terminating decimal. This in champion.
So in this fraction this is the deciding factor. This decides deciding factor.
Yes, my dear champion, denominator is the deciding factor. Now you might be thinking sir, how how denominator is the deciding factor whether the given fraction is terminating or repeating? My dear champion, if the denominator listen if the denominator if the denominator is having only only if the denominator is having only two and five 2 and five as prime factors, then it indicates that the given fraction will be terminating.
If only two and five is there, the given decimal will be terminating. Please make sure very important line. If the for example, if I if I talk about 7 by 20.
Now if you do the prime factorization of 20, if you do the prime factorization of 20. So my dear champion here 20 can be written as 2 into 2 into 5. Oh my god.
It implies that the prime factorization of 20 contain 2 to power 2 into 5. Only 2 and 5 is there. So if only 2 and 5 is there, it indicates that the given fraction will be terminating decimal.
Yes, my dear champion. If there is one more, if there is another number as prime factor apart from two and five, three is there, seven is there. So that will be repeating decimal, not terminating decimal. So this is the beauty. Okay. So if apart from apart from 2 and five other other numbers are there.
So this is this will be repeating non repeating terminating will not be there.
So without doing the long division we can identify for example the first question will be the prime factor of denominator is what 2 2 into 5. So only 2 and 5 is there. It means that it will be it would be what terminating. Let me let me verify let me verify by long division. So divide 7 from divide 7 from 20. So we required zero point. So this is zero. Now 23 is a 60 01. Now we can now 20 oh my god it has divided fully it has divided it fully the remainder is zero so this is called terminating terminating and that's why I have taught you here so denominator is the deciding factor after after decimal if only two is there yeah if only two is there obviously but it dep V sir can you explain the question express the following the form of p by q where p and q R integer not equal to guys I will I will be solving the these doubts for you don't worry but first solve these questions because I have brought up here few quality questions important questions after decimal there are two number huh if after decimal there are two numbers and which is repeating in bunch all together so you have to multiply by 100 to shift the decimal behind that repetition hi there sir when is the class 10th very soon champion don't worry clear to you all clear to you all now check whether question number two whether it is repeating or terminating so this is 4 by 15 and which is the deciding factor 15 is the deciding factor so my dear 15 can be written as three oh my god 3 into 5 2 is not there two two should be there for terminating but two is not there so this is not terminating this will be repeating clear everyone for terminating two should be there and five should be there both and third is third is 13 by 250 now you guys check now please you guys check whether this fraction will be terminating or repeating you have to write the prime factorization of 250 I hope you guys can do I'm doing giving you one question as homework moving ahead find three number guys please the student who is watching this live session for the first time on this channel guys please don't forget to hit the subscribe button. This is your Sony source request. Don't forget to hit the subscribe button. And who have already subscribed the channel, do let me know in chat section. Yes sir, I have already subscribed. I have already subscribed. Write in chat section. And I'm telling you guys for your own benefits because I will be guiding you.
I'll be giving you lot of you know competency question sessions will be there. I'm planning in future.
Okay. So please stay connected with this channel. Okay. Find three rational numbers lying between this and oh what a question. What a question. Of course this is terminating decimal because dot is not there at the end. Here dot dot dot not is not there at the end. So this is terminating decimal. If there will be there would be dot dot dot dot dot dot it means non-terminating and so on.
Okay. Already subscribed sir. Okay. So I'm requesting to all the student who have not subscribed the channel yet. You guys are making big mistake there. Okay.
So the first number is 3415.
This is the first number. And the second number we have is 3.1416.
Right? Listen. And we have to write here three rational numbers between this R1, R2 and R3.
Now let's use your brain. Let's use your brain. If you have brain, I know you guys have brain. So let's use this. My dear champion, tell me one thing. If I'm writing 4.6, right? And if I'm writing 4.60, is there any difference? No difference.
No difference. 4.6 4.60. If there is nothing, it means zero is there.
Yes, my dear. If there is nothing, zero is there. So similarly we can write this number as 3.145 and put here zero also 3.14160.
Now look at both the numbers. Now look at both the numbers after decimal 14150 and this is after decimal 1 1460.
It means that this number is in counting from here to here. Now we can write any number. The first rational number will become the first rational number will become 3.14151.
Second rational number 3.1415 2. Third number will become 3.14153 and so on till 60 till 60. So these are the answers. Wow sir you are magician.
Is it clear to you all? H yes yes hi Patel hi JS Patel superb super super good so the student who is new here who haven't subscribe the channel yet go and hit the subscribe button everyone so that you guys could not miss the ch I mean quality sessions you know powerful sessions ahead in future because this channel is dedicately designed for all class 10th and ninth champions moving ahead this is most I have solved This question trust me my dear I'm not telling lie I have solved this question almost six to seven times in this session because this question is common in both the classes class 10th and class 9th and again this question has arrived in front of me very easy question please take my name hi kid to dancer I don't know your correct name but yeah k2 dancer hi okay prove that root five is an irrational number how to prove root five is an irrational number So this is very common question and most probably there is huge chances that you guys will be getting this question in your exam whether it is in midterm exam periodic test exam or it is in I mean your final exam this question is very common okay so how to solve how to everyone knows root five is an irrational number because when you write the decimal of root five it would look like 2.260 1 2 0 6 06 something number is there. So you you can see after decimal after decimal the repetition is very unsistatic. Sometime two is coming sometime three is reflecting again six has come then zero then six and what kind of repetition is this? So this repetition is basically non-terminating non-re repeating and non-terminating non-re repeating decimals are always irrational. So we know we know root five is irrational number but but we have to prove mathematically it's all about marks and marks matters a lot in exam right we know root five irrational but we have to prove mathematically so now we are going to prove this mathematically and there is a method called contradiction method please everyone class 9th champions please focus this question is very common and important very important and this question will be reflecting in class 10th also so this This is very common question for all who is there.
Should we prove through contradiction?
Yes. Yes. I'm using the contradiction method here. Subscribed please. After this question, can you solve I? Okay.
So, sure. Uh if I'll be getting time, I can solve the question for you. Don't worry. Okay. So, this is contradiction method. What is called contradiction method? First understand, dear champions, first understand the contradiction method. So, contradiction method is what? When our assumption whatever we are assuming in the beginning of the solution our assumption and whatever we are getting as a result both are not matching. Our result is not as per the assumption. Both are not matching. Both are not synchronizing.
Result is something else and our assumption is something else. Both are different from each other. When the result is not synchronizing with our assumption, this condition is called contradiction. Some error has occurred, some problem is there, something is there, something some error is there.
Such condition is called contradiction.
Okay. So now we are going to start the solution. So how to start the solution?
The solution starts from assuming root five.
Assume root five as a rational number.
So assume assume assume roo<unk> 5 is a what? A rational number. This is our assumption. But we all know root five is an irrational number. But still we are assuming it. Okay. Assume roo<unk> 5 is a rational number. My dear champion, if we have assumed roo<unk> 5 is a rational number, it implies that we can write<unk> 5 is equal to p by q, right?
Because roo<unk> 5, we have already assumed root 5 is a rational number. So it can be written as p by q. My dear, don't forget to write this information that I'm going to write now. This is very important in this solution. Here here P and Q P and Q are are co-prime pairs.
Wow, this is very important to mention.
What does it mean by co-prime pairs?
Co-prime pairs means P can be divided by one and Q can be divided by one only. It means that both P and Q are having only one common factor which is 1. For example 3 and 17, 11 and 23 these all are 19 and 17. They all are co-prime numbers. So P and Q are co-prime numbers. If you sometime I mean due to exam pressure due to I mean lot of stress you might be you know forgetting this uh line in your examination hall.
But make sure to write this also. You guys can write this. HCF is equal to 1.
It will also work there. Don't worry.
Right? So if you write this line, this is very good. If you write this, no problem. It will work. HCF is equal to 1. So P and Q are having only one common factor which is one. Right? Okay. Now do the cross multiplication. So if we cross multiply roo<unk> 5 into Q will be equal to P. Now squaring both sides.
So if we do a squaring squaring both sides so my dear root will remove.
So this will look like 5 Q² is equal to P². Oh my god. Can you see this equation champion? Five is in multiplication with Q. So if the five will go there, if the five will go right side, it means that QP² will be divided by five because five is in multiplication. If five will go there right side so P square will be divided by P. It indicates now because the five will go in division. So here here P² is divisible by five. So P is also P is also divisible by five.
It look like that there is some in contradiction there is some error because our assumption was P and Q are co-prime numbers. So let's do so if P is also divisible by five. So we can write one more equation here. P is equal to 5 into anything. Five into anything. At the place of M, you can take Delhi, you can take America, you you can take whatever you can take Ronaldo, you can take Messi, you can take Mbappe, you can take anything because when five is in multiply with M, when five will go there, it means P is dividing by five.
So make sure to write one more new family. This is your second equation and this is your first equation. I hope this is clear to you. If any one of you is wants to take here anything at the place of m you can take anything. Now putting or substituting what happens putting second in first. So where is first? This is first. So at the place of P 5 M will come. So 5 Q² and at the place of P 5 M will come.
So this is 5 M² because P is in square form. So solve this 5 Q² and 5 M² will be 25 M². Oh my god. Now we have this equation Q² is equal to 5 M². Now five is in multiplication with M. So when this five will go there, when this five will go there at the left hand side, it means that now Q² is divis dividing by five. Now Q² is divisible by five. If Q² is divisible by five, means Q is also divisible by five. Here P is divisible by five. Here Q is divisible by But what was our assumption? Our assumption was P and Q are co-prime. So they can be divided by only one. But what we are getting here one more factor. So please write everything in detail. Q² is divisible by five.
So Q is also divisible by five. Oh my god. P can be divided by five. Q can be divided by five. But what was our assumption? Our assumption was P and Q are co-prime. So something is missing here. Some error is there. So it is contradicting. Hence it is contradicting.
This is the contradiction.
The the result and the assumption both have clashed. Both have collided. Our assumption was P and Q are co-prime. But what we are getting here? So it is contradicting.
It is contradicting our assumption our assumption that that P and Q are what co-prime and why this uh contradiction has occurred because we assumed that root 5 is a rational number so root 5 can't be a rational number root 5 has to be irrational number So hence roo<unk> 5 is and what champions irrational number irrational number.
Wow. Finally, finally this is my eighth time. This is my eighth time I'm solving this question. Same question in this session 2026 27. So it is Oh my god. Who is writing complete solution in Bab single? Sir, I can help you. P square is divisible.
Oh my god. Who is this guy? Oh my JS Patil. Thank you so much guys.
Wow. Moving ahead. I hope this is clear to you all my dear champion. This is the most loved lovable question of all math teacher. So please take a screenshot.
This question might be coming in your exam. Moving ahead.
We cannot prove it is a rational number.
Why we cannot prove it is a rational number? Okay. Yeah. So it is contradicting of course because our assumption was P and Q are co-prime numbers. Now so they have both P and Q had as one. But what we have proved P is divisible by five. Q is divisible. So this is contradicting our assumption.
Not possible. Now yes. Who is Jasri Patil?
Superb. Superb. Heads up. Superb. Moving ahead to the next question. Ancient Indian. ends in Indian counted using the three joint of each fingers finger with the thumb as a pointer in such a way you might have seen your elders look 1 2 3 4 5 6 7 8 9 10 11 12 this methodology so ancient Indians counted using the three joints of each finger with the thumb as a pointer how many can you count on one hand and how does this relate to ancient base 12 counting system this is very Easy question. Literally my dear champion. If you use thumb as a pointer, if you use thumb as a pointer, so this will be one because in every finger we have three joints. One joint 2 3 4 three joints. So we can count count maximum 12. 1 2 3 4 5 6 7 8 9 10 because thumb is working as a pointer. Right? So in ancient time if thumb is pointer, thumb is pointer. So we can count three number in single time. I mean on one single finger there are three number because three parts are there right?
Three joints are there. So my dear champion four into three joints because four fingers we have except thumb ex because thumb is working as a pointer.
So how many numbers we can count maximum 12 12 and that is the reason why in ancient time people used base 12 ancient time people used base 12 counting system base 12 counting system and that is the reason why you know dozen 1 dozen 12 hours these all are you know based on 12 because of this ancient terminal ology.
So 12 counting system and that is the reason why we have one dozen one dozen which is equal to 12 bunch of 12 is called one dozen and in clock also in clock 12 hours is there no 1 hour 2 hour 3 hour 12 hours 1 to 12 12 is the maximum so because of this only these are the Yeah service don't worry I'm planning for the for that in future don't worry service if you have subscribed the channel very good stay connected with me stay connected with me thank you so much so finally finally the live is done and uh I have solved lot of question today of both the chapters uh introduction to linear polomials and and yeah this is the world of numbers as the thumb is pointer jas is giving explanation of every question. Very good. Yeah, Jas. Superb guys.
Thank you so much for having me. Thank you so much everyone. Class 9 champions for joining me in this live session. I'm having exam Thursday. Okay. So before Thursday, I'll be trying to come on live again. So champions stay blazed, stay sparkling and uh may you have so many so many successful years ahead. Take care of yourself and take care of your parents as well champion because parents are very important. They all are priceless for us. Okay. So, thank you for so much for having me and stay stay smiling like this like this always. My blessings are always with you. Keep smiling. Thank you so much for joining me. Stay blessed.
Bye-bye everyone. Take care of yourself and take care of your parents also.
Bye-bye everyone.
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