IGCSE/O LEVEL Add Maths - Cubic Graphs and Inequalities - Concepts and Past Papers

追加: 2026-05-11

[00:03:19]Asalam everyone.

[00:03:36]Just give me one minute. Take ma'am.

[00:03:37]Start.

[00:04:21]Just a minute.

[00:05:28]Okay, so let's begin. Apologies for the delay.

[00:05:34]Okay, I'm reading the chat now.

[00:05:36]Uh, sir, your math videos help me. I'm glad they did. Uh, wikum assalam walaykum salam. I'm good. Add math mock inshallah. I'm thinking this Sunday. I'm thinking this Sunday inshallah we will have our first live admat mock of the year 2026.

[00:05:57]Okay. Any chances of maths P4 being reconducted? I don't think so. I don't think they have acknowledged anything about math. So I don't think that they're going to reconduct that.

[00:06:13]level took us I don't know how many emails they sent next time here's my advice irrespective of whether you're sure or not the problem is that students just wait for the paper for the actual paper so that they can cross check whether the paper was genuinely leaked or not and then they report and by then it's too late Okay, because it's happened so many times with math already. So I I guess math students have now just been trained what to do. Although that's not a good thing but u that's how it is.

[00:06:55]What they did was they reported it immediately and because they reported it immediately that's why they took action.

[00:07:01]Yeah, sadly what people do is what Ole students and a lot of A-level students also what they do is they just wait cross check if it's the actual paper or not and then they report and by then it's too late you know it's it's already difficult to get them to admit anything and then reporting it after the paper makes it even worse sadly okay now let's begin so PNC key support yeah we'll do permutation combination also attention topic here because we had already touched upon polomial. So I thought might as well finish it completely before we move on to the next one. Okay, next one that next topic that I will do will most likely be either APGP or binomial and then I'll do circular measure also. Okay, so my aim is inshallah to do APGP, binomial, circular measure, functions, vectors, chyntatics, differentiation, integration. Okay, I hope I haven't missed anything. Yeah, differentiation and of course permutation combination. Okay, of course permutation combination also. Anyway, so let's begin. So before we begin, let's first have a look at the concepts briefly. Let's see what the concepts look like and then we will solve some past paper questions.

[00:08:17]So bear in mind that this sketching of cubic curves and cubic inequalities was something that was added in 2022.

[00:08:26]I think 20202 but it's something that has been added to your syllabus and it it hasn't always been there when we the prerequisite the prerequisite of this is that you guys already know how to sketch a quadratic curve I think I've already done a stream on quadratics so if you want to review that you can do that later also a cubic curve can be one of three types okay it's possible that it has one solution in which case you can see x - p x= to p is one solution and the other is a quadratic equation which cannot be further factorized. Okay, we did a question like this yesterday also. Okay.

[00:09:06]Now I haven't shared these notes. I'll share these notes right now. And then there is a possibility that you have a cubic curve which has only two solutions. Okay. And what will the equation look like? The equation will look like x minus p. And when where the curve is tangent to the x-axis, there will be a square over there. Okay. Okay.

[00:09:23]And then there is a third possibility which is that a cubic curve has three solutions. So x= to p x= to q x= to r.

[00:09:30]Now what will it look like before we get the values of pq and r before we get these three values of x? It would be x - p x - q x - r. Okay. And what is k here?

[00:09:40]K here is representing the coefficient of x cq. All these types can also be like this also. Okay.

[00:09:48]It doesn't necessarily have to be a maximum minimum curve. It can be the opposite. It can be a minimum maximum curve. Yeah, baby. It doesn't necessarily have to be like this. It can be a minimum maximum curve. Here also, it doesn't have to be a maximum minimum curve. It can be a minimum maximum curve also. Okay. Now, how do we decide whether it's going to be a maximum minimum curve or whether it's going to be a minimum maximum curve? Let's have a look at that. And by the way, this is something that's part of math also. So, if you guys have given math this year or even the last year, you would have done this in math as well. I'll share these notes. You'll find these notes in the same folder.

[00:10:27]Take a now how do you sketch a cubic curve? Now what are all the things that you need in order to sketch a cubic curve? There are three things. You need the x intercepts perhaps the most important in a cubic curve. You need the y intercept and you need the nature. How do you get the x intercepts? In order to find out the x intercept of anything, whether it's a curve, a cubic curve, a quadratic curve, a straight line, what you do is you set y equals to zero and you solve for x.

[00:10:56]Now, how do you solve for x? We've learned that. How do we solve a cubic equation? There are multiple ways with the help of which you can solve a cubic equation. Okay? You can do middle term breaking. You can do sorry middle term breaking. You can't do middle term breaking. You can do synthetic division.

[00:11:10]You can do uh identities. And then there is also trial and error with the help of which you find out the first vector irrespective of the method that you use.

[00:11:18]But this is basically how you find out the x intercepts. The main part is that you set y equals to 0 and you solve for x. Then how do you find out the y intercept? y for y intercept you set x equals to 0 and then you solve for y.

[00:11:31]Then there is the nature. Now a cubic curve can either be a maximum to minimum curve. Okay? Notice that this nature of turning point is maximum and this nature of turning point is minimum which is why we call it a maximum to minimum curve or it can be a minimum to maximum curve.

[00:11:46]Notice that this nature of turning point is minimum and this nature of turning point is maximum. That's why we call it a minimum to maximum curve. That is decided by looking at the coefficient of x cq. If the coefficient of x cube which is k which is what I'm referring to as k is positive then it's a maximum to minimum curve. If the coefficient of x cube is negative, it is going to be a minimum to maximum curve. Okay. Now let's have a look at some examples.

[00:12:14]So here we have a cubic equation which is already factorized and what we have to do is we now have to sketch the curve. So first thing that we're going to do is we're going to find out the x intercept. How will you find out the x intercept? One by one we will equate every expression equal to zero. Okay? Every factor equal to 0. So x -1 = 0 x = 1 x - 3 = 0 x = 3 x - 2 = 0 x = 2. So you have the x intercepts 1 2 and 3. Then how do you find out the y intercept? Y intercept can be found by either simply multiplying the constant. So that's -1 into -3 that's 3. 3 into -2 that's -6.

[00:12:54]Okay? Or you can think of it this way that you plug in 0 in place of x and you multiply the values that we get. 0 - 1, 0 - 3, 0 - 2. Multiply them, you get y = to - 6. Okay. So, what do we have now?

[00:13:06]We have the x intercepts. We have the y intercept. Now, what's left is for us to determine the nature. To determine the nature without expanding, just multiply all the terms that have x in them. So, x into x is x². x² into x that's positive x cub. Which means this is going to be a curve which has the coefficient of x cub as positive. That means it's going to be a maximum to minimum curve. Okay. So once you've marked all the points, okay, the x intercepts, the y intercept, the kind of curve that you're going to draw is going to be a maximum to minimum curve. So an important thing to keep in mind is that we never find out the turning point of a cubic curve unless the question categorically says so. If the question categorically says so, then you do it otherwise you don't. But one thing that we keep in mind is that the turning point of a cubic curve is always somewhere in between, not exactly in between, but always somewhere in between the two x intercepts. Okay? Okay. So if you have x intercepts let's suppose 1 and two. So the turning point will be somewhere in between 1 and two. That does not mean that it's going to be exactly in between 1 and two and it's going to be 1.5. No they are not cubic curves are not perfectly symmetrical.

[00:14:11]Okay. But they are somewhat symmetrical.

[00:14:13]Okay. So this is something that you keep in mind and something that we we use this this is a concept that comes in handy especially when you have one x intercept on the left hand on the negative side and one on the positive side. Okay. So where are we going to turn the curve for which we use this concept? Okay. Now let's have a look at another equation. So here's a cubic equation and we're supposed to draw a cubic curve. First things first find out the x intercept. So x + 1 = to0 that means x= to -1. x -2 = to0 that means x = 2. x - 4 = to 0 that means x = to 4.

[00:14:47]So we have the x intercepts. For y intercept simply plug in 0 in place of x. So 0 + 1 is 1. Then we have -2. Then we have -4. Multiply all of them together, you get the y intercept, which is equal to -16. Then what about the coefficient of x cub? So x into x is x².

[00:15:04]x² into x is x cub. Then when we multiply it by -2, we have - 2x cube. No need to expand the whole thing. Okay. So - 2x means that the coefficient of x cub is negative, which means it's going to be a minimum to maximum curve. Up they go. One x intercept is -1. The other is 2. And then the third one is four and the y intercept is -6. Notice where is this curve turning? This curve is turning somewhere in between minus1 and 2. And what comes somewhere? What is exactly in between minus1 and 2? Exactly in between is -0.5. But you don't have to make it exactly in between the two x intercepts. Somewhere in between.

[00:15:44]Make sure the curve turns somewhere in between the two x intercepts. Not exactly in between, but somewhere in between. And we do not waste our time finding out the x intercepts because we're not uh sorry finding out the turning points because we're never asked to find out the turning points.

[00:15:59]Let me look at some questions. Yes, I will do trigonometry and I've shared these notes.

[00:16:05]Sir, I'm going to take A levels in math.

[00:16:08]You're going to take A levels in math or you going to take math in A levels? Uh should I study admats and summer vacations or A? No, no, there's no need to study admats. If you want to take maths and A levels, it's best that you start studying maths and A levels. It's best that you start studying A level math only to give yourself a head start.

[00:16:24]Okay. Um, welcome.

[00:16:29]So, is it possible to find the turning point? Yep, it is possible. You know differentiation, right? So, you can find out the turning point through differentiation. Okay, but it's it's a total waste of time. If you find out if you're not asked to and you do it, it's a total waste of time. Okay. Modulus we will not cover it in this stream.

[00:16:47]Modulus we will cover it in another stream. Will it be incorrect if the curve is turned in the wrong quadrant?

[00:16:53]Of course.

[00:16:55]Of course.

[00:17:00]Let's do some questions and where we have to sketch the cubic curve. Then we will do some questions where we have to we will do some questions where we have to find out the equation from the sketch. Okay. So let's begin. On the axis below, sketch the graph of y = x - 2, x + 1, 3 - x, stating the intercepts on the coordinate axis. I should read the question carefully. The main requirement are the intercepts. Okay. The main requirement are the intercepts on the coordinate axis. Okay. So one by one, let's find out the intercepts. So can you do exponential graph? Yeah, that I will do with logs. Okay. Exponential or logarithm accepted stream. We will do that with functions inshallah. So x= to 2, x = to minus1 and then we have another x intercept which is equal to 3.

[00:17:50]Okay. Now even though it's a sketch but we will still try to make sure that we are as accurate as possible. So 1 2 3 and then we have minus one. Okay mock inshallah this Sunday I'm thinking to schedule a mock. Okay. Now let's find out the y intercept. How will I find out the y intercept? For the y intercept, we will plug in 0 in place of x. 0 - 2 that's - 2.

[00:18:18]Then 0 + 1 that's 1. Then 3 - 0 that's equal to 3. So 3 * -2 that's equal to -6. Okay. So y intercept y intercept is -6. So it's go mark roughly somewhere over here.

[00:18:36]Okay. Okay. Now what about the nature?

[00:18:40]The nature of this curve is x into x that's x² x² into - x. So that's - x cub - x cq means that we are looking at we are going to look at we are going to get a minimum to maximum curve. Okay we are going to get a minimum to maximum curve.

[00:18:57]So Sunday please mark cuz Monday is business Tuesday is bio. Oh okay. So let me see if I can do it tomorrow. Tomorrow will be a very short notice but so let me see if I can do it tomorrow.

[00:19:08]Now turning point is not exactly in the middle. It's somewhere in between.

[00:19:13]Let's mark the x intercepts. The x intercepts are minus1 2 and 3. Okay? And the y intercept is minus 6.

[00:19:24]What you guys should do is you guys should just shadow sketch. Shadow sketch means just hover your pencil so that you have an idea of what it is that you have to sketch. Okay? So just hover your pencil through the point so that you get an idea of the shape of the curve. Okay.

[00:19:42]So here I am just hovering my pencil over the points just adding the shape of the curve to my muscle memory gradually.

[00:19:51]Okay. Notice that the x intercepts are minus1 and 2 and somewhere in between -1 and 2 is -0.5. And I'll make sure that the curve turns somewhere around that.

[00:20:01]Sorry notus 0.5 positive 0.5. Okay. So minus one or two average. So that's positive 0.5. So we'll make sure that the curve turns somewhere around that.

[00:20:11]Okay.

[00:20:12]Y intercept. How do you find out the y intercept of any equation? Just plug in zero in place of x. You get the y intercept.

[00:20:19]Okay. So now that I've done multiple I've hovered my pencil over the point multiple times. I've made multiple shadow sketches. Now I'm going to go ahead and make the actual one.

[00:20:32]And that's it.

[00:20:34]job done.

[00:20:35]Okay, not complicated at all. How do you say x= to 0.5 for turning point? I did not say x= to 0.5 is exactly the turning point. I did not say that.

[00:20:49]Tell me if the curve that I'm drawing is correct.

[00:20:55]Is this correct? Yes or no?

[00:21:00]Do you think this is correct?

[00:21:03]mock after Eid.

[00:21:08]No, this is incorrect.

[00:21:11]This is incorrect. Okay, this is not right.

[00:21:16]This is incorrect, guys. What makes you say that this is correct?

[00:21:22]Three people are saying yes. No, this is incorrect.

[00:21:26]Look at where this curve is turning. Is this curve supposed to turn at the y intercept? No.

[00:21:32]What are the two x intercepts? The two x intercepts are minus1 and 2. Which means that it will turn somewhere in between the two x intercepts. Okay? Somewhere in between means somewhere around 0.5.

[00:21:45]Okay? Somewhere around 0.5.

[00:21:48]So the correct sketch would be this one right here.

[00:22:11]There you go.

[00:22:14]Go get Let's just do it one more time.

[00:22:24]Now it's better. Okay. compared to the previous one where we just turned our curve at the y intercept. That is wrong.

[00:22:34]Okay, that is wrong. So just make sure that you start finding out the x coordinate of the turning point and you make sure that the curve is turning exactly at that point. No, it's supposed to be somewhere around somewhere in between the two x intercepts. That's it.

[00:22:54]Okay, not exactly in the middle because these curves are not perfectly symmetrical. Let me show you up. They go.

[00:23:02]Look at this curve right here. -2 -1.

[00:23:05]Where is the curve turning? Somewhere in between. Somewhere in between the two x intercepts. Then look at minus1 and 5.

[00:23:12]Where is the curve turning? Somewhere in between these two x intercepts. Not perfectly symmetrical. Okay. Not perfectly symmetrical, but somewhere in between. Okay.

[00:23:23]All right. Now let's do another question.

[00:23:29]This is modus. We'll do this later.

[00:23:31]There's then there's this. Hence write down the values of x for which x - 2 x + 1 into 3 - x is greater than zero. Okay.

[00:23:39]So what's the main focus over here? The main focus over here is greater than zero. Meaning we have to write down the range of x where this curve is greater than zero. What's your question better?

[00:23:48]What is your question? Instead of saying just answer your question, it's best that you send your question again so I don't have to search for your question in the chat. Okay. So greater than zero means that which part of the curve do we want? We want the part of the curve which is above the x-axis. Okay. So whenever the question says greater than zero, greater than zero means that it wants the range of x where the curve is above the x-axis. Will we lose mark if you do that? Yeah. If your turning point, if the shape of the curve is inappropriate, you will lose marks.

[00:24:20]Simple as that. Okay, if that's what you're saying. So, this means that we are looking for the range of x for which the curve is above the x-axis.

[00:24:34]Okay. Now, let's identify parts of the curve that are above the x-axis. There's this part.

[00:24:40]Now, what is the range of x for this part? This is on the left hand side of minus1 which means that x is less than minus one.

[00:25:01]Here we put a comma. We do not write and we just put a comma and we say x is greater than 2 and less than 3.

[00:25:12]That's it.

[00:25:14]Okay. So I repeat this expression is the same as the cubic curve that we have just sketched. So that means the curve is in front of us.

[00:25:23]We don't have to do anything to solve this inequality cubic inequality. And that's the reason why the question says hence that means we have to use our sketch. Now it says x - 2 x + 1 3 - x is greater than 0. That means we want the range of x for which the curve is above the xaxis. Now what if it was less than 0? We will look at that later. So first thing you need to do is you need to identify the part or parts of the curve where it's above the x-axis. So here is one part. What is the range of x over here? Since since it's on the left hand side of minus1, it's going to be x is less than minus1. Then there is this part of the curve which is between 2 and 3. That means x is greater than 2 and less than three. That's it. Aa I'll look at your questions.

[00:26:07]Uh sir, how do you practice for paper one like in previous? Okay. If you're practicing for paper one, for paper one, what you need to do is you need to minimize the use of a calculator. If you're solving a paper prior to the syllabus change, like for example, if you're solving a paper from 2020 or 2021 or 2022, then do not use a calculator in every question. Use your calculator when you reach a point where you know that this question is designed to be done with a calculator. Okay? Because even prior to the syllabus change, there were majority of questions that could be done without a calculator. It would be nice to have one, but you could still do them without a calculator. So, minimize the use of a calculator until you reach a point where you absolutely have to.

[00:26:45]Okay.

[00:26:47]So, will you do streams of ASP1 again? I don't know. We'll find out. Yeah, I will stream functions. If question states to write stationary points though, we will differentiate of course. Will we lose mark if we do that? Do what? I don't know. So, how to practice for paper one?

[00:27:00]I've answered that. Said is doing too much questions bad. It seems like I'm memorizing them or should I stick to the concept? Of course, you should stick to the concept. But there are some things that if you memorize, you can end up saving some time. Okay. So, memorize only for the purpose of saving time.

[00:27:14]Make sure that you always prioritize understanding over memorizing.

[00:27:19]So, what do you expect? Great. I don't know. Great threshold. That's an irrelevant question. Total waste of time. P1 my calculator is not allowed in admat P1 calculator is not allowed guys should please do velocity vectors again okay I'll do velocity vectors again why is x is less than minus1 x is less than minus1 because the curve is greater than zero for as long as x is less than minus1 that is why it's less than minus1 okay why does the curve not turn in the center because this is a cubic curve this is not a quadratic curve quadratic curves are perfectly symmetrical Cubic curves are not perfectly symmetrical.

[00:27:58]Okay, so I hope I've answered all questions. If you still have more questions, then wait. Let me do one more question, then I will look at your questions again. Okay, let's do this one. On the axis below, sketch the graph of y = -1 upon 4, 2x + 1, x - 3 into x + 4. So guys, for as long as I'm solving this question, I'm not going to be looking at the chat. Okay? And I want you guys also to pay attention. If I can't multitask, then I'm pretty sure you guys can also not multitask. So pay attention.

[00:28:26]So first thing that we're going to do is we're going to find out the x intercepts. So always look at the amount of space that you're given. So we're given plenty of space but still let's make sure we utilize it efficiently. So 2x + 1 =0 which means x = -1 upon 2.

[00:28:40]That's 1 x one x intercept. x - 3 =0 which means x = 3. That's another x intercept. Then we have x + 4 =0 which means x = -4. That's another x intercept. Okay. To find out the y intercept, simply plug in 0 in place of x. So this is going to be 1.

[00:29:00]Don't forget the minus 1 upon 4. - 1 upon 4 into 1 into minus 3 into 4. Now bear in mind that this is a question which is prior to the syllabus change.

[00:29:09]But I'm still not using a calculator.

[00:29:10]Okay? And I will avoid using a calculator throughout so that I can practice for non-cal.

[00:29:15]Four and four cancel. Say -1 into -3.

[00:29:20]What's that equal to? That's equal to 3.

[00:29:21]So that means the y intercept is equ= to posit3. Okay. Now if you look at the coefficient of x cub so 2x into x that's 2x² 2x² into x that's 2x cq - 1 upon 4 into 2x cq is equ= to - 1 upon 2x cube which means that this will be a minimum to maximum curve. Since the coefficient of x cube is negative that means it's going to be a minimum to maximum curve.

[00:29:49]Okay. Now we've gotten all the ingredients. Now it's time to nicely put them together. Okay. So just having the right ingredients is not enough. It's about the order in which you mix them that is also very important. Okay. So we have all the ingredients. Now it's time to put them together. And let's put them together as nicely as possible. Okay.

[00:30:12]So like I said, even though it's a sketch, but still make sure that you try and be as accurate as possible. So let's begin by marking the x intercepts.

[00:30:22]One x intercept is -4. The other is min -/ and then there is -4 -/ and then there is 3. Take okay now then what about the y intercept? The y intercept is 3.

[00:30:42]Take say -/ pos3 -4 and the y intercept say okay now what kind of a curve are we sketching we're sketching a minimum to maximum curve up they go this curve will turn somewhere over here and then as far as these two x intercepts are concerned.

[00:31:15]Okay. So, first let's do some shadow sketching.

[00:31:22]Okay. So, here are some shadow sketches.

[00:31:30]I'm shadow sketch. We don't hit the sweet spot. So, I think we have hit the sweet spot. No.

[00:31:39]What you can also do is after you've made the sketch, if after that you feel like you want to tweak some values, you can do that also.

[00:31:48]Yeah. So I think we have hit the sweet spot. Now we've added the shape of the curve into our muscle memory. Now it's time to why will we not multiply all the x intercepts with 1 upon 4? When you equate it to zero, what happens to 1 upon 4? 1 upon 4 you will take it across the equals to sign 0 gets divided by 1 upon 4 doesn't matter whether it's positive or negative you will it will all it will just disappear just like you know when you take something common from a quadratic equation what happens to the constant that you factor out you take it across the equals to sign and then it just disappears okay all right now time to put this all together.

[00:32:35]That's it. Job done.

[00:32:37]Why will we not multiply? Okay, I've already answered that we do not multiply the coefficient of the x intercepts with one upon 4 equation equals to 0.

[00:32:48]When you let's suppose you have - 1 upon 4 into 2x + 1 into x - 3 into x + 4 equals to 0. When you equate it to 0, what happens when you take this constant across the equals to sign? It just disappears.

[00:33:13]Okay? And the reason behind it disappearing is because you do 0 divided by the constant and 0 / anything is equal to 0. Okay. So that is why we don't multiply the x intercepts by 1 upon 4. Reason why we multiply the y intercept with 1 upon 4 because when we plug in 0 in place of x, nothing happens to this constant. That's why we multiply all the constants together to get the y intercepts. Okay? And that's it. How many marks do we get? we get three marks. So, three easy marks. Not too difficult. Three easy marks. Okay. I should I said I have a problem in calculations. I know how to solve a question but I lose marks in calculations.

[00:33:55]The better fix your calculations. That's all I can say. I mean when you know what what is wrong then you know what to fix in math paper if we have just shown ratios and solved the rest in calculator and given the answer.

[00:34:11]But I don't know. Okay. I just by you explaining, I can't really tell. It depends on the question. There are so many things that I don't know, which is why I can't answer that question.

[00:34:22]If we already know the y intercept, then what's the need to decide if it's minimum? A designer? That's a good question. If you know the y intercept and if you have the x intercepts, then you're absolutely right. There is only one shape of the curve that you can draw. But if you're able to figure that out, that's great. But I'm afraid not a lot of people will be able to figure that out. Okay. But yeah, that's that's great and that's actually very smart.

[00:34:45]How many mocks can you conduct? I'm thinking maybe three mocks each. Let's see. Okay. Okay. Let's do another question. 5 into x + 1 into 3x -2 into x -2. Okay. With which chapters to prioritize? Again, don't be lazy. Solve enough yearly past papers. You will figure out which topics are important, which topics carry more marks and which topics don't. Okay?

[00:35:12]If you have done enough past papers by now, you would know exactly which topics carry more weightage, which topics don't. Okay? If you're looking for shortcuts, if you're looking for just doing the bare minimum, then I'm afraid I can't help you. Okay? Okay. So, here we have a cubic equation once again. So, I'm working to short because we already know how to find out the x intercepts. We already know how to find out the y intercept. Let's shorten our working as much as possible. Okay.

[00:35:43]So when we equate this whole thing to zero, we will get one x intercept to be minus1. We will get another x intercept to be equal to 2 upon 3. And then we will get another x intercept to be equal to 2. Okay. Now let's find out the y intercept. For y intercept, we're going to plug in 0 in place of x. So 0 + 1 is 1. 3 * 0 is 0. 0 - 2 is -2. Then once again 0 - 2 is -2. And don't forget to multiply it by 5.

[00:36:12]Okay, so 5 into 1 is 5. 5 into 4 that's equal to 20. So we have the y intercept also. Okay, now we have the x intercepts. We have the y intercept. Now what is the shape of the curve that we're looking at? x into 3x is 3x². 3x² into x is 3x cub. So 3x cub the coefficient of x cub is greater than zero. That means we're looking at a maximum minimum curve. Okay. Now, so here's minus one, here is one, and here is two. So, let's space it. Let's space it out a little.

[00:36:51]Okay. So, let's mark one over here.

[00:36:54]Let's mark two over here because the values are close to each other. So, let's space them out a little. Okay. So, let's mark the x intercepts. Here is the 2 upon 3 mean 0.6. So that's going to be somewhere over here. Okay. And then you have two which is going to be somewhere over here. What's the y intercept? The y intercept is 20.

[00:37:19]Okay. So the y intercept is 20. That means somewhere over here is the y intercept 20. Okay. Okay. Now remember we are making a maximum to minimum curve. That means the curve that we're drawing is going to be something like this. Okay? The curve that we are drawing is going to be something like this. Okay? In fact, exactly like this muscle memory.

[00:37:52]Let's go ahead and sketch the actual thing. Okay. So, let's put the actual thing together now.

[00:38:00]And please make sure that you sketch using a pencil.

[00:38:14]There you go. Job done. S1, there's no S1 stream today. I will do an S1 stream tomorrow inshallah. Tomorrow I'll try to do multiple streams of S1 since we're short on time and we have to complete the syllabus. So inshallah I'll try and do multiple S1 streams. Okay. Now we have a cubic inequality question. Hence find the values of x for which 5 x + 1 3x - 2 x -2 is greater than 0. So what's important here is the fact that it's greater than zero.

[00:38:40]Turning point they go -1 or 2 some positive value. That's why I made sure that the curve turns over here.

[00:38:50]Here also I will make sure that the curve turns somewhere in between the two x intercepts. Okay? I'm not going to waste my time trying to calculate the turning point. I will just make sure that it is in between the somewhere in between the two x intercepts. That's it.

[00:39:13]And then we label the intercepts. That's a good idea.

[00:39:17]So that we first sketch the curve and then we label the intercepts. But make sure that you don't lose track of what the x intercepts are and you also don't lose track of what the y intercept is. Okay. So let's do that. It's a good suggestion.

[00:39:45]Okay. So what's the y intercept? The y intercept is 20.

[00:39:51]What is the first positive x intercept?

[00:39:53]2 upon 3 then negative x intercept minus one then the next x intercept 2 okay that's also a good idea so isn't the nature 15 x cube even if it's 15 doesn't matter you're right it's 15 x cq the idea is that it's positive okay what's important you're right it's 15 that's absolutely true but the fact of the matter is that the coefficient of x cube is positive which means that it's a maximum to minimum curve okay so in advance if you do one part incorrect.

[00:40:24]No, no, you do get something known as follow through, but it it it's not that it applies to every single question.

[00:40:29]It's where you use the answer to the previous part to answer the following part. Okay. Okay. Now, set of turning point in positive x then to let it be in positive. If if in this curve you make the curve turn on the positive side, then that is wrong. Okay?

[00:40:47]Then that is wrong. Okay? It has to be somewhere in between the two x intercepts.

[00:40:54]I don't see why that is so complicated.

[00:40:56]I I honestly don't understand.

[00:41:06]These are our next pair of x intercepts.

[00:41:07]We're going to make sure that it turns somewhere in between.

[00:41:13]instruction though. I don't see why this is such a big deal. Just make sure that it turns somewhere in between the two x intercepts. And if you're doubtful, what will be in between the two x intercepts?

[00:41:26]Then just use your calculator, find out the midpoint. And then make sure that it turns not exactly at that point. Okay, no need to make sure that it turns exactly at that point. That won't even be correct. But somewhere around that point. That's it.

[00:41:39]Okay. Now, what do we want? We want the range of x for which the curve is greater than zero. Okay, we want the range of x for which the curve is greater than zero. Greater than 0 means we want the range of x where the curve is above the x-axis. Now let's identify that range.

[00:42:00]So that's this range which is in between -1 and 2 upon 3. Okay.

[00:42:09]X says sir isn't the question asking for values then range it is asking for values but the question is an inequality so answer has to be in range form so x is greater than minus1 and less than 2 upon 3 okay so that's one part of the curve so we put a comma and then the next part of the curve is this where the curve is above the x-axis that is greater than 2 so we put a comma and we say x is greater than That's it. Take okay now where we are given the equation where we're given the curve and we have to find out the equation. I will try to cover at least all the important topics before 18th of May in two x is greater than two. Why not?

[00:43:06]A okay.

[00:43:11]Yeah. Sorry. X is greater than -1, less than 2 upon 3 and X is greater than 2 upon 2. So yeah. Okay. X is greater than -1 and X is less than 2 upon 3 and then X is greater than 2. Okay. Thank you for pointing that out. All right. Now let's circle back to the very first question.

[00:43:31]So now I'll give you guys a minute to first try and figure this out yourself.

[00:43:37]0580 because 0580 is done.

[00:43:43]0580 is math which is done. So when do we use greater than equal to or less than equal to? If in the inequality like for example here the inequality by default is less than or equal to zero then when you're giving your answer you will also write it as less than or equal to zero.

[00:44:07]mock of admats I'm thinking this Sunday inshallah till I solve this I'll give you guys a few minutes then I'll be back on my Okay. Now, so find an expression for f ofx.

[00:46:49]They go. There's a pretty standard way of solving this and I would advise you guys to do the same.

[00:46:56]So here's what we do. First of all, we look at the x intercepts. Okay? So the x intercepts are -2, -1, and 5. Now we reverse engineer x = -2, x = -1 and x = 5. Okay. Now we're going to reverse engineer S1 stream. Inshallah we will do it tomorrow. Okay. So this now becomes x + 2 into x -1 sorry into x + 1 not x - 1 into x - 5.

[00:47:31]Okay. Now we still don't know what is the coefficient of x cub. So we're going to put a k over here and we're going to say that this is y= to y= to yeah f ofx= to see okay so we're going to say that this is y= to ke f ofx= to ke that's up to you now in order to find out the value of k what you want is another point through which the curve is passing okay what you want is another point through which the curve is passing other than the x intercepts. Okay, other than the x intercepts because the x intercepts are not going to help. Are we given such a point? Yes, we are. Notice that there is a point that we're given which is 0a 5.

[00:48:19]Okay. So, what we're going to do now is we're going to plug in five in place of y and we're going to plug in 0 in place of x. So 0 + 2 becomes 2. 0 + 1 becomes 1. And 0 - 5 becomes - 5.

[00:48:36]Okay. So what do we have? We have - 10 k which is equal to 5. Which means k is equals to - 1 upon 2. Now what is the final equation? The final equation is f ofx equals - 1 upon 2. Okay. And then all of this as it is.

[00:48:59]There we go. And make a habit of making a box around your final answer.

[00:49:12]There we go. Okay. Now then the question says, solve f ofx is less than or equal to z. Notice the question did not use the word hence. That means you don't really need your answer to solve this.

[00:49:24]You can still solve it. So lesser than or equal to 0 means that you need the part of the curve which is below the x-axis. Okay. So lesser than or equal to 0 that you are looking for the range of x where the curve is below the x-axis.

[00:49:39]And in your answer when you're writing down the range you will also include the equal to sign. That's it. Okay. So this is the only way you figure out when to include the equal to sign and when not to. So below the x-axis there is this part and then there is this part. So for the first part, x is greater than or equal to minus2 and lesser than or equal to 1. So that's one part of the curve. And then you put a comma. The other part of the curve where it's below the x-axis is greater than five. So we're going to say greater than or equal to 5. So x is greater than or equal to 5.

[00:50:19]There we go.

[00:50:22]Okay.

[00:50:23]So that's it. Job done.

[00:50:27]Now let's do another equation. Minus one.

[00:50:34]Ohus.

[00:50:48]Okay, let's do this one.

[00:50:51]I know I know it was minus one. I fixed it. Find the set of values of x for which f of x is less than 0. Okay, less than 0 means this part and this part.

[00:51:04]Very similar to the previous question. X is greater than minus 3 less than 1. And then you have x is greater than 5.

[00:51:16]That's it. Okay. Find an expression for f ofx. Okay. So let's reverse engineer.

[00:51:21]Let's start from the x intercepts. The x intercept is minus3. Then another x intercept is 1. Then we have another x intercept which is equal to 5. Okay. Now let's reverse engineer. Let's double check - 3 1 and 5. Yeah. So now when we reverse engineer we get x + 3 into x -1 into x -1 into x - 5. Okay. But this is not complete. Okay. So this is f ofx but it's not the complete f ofx because we don't know the coefficient of x cub yet.

[00:51:55]So we're going to put a k over here.

[00:51:57]Okay. Now for which we need a point through which the curve is passing and we have just that and there's very good reason why the question has given us another point explicitly as 0 and minus 5. So let's plug in 0 in place of x.

[00:52:09]Let's plug in -5 in place of y. So -5 is equals to k 0 + 3. What's that equal to?

[00:52:16]That's equal to 3. 0 - 1 what's that equal to? That's equal to -1. 0 - 5 was that what's that equal to? That's equal to -5.

[00:52:28]Okay. So - 5 - 5 cancel though k is equals to - 1 upon 3. Basically 1 is = - 3k which means k is equals to - 1 upon 3.

[00:52:50]Okay. Now, now that you have the value of K, the equation is now complete. This is what it was before we figured out the value of K. But now that we know that the value of K is - 1 upon 3, here it is the complete equation.

[00:53:06]There we go. Okay. So, let's see if we can find another question. This modus related questions we are k is equals to minus 1 upon 3 beta. It's a minimum to maximum curve. A minimum to maximum curve should have a negative coefficient of x cube. Okay. A minimum to maximum curve must have a negative co oops negative coefficient of x cube. And if it's not negative, that means you've done something wrong.

[00:53:44]If the examiner says the point passes 1A 5, what do you mean the point passes 1A 5? You mean the curve passes through the point 1a 5.

[00:53:58]Correct?

[00:54:02]All right, let's do this one.

[00:54:09]Yeah, you you substitute 1a 5 they go to find out k. You just need any point other than the x intercept to find out the coefficient of x cube.

[00:54:21]Okay, you just need any point other than the x intercepts.

[00:54:29]Okay, solve. The diagram shows the graph of the cubic polomial y= to f ofx. Find an expression for f ofx in factorized form. Write each linear factor with its coefficients as integers. Important that you have to write its coefficients as integers. Now let's reverse engineer.

[00:54:51]Let's reverse engineer. So we have one x intercept which is -1 upon 3. Then we have another x intercept which is 1. And then we have another x intercept which is 5 upon 2. Okay. Now what must have happened before we got these values of x? This must have been 3x = -1. This must have been x -1. This must have been 2x = 5. Which means before all this it must have been 3x + 1 into x -1.

[00:55:23]Okay. into x -1 into 2x - 5. Okay, so this is what it must have looked like before we got the values of x. Okay, now we still haven't completed the equation.

[00:55:36]This is still f of x. Okay, now we have to find out the coefficient of x cube.

[00:55:42]Okay, we have to find out the constant with which the entire equation is being multiplied. Now for that we need a point through which the curve is passing and that point is 0, -15. any point other than the x intercepts. Okay.

[00:55:56]So let's plug in -5 in place of y. Let's plug in 0 in place of x. So 3 * 0 is 0.

[00:56:02]0 + 1 is 1. 0 - 1 is - 1. 0 - 5 is -5.

[00:56:12]Okay. So do we find the constant? Yes, of course you find out the constant in every question. So this becomes positive and k is equals to minus3.

[00:56:25]Okay. So now we complete the equation with the value of k that we have found.

[00:56:30]A you should be a few steps ahead of the question. This is a minimum to maximum curve. When do you get a minimum to maximum curve? What should be the coefficient of x cube that will give you a minimum to maximum curve? It has to be a negative coefficient of x cube. Okay.

[00:56:45]So now you should know you should anticipate a negative coefficient of x cube. And does minus3 make the coefficient of x cube negative? Yes, it does. Minus3 does make the coefficient of x cube negative which means this is correct. Okay?

[00:57:00]Because we are looking at a minimum maximum curve which like I said you only get when the coefficient of x cube is negative. Then it says write down the values of x such that f ofx is less than zero. Again less than zero means that it's below the x-axis.

[00:57:16]Below the x-axis. Now what is the range of x where the curve is below the x-axis? Between - 1 upon 3 and 1. So x is greater than - 1 upon 3 and x is less than 1. That is 1. And then you have x is greater than 5 upon 2.

[00:57:39]There you go. No, you cannot write x intercepts as 1 upon 3, minus 1 and 5 upon 2. Why? Read the question. I guess you did not read the question carefully.

[00:57:51]Normally you can but here you cannot.

[00:57:54]Why can you not do that? You cannot do that because because because because it says write each linear factor with its coefficients as integers.

[00:58:04]So the linear factor should have coefficients as integers.

[00:58:12]Take there we go.

[00:58:17]I think we have done this tricky.

[00:58:30]Okay, let's do this.

[00:58:45]I have a class anyway. I need to pray.

[00:58:48]So, let's do I think we're not going to do that. We're not going to do this.

[00:58:58]I have a class at 6:30. So, let's do two more questions. Let's do this one. Let's do one more and then we will end. Okay.

[00:59:06]any stream tomorrow. Tomorrow is Saturday. Tomorrow maybe I'll take a mock because you guys are saying the Monday you guys have multiple exams. So maybe tomorrow I'll take a mock of admats.

[00:59:17]Let's see.

[00:59:19]Okay. So let's this is the equation right here.

[00:59:26]Okay.

[00:59:28]So the x intercepts are -2 -0.5 and 1.

[00:59:33]Okay.

[00:59:36]So let's find out the equation. x = to -2.

[00:59:40]Then we have x = -1 upon 2. Okay. Since this is -0.5 and then we have x = to 1.

[00:59:47]Now before x before we got x= to -2, this must have been x + 2. Before we got x = -/2, this must have been 2x + 1.

[00:59:56]Before we got x = 1, this must have been x -1. x + 2 2x + 1 x -1. Yeah. Then don't forget that we want the coefficient of x cube. So notice that the coefficient of x cube as per the given equation there isn't one. Okay. So we're not going to waste our time finding out the coefficient of x cube.

[01:00:16]Okay? You can see from the equation that's written that it's one. So we're not going to waste our time doing that because we already know that k is equals to 1. Okay? If you want you can plug in the value and then figure it out. But you can see from the equation since part of the equation is already given and you can see that there is no constant with which the entire equation is being multiplied. So that means it's equal to one. Let's write it the way the question has written it. It has written it as 2x + 1.

[01:00:44]Okay. Then you have x + b x + c. So that means x + 2 and then x -1.

[01:00:53]Okay. So a is equals to what? A = 1, B = 2 and C = -1.

[01:01:02]You can have A= 1 and B = -1 or and C= to 2.

[01:01:12]There's no hard and fast rule.

[01:01:15]Value of A B or C.

[01:01:20]Value of A must be fixed. Value of B and C can interchange. Use the graph to find the values of x for which y is greater than or equal to 2. So use your graph to find the values of x for which y is greater than or equal to 2. Use your graph means that we have to first draw a line at y = to 2. Okay. Now y is greater than or equal to 2 means that we want we're going to include this part of the curve. Okay. This is where the curve is greater than 2. And then this is also another part of the curve where it's greater than two. Okay. So we want the range of x for which the curve is above the line. Okay. So this is the curve.

[01:01:56]This is the line. We want the curve above the line. Okay. So that's the range of x that we're looking for. We're looking for the range of x for which the curve is above the line. And because the curve is already drawn, there will be little to no margin of error over here.

[01:02:12]Okay? Because curve already. So there will be little to no margin of error.

[01:02:18]Okay.

[01:02:22]So this is -1. That's quite clear. This is - 1.5 - 1.6 - 1.7. Now since this is in between - 1.5 and - 1 - 1.6 and - 1.7 beach. Let's just double check make sure that we've done it correctly.

[01:02:47]So -1 - 1.1 2 no wait a minute 2 4 6 right - 1.5 - 1.6 - 1.7 so we're going to go with -1.65 okay so this is as per my graph - 1.65 65. Okay, I'm not going to be too casual about it. I will make sure that I keep it as exact as possible.

[01:03:13]We'll have to compromise because this is in between 1.1 and 1.2, but it's not exactly in between. It's actually very close to 1.2. So, we will take this as 1.2 only.

[01:03:26]Okay, we will take this as 1.2 only. But this since this is exactly in between the two x values - 1.6 and - 1.6 7 we will take this as min - 1.65 and this is pretty exact already this is minus1 okay now so what is the range of x for which the curve is above the line we have x is greater than or equal to -1.65 65 and lesser than or equal to minus one. Okay, so that's one. And then what do we do? We put a comma and then we say x is greater than or equal to what is this? This is 1.2 1.175 will be way too much. Okay, we we're not that accurate.

[01:04:22]Okay, as per the grid that's given to us, we can't really anticipate that it's going to be 1.175.

[01:04:28]But again, we try and be as accurate as possible. So there you go, that's the answer to part B. Okay, sir. If question is asking specifically for turning points though, we will differentiate.

[01:04:41]Yep, that is correct. Sir, will you do all topics for P1? P1 P1. What do you mean? You mean add maths paper one or a level math paper one?

[01:04:57]So if the question said total length of all edges in cuboid I said total length of all edges in the net as well but I don't know usually avoid doing that because I might answer it incorrectly and then people will start panicking.

[01:05:16]Add math paper one. Yeah, I'll do a syllabus run through. Don't worry sir, this is a three mark question. So don't we substitute the value of y2? No, no, you do not substitute the value of y2.

[01:05:25]Use the sorry read the question. Con read the question. Notice it says use your graph. If it says use your graph, then obviously you're going to use your graph.

[01:05:35]Why would you not use your graph? So finding domain and range of composite function. No, that they go only a part of it is part of her syllabus. And what part is it? We will solve. Okay, we will do that. Don't worry.

[01:05:48]Okay, let's solve this.

[01:05:51]The diagram shows the graph of y= to h of x where h of x = to x + a. The whole thing square b + cx and a, b, and c are integers. The curve meets the xaxis at the points - 2, 0. That's this right here. And then there is this which is 1.5. Find the values of a, b, and c.

[01:06:14]Let's find the equation of this from scratch. Okay, let's find the equation of this curve from scratch.

[01:06:20]So we can see that we have the curve at tangent to the xaxis when x = to -2. So that means if x is equ= to minus2 this must have been x = -2 x= to -2 x= -2. So wherever the curve is tangent that is where you have a whole square. So this must be x = -2. So before we got get this before we got x= to -2 it must have been x + 2 the whole thing squared. Okay. So then there is another x intercept which is x = 1.5.

[01:06:57]Now x = to 1.5 basically means that x is equ= to 3 upon 2. Okay. And x= to 3 upon 2 basically means that before it became x = to 3 upon 2 it must have been 2x - 3. Okay. So we have 2x - 3. Why square?

[01:07:17]Because the curve is tangent there. Why do we put a square here? Because the curve is tangent. So wherever the curve is tangent, that is where the square is going to be.

[01:07:27]Okay? So wherever the curve is tangent, that is where the square is going to be.

[01:07:34]Notice here the curve is tangent to the x-axis. So we're going to have a square over here.

[01:07:41]Okay? curve is tangent. So we have a square. Okay.

[01:07:47]Okay. Now equation.

[01:07:52]So what were we supposed to do? Again you can see that there is no constant with which the entire expression is being multiplied. So that means this is our final answer except that we're just going to write it nicely.

[01:08:06]Okay. Except that we're going to write it nicely. So h of x is equals to x + 2 the whole thing squared. So that we're going to keep as it is. And then instead of writing it as 2x - 3, we're going to write it as - 3 + 2x.

[01:08:25]Okay? Instead of writing this as 2x - 3, we're going to write this as - 3 + 2x.

[01:08:31]Okay? What does the question say? The question says find the values of a, b, and c. So if I compare this now with A, B and C problem here problem identify.

[01:08:53]No, you can't write X - 1.5. You're not reading the question. You can't write X - 1.5. Read the question. The question in the question, the coefficient of X is not equal to one. There is a coefficient.

[01:09:05]There is a problem, guys.

[01:09:08]Is this the appropriate equation? Yes or no? Is this the appropriate equation as far as the curve is concerned?

[01:09:25]No, it's not. This is not the appropriate equation. Why is this not the appropriate equation? Okay. The reason why this is not the appropriate equation is because the nature of the curve here is minimum maximum for which the coefficient of x cube should be negative whereas here we have the coefficient of x cube which is positive. Okay.

[01:09:52]Or you can say the y intercept does not match. Okay.

[01:09:59]If I find out the y intercept using the equation that we have found.

[01:10:04]So that's going to be 2² into minus3 which is equal to what? Which is equal to -12. And that is incorrect. Our y intercept should be pos2. Okay. We have two options. We have two options.

[01:10:17]We can take minus common from here.

[01:10:22]Okay. What can we do? We can take minus common from here. Okay.

[01:10:27]or or that's one option or what we can do is we can find out the y intercept.

[01:10:40]Okay, we can find out a constant with which we need to multiply the entire equation.

[01:10:48]Let's find out the constant with which the entire equation gets multiplied.

[01:10:54]So make sure that you use the space efficiently.

[01:11:00]We need a point through which the curve is passing. We are given that point that is 0a 12. So let's plug in 0 in place of x and 12 in place of y. So we have 12 which is equal to k * 0 + 2. So that's 2 ^ 2 and then - 3 + 2. So that's equal to - 3 + 0. So that's equal to - 3. Okay?

[01:11:22]So 22 is 4. 4 into - 3 is -12. So that means k is equ= to -1. This means that the final equation looks like this. H of x = -1 into x + 2 the whole thing squared. Now I know some of you have already anticipated k value minus1 and you don't plan on doing all this working. Okay?

[01:11:46]But if you didn't anticipate then this is the way out. Okay. Now what am I going to do with this minus one? I'm not going to keep this minus1. I'm going to multiply this minus1 with this linear factor. As you can see in the equation, there is no minus1 outside. Okay, in the equation that we have over here, there is no minus1. So, we're going to multiply minus1 with the linear factor.

[01:12:07]And then in the end, the equation that we end up with looks like this.

[01:12:11]X + 2 the whole thing squared. And then if you multiply -1 with 2x - 3, it becomes 3 - 2x. Okay. And now you can safely conclude the values of a, b, and c. a is equals to 2, b is equ= to 3 and c is = to minus2. Yes, beta, I will do functions. I can't tell you when, but I can tell you that I will do them. So stay tuned. I whenever I'm doing them, I will announce on my Instagram. Okay.

[01:12:40]Okay. Now I'll summarize this because it's a tricky one. That's the reason why I wanted to solve this. It's a tricky question. Take.

[01:12:48]So when we start we see that this is a regular question. Okay, we're given the x intercept where the curve is tangent.

[01:12:55]So wherever it's tangent you put a square there and you have another proper x intercept which is 1.5. So that means before we got x= to -2 it must have been x + 2 the whole thing square y square once again that's because it is tangent.

[01:13:09]Okay. And then you have another x intercept which is 1.5. So before we got x = to 1.5 which is x = 3 upon 2. it must have been 2x - 3. Okay, now you shift all the terms towards the left hand side and you have the two expressions. But wait a minute, you might think that this is the answer, but this is actually not the answer. Okay, and you have to be vigilant. Why? Because the coefficient of x cube in this expansion is positive, which means that it's not going to give us the curve that we see. So there is something wrong. So what we do is we find out K which is the term with which the entire equation gets multiplied and that is minus1. And then we see that they haven't written that constant separately here. They haven't written minus1 separately over here. So that means what are we going to do with this minus1? We're going to multiply the minus1 with this linear factor. So when we multiply the minus1 with the linear factor, this is what it looks like. And now does it look like the equation that is gi that was mentioned in the question? Yes, it does. And now all we have to do is compare it and find out the values of a, b and c. Okay, I know better it's not + 2, it's -2. A is equals to 2, b is equals to 3 and c= to -2. So then finally it says use your graph to solve the inequality h of x is lesser than or equal to 9. So what we need to do now is we need to draw a line at 9. And h of x is lesser than or equal to 9 means we want the part of the curve where it's below the line. So first find out the x intercepts known as the critical values. Okay. The point where the curve and the line intersect. These are your critical values.

[01:14:41]Okay. So they are minus3 minus1 and 1. Okay. Now remember what do we want? We want the curve below the line.

[01:14:56]Okay. So what's the range of x where the curve is below the line? That's from -3 to -1. So that means x is greater than or equal to -3 and x is lesser than or equal to -1. And then there is x is greater than or equal to 1.

[01:15:24]And that's it. Three easy marks.

[01:15:32]Oh, - 0.5. Yeah, it's - 0.5. Sorry. - 1 upon 2 physics. Parallax error.

[01:15:44]Correct. This is something that's called paral.

[01:16:02]Sir, I'm really good at maths. Probably going to get an A star. The better. I don't see a question.

[01:16:07]What's your question? Practice admiration of math P1, I wrote the value of a plus minus3.

[01:16:25]I don't know. I don't I didn't make the paper, so I don't know. I don't know exactly what the question was like, so I can't say anything. But what's done is done. There's no point in crying over it. Okay. Okay. So, here we are.

[01:16:42]I will encourage you guys to practice the remaining questions. I've shared this worksheet with you guys. So, we will solve this worksheet later on. For example, when we're doing all the modus related questions, modus equations and inequalities that we will do these questions. Now, I will encourage you guys to solve the remaining questions.

[01:16:58]Okay? But for now, this is it. Another stream inshallah. Okay, I'll summarize.

[01:17:04]We will have our next stream. I can't tell you when. I can't tell you which topic. But what I can tell you is that I will summarize. It's not going to be a surprise stream. I will summarize and I will announce whatever stream it is that I'm doing on my Instagram and I'll post on YouTube also. Okay. But the topics that I will be covering, I'll tell you those. I'll be covering APGP, I'll be covering binomial, I'll be covering functions, trigonometry, differentiation, integration, vectors, chynomatics and functions if I didn't mention that already. Okay. So, ESR topics cover.

[01:17:34]So, yeah. Now, where were we? Use the graph to solve the inequality h of x is lesser than or equal to 9. So you draw a line at 9 lesser than or equal to 9 means you want the range of x where the curve the values on the curve are lesser than the values on the line. That means the values of the curve are less than 9 or in other words the curve is below the line. So one range of x where the curve is below the line is this from minus3 to -0.5 and the other range of x where the curve is below the line is when x is greater than one. Okay? So there you go.

[01:18:08]That's your answer.

[01:18:13]Okay. So that's it fellas. We will keep it till here. Only the next class I will be doing some other topic and whatever topic I'm doing I like I said I'll announce on my Instagram. Okay. So that's it for today's class. Take care everyone. Have a nice weekend. Allah hop.