Here's the six different kinds of work the first one the first working genius is called wonder Okay, most people don't even think this is a working genius This is up at 50,000 feet by the way And this is somebody that loves to naturally my wife does this loves to sit around and ponder things and think about things and ask Questions like why are things like that? Is this really the best it can be? Is there another way to do this that is an absolute genius? God gave some people the ability to be great at that and other people never do it Johnny's blushing That per hour conversation before this guy had started I was talking about a lot of questions that I was pondering as of lately through out the last few years Because that is the natural state that I'm constantly and asking questions Looking for answers pondering it in my own life And I'll go out on a run and then I will come back by AJ. I got I got some ideas. I got to run by Yeah, I gotta be sitting down for this I'm gonna tell you something I was walking out of the house today We she and I were exercising and praying and talking and I was walking out of the room and she goes Oh, wait, wait. Hey, I have a question for you and I said hey Is this one of those wonder things because if it is probably do it later because I gotta go She was like, yeah, it is and so we okay. We'll talk about it later Because you're not always in the right time to wonder But it's critical in every activity. It's also okay to say okay I can set my wonder aside for a moment my favorite one Which is the second genius which is called invention There are people who wake up every morning and love to come up with new ideas Even if it's not necessary That's what I do And it's it's a god-given talent and it's not more important than the others people think that inventors are gene Everybody's a genius, but I like to come up with new ideas And so I have a ton of them and anytime somebody comes to me with a problem I love to solve it and I love to do it in some unique Novel way and I like not having a lot of context. It's like, hey, let me just take a shot at that So invention is really important and these first two wonder and invention are what's called ideation This is where new ideas come from. So that's the first two Wonder and invention, but you can't stop there because after invention comes the next one, which is called discernment This is people that have good gut feel instincts intuition. They're good at pattern recognition and what you call integrative thinking It's not based on expertise specific knowledge or data. They just have a good gut Tracy a woman in our office who I've worked with forever She said even as a child people would come to her and ask her for advice And my wife will say to me it I'll say to my wife. Hey, should we write refinance our home? Or do you think we should go on vacation here? Or do you think this looks good at what I'm wearing and she'll say the same thing no matter what I ask her Ask Tracy Tracy has great instincts And everybody knows it and it's one of those skills It's a it's a true talent when an inventor comes up with a new idea You take it to somebody with great discernment and they're they can immediately look at it and go I think this is really great. This is really promising or Oh, my gut tells me there's a whole bunch of problems here and they are good at giving feedback and vetting an idea And helping tweak it. And so the inventor and the discerner talk back and forth Well, if you don't know that's your skill We had a guy write to us and say I thought my wife didn't like me and we were like what and he goes I'm really mean it. He laughed. He goes he because we finally talked to me said I thought she kind of hated me We're like why he goes I come up with new ideas all the time and every time I do She tells me what's wrong with it. I thought she was just trying to Bust my you know, whatever on their anniversary. They took the assessment She was a discerner. He was an inventor And she said no, that's how I help you honey And he was like so you don't you just don't want to crush my dreams He goes no I want to make sure that would actually work And he said that's the day they went from thinking we were working against each other To we were actually just doing what we're naturally best at after that comes galvanizing. That's some people love this I don't and I was doing this almost every day at work. And that's how this whole thing came about I was frustrated because every day I'd come in work ready to invent and I was constantly galvanizing Galvanizing is people who wake up in the morning and love to rally the troops keep them pushed forward Keep inspiring them reminding them. Hey, let's do it. Come on. We can do it now I can do it once maybe twice, but I don't love doing it I like to move on to the next thing and there are people there are galvanizers out there that just love that process And it is an absolute inherent god-given genius And some and it's like their best thing in the world to do one man's trash is another man's treasure galvanizing Those middle two we call activation. That's how you take the ideation and figure out Is this the right thing and how do we get people on board? The last two are called implementation and that is the next genius is Enablement now. I know people we struggled with this word because people said well, that sounds like drug addiction or alcoholism We're like no enabling somebody to do something good is really good And there's some people that are naturally good at hearing somebody say I really want to do this and they say Ooh, let me help you. What do you need? How can I be of assistance to you? It's not because they're nice It's not because they're easily manipulated people that have enablement as a genius tend to think well It's not a genius. I'm just helpful. It's like no no no Some people wake up every morning and say if somebody would just ask me to help them I would be the happiest person in the world. Let me know what you need and I will do it I'll probably do a little bit more I'll look around the corner for you and I'll anticipate your needs and they just naturally do it now I should say this though Everybody has to do things they don't love sometimes so sometimes I just got to stand in the corner and do What she asked me to do None of us get to go to work or be at home and do things that we don't love But we don't want to put ourselves in a position to do it a lot We certainly don't want to be in a position to live in our frustrations because that leads to burn out Really difficult things in life And god did not make us to be miserable in our work or in our lives. He gave us talent so we could use them It's wonderful and they're all needed They're all needed. So the last the last genius is beyond enablement It's one thing to want to help people. It's another thing to want to finish things. This is called tenacity I haven't have none of this. These are people that wake up in the morning and say, please let me finish something I want to push through obstacles. I want to wrestle things to the ground I want to prove that it can be done and I want to make I want to do it on time. I want to meet the standards I want to say yes We finished so it goes like this wonder at a high level asking the question invention solving the problem in a unique way discernment Evaluating it and making sure it's right galvanizing getting people on board enabling coming alongside and helping and tenacity finishing There is not a project in the world doesn't need all of those things All human beings have talents that are necessary Find out what they are so you can do the things you're meant to do and your success and your joy Will be greater than ever


He actually got behind me. No, I couldn't block the simple overhead. That was fast. He's got loops He's proven it before but he is consistent with them. That is that is really a hard to do I Will never stop being impressed by base Vegeta. How's it going everybody? Dachodoi here back with some more Dragon Ball fighters ranked matches this time playing a team of Broly Krillin and Yopcha And specifically we're going for some new Broly stuff in the corner Thanks to the season 3 patch some frames have changed so that when you use your barrier in the corner You can actually set yourself up for a safe jump with your heavy, which is amazing I should add on that the barrier is kind of just there it makes it a little harder to see it helps time the safe jump You know things like that, but really I just like going for also this guy's playing team top tier Top tier no remorse is what I call this team now. You could argue that third slot You know Vegeta Bardock and then the third slot can be whoever, but I'm up for my money Trunks is so good this patch. I've said it before I'll say it again. I'm a huge Trunks fan and I definitely have been mating him since day one, so if I start mating him nobody If I start mating him we have to admit that uh, you know, I'm just a huge Trunks fan All right, do I guard cancel? I'm gonna do it. Yeah guard cancel right there. Make sure we're not getting hit by That was a great. I didn't think he'd click the light there. I thought he'd be afraid paying gloobs okay Base Vegeta is nasty You guys remember when base Vegeta was better than he is now? Well, I don't know if better was the word But you remember when he could vanish after his knee and it would be true That was crazy. Those were crazy times We're gonna spark here if he tries to put us into any sort of mix-up scenario Yeah, we out That was great. That was a great angle. This is uh, these are all trans thing players We're dealing with now we're where we've moved up from uh, Super Saiyan Rose to Pride troopers So the average like the average opponent we get I Ironically it's still all over the place, but it seems to cut out and don't on it I didn't even though he down heavied. I didn't think that one would hit That was that honestly looked insane. All right, Broly's back out. We lost spark. Oh, let's go All right, what do I want to do here though? Do I just want to go for the grab? I think we do right? Now let's throw a level three on there We've got to start making some progress somewhere. So if we can maybe get Bardock, maybe he doesn't spark He gets hit with something Like right there, that would be great. Oh, where you going trying to click those buttons Oh, he actually got behind me. No, I couldn't block the simple overhead. That was fast He's got loops. He's proven it before but he is consistent with them. That is that is really hard to do I will never stop being impressed by base Vegeta. I I mean I can play him and you can do you can do decently with him even without knowing the loops But the loops just make every hit hurt if they if you got him in the corner Those loops are going to turn some some scenarios into Crazy wins for you We're not guard canceling again. It's way too much.   Thank you All right, we got a dragon rush to connect.  I thought I I told you guys I I make that mistake so much I thought vanish was a half bar, which even then I might have been wrong to think Wow, he that was that was a pretty good down heavy Now he's in a situation where level 1 sparking might run out So if we can hold off the storm for long enough we can make this game pretty even Oh my god, what kind of mix was that? Just dragon rush already dude. It's obvious. I could feel it in the air. You're thinking about it You're thinking about it. I know you're thinking about it. This is a thing Okay, we're just gonna keep we're just gonna get him into the corner. I why do I vanish there? I don't know. I don't know why vanish there. It's just a panicky sort of thing to do seven bars Just gonna style on me for a little bit try to keep last maybe a jab would have been better Uh, can he kill you the damage actually doesn't look too crazy, but he can do two level ones into a level three Yeah, we're dead. Oh, that's damage good stuff though. I I made too many mistakes Wow, I was I was wondering what was happening. It's been a while since that dude What a beautiful destructive though easily the best in the game second only the trunks Can we talk about those mistakes? I made all right one. What was it? I What was the first mistake? The second one was that I I just kind of froze up with him. Uh, and then I tried a key blast when he went in Uh, do we go back? All right, good stuff decent stuff Um, oh my goodness. He dashed out of that corner D said he just got to keep putting this pressure on like this At ex that make sure we keep bar dock in the corner No, he just got me with the lights. I gotta stop jumping like that. They can escape through dashing. Maybe hold on How how do you do you think jumping and then back dashing would work because that would be wild Man, I really hope we get it and that was me trying to force something We haven't done the actual broly broly thing. I wanted to do because that it it does okay Uh, okay another situation where we're gonna spark. He might read this though Because we sparked early last time in this same situation. No, I mean, what would he have done? Okay, get that a waste of time Let's go He stopped me He hit the first one and actually stopped. He's a god Nobody I don't do that. I hit it and then I just start spamming buttons crying Usually the tears are already falling out of my eyes at that point Level two level three level threes. Does it kill? Maybe maybe we live with a sliver now bro. He's dead. But yeah, that was basically the barrier thing Uh, to a lighter degree. I obviously I didn't actually connect with it.  So cheap. Oh, that was pretty cheap too He delayed the super. Oh, that was so nice. No We got after image two. He doesn't want to click a button. He's very uh, safe on defense Which is, you know, a good trait to have. I wish I had that trait, but I don't I'd be mashing sometimes. I'll fully admit All right, cross up work didn't throw out any buttons beforehand Please hit. Please hit. Please hit. Please hit. Please hit.  Now that we're in pride. Sure How am I gonna do this with random teams? Nice tech Gotta stop jumping. Gotta stop jumping. Oh, I got hit and couldn't reflect afterwards. Nice Three bars as well level three Oh, he's going for it. All right. We gotta block this mix if we block this mix We have a high chance of winning if we don't it's the opposite. Good. I keep getting hits Oh, he's okay. I'm starting to crack that jab was a pure Nice that reflect it missed give me give me an arrange This should be range Oh, it could have been two bars.   All right play it snafe does he bar doc Back him up. Oh, that was a crucial moment Wait every time it's like conquerors hockey, dude every time I go in it just it just what the Don't don't dragon rush That that's pretty cheap online, but I'm not I feel so bad. What was that in the clutch, dude? He hit me with the like He hit me with the aura. He was like like It was like walking up to his why am I making so many anime references? Let me stop. Let me stop. Let me stop. But you guys know what I was gonna say. So. Oh my god, dude You can tell I'm getting into it by the way. I'm losing everything that overhead is so cheap. Come on.  I I love base Vegeta players, but You guys be a little too talented sometimes. I gotta keep it honest with you All right, let's go Don't drag and rush me. I will take it. I thought cut me in half Don't drag and rush me. I'll take it. I swear. I swear. I will don't even try it Don't do it tired of blocking No, I could get this great match, dude.  Why are not why are more people not playing Krillin? He's sick. He's genuinely too good Oh, he got through that with superdash. That was smart. That was good stuff He might have forgotten rocks beat superdash, but I'll give him he's obviously a great player So I'm gonna give him credit. I'm gonna say I'm gonna say he he knew the superdash wouldn't hit the rocks Level three. I guess you needed it. I guess he did need it actually Hmm, did he use spark it? Have I used spark?  I'm gonna forget the whole rest of the match It was a little too stressful and we didn't even block for 30 minutes, by the way I should I should add that Okay, I don't know the region. I haven't practiced. I do know the region But I haven't practiced it in so long that I'm not gonna go for it Not in a set like this All right, that worked That sucks that there had to be a better way. I using three bars and that that's that way Dude, this went from like a oh, I'm gonna be you know happy. I'm gonna talk about Talk about all these things I want to do with Broly too. I'm going to be very silent and play fighters now Oh, let's go. Oh, I'm so so fast. I couldn't react to that if I tried. Um, I I didn't get armor for that That's why that's what I had to hear Well, look at the damage dude That's what I was saying earlier If you know those you can get some scenarios where you can kill That look I wouldn't have been able to kill there as base Vegeta I would have had to hold that. Oh, wow. He beat me. I tried to jump in button too, but didn't work Oh Gotta get gotta keep corner at all times. All right, we should we should have sparked yesterday Oh, I gotta be way more careful against trunks and neutral than I'm being Trying to get him with an ex will think fist and then a raw super dash ain't never gonna work He beat me on that. I thought I had the I thought I was full advantage. Oh, we're losing And the sad part is with green life sparking limit break All right, we got a we got a hit, uh, which it may have been caused by an unfortunate lag, but you know Uh, you know All right, what's he gonna? Is he gonna go behind me? Oh, it's my sparking gone. No Oh, dude. Oh, I wanted to level three two What is that two one? Oh, it's two one Me man, I'm angry at that not angry at him. I'm angry at me We choked so much with yamchin sparking All right, hold on. Let's bring it back. Oh, I'm getting I'm gonna I'm losing neutral as Broly. What is this? How am I losing neutral as Broly? All right, here we go barrier mix Oh nice held it to make sure he got a combo off of this. Can he he better not be okay. I thought he was gonna Oh, that sucked overhead at the last second Dang, he's got me in the corner. My sparking was complete waste. It didn't blow him away. Oh We out we lived and it's all thanks to sparking, baby Uh, do we have to do this?  Just throw all the supers Yes, all right We got base Vegeta out of the way the sparking was not a waste now. It did it was because he dropped the combo All right, we landed behind him. We should probably let krillin take the reins from here He's just an overall better point character Sometimes the simplest stuff is the best, baby It's just the second double jump to make sure his reflect the way I've found me Oh That was a read and a half. It was fast. Okay. That was it. I had to go silent there All right, another good hit another good hit. We are well on our way now boys I'm level three.  This is intense. What do I even time not talking? This is a terrible video Oh, no. All right. We are getting no right points. It's so hard to rake up, dude But we are we are almost halfway to altruistic Let's go. Oh beautiful 2l the call out. Oh, it's lagging. It's it's weirdly It's not bumping up the frames. We're just getting like seconds of freeze Hold on gotta go quiet in defense time overheads. Yes. Hey, I thought it was gonna be it's lad, dude Oh, there's a frame. Never mind. I was wrong He's by the way still got the loop This man is good. Oh again.  He's got the he's got the neutral. I can't play neutral But Broly has a character Ken. So that's what carries me Oh that assist is so good. I didn't block overhead Defense is really I might buy my defense like down the drain All right landed a hit with yamcha assist critical I don't want to waste the bar, but I'm just gonna do it We need to get we need to get some form of momentum back. I just need him to sit down and have to block these Oh, well that there goes that And it caught the slide Oh, he didn't run up at all or he might have micro dashed up But I was trying to call out his read on my back dash to be fair He might that might be a cover all option or something like that, but oh that was so good Two bars looks like it's not gonna happen here. That didn't kill. Come on. All right, whatever Wolfgang fist did get the job done eventually. All right decent We did a decent job of getting back into the game beam assist Don't lag overhead to the cross up. Oh Beautiful stuff. It doesn't have any assist to continue this though. I'm that's fine He's probably gonna level three to make up for that.  It's definitely sparking time. The question is what's the read Did you what was he reading now? Another good hit. Let's go keep getting them to the corner We're gonna do it with yamcha here because I want to read the sparking the sparking is coming The question is when when when when when was right away? All right, we're out Oh reflect and he continues it with the sparking when he dropped it overhead. We blocked those Oh, I couldn't block the low.  Does he need another one? Yes, he does. Okay, three ones spent. How much help did yamcha have again? Ooh, I don't know if that's the comeback zone But with limit break anything is possible. He's gonna jump in for some buttons I'm just kicking at the air, bro. I'm just kicking at the air. Oh, okay. That's it Wasn't the most climactic of final touches, but if he drops it we wanted to give you a second chance. Let's go That was a climb back. You know what? Thank you He knew I was filming a youtube video. He said climactic. Oh, it wasn't climatic. I got you, bro But I appreciate it the great games That is such a scary team be on the lookout for this team guys If you want my advice if you're in the comments, you're new to the game and you say And you're like, hey, I just want to learn this game. I want to know what your opinion and the best team is I really really like base vejita and bardock as a shell now. You got to run the b assist in my opinion with but No, we're gonna end it there. It's 21. I checked the time. I was like if it's if it's not over 20 I'll do it again But yeah, you got to run the b assist with bardock in my opinion. It just it gives you so much mix there Uh, but yeah, let me let me stop. Thank you so much for watching this video If you did enjoy this more intense one definitely go down below in the comments Let me know by hitting that like button and subscribing to the channel If just in general you like the stuff here other than that if you want to watch more videos right away There should be some on your screen. So give those a watch if they catch your eye I have been doctor today. Thanks so much for watching this one all the way to the end And I will see you in the next video


That's why I want children out of schools. Even 50 years ago, the students that were coming out of high schools were able to be creative in ways that they are not right now. We're killing that spark of creativity. And along comes chat GBT. It's shifting, making us more into robots and making the AI more human-like. Parents have got to step up and say, you know what? That's not for you. I want you at home. I want you to read, write your own papers.  Because once you tap into your creativity, it's like a fire that burns inside.


Hey everyone, are you ready for some great news today? We have updates on the sales of the top 10 Nintendo Switch games out there, including Animal Crossing New Horizons, literally shattering records. It's insane. We have total Switch sales. We have a whole bunch of news to drop your way because Nintendo had a nice big financial briefing over the sleep period of last night for me. However, before we get into that, I want to remind you to enter our giveaway for Final Fantasy Crystal Chronicles Remastered. We will be giving away a copy of it at the end of the month to enter. All you need to do is comment down below, like this video, subscribe to the channel and hit that bell icon to set it to all notifications. Also, we are on our road to 50,000 subscribers and we can hit 50k before November. The November giveaway will be, well, it'll be a PlayStation 5 and Xbox Series X or a Nintendo Switch. It'll be up to the winner's choice. But hey, you know what, we're not there yet, so let's get into the big news we have today. So before we get into the numbers, because the numbers are a big deal and they show why Nintendo just doesn't care about launching Nintendo Directs to get us hyped for future stuff because they are doing so well in the present, it's insane. And this also explains why maybe the stock prices aren't dropping as quickly as people might think they should. However, let's talk about stock in terms of the Switch itself because we all know it's been very hard to get a Switch this summer and really leaking into last spring. Well, according to Nintendo, it has reached a point now where stock is almost back to normal numbers. The company highlighted problems it faced thanks to the coronavirus and its three months earning report published earlier today. The report reads that COVID-19 created some difficulties in procuring the parts required for manufacturing Nintendo Switch consoles, but the overall production situation has almost recovered. Now the thing is, they're saying almost recovered. That means it's not actually back to full production. And we have seen things like the animal crossing Nintendo Switch has reappeared somehow out of nowhere in Australia for all of 10 seconds. So it's interesting to see that production is ramping back up. And I've said this before, at some point in August, back when Nintendo said that they were almost back, this is like over a month ago now, Nintendo said they were getting back to normal production levels. I told you guys then. I've seen some comments on that video where it's like, oh, see, you were wrong. And I'm like, did you listen to the video? I said by some point in August, the middle of August or so is when you would start to see switches end up back on store shelves because it takes a while just because they're back to normal production levels. They're shipping involved and shipping kicks a long time. You got to get it on the boats to go overseas back to the US until their warehouses then out to all the various retailers. It could take a month to two months for that to happen. So yes, expected to be fairly easy to get a switch soon. How long they can stay into that production level? I have no idea. Hopefully for the rest of the year. Now let's get into the numbers. And let's first off just go with Nintendo's net profits. They reached $1 billion in the last quarter and have a 541% rise year on year compared to quarter one of last year. Nintendo's net sales have more than doubled this time around covering the period between April 1st and the 30th of June 2020. Those net sales reached a total of 358.1 billion yen up from 172.1 billion yen last year. It also saw a significant rise in operating profit and ordinary profit with numbers rising across the board. The 541.3% rise we mentioned translates to a quarter one net profit total of 106.4 billion yen or just over $1 billion USD. Over the same period last year, Nintendo's net profits reached 16.6 billion yen, which is about $157 million USD. And now we're going to see why that happened because people have been telling me all year old because of the lack of announcements and this and that, switch stocks should be going down. And I kept telling people the switch stock prices really shouldn't be moving anywhere because switch is in the midst of their best year yet. And well, as you're about to see, the numbers kind of show why. So first off, let's get into the top 10 best selling Nintendo Switch games. And you're going to see some crazy numbers here. So Mario Kart 8 Deluxe is still holding onto that number one spot. However, it might not last past 2020 because at number two is Animal Crossing New Horizons. And guess how much that game has sold? Remember, that game came out just a handful of months ago. 22.4 million units. It is the fastest selling game period on Switch. We have not seen the game sell this quickly on Switch at all. It's insane. It hasn't even had a holiday period hit yet. Animal Crossing New Horizons is likely going to go across over 30 million. It is very clearly the best selling game in the series. I am just, oh my gosh, 22.4 million. Can you view that for Animal Crossing? That's just the number two game. At number three, we have Super Smash Bros. Ultimate, which is basically at 20 million. It's at 19.99 million. Probably by now in August, it's probably at 20 million. Yeah, Super Smash Bros. Ultimate, I don't know if we'll ever see another fighting game sell these kind of numbers, even another Smash game. Insane. Wow. Next up at number four, still holding on is Breath of the Freakin' Wild at 18.6 million. Wouldn't be surprised to see that over 19 by the end of this holiday period. But yeah, Breath of the Wild just keeps on selling. If you ever wonder why Nintendo doesn't lower the prices of games, just look at the sales of their games. Next up, this is really cool. Pokemon Sword and Shield has sold 18.22 million at the number five spot, 18.22. To put that in perspective, it is now the best selling game in the Pokemon franchise since Silver and Gold, Gen 2. It is the best selling game in the franchise since those days. By the end of the sales of this game, it could end up being only comparable to Gen 1. You know, red, blue, green, etc. So yeah, this is absolutely nuts. And I just, Pokemon Fever is just going anywhere. Next up, we have Super Mario Odyssey, which has crossed the 18 million mark as well at number six at 18.06 million. Pokemon Let's Go Pikachu and Let's Go Eevee has crept above 12 million as well at 12.2. So that game is getting some long tail sales there because it was stuck at about 10 to 11 million for a while, but maybe it's possible Sword and Shield gave it an extra million boost over the last year. Super Mario Party is at number eight with 10.94 million. That's going to be at 11 million here pretty quickly. We have Splatoon 2 at number nine at 10.71 million. And then the only game not over 10 million at the number 10 spot is new Super Mario Brothers U Deluxe. Another Wii U port at 7.44 million units. We also know that Xenobite Chronicles, definitive edition, has sold over a million units as well in its short time on the market. I think it's actually already outsold the original release back on Wii. So good for that. Now, the Nintendo Switch itself. Here we go. Get ready. The Nintendo Switch itself has sold 61.44 million units. And if you look at the hardware ranking sales, it at this point would have passed the NES. The NES, at the time of Nintendo's official report, was still about 500,000 units ahead. But again, this was back in June. It is now August. It is highly likely the Nintendo Switch has surpassed the NES, meaning the only home console left on Nintendo's radar is the Nintendo Wii. So we'll see if it can get that extra 40 million in sales to pass Wii. What's interesting too is software sales are at 406.67 million units. That's still about 100 million or so behind the NES. So it's very interesting to see that although the hardware sales are similar, the software sales are a little bit off pace with the NES. Although it is ahead of basically all of Nintendo's other home consoles. Beyond Wii, of course. Wii sold almost a billion units of software. So we'll see what happens when Switch gets up to 100 million or if it does. That means it sold over 5 million in the last fiscal quarter for those that are wondering on the exact sales of Switch. This is what Nintendo also had to say about Animal Crossing New Horizons. In case you're curious, it said the number of consumers who started playing on Nintendo Switch because of this game continued to increase during the first quarter. From April to June of 2020, of all systems in the Nintendo Switch family that were played for the first time during that period, over half were used for Animal Crossing New Horizons on the first day. So since the release of Animal Crossing New Horizons, half of the people that bought a Nintendo Switch played that game. So talk about a system seller. Oh yeah.  So at this point, it took us all this time just to get through all of the actual data. There's going to be more information that comes out eventually. You're going to see a Q&A that we're going to answer more specific questions. And we'll cover that as well if there's anything interesting. But I just want to take a moment to admire, I guess, these sales figures. And I get that for me personally, it's not like these sales figures mean anything to me. I'm not making any money off of these sales figures. They're not sitting here handing me checks because Pokemon Sword and Shield over $18 million or because over $22 million of animals. Like these are insane numbers. $22 million for any game is insane, let alone Animal Crossing. And I just think that Nintendo, when we've talked about the lack of directs this year, when we've talked about the only knowing one game at a time, like we knew what Paper Mario was coming then. We didn't know for 17 days what we're next. Now we know one game is coming and Pikmin 3 Deluxe. And I know it might be until that game comes out that we find out anything else. I think that while it's frustrating and infuriating and it feels like maybe new, new games, because if people want to talk about just new games, technically only two have released from Nintendo this year that it's a little frustrating. Nintendo's in the midst of the best year of Switch by far. And investors should be extremely happy at the amount of money and likely dividends coming their way on their stocks. I think that at this point, Nintendo can almost do no wrong. And that's saying something as we head into the Xbox Series X and PlayStation 5 coming out, which everyone is making kind of an assumption it'll affect sales when Miles Morales hits and Halo Infinite. Like, you know, Nintendo is going to be in trouble. But again, these systems don't really replace Switch. They're amazing platforms. I honestly can't wait to see what they're going to cost.  But I think to be affordable, they're going to both companies will be taking big losses per unit sold. But honestly, it doesn't matter what these 4K, 8K, 120 FPS systems can do because they don't they're not portable. You just you simply just can't take them with you in a convenient way and you can switch. So I'm not that worried about it. And I don't think Nintendo is that worried about it either, at least in the short term. I do think it'll affect third party games. I don't think we're going to start seeing Doom Eternal and Witcher 3 kind of ports next year or the year after for Switch. I think that kind of sports going to be dropping off. And that's that's a shame because we never really got to get full third party support at any point in Switch's life. But, you know, Nintendo took a huge step in the right direction. This gen and they're setting themselves up well for the next generation whenever they decide to launch that. But yes, Switch is killing it. Nintendo is killing it. I'm just baffled by these sales right now. My mind is swimming in data and I love data. Let me tell you, I'm going to keep swimming in it. And if I find anything else interesting out of all the data that's out there, you'll be sure to know that I'll make a nice video about that. Anyways, enjoy. I hope you guys have a lovely start to your day today and I'll catch each and every one of you in the next video.


Hello, good afternoon getting from sender academy. Welcome to the J main problem solving session So I would request the students who have joined in please type in your names in the chat box So that I know who all are attending the session Can all of you see my screen? Can all of you hear me clearly? Okay, great. All right, so we are starting from the first question and I think most of you have started answering it Great, so which of the following is not true Which of the following is not true For any of the two statements P and Q of course Statement one is going to be correct Statement two is not correct because it is Contradicting de Morgan's law. It is contradicting the de Morgan's law. Okay So we all know de Morgan's law negation of P or Q is Equivalent to saying negation of P and negation of Q. So this symbol should not be the symbol of This should be the symbol of and not for the symbol of all. Okay, so option number two becomes correct in this case All right, so long. Let's move on to the next one Let R be on the real line consider the following subsets of the plane of R cross R so S is a Set of ordered pairs X comma Y Such that Y is GIF of X X belonging to zero to four and T is all ordered pairs X comma Y So is that X minus Y is an integer Which of the following is true? Please give me a response after stating the question number so that I know which question number You're trying to answer. Good afternoon Rohan Aditya Sanjana Okay, again, I'll wait for three people to respond at least Okay, so all of you are saying option number one. Okay, let's check it out So the options are saying T is equivalent solution on R, but S is not of course, you can see if you take an element like 1.2 comma 1 it clearly belongs to S But the moment you flip the position that is 1 comma 1.2. This does not belong to us Okay, the reason being 1 is the GIF of 1.2, but we cannot say 1.2 is the GIF of 1 so this relation is not correct So the symmetric relation is violated by the set S and therefore S cannot be an equivalent solution So the statement which is true is going to be having the fact that S is not an equivalent solution Now X minus Y is an integer will always be an equivalent solution because it will be reflexive symmetric Transitive and hence equivalent so 1 is correct because it says T is an equivalent solution, but on R but not S So the first one to answer this was Sondarya. Well done. Good. Let's move on to the next one The complete solution set of mod X minus sine X less than 1 in the interval 0 to pi Okay Alright, so let's discuss this So we know that in the interval 0 to pi your mod of sine X is going to behave as sine X itself right and From pi onwards you can say pi to 2 pi it will behave as negative of sine X right So this is the definition of mod of sine X. So if I take this interval Then in the interval 0 to pi you would get something like this 0 Less than 1 0 less than 1 is obviously true That means all the values in the interval 0 to pi would be your answer. So this is an accepted Region so basically in 0 to pi all values of X are going to satisfy this given inequality Now the moment I am between pi to 2 pi so in the interval where X belongs to pi to 2 pi This inequality will convert itself to minus of 2 sine X less than 1. That means Sine X is less than negative half Okay, sorry sine X is greater than negative half. Remember the inequality sign is going to flip So when you say sine X is greater than negative half and you're looking from the interval pi to 2 pi So this is your negative half line Right and you want your sine X graph to be above that So it has to be in the zone where This part of the graph is line. So it has to be above the graph of y is equal to negative half Okay, so this point here is 7 pi by 6 undoubtedly this point here is going to be 11 pi by 6 Okay, so I can say in this interval your solution zone will be from pi to 7 pi by 6 Okay union 11 pi by 6 To 2 pi Right now finally what I have to do I have to take the union of 0 to pi with this interval as well Right because there were two cases this was your first case Where you had taken your x between 0 to pi and this was your second case where you had taken your x between pi to 2 pi So if you combine it your answer will turn out to be from 0 to 7 pi by 6 union 11 pi by 6 to 2 pi Which is clearly your option number 3 option number 3 is going to be correct in this case and the first one to answer this again was Why should we this Monday very good? Keep it up by should be Let's move on to the next question, which is your fourth question. So we are going at a good rate It's almost like three minutes Three minutes per question. We are taking almost right Ayantika. Correct Yeah, now again a easy question if a express B y equal to 1 touches the circle x square plus y square is equal to p square Then the locus of a comma B is Okay, I need one more response before I start solving this alright, so basically it's a case where there's a circle of Radius p units having a center at the origin and There is a tangent and the tangent a x plus B y equal to 1 which touches the circle So it's very obvious that the distance of the center from this tangent should be equal to p units Okay, so the distance of the center from the tangent that is going to be mod of 1 by under root of a square plus B square and That should be equal to p.  You are going to end up getting a Circle of radius 1 by p So option number 4 becomes the right option in this case absolutely correct again an easy question So without much waste of time will go on to the next one question number 5 if Integral of x times 1 minus x square to the power of 1 third into x to the power minus 4 dx is a Times x to the power b minus 1 whole to the power c plus k Then which of the following option is correct So this is how you may write it 1 minus x square whole a to the power of 1 third By x to the power 4 dx. This is given to us as a times x to the power b minus 1 whole to the power c Plus a constant. Okay, Shantanu says the third option Let's wait for fewer people to respond Okay, so Roger and thinks differently. He thinks option number 4 is correct Okay, so mostly people are saying option number 4. Let's check this out First of all, I can write this expression as x minus x cube. There's to be power of 1 3rd let me divide this x to the power 4 into x and x cube if I introduce this x within the Cube root symbol it will become x minus x cube by x cube all to the power of 1 3rd times 1 by x cube dx right Now this is clearly Integral of 1 by x square minus 1 Whole to the power one-third into 1 by x cube with respect to dx, right? Now let us do one thing. Let me take 1 by x square minus 1 as t So if we do that negative 2 by x cube dx will become dt That's nothing but dx by x cube will start acting as negative half dt Right, so this entire integral will convert to negative half t to the power one-third dt So that's going to be negative half t to the power of 4 by 3 Okay, divide it by 4 by 3 so it will become a 3 by 4 on top Okay plus c and If you replace our t back, this is what we are going to get Okay, so from this expression. It is very clear that I Can write 1 by x square as x to the power negative 2 as well So it is very clear that my a is negative 3 by 8 My b is negative 2 and my c is 4 by 3 Right, so ideally speaking neither 1 is correct nor 2 is correct nor 3 is correct And let's check a ta plus b plus 3 c plus 1 a ta is going to be minus 3 B is going to be negative 2 3 c is going to be 4 and of course 1 is going to be 1 So when you add a 1 you will definitely get a 0 So option number 4 is the right option in this case and The first one to get this correct was I guess Ramcharan was the first one to get this correct. Okay Well done. Very good. Let's now move on to the sixth question Six questions says the equation of the circle which cuts the circle given to you on your screen and the lines Lines means pair of straight lines Orthogonally is which of the following? So there is a circle which cuts the given circle as well as these pairs of lines Orthogonally, Orthogonally means at 90 degrees So which could be that possible circle? Okay Got the first response So I also backs him up anyone else I Need one more person to answer Now that doesn't mean you'll type anything so that I start solving it so please try it out All right, guys So first of all just a question in general to all of you that if a line is cutting a circle Orthogonally, so let's say this is a line which cuts the circle orthogonally What are the meaning of that? The meaning is very clear and simple that the line must pass through the Center of the circle then only it can cut the circle orthogonally in that case We actually do not use the word orthogonally. We use the word normal Orthogonally is generally used when we are talking about two curves when you're talking about a line in a curve We use the word normal and when we talk about two lines being perpendicular, we use the word perpendicular Anyhow, so they are pair of lines That means there are two lines and both of them are basically cutting the circle orthogonally means both are normal to the circle and That can only happen when the meeting point of these two lines happens to be the center of the circle So whatever circle I'm going to find out that's the circle of the center of that circle must lie at the point of Intersection of these two lines Which are collectively expressed by this expression Correct. I can easily find that out because if you take the first and the third term you get something like this and You take minus 2 common you get x minus 1 equal to 0 That means y minus 2 times x minus 1 equal to 0 is the two lines which comprise or which constitute this pair of straight lines Now this clearly implies that the meeting point has to be 1 comma 2 right Now unfortunately all the options given to us have centers at 1 comma 2 So this is not going to help us to that extent So we need to now move on to check What should be the radius of that circle? now The radius of the circle that we are looking for Should be such that It cuts this circle Okay, so basically The tangent drawn at the point of intersection Must pass through the center right So basically the tan this line or any one of the lines Sorry from the center of the circle. We are going to draw a tangent to this given circle Right because if they're cutting orthogonally Let's say these two circles are cutting orthogonally then the tangent drawn at the point of intersection Right must pass through the center of the other Okay, so this line must pass through the center of the required circle. So this circle is required circle This circle is given circle Okay, okay So this point is 1 comma 2 we have already figured out over here So in order to find the radius I'm just going to find the length of the tangent drawn from 1 comma 2 To the given circle that is x square plus y square plus 2x plus 4 y minus 4 That length would be under root of s1 So your radius r will be under root of s1 which we call as the power of the point with respect to the circle Right so under root of s1 means under root of x1 square y1 square 2x1 plus 4 y1 minus 4 and your x1 y1 is nothing but 1 comma 2 so that would be 1 plus 4 and This will be 2 plus 8 minus 4. That's going to be root of 11 Okay, so whichever of these circles has a root of 11 That would be your desired circle. So if you see The second option that is 1 square plus 2 square plus 6 This actually gives you root of 11 So it is satisfied by the second equation and none of the other equation satisfied So option number two becomes the right option and the first one to answer this correctly was Kushal Okay Awesome great. So hope you have recalled through this problem The concept of length of a tangent and when does two circles meet each other orthogonally So when two circles meet each other orthogonally the tangent drawn at the point of intersection Will pass through their respective centers. So it will pass through C1 and C2 Respectively, okay. So this was the concept which we used All right, let's move on now to the seventh question for the day Seventh question says there are four points in a plane of which five are collinear. So five of them are collinear Barring these fives no three are collinear Then the number of quadrilaterals that can be formed using these points in an easy question should be you should be able to do This within one minute. Yes. Yes any response All right Okay option D So I say the option three. Okay, let's discuss this out now if I have to make a quadrilateral I have to choose all the four points in such a way That not more than two of them should be collinear, right? So I have seven of them which are definitely non collinear Okay, and five of them are collinear I can make a quadrilateral in following ways I can choose all the four points from the non collinear ones, which will be seven C4 in number Correct. So all the seven points Which are non collinear from there I can choose for such points and make a quadrilateral so that will be seven C4 ways or It can happen that I can choose three points from these non collinear ones and One point from these collinear ones that will also lead to a quadrilateral Okay, third case could be you can choose two points from the non collinear ones Correct and again two points from the collinear one. So that's going to be five C2 Okay, so these are the only cases that you can form Seven C4 would be Five into six into a seven by six that's going to be 35 in number Okay, and Seven C3 will be again 35 into five That's going to be 175 in number And seven C2 is going to be 21 into 10 that's going to be 210 in number. So when you add it you get 420 quadrilaterals possible which is clearly your option number four that is correct, right? I thought it was a very simple question and you would answer this within well within one minute Okay So great so first one to answer this was Ramcharan again awesome great Let's not move on to question number eight the first sum of an infinite geometric series is two and Its sum is denoted by s Says that mod of s minus two is less than one by ten Then the two set of the range of the common ratio of the series is Okay Psi son area they have already given the response Great. So let's discuss this we all know that the sum of infinite series for this case will be two by one minus r R being the common ratio. So this is less than one by ten. So two Minus one bar minus two should lie between minus one by ten to one by ten That means two by one minus r should lie between two minus one by ten to two plus one by ten That's going to be nineteen by ten to twenty one by ten Right now. We all know that if this has to be an infinite series Mod of r has to be less than one Correct that mean r has to be a fraction that can be between minus one to one So one minus r will always be a positive term. So let me solve this as two separate inequalities one is This now in this case, I can always write this as one minus r is less than 20 by 19 the reason I could do that is because I know one minus r is a positive term Right that means r should be greater than minus one by 19 Okay, in a similar way, I can stick a two by one minus r less than 21 by 10 Which again means one minus r is greater than 20 by 21 So r is less than r is going to be less than one by 21 and of course r cannot be zero Okay So in this case in a geometric progression, we can never have the common ratio as zero So your r should belong to the interval minus one by 19 to one by 21 Excluding zero r cannot be zero. So obviously option number four is going to be the right option Simple question just was testing your concept of inequalities. That's it So without much ado, we'll move on to the ninth question two equal sides of an isosceles triangle with fixed base six centimeter Are decreasing at a rate of three centimeter per second? How fast is the area decreasing when? Two equal sides are equal to the base Okay, I'm getting a mix of spawns kushals is for Vashnavi says option three Alright, so this is because this so let's say these two equal sides are of length xx each, okay? and Let's say this is my a b and c sides and I drop a perpendicular When I drop a perpendicular, okay, this is going to be three the perpendicular let's say ad is Is going to be under root of x square minus nine So area of this triangle is half base into height That's going to be three times in the root x square minus nine This is area and if you want the area to be How fast is the area decreasing when two equal sides are equal to the base that means I want to find dA by dt The dA by dt will be linked To this expression by two under root of x square minus nine times two x into dx by dt so that's going to be a Three x by under root of x square minus nine into dx by dt So dx by dt is given to us as Negative three because it is saying that it is decreasing at a rate of three centimeters per second So when we put over here, I get dA by dt as three x Now x has to be equal to six So 36 minus nine is going to be five into negative three, okay? Sorry, it will be equal to 36 minus nine is going to be 27 small mistake here So that's going to be a root 27 root 27 is three root three Okay, so root three and this will go off giving you the answer as negative six root three centimeters square per second So it's always saying decrease. So the negative sign is already accounted for so option number four is going to match The required result So the first one to get this correct is I Think same here No, sorry kushal kushal kushal got this correct great Let's move on to the next one question number 10 in a complex way in the point a and b are at z1 and z2 and There's a point P which moves in such a way that mod of z minus z1 is twice mod of z minus z2 Then the maximum area of Pab anyone and response Nobody So these kind of questions are going to you know take a bit of your time if you are not aware of the concept here so guys all of you would have you know come across You know expressions of this nature right more of this is equal to k correct Remember here that if k is equal to one it actually is the locus of z is actually going to be a Line which is exactly bisecting the joint of z1 and z2 so basically if you join this Then z would be lying on the bisector only if k is equal to one right so z can lie anywhere on this line Right because when case one it is just like saying z minus z1 is equal to mod z minus z2 That means the distance of z from z1 has to be equal to the distance of z from z2 That can only be possible if your z is located on the on the Perpendicular bisector of the line segment joining z1 and z2 However, if k is not equal to one then what happens it actually results in a circle which we call as the aplonious circle Okay, what is a circle Aplonious circle is basically a circle In this case which you'll find that Has diameters Let me draw a circle just to you know make it more clear to you. So it would be a circle Which would pass in such a way that the extremities of the circle The extremities of the circle Let's say point a and point b would be in such that a divides a divides z1 and z2 in the ratio of k is to 1 write the same k internally internally and B divides this point B as you can see over here this B would divide the join of z1 and z2 in the ratio of k is to 1 again But this time externally in the ratio of k is to 1 But this time externally this is very important and it saves a lot of time So right now in my given problem I can see that z1 z minus z1 mod by z minus z2 mod is equal to 2 Right. So k divides sorry a divides z1 and z2 internally in the ratio of 2 is to 1. So this is 2 is to 1 So if I know the coordinates of z1, which is going to be 5 comma negative 2 and Z2 coordinates, which is 1 comma 1 Can I find out the coordinates of a? Of course, we can we can use a section formula to get the coordinates of a so that's going to be 2 into 1 1 into 5 that's 7 upon 3 comma 2 into 1 and 1 into negative 2 that's going to be 0 so a is going to be 7 by 3 comma 0 a is going to be 7 by 3 comma 0 Similarly, we can easily find out the coordinates of point B by using our external division formula You just have to take your 2 as negative 2. So it's going to be a negative 2 Okay, so negative 2 into 1 again a minus 2 by negative 2 plus 1 comma and again negative 2 negative 2 I'm sorry negative 2 Plus 5 there would be a 5 over here. My mistake. I wrote a 2. I'm sorry. That's a 5 and you'll have a negative 2 Into minus negative 2 by minus 2 plus 1 Right So if I'm not wrong the B coordinates is going to be negative 3 comma Negative 3 comma 4 is that fine? So if you found A and B what is going to be the What is going to be the? Radius of this circle AB is the diameter. So what is going to be the radius? Let me find some place to write so radius would be nothing but half of the distance between A and B, right? So the distance between A and B would be 7 by 3 plus 3 square and 4 square. That's going to be 9 9 to 7 16 16 by 3 that's 16 square by 3 square plus 16 under root Let me pull out a 16 out as 4 So inside I will have a 16 by 9 plus 1. That's going to be 5 by 3. So 4 into 5 by 3 That's going to be 20 by 3 and half of that. I'm sorry. I forgot the half everywhere So half of that would be 10 by 3 Correct So now you are going to make a triangle with P AB. So P is basically moving So when do you when do you think your triangle will have the maximum area when its height is maximum because base is fixed Correct. So if you are able to draw a triangle like this, that triangle will have a maximum area So its height is going to be 10 by 3 And its base is going to be the distance between these two points So I think the distance between 5 comma negative 2 and 1 comma 1 would be under root of 4 square plus under root of 3 square that's going to be 5 So the answer is going to be half base into height Half base into height that gives you 25 by 3 which is clearly option number 2, which is correct Well, we learned a lot of things over here in case you have not practiced a lot of problems of this type Let me tell you this is a commonly seen expression and it will always result into an a plonious circle Whose diametrical opposite ends would be lying at points A and B where A divides the join of Z1 and Z2 in the ratio of case to 1 internally and B divides the join of Z1 and Z2 in the same ratio case to 1 but externally So the moment I know this concept I could actually find out the points A and B and Then after I could get the you know You can you can get the radius by halving the distance between A and B and For the area of this triangle to be maximum since the base is fixed Its height has to be maximum and height can at the max be the radius of this a plonious circle and therefore the result Right. So good learning in case You were not aware of this So none of you could answer this question. So let's move on Let's move on to the next one, which is question number 11 for the day For the system of equations as you can see on your screen, which of the following is false Which of the following is false Okay, so mostly I don't know why all if you are saying One because if you see if your P is equal to Q is equal to R this thing will become 0 0 0, right? That means it will result into a system of homogeneous equation, right and Homogeneous equations are always known to have a solution Either the trivial solution that we all know as x equal to 0 y equal to 0 or z equal to 0 Or they have infinitely many solutions So the fact that they have solution makes them consistent. So this statement can never be false This is correct statement because If it is consistent, that means it has a solution, right? So even P is equal to Q is equal to R. It will have a solution. So this is a true statement So my question is when when is the statement false? Please do not, you know, get confused here. It's very important all homogeneous system of linear equations are consistent So one definitely cannot be your answer. Okay? So Andrea says option number four. Okay, again a equal to b equal to c is not the decider because if these numbers are all zero Okay, you can have infinitely many Solutions for that. So it can be consistent even in case of fourth But if you look at option number two, if you make your ABC equal and your PQR are distinct Okay, then what will happen? It'll actually result in let's say if you take PQR as distinct So it'll actually let's say I take ABC all equal to one Okay, so at one case, I will get something like this at another case I will get something like this at another case. I'll get something like this Okay Now unless until P minus Q is equal to R minus P is equal to P minus R Only in that case both will represent the same equation as they would represent parallel planes Right? If they represent parallel planes, then they will know there'll be no solution Then in that case it can become Inconsistent so it would be wrong to say that it is consistent for all such cases Right, so I could figure out a case where in this case it would be non consistent or inconsistent whatever you call it So this Option number two. I do not agree with so this is a false statement So two becomes your right answer because they're asking us which of the statement is false So absolutely, I think Ramcharan gave the answer first Okay, so guys be very very clear with your idea of Solution of system of linear equations and what conditions are responsible for consistency and inconsistency Right, so without much I do we'll move on to the next question, which is the 12th question a simple one. I think You would have come across this question before as well Okay, so Ganesh has responded Okay, so mostly people are going for option number two. So let's look into this First of all we can write our dy by dx as xy upon x square plus y square It's a clear-cut case of a homogeneous differential equation So let me take y as vx implies dy by dx will be v plus x dv by dx so when you make these substitutions we get v plus x dv by dx is equal to v square by 1 plus v square so Sorry v by 1 plus v square So let me take the v on the other side. So this we let me bring it to the other side Okay, so this will become negative of v cube by 1 plus v square. This is x dv by dx Right So you'll get 1 plus v square by v cube dv is negative dx by x. Let's integrate both sides Okay, so this one will give me integration of 1 by v cube Integration of 1 by v cube is going to be negative 1 by 2 v square plus integration of 1 by v is going to be ln v minus ln x plus c Okay, that's like saying minus x square by 2 y square Plus ln y by x is equal to Negative ln x plus c now we can use this initial value condition that when x is 1 y is 1 So when x is 1 y is 1 gives me 0 is equal to negative ln x Sorry, this is also going to be 1 So your c is going to be negative half Okay, so putting back over here. I Get minus x square by Nick by 2 y square Plus ln y by x is equal to negative ln x minus half Now they're saying that when x is x naught your y is e So if I substitute my x with x naught and y with e By the way, I can write this as ln y minus ln x that will get cancelled over there Okay, in this case, this is going to be e and Lne is going to be 1 so this I can replace with 1 okay one goes on the other side it becomes Negative 3 by 2 negative negative on 2 to go on so x naught square is equal to 3 square So x naught is going to be a root 3 e in fact Plus minus root 3 so one of the values They should write x naught can be x naught can be plus root 3 E which is option number 2 which is correct, okay? Absolutely correct. The first one to answer this was Rithvik Next is your question number 13 This says if f of x is this function, then which of the following is correct Okay. All right, so let's do this question. So derivative of this I Can simply follow product rule Now if this is 0 first of all we can take e to the power 2x minus 1 common Okay, so that would leave us with 2x minus 2 plus x square Minus 2x plus 1 so e to the power 2 x minus 1 and This will give me x square minus 1 equal to 0 so definitely f dash 1 is going to be 0 So if I put 0 I get 2 points 1 is plus minus 1. I hope I've done this calculation correctly. So let me just quickly Check this up. So x square minus 2x plus 1 so Those off it becomes x square minus 1. Yeah, that's absolutely correct. Oh There's a into 2 as well into 2 as well. So there's a slight error over here So if I have to differentiate e to the power 2x minus 1 a factor of 2 will definitely come inside So that would make it 2x minus 2 plus 2x square minus 4x plus 2 Right, so that would give you actually 2x square minus 2x Yeah, but anyhow When I put this to 0 I get two critical points One is x equal to 1 other is x equal to 0 Okay. Now, let me make the analysis of the nature of these critical points through the wavy curve sine scheme Now this is always positive, right? This term will always be positive So if I draw the wavy curve for 2x square minus x That means I draw the wavy curve for x into x minus 1. This will be positive negative positive right So at 1 you see that the curve is coming from negative slope to positive slope Negative slope to positive slope that means there's a formation of a minima there Okay So of course f dash 1 cannot be less than 0. So this is a wrong option now Zero is a point from where it's coming from positive slope to negative slope correct So it is a point of maxima in a point of maxima the double derivative is going to be negative so 4 is also not correct and Of course minus 1 is not a point of extrema. So 1 is also not correct and Of course 2 is going to be correct because we figured this out Now why I'm doing all these things is because if they ask this in a multiple option correct question You should be aware of all the possibilities But since you know here is one only one option correct. That's why you can save your time the moment you realize that minus 1 sorry x minus 1 is appearing as a factor for this Okay Great, so let's move on to the next question now The mean and variance of n observations x1 x0 till xn are 5 and 0 so x bar is 5 and Variance is 0 Summation of x i square is 400 find the value of n You may expect one statistics question to be asked in your j main However, it's a very easy chapter, but many people ignore this because they don't find it interesting enough But it's very scoring So mathematical reasoning and statistics, please do revisit the formulas at least the basic ones which you have in your ncrd textbook So this is your question number 14. Okay. All right. So in this case, it is a very easy question We all know that the variance formula is 1 by n summation x i square Okay, minus the mean square and this is given to us as 0 Right. So 1 by small n summation x i square is going to be 400 minus 5 square is going to be 0 So n is going to be 400 by 25, which is 16. It's a hardly 10 second question Okay, so please be aware of your formulas or type of formulas There are actually four formulas for finding the variance or the standard deviation one of them is the one which involves the Summation of the square of the data itself, which is this formula in front of your screen next question number 15 for the day Okay anybody else So far only Purvig has responded. Okay. So in this case first of all we'll go for f of 1 by x When you say f of 1 by x your upper limit is just going to be replaced with 1 by x Now if I want if I see the options are all in the limit 1 to x 1 to x So what I'll do is I'll put an obvious substitution Let 1 by x be Let's say you Okay, that means x is 1 by you which means dx is going to be negative 1 by u square du So in that situation your entire expression becomes in of 1 by you 1 plus 1 by u into negative 1 by u squared u Okay, sorry. In fact, I've put t as 1 by you not x t as 1. Okay So if you simplify this expression you get Okay, by the way limits will be 1 by 1 to 1 by 1 by x which is going to be x again So it's 1 by x negative ln u This will become 1 plus u into Minus 1 by u square into you Right, so negative negative will take care of itself. So it's going to be 1 by x ln u by u 1 plus u du now there's nothing in the name you can replace your u back with t So it's going to be ln t by t 1 plus t dt and that's going to match with your option number 2 Okay, however, if you treat this as your multiple options correct question Okay, then you should always check one of the three and four because this cannot be the possibility. Okay So always see the different types of questions linked in the multiple option correct So here you can see they're also inquiring about f of x plus f of 1 by x Okay, so let's try that out. So having figured out that 2 is correct I'm just going to spend some time verifying whether any one of 3 and 4 is correct or not. Okay So I'm just going to clear this up So f of x plus f of 1 by x Is going to be 1 to x ln t by 1 plus t plus ln t by t times 1 plus t dt Okay, if you take ln t by 1 plus t common It becomes 1 plus 1 by t which is 1 plus t by t That cancels off 1 plus t 1 plus t now it becomes a simple integral to do Where we can take ln t as let's say z then 1 by t dt will become dz Okay, so the limit of integration will become 0 to ln of x Right and this becomes This becomes Z dz which is going to be z square by 2. So the answer is going to be ln of x square by 2 So neither option 3 matches neither option 4 matches only option 2 is going to be the right one and hence only I think poor week Who it was the first one to give the correct answer for that? So we'll now move on to question number 16 This is a binomial theorem chapter especially related to the properties of binomial coefficients Okay. Yeah, so we have done this question Many times so if you expand this term you get C0 plus C1 x plus C2 x square C3 x cube C4 x 4 on the way till Cn xn right and If you write the expansion of 1 plus x to the power n we get C0 x to the power n Cn x to the power n minus 1 C2 x to the power n minus 2 C3 x to the power n minus 3 C4 x to the power n minus 4 and so on Till we see it. Okay now we actually looking for the The sum of these series and the series is obtained when you see that these two terms actually multiply So C1 C taught C0 C2 is obtained when these two terms multiply Similarly C1 C3 is obtained when these two terms multiply and Similarly C2 C4 is obtained when these two terms multiply. You can see that in all these cases C1 C2 Sorry C0 C2 C1 C3 C2 C4 etc It forms the coefficient of x to the power of n minus 2 Right it comes along with x to the power n minus 2 So basically what I'm going to do. I'm going to find the coefficient of X to the power n minus 2 In the multiplication of these two terms, which is nothing but 1 plus x to the power 2n Right, which is obvious. It's 2n Cn minus 2. That's nothing but 2n factorial by n minus 2 factorial n plus 2 factorial and We also know that in general if basically this is what is happening CR plus 1 C2 CR plus 3 etc Then in that case my formula becomes 2n factorial by n minus r factorial n plus r factorial So it's very important that you know this so that you can save a lot of time If you know this you can do this within two seconds. So option number two is going to be the right option Okay, great We'll not talk about the next question, which is question number 17. Can you see the screen now? What Yeah, now I can now I can see the screen I guess Are you able to see the screen guys, and now I think some movement has started to happen Wait wait wait, yeah okay, so Sondarya says 17 Option 2, okay. All right, so let's discuss this. So first of all I can say that If x plus y plus z is equal to a that means mod of x plus y plus Z is going to be Mod a mod a is going to be 2 right So if you square this it's going to be 2 squared Correct and if I expand this I'm going to get mod x square plus mod y square mod z square plus twice of x dot y y dot z Z dot x that's going to be four Okay, and since x y z are unit vectors each of them are going to be one one each So I can conclude that x dot y y dot z Z dot x is going to be 3 by 2 Sorry 1 by 2 not 3 by 2. I'm sorry. So it's going to be 1 by 2. Yeah, is that fine? Okay, now if you look at the options the options are all hinting towards How are the dot products of these unit vectors related to one another that means? First one is basically trying to hint whether y dot z is 0 or not second option is just trying to hit whether y dot b is 0 or not Third option sees whether x dot y is positive or not Fourth option is like whether x dot z is negative or not So basically it's all about knowing the values of these x dot y y dot z and z dot x from this expression So we'll try to tackle it one by one. So first of all From this expression if I take the dot product with respect to a Because I have been given information about a dot x and a dot y I would get a dot x plus a dot y Plus a dot z is equal to 2 is equal to 2.2 2 into 2 which is 4 So let me write 4 over here. Okay now Does this give us any information about any one of these required options? So we know that first a dot x a dot x a I can write it as x plus y plus z So a dot x is given to us as 3 by 2. So this is not helping us out in any way So I'm going to just erase this off So this gives you x dot x which is mod of x square x dot y Plus x dot z is equal to 3 by 2 that means x dot y Plus x dot z is going to be 3 by 2 minus 1 which is going to be half So these two expressions. I know which clearly implies Y dot z is going to be 0. Okay That means y and z are perpendicular to each other. That's obvious So option number one cannot be correct because they're asking us Which statement is false? So statement one is true So it's not going to be not going to be the correct option Right in a similar way. I know that a dot y is equal to 7 by 4 So if I take the dot product with y I should get 7 by 4 that means x dot y plus Y dot z is equal to 3 by 4 So we already know x dot y plus y dot z plus z dot x which is equal to half so from here it is Evident that z dot x would be half minus 3 by 4 right half minus 3 by 4 is going to be negative 1 by 4 So the angle between z and x Has to be obtuse therefore option For which says angle between x comma z is obtuse is the right statement and hence this also cannot be my answer Now I haven't yet used this particular expression yet So we all know the vector triple product So when we say a cross b cross c Right, what is the formula for that? a dot c times b vector minus a dot b times c vector Right, so in this case I can write this as negative of z cross x cross y Okay, let me make some space for myself Let me use this space over here Yeah, so this would be if you expand this it's going to be a z dot y Times x Okay minus x dot z times y is equal to negative of b we can say here Now z dot y how do I get the value of z dot y? I think we figured out the value of z dot y earlier That's was zero actually because they were perpendicular So zero and x dot z also was negative one-fourth Okay So this gives us minus b vector as one-fourth of y vector which implies Y vector is parallel to b vector So why cannot be perpendicular to b? So in this case option number two becomes our correct answer because it is false Okay, so we need not look any further so option number Yeah, by the way, we can also find out x dot y the moment. We know y dot z By dot z was zero That dot x was negative of one by four. I Can use this to find my x dot y Okay, so x dot y will now become three by four Which is a positive quantity and hence the angle between x and y should be acute so option number three is also a correct statement So three also cannot be my answer in case we are actually solving a multiple option correct question So option number two is the right option and yes, Sondarya Purvik. You are correct Let's now move on to the next question, which is question number 18 We need to find integral of Ln Under root of this whole thing is under the Ln. Okay, so Ln is operating on the whole thing Okay Wow, so most of you are going with option number three So let's discuss this first of all, let me call this as I and I'm going to apply The King's property on this integral so when I do that I get I is equal to integral from zero to four a lawn of replace x with four minus x so that's going to give you under root of Four minus x minus two. Okay, that's going to be two minus x Square plus four and this is going to be four minus x minus two again going to be two minus x Okay So since this is subjected to the Square you can actually rewrite this as x minus two whole square plus four But here I will have to have a negative sign right Now the obvious step that comes in our mind is let's try to rationalize this term on the in the inside of Ln term So you would get x minus two whole square plus four plus x minus two Okay, and here you'll have x minus two whole square plus four negative x minus two whole square So this and this will get cancelled off Okay So this will leave you with I is equal to zero to four Ln four DX minus zero to four Ln X minus two square plus x minus two DX This is I again So two I is equal to zero to four Ln four that's going to be four Ln four Correct drop the factor of two. So I will be two Ln four, which is again four Ln two So option number three becomes the right option in this case So well done guys all of you have answered correctly Let's move on to the 19th question. Okay, Vaishnavi All right, so let's say the roots are alpha and beta So you want the sum of the squares of the loose roots to be leased so this should be leased Okay, so we already know that alpha square plus beta square you can actually express it as alpha plus beta whole square minus two alpha beta so alpha plus beta whole square will be 4a minus 6 whole square minus two alpha beta is going to be plus Two times four a plus four Okay, you want this expression to be leased Right, so if you want this expression to be leased you can use the derivative of this with respect to a that would be a faster Or you could complete the perfect square with respect to a So when we do that we get two times four a minus six times four plus eight is equal to zero That's let me just check if something everything is fine So two times four a minus x is eight Okay, so in this case I would get four a minus six plus one is equal to zero So four a minus five is zero that means a has to be five by four Okay, and since this is a quadratic expression in a it has to be Opening upwards opening upwards will have a minimum only it cannot be a maxima point So no need to do a double derivative test So option number two becomes the right option in this case I think most of you got this Let's take up the next question, which is the 20th question. So there's a function defined like this Which of the following is false? Again focus on the question. They're asking you which of them is false Okay So these kind of problems are best solved by plotting graphs So for x greater than equal to zero you have two functions one is x and other is x square So I will draw them in different colors. Let's say x is this line okay And x square is like this And when x is less than one But we have to take the minimum of the two minimum of the two means the part of the graph which is the minimum most So we'll have this part And then this part okay Similarly for the right-hand side x square minus one graph would be like this And y is equal to 2x graph will be like this correct Again, we have to take the minimum. Sorry the maximum of the two so maximum of the two would follow this graph And then this graph right so ultimately your figure is going to be Obtained like this right now This point is clearly going to be one and there is a King getting formed over here. So it is not differentiable at x equal to one So The fact that it is differentiable at x equal to one is a false statement. So this is my correct question answer By the way, we'll just quickly run through other options. This cannot be my right option because it is two It's continuous at zero It is not differentiable at exactly three points. That is also correct Because you can see that it is not differentiable here It is not differentiable here And of course at one so three points are there. So this is also a correct statement. So cannot be my answer It is continuous everywhere. That's correct. So this also cannot be my answer. So option number two is only The wrong statement and hence the right option Well done So we'll now move on to the next question question number 21 find the area bounded by y is equal to under root of five minus x square and y equal to mod of x minus one Yes So Rohan has responded. So this is the A graph for the given scenario Note that when they say y is equal to under root of five minus x square We we cannot make the full circle. We only have to take that part of the circle which has got y as positive Okay Mod x minus one is the graph of mod x shifted one units to the right So we can just take a shortcut. We can see that there are two small triangles formed over here, right? So let's say this triangle Let us find out the area of this triangle. Let's say I call it t1 triangle and t2 triangle Okay, so for finding the area of the t1 triangle I would need to know the x and the y coordinates of this point So I can say under root of five minus x square This line is going to be y equal to one minus x. This line is going to be y equal to x minus one So let's try to compare this. So five minus x square Is one plus x square minus two x That's going to give you two x square Uh minus two x minus four equal to zero. That's x square minus x minus two equal to zero This could be factorized as x minus two into x plus one equal to zero And this has to be x equal to negative one point Okay So this is negative one Okay, and this if I solve it Five minus x square is x square minus two x plus one I will get two x square Minus two x minus four equal to zero. That's x square minus x Uh minus two equal to zero same equation we'll get in fact So the other root which is x equal to two is the one which is uh to be used over here So x equal to two over here Now this triangle t1 the area would be half base into height Now the moment you put x as minus of one the y value is going to be two And here if I put x as one the y value is going to be one So area of t1 is half two into two which is two Area of t2 is Half into one into one That's going to be half So these two areas have to be subtracted that means Uh, I need to subtract Two plus half that is five by two from the area within the circle So for area within the circle, I can directly use my integration. So I'll just erase this part integration from minus one to integration of minus one to two of under root five minus x square dx That's going to be x under root x by two five minus x square Plus five by two sine inverse Sine inverse x by root five Is that fine? So if you substitute two Yeah, if you substitute two you will get uh one into one And this will become five by two sine inverse two by root five Correct and if you substitute a minus one you get a minus half Uh, this will again be two Okay, and it'll give you uh minus five by two sine inverse one by root five Okay, so when you Open the brackets you already get one plus One plus and this side will give you again a one Okay And you'll have five by two sine inverse two by root five Plus five by two sine inverse one by root five Is that fine? So let me make some space for myself Yeah So two plus five by two you can use the formula of sine inverse x plus sine inverse y for this That's going to be sine inverse x into Under root of one minus One by five that's going to be four by five So it'll be two by a root five again Okay, plus one by root five Into one by root five Okay, so that's going to be a four by five plus one by five that's going to be one So that's two plus five by two into five by two and you subtract a five by two from here So it's five five by four minus half That's five five Minus two by four that's option number three is going to be correct So a bit of formula that is used over here from inverse signometry That's sine inverse x under root one minus y square plus y under root one minus x square Okay, great. So most of you got this correct. So we'll move on to the next one That's a question from probability question number 22 Please read the question very carefully Okay See, uh, if you read this question a coin is tossed 10 times and the outcomes are listed listed H i be the event that ith outcome is a head And am be the event that the list contains exactly m heads So when you toss a coin 10 times you get exactly m heads in that That is your am Okay H i is basically the probability of getting the ith Outcome as a head Which is very simple. That's going to be half right any outcome head is half because it's a it's a fair coin But what is p a m? So if you want exactly m heads to come that means 10 minus m tail should be there, right? So, of course, you will say 10 cm half to the power of m and half to the power of 10 minus m correct So in both in all the possibilities, this is going to be 10 cm half to the power of 10 So i've done almost 60 of your solution here After that you proceed I think Rohan and Ramchiran have already responded to this All of you please remember that if An event is independent Then the product Let's say a and b are independent Then the product of a into p a into p b is product of is the The probability of a intersection b is the product of probability of a and probability of b So this is when your a and b are independent events Okay, so if I say what is the probability of h i intersection a m That means you want your ith outcome to be head And you want exactly m heads to appear that means If ith outcome is known to be head that means the remaining Nine throws should have given me m minus one heads correct So if the ith outcome is the head That means remaining nine minus remaining nine throws would have given me m minus one heads correct So probability for that to happen would be choosing any m minus one throws from the nine which will give you a head That's going to be this And the ith throw which is going to be a head is going to be half That's nothing but nine c m minus one into half to the power of 10 If you want this to be independent that means this expression that is nine c m minus one into half to the power 10 it should be the product of This and this separately so 10 c m half to the power of 10 into half So I think we can cancel off uh this term from both the sides so nine c Let me make some space So nine c m minus one is half times 10 c m Right, so I can use this formula n cr is n by r into n minus one cr minus one So I can use here 10 by m into nine c m minus one is equal to nine c m minus one. So this and this gets cancelled So I finally get five by m is equal to one. So m has to be five so a in independent of i a five A five will be independent with anything correct, so a five is independent Of any event so this option three will be correct Okay, it is independent of i i can be anything so only dictating factor is the five Which is only present in option three So unfortunately nobody could give the right answer to this Which part purvic you did not understand See i'll repeat this the entire process once again first of all P a m is clear that means if you want to get Exactly m heads out of 10 throws it will be a case of binomial distribution And getting a head on the ith throw would be just half That's also understood So p a m and p h i are clear Okay, now what about p h i intersection a m that means you want the ith throw to be head as well as you want m heads to appear in total out of 10 times that you throw so if you have already account for one head in the ith throw that means The remaining nine throws should have resulted into m minus one heads So this is what I have done the remaining nine throws should have resulted in m minus one heads Not exactly this it is without the half expression Okay This expression and this half is because you want the ith throw to give you A head so again a multiplication of half is required So the first part deals with getting uh m minus one heads in nine throws And the last half denotes getting a head on the ith some ith I can be anywhere in between Okay So this expression multi this expression together should be the product of these two separately So I saw them got the value of m So they would be independent only under the situation when m is five great great So let's now move on to the next question. That's question number 23 Okay, so most of you are going with one. So let's discuss this So if I have to find the inverse of this Okay, uh, we all know that we can use the substitution x equal to cos of let's say theta Okay, but remember the moment you say your x lies in the interval minus one two minus half Right Your theta will actually belong to the interval Two pi by three to pi Right So three theta will belong to the interval two pi to three pi So when I make this substitution over here, I get four cos cube theta minus three cos theta Which clearly is the formula for cos of three theta Which is nothing but saying Now be careful about the intervals of x Okay, let's say I want to take cos inverse on both the sides. I take on both the sides Now be very very careful here because Your three theta is not in the principal value branch. It is in the branch where your x is between two pi to three pi And you would remember the graph of cos inverse cos theta Cos inverse cos theta graph versus theta. It's like this zero two pi three pi So you are actually dealing with this branch You're actually dealing with this branch correct And this branch equation is going to be x minus two pi Or theta minus two pi whatever you can call it. So this is going to give you three theta minus two pi as your answer right cos inverse pi And cos inverse y is three theta three theta means three cos inverse x minus two pi correct, so let me take the Two pi on the other side. I have to make x the subject of the formula. So it becomes two pi plus cos inverse y So cos inverse x will be two pi by three plus one third cos inverse y So you can take cos on both the sides freely So your answer will be this and this is your f inverse y So f inverse x is going to be cos two pi by three plus one third cos inverse x And to the surprise of all of you option number four is going to be correct Okay, so I think after ramchar and people started realizing That option four is correct and not three. So these values should not be ignored when you are dealing with inverse signometric functions They're very very sensitive to the interval of x that you're dealing with Okay, so this question is an eye-opener to most of you. Please do not take inverse signometric functions very lightly They are very sensitive to the interval of x in which you are working with next Question number 24 Okay Undefined, this is by the way minus two Now please change your answer. Oh now everybody started taking domain very seriously In the last question what had happened fine. So let us first make uh dy by dx the subject of the formula over here So it's a x cube minus two x square minus x plus two Is equal to uh f of x times x square minus four Okay, so I can say that f of x is actually x cube minus two x square minus x plus two by x square minus four Okay, now the moment you put a two you would realize that it's actually giving you Zero by zero right so eight minus eight minus two plus two okay, but Two is a part of the domain because I'm just excluding negative two So there has to be a value for two and that value has to be nothing but the limit of this function as x-institute correct And for that I can factorize this Okay Cancel out a factor of x minus two from both the places and the moment we put two we get three by four That's option number two. That is correct. Again, none of you got it correct Surprisingly, is that fine? Did you find your mistakes? Great. So let's move on to the next one That's question number 25 So there is a function from r to r which satisfies this functional equation That is f of x into f of y minus f of x y is equal to x plus y for all x belonging all x and y belonging to real numbers f of one is given to be positive H of x is the product of f of x and f inverse x Then the length of the longest interval in which The function h of sine x plus cos x is strictly decreasing Yes, guys any idea anyone First the full focus should be on getting your function itself find the function And when when you know the function you can always find its inverse And then you would know your h of x Okay, so he has come up with one answer anyone else Okay, so see here first of all in order to figure out what is my f of x Uh, I would like to first find out what is the value of f of one. So let's put x and y both as one in this So f of one square minus f of one is equal to two Right, okay So this gives me a quadratic in f of f f of one. So let's say k is f of one So it gives you quadratic k square minus k minus two equal to zero which is factorizable And this is factorizable as x minus two times x k plus one that's zero So k value is two or k value is negative one But since they have given that f of one is positive this value is rejected. So f of one is going to be two Now, let us do one thing. Uh, let's put x as one sorry x as Sorry, let's put y as one Let's put y as one and let x be as it is so f of one minus f of x Is equal to x plus one So f of one is already two. So two f of x minus f of x is equal to x plus one That means f of x is going to be x plus one Right, this is a very big achievement for us because now we know f of x So f inverse x is also very simple. That's going to be x minus one, right So the product of f of x into f inverse x That's your h of x is going to be x square minus one. This is something which is important Now in this function, if you put your x itself as sine x plus cos x That means this will give you sine x plus cos x whole square minus one And I can actually write that as sine of two x Now they're asking you which is the longest interval in which sine two x is decreasing Okay, so sine two x if I just draw the graph of sine two x Any interval you can draw, okay You can see that sine two x. Let's say I take this as pi by four Okay, this will be pi by two. This will be three pi by four So the longest interval in which you will find it decreasing Will be from let's say I take an example three pi by four to pi by four interval That's going to be of length pi by two itself So option number two is going to be correct in this case. Is that fine? So, yeah, now I think sigh me here Got the right answer Okay Great. So we'll all move on to the next one. That's question number 26 This is a single integer type question If x plus y plus one equal to zero is a common tangent to the parabola y square is equal to four x and x square is equal to lambda y these two curves This is a common tangent Then find the value of lambda So please type in your answer in your chat box Okay size says four Okay, so mostly people are going for option four So let's check this out. So if this is a tangent to this of course, uh, let's see Let's take the fact that We are trying to solve these two simultaneously. So when we solve them simultaneously ideally I should be getting a quadratic in, uh, let's say Let's try to make a quadratic here in x So why I'm going to replace with, uh negative one negative x So I'm getting x square plus lambda x plus lambda equal to zero So this is a common tangent This must always give me the discriminant as zero discriminant means b square minus four ac should be equal to zero So lambda can either be zero or lambda can be four Now if lambda is zero this actually becomes the y-axis Now this cannot be a tangent to the y-axis because it is definitely crossing y-axis. So lambda has to be four So very clear cut option. Uh, the answer is going to be four Okay, let's move on to the next question. That's uh, sorry Question number 27 Yes done It's pretty simple if the same vector is If the same vector x is perpendicular to three vectors simultaneously means abc should be coplanar okay that means the Scala triple product should be zero. So absolutely that was a simple question Should have not taken you more than two seconds to answer this Next is question number 28 a circular sector is formed by a thread of length l. What is the ratio of the maximum area of the sector? To maximum area of the rectangle formed by the same thread Two plus Yes, yes, it's an integer type Okay, I think you're taking a lot of time in this. So first of all the area of the sector Let's say I consider that this sector of radius r Okay, we know that half r squared theta is the area of the sector So we also know the fact that the total length of the wire will be equal to two r plus r theta So I can say it's half r square in place of theta I can write it as This which is half r times l minus two r Is that fine? So, uh, if I differentiate this With respect to this is my area of the sector if I differentiate this with respect to r.  That's going to be l square by 16 itself. Okay, and if you want the Rectangle to the maximum area We all know that it has to be a square a square is basically a rectangle with maximum area So in this case it will be again l square by 16 So they're asking you what is the ratio ratio is going to be one in this case Is that fine? So guys two more questions are there. Uh, I would just Leave them for your homework Uh, not very difficult ones. One is the limit question other is a Shorter distance between skew lines ocean. So I would leave them as your homework so We'll call this session end Thank you so much everybody for coming online Uh, I think on second of april most of you would be having your board exams So there would not be any class on those days Okay And in case you have any doubt Or you want to solve more mock papers do connect with us Thank you so much over and out from my side Have a good day and best of luck