Full transcript of “CAT 2024 Algebra Marathon | 10hrs+ Algebra Revision | Important Questions for CAT Quants by ALP”. Click a timestamp to open the video at that moment. Need another video? Use the YouTube transcript generator.
0:10 good morning everyone okay so I can see everyone is here am I audible confirm the
0:57 audio copy confirm the audio everyone I think everyone is excited isn't it CH perfect so let's start okay there are certain rules I'll write that point number one
1:43 ask very few questions okay so because uh the assumption is that you have gone through entire algebra there's a final revision and in this we I won't be expecting people be asking me very basic concepts okay so that won't be there is that clear that is one thing the other thing is that after every 2.5 hours we'll take a break of 15 minutes so we'll make it realistic without you know going for 10 hours Full Throttle I don't think that will be a great idea for you or for me or for anyone any human so we'll just you know after 2.5 hours we'll take a 15 minutes break all right and all the questions we'll be doing all the questions we won't be doing much
2:28 Concepts over here so for every question you will get 2 minutes to solve and after that I'll be solving in 3 to 5 minutes depending on the question's difficulty is that clear so these are the rules and we will start with the first question Studio hi
3:15 Wades rise in special of course that is the only way Cruise of course question see smart guys are there Einstein will be doing questions and in the questions we'll be having some Concepts which I'll be discussing not the entire algebra of course because that is not possible I won't be doing
4:01 the basic concepts take so this question is simple um see in this question what happens whenever you have a question like this in which you know if you change the variables it doesn't really matter we call them symmetric Expressions when symetric expression just do one thing so this thing will be true for all sorts of
4:47 values of the variables a b CDE e okay what I'm saying what are symmetric equations if you change if you change you know um the if you interchange the values of variables expression doesn't change in that case so a lot of you will be asking me that what is induction and where we use it we'll use it only in symmetric Expressions I'm repeating whenever you interchange the variables things do not change if you make a as b b as C and whatever it won't change okay so in that case you can use any value for ab bcde e the simplest value which we know is one isn't it let's put all of it equal to 1 and get the answer 1 + 1 2 C + again two so we'll get this isn't
5:33 it 5 * 2 upon 1 into 1 into 1 into one into one and that is how you get the answer that is a kind of a shortcut technique which will be very much helpful in algebra questions and just pray that something like that cat comes in cat and it actually comes in cat is that clear everyone AR no understood tell me everyone that you understood it how and where to use induction cool now coming to next question 2 minutes for
6:20 you there's some J aspirant over here it will be useful for them as well the questions are kind of relevant to both okay distinct values yeaha if they're distinct values then you can uh just put in actual listing values 1 2 3 5 okay but that question wouldn't be coming in this
7:05 form it will have it will be having a different form altoe all right I can see here people are giving the answer so fast have you guys gotten smarter what's happened with you we need to find the least value of this K here anyone would tell
7:58 me m one everyone has put in one is that right here you can't use induction see X is greater than zero that is there moreover it's not a symmetric expression it is not a symmetric expression not really because we have only one variable that is X okay so if we we cannot interchange with it with anything else it can less than one also exactly can be fractional greater than zero
8:43 fraction isn't it all right so listen to me very carefully whenever normally what happened if you have questions like this least value of something if it is a quadratic function then we know that the average of the roots at the average of roots you'll find the Minima or the maximum isn't it tashn through the questions uh I will discuss the concepts as well Cruise way symmetric you can interchange the variables all right well so in other cases there is a high possibility that you will use the inequality that is am and g m do you
9:31 know this inequality so I I will revise this concept because this is very important concept okay whatever is important I'll just revise that differentiation can also be done calculus is not in the syllabus I'll do it by normal algebraic method so let's say we have two things everybody pay attention and also harmonic means so everybody needs to pay attention over here this is very important concept are you writing it note these things down so let's say I have two numbers A and B okay it's arithmetic mean will be a plus b arithmetic means nothing but average add the numbers divide by number of elements okay geometric mean will be you multiply those numbers and take
10:17 the nth roote so here we have two numbers we'll take the second root if we had three numbers we take the three roots and so on the harmonic mean in this case will be 2 a Upon A + B in this case but if we had three if we had three three values then it would have been 3 a c Upon A + B C plus CA is that clear everyone all right everyone yes if you know um differentation then
11:05 of course tell me is this part clear or not this is going to be very much helpful in the questions of algebra well so I want to do something with this how can I do it what needs to be done is there a way let me break it as 13.5x Cub plus see here we have 3 X's here we have only one I somehow want to have 3 X's so can I write it as 1X 6X + 1X 6X + 1X 6X is that a problem I can rewrite FX as
11:52 this isn't it yes moit it will be saved on YouTube sh it's not recorded I'm answering to your query so of course it's not recorded looks good everyone now you'll be asking why I took this because here I have three times X's here have only one I made it three because when I try to find a geometric mean all the X's will be canceled out clear so arithmetic mean here will be let's add all of it add why to waste time and space arithmetic mean will simply be FX upon 4
12:41 is that right and the geometric mean will be I multiply all of it so I'll get 13.5 upon 216 car 4th root is that clear everyone looks good perfect now for from the inequality amgm inequality we can just write by the way you can solve this also 13.5 double 27
13:27 27 8 * 216 this is 1x 16 to the power 1X 4 which means this is 1x2 isn't it so am must be greater than or equal to GM which means FX by 4 must be greater than or equal to 1x 2 see how cool is this so FX value should be greater than or equal to two we want to find the least value answer is two two very good pronoun there's a very important uh you know uh I would say theorem amgm
14:16 inequality it's lovely isn't it cool let's go to the next question we won't be using calculus today with this is purely algebra okay if I take calculus someday
15:03 we'll be doing calculus as well D archit all right homin okay W 2 minutes for solving 1 minute for noting Down 2 minutes for me solving we'll be aiming to solve one question completely in 7 to 8 minutes
16:11 okay okay
16:56 luxes answer to do for
17:56 all right do 2 minutes over let's discuss p and Q are positive integers there are certain things you know which we must keep in mind during the exams like if you have something like in this format where you see that two variables are getting added and then their product is also getting added listen to me carefully if you see two variables and I hope you're noting these things down the concepts that are there which will be very useful not the basic ones but the ones which will be helpful to you so if you have two variables and those are getting added and the product is also getting added so ideally you know what should you do you should add one on both the sides now why I'm doing that because this is
18:41 how I'll be getting a proper factorizable expression so I can take if you see here I getting 1+ p in bracket and then from here I can take Q common I'll again get 1 plus p in the bracket is that clear here I can take 1+ P common so all I'll be left with is 1+ p * 1 + Q is equal to 72 this is going to be a lifesaver in expressions like this clear cool okay also can I say can I say that these two are factors of 72 isn't it
19:30 yeah so we can make a table and then try to find things out 1+ p p 1 + q q for the values of 1+ p 1+ q we can find the Val p p and Q as well we have to find what do we have to find to find this p² + q s that is what we need to find out okay all right so I'll also write P Square here I also write Q Square here and then I'll write p sare + Q sare over here by the way there's one more information given to us from this second information we can take PQ common I'll be left with p +
20:16 q that is 8 8 0 all right okay we can put the values JJ is that clear to factorize the equations okay now it's it's very simple 72 factors 1 into 72 by the way we cannot take 1 because if I take 1 plus p is 1 P will be zero doesn't make sense so 2 into 36 2 into 36 is 72 which means P will be 1 Q will be 35 all right similarly 3 into 24 does that make sense 3 24 is all 72 isn't it and so is
21:05 4 into 18 6 into 12 8 into 9 so P value three either 17 five either 11 7 either 8 all right P Square 1 2 square is 4 3 square is 9 5 Square 20 5 7 Square 49 Q Square 1 22 5 23 square is 5 29 17 square is 289 11 square is 121 here it is 64 cool
21:53 take P plus Q p + Q 1 + 35 is 36 2 + 23 is 25 3 + 17 is 20 this is 20 5 + 11 is 55 sorry 5 + 11 is 16 what I'm writing and 9 + 7 + 8 is 15 take cool PQ into p+ Q
22:40 880 I'm making it very simple the thing is you don't have to do this much you can do it mentally as well 2 into 23 is 46 3 into 17 is 51 5 into 11 is 55 I think I got the answer this is the correct value 16 into 55 is 880 isn't it this is our correct value I'll just encircle everything here 16 * 55 is actually 880 find p sare + q sare p 25 qare 121 add them you get 146 as the correct answer lot of you must have got it
23:46 clear yeah lakia of course I'm explaining the entire thing so everybody gets it probably it's just that explaining each and everything okay it's for everyone no a lot of people watching it afterwards man complete let's go to question number four [Music]
24:32 you can do that as well MB self repair yeah of course that's not a problem you can do both the ways in this equation you have to find the quadratic in this question you find the quadratic equation so of course the answer would not be in terms of numbers am I'm good good morning yeah as I said quadratic
25:51 a sure because if you assume P plus Q to be the sum of roots and PQ to product of roots that is how you can to do it all right [Music] Karen J that was clever right in the form of uh you know [Music]
26:37 ABC quite smart karanjit you need to find the value of P here and put it there n of course okay let's do this one first um let's set Q is equal to P - 1 p and p + 1 consider quadratic equation x² - a x + B is equal to 0 and x² - A + 2 * x + C = 0 the roots of each of the quadratic
27:25 equations are distinct and part of set Q Okay so set Q we have three elements isn't it so out of these three any2 will be the roots of these two equations after that the roots of um find the quadratic equation with the roots p minus1 and p + 1 so obviously third root that we need to find out will be equal to x² - 2 PX + p² - 1 is equal to 0 isn't it if someone for has forgotten it I'll just write s o r is sum of roots p is product of roots if you know the roots this is how you can frame the equation okay now
28:12 having said that so we have two Roots at P minus one and P + 1 okay one will with p and p minus one other will be with p and p+ one is that right there can be three combinations one of which is asked in the question itself remaining two are here looks good so I'm calling them equation number one equation number two for equation one the sum of roots is equal to how much a and for equation 2 the sum of roots is how much it's a + 2 so obviously sum in equation 2 is
28:57 greater that means we'll take p and p +1 over here and p and p minus one over here does that make sense we're not sure that where to take p and p minus one and where to take p and p + 1 but because the sum we look at the sum and now we realize that this is how it should be taken perfect take okay so well so a is 2 p minus one and here same it will be same thing okay what else can be calculated her humate
29:44 cases saki what happened the sum of roots and to find the product of roots P into P minus one this will be equal to how much here B and in this case P into p + one will be equal to how much be equal to C clear so now we have some you know equations in terms of p b and c so we need to somehow eliminate P from here and write in terms of b c a whatever is possible is that right okay this can be and I expand it I get p² - P
30:32 isal to B here I get p² + P isal to C obviously we can find the value so if I call it third equation and this is fourth equation to find the value of P can just subtract three from four isn't it the moment I do it P Square gets canceled all I have is 2 p is equal to C minus B what I can say p is equal to C - B upon 2 put the value of p over here and get the correct answer all right perfect you can do it after this right
31:18 let's put the value of p and cus V by2 in this equation and get your answer by the way if this question comes in the real cat you'll have the options in terms of Expressions so this question will be very easy it looks difficult but it will be very easy because you cannot have a TAA question in this form isn't it that is why I say algebra is very easy by the way all of you try to understand that algebra is the easiest and quickest subsection of quants hush depends on what the options are all right let's go to the next question um this is I would say I would rate this question as um a difficult question this is difficult the first question others were
32:05 moderate to difficult but this one is difficult I'm giving you 2 minutes completely yes yes lxa we can get multiple answers brother John you can get multiple answers all right it all depends on the options options are not given it's us that you understand the method that's allometry
33:02 multiple ways all depends on what the options are so let's do this one this is you will you will love this question is that I'm giving you prop 3 minutes to this question this will not be an easy thing yes Isha recorded will be [Applause]
33:49 there shagun smart shagun is smart shagun I think you're the same person yesterday when I asked you you join the course recently [Music] right all right arendra I'm seeing you for the very first time
34:40 [Music] the things that are doing today I'm kind of uh I won't say completely sure but somehow I feel that the question that be doing today high chances
35:30 90 seconds to go n PDF session record
36:34 go I'll I'll see that c what happened table AA okay tell let's try to solve it a function fxy is defined such that
37:20 fxy is equal to fyx okay and FX y + x is equal to x + y y upon x * by fxy great and FXX is equal to X it's basically Loops you know step one then step two that's how it it it happens um with X and Y positive integers the value of f58 13 by the way if you look at the first statement can you say that f583 is equal to f13a 58 would that make sense
38:11 looks good okay and try to reframe this in this format so can I say this will be equal to F of 13 comma 13 + 45 perfect which will be nothing but 13 + 45 upon 45 x + y upon oh sorry Loop
39:00 B multiplied by F of 13A 45 that is what we will be getting all right is that is that clear everyone
40:07 what happened x + y upon X oh the person who has made type this is why actually otherwise it won't be able to you will be able to solve it h say it's y it's y the denominator it's y that is how this function be made the recursive function
41:15 all right now huh up take okay split split because it's given in the question no that whenever you have something like this FXX we have the option to calculate it that is why I split okay CH so f345 now everybody pay attention this f345 what can be written what can I write it this is f 13 and then 13 plus + 22 isn't it is that
42:02 right shall be 13 + 45 is 58 no 58 minus that's why I got 45 clear is that clear a no X is not 58 anymore AA the moment you that's what this function means in these functions what happens moment to convert it to convert it okay 58 okay 13 + 32 okay this will be equal to how much F of 13 +
42:50 32 13 + 32 upon 32 all right multiplied by F of 13 comma 32 is that clear is that clear similarly let's go ahead so F of 13 comma 32 this can be written as F of 13 comma 13 + 19 all right so now I'll get this will be equal to how much 13 + 19 upon 19 multiplied by f13 comma 19 we'll keep
43:38 on doing that interesting isn't it after that now we can write this f13 comma 19 as F of 13 comma 6 + 13 all right Kik say which will be equal to will be equal to 13 + 6 upon 6
44:27 multiplied by F of 6A 13 what next 58 45 13 plus 45 14 up 13 45 32 difference 13 + 32 upon 32 difference 19 upon 19 difference is that right so this is the pattern that we can realize so F of
45:16 613 this is nothing but F of 6 6 + 7 which will be equal to how much do you get it 6 + 7 upon 7 63 and see whatever write it just that see whatever write doesn't really matter what matters is that what is the difference seven 19 so whatever the difference is that is coming under the
46:02 denominator denominator and this part is going to the numerator that is how this entire function is being made do you realize that F of 6A 7 difference is of one do you understand that part okay so the board is filled up now you can do it again bua directly ultimately what we will get what are we going to get in the end let's write that so we will get 58 upon
46:50 45 after that 45 upon 32 after that 32 upon 19 now you can see the pattern 5 45 the difference was 13 45 32 difference okay so that's how we are breaking it again and again 32 by 19 after that this will be 19 upon different this is 13 oh sorry H take it after that 13 upon
47:37 7 after that 7 upon 6 6 upon 5 5 upon 4 4 upon 3 3 upon 2 and in the end you'll get 2 upon 1 that's it that's it that is recursion gag exactly things are getting canceled out is that cool
48:43 is 1 upon 7 so whatever things are getting canceled out 7766 55 44 3322 uh what is left with us 58 multiplied by 13 what's the problem yeah you you can just expand that right and this is what we'll get so answer is 754 upon
49:28 7 + 6 differen is of one differen is of one no one right okay CH you calculate that you got the approach isn't it let's go ahead take some H right right right basically you got the approach just calculate it you'll get in this this in the end uh let's go to the next question
50:46 she answer 754 sh exactly huh problem is see this is recursive function fxy FX
51:38 Kik we should not calculate that last at till last by looking at the pattern you can directly calculate it that's what did in the end end looking the pattern 58 13 okay Kik exactly Z okay just remind me after uh 10 questions are done all right in case you want me to write it again
52:45 arithmetic revision Joy for
53:54 pram uh will'll take a break after 2.5 hours like first break will be at what 12:30 here I'll rewrite it at 216 * by 36 to the power X upon 64 to the^
54:46 XA is it visible now for for
56:13 the answer is 216 [Music] exactly how many of you unable to approach this question right the 6 power 2x is 2^ 6X isn't it so I'll get 216 in the left hand side and sending everything to the right hand side this is 6 the^ x this 2^ 6 x all right so I'll just write that 6 ^ x multiplied by 2 ^ 6 x
57:00 upon 6 ^ 2x multiplied by 2^ 9x looks good which also means that 216 will be equal to 6 power - x multiplied by 2 power - 3x all right I can say if I do it you
57:49 know- 3 here okay so I can write it like this 216 is equal to 1 upon 6 ^ x multiplied by 1 upon 8 to ^ x would that look good yeah that is 48 to ^ x is equal to 1 upon 216 perfect now let's take log on both sides exponent log with what log with base base
58:35 con let's take base 48 okay so in that case I'll get all right this will be equal to minus log 216 base 48 looks good so far so good now what we need to find this is what we need to find
59:23 [Music] this is log 48 base 48 is one only right so I can directly write X is equal to minus log 216 base 48 isn't it that is what we have with us okay all right satu what
1:00:17 happened okay now we'll put the value of x over here this value of x over there what we need to find out why minus this 1 216 2 6us okay right okay let's put there what we need to find 8 to the power - 1 + log 6 base 8 and X is this minus minus multiply plus and we'll get log 216 base
1:01:03 48 all right okay now when you have something like this log 6 base 8 one is I can write in terms of log 8 isn't it I can write this as log 8 base 8 okay so let's write it this as 8 in the bracket this becomes log 8 base 8 or I can say log 48 base 8 multiplied
1:01:50 by log 216 base 48 now this is something in the powers if I put the change of Base then what's going to happen log 48 isn't it the log 48 will be cancel out so 8 log 48 upon log 8 multiplied by log 216 upon log 48 this gets cancel out so all we are left with is 8 log 216 base 8 which will be equal to 21 six that's
1:02:38 it last step P sha this one there one and what is one log of something to the same base something which is question only see I'll just write it this one no this one I written this one as log 8 base 8 I can add these two the log 8 plus log
1:03:25 6 will log 8 into 6 that is the reason now it's clear okay after that all of us know that log a base B can be written as log a upon log B to any base so it is log 48 base 8 wrot it as log 48 upon log 8 log by using this this formula so keep on noting these identities this one and this one I have used these two to solve
1:04:11 it clear uh this is I think just that the properties of log should be in mind I need the long way the proper mathematical way so that everybody understands one last property
1:05:05 last step what is the problem this it's a property no it's like a to the^ log X base a is equal to X this this and this these three properties have been used here Soo let's do this question this can be difficult for many
1:06:05 last property that is very common anyways this question I can say it's a difficult question one of the difficult questions of this session so time 3 minutes yeah [Music]
1:06:54 for for for
1:08:11 okay uh those who are not able to understand this question FN FNX is given as F of FN minus1 X f2x that will be F of f1x x f of FX of FX so f1x is only FX f2x is f of FX if I say f3x will be F of f of FX is that clear nested functions
1:09:21 foreign spee okay like that now what does this mean if I write f2x this will be F of FX if I write f3x this will be F of f of FX is that clear now
1:10:12 something a lot of people are solving it for
1:11:13 nested functions are exactly over here Sam is getting 243 as the answer okay Sam let's see for
1:12:31 no goik Chell okay let's solve it now time up H I will give you a
1:13:17 very interesting thing over here which I have not given you yet in the the theorem in the live class that is whenever you have this kind of function note it down somewhere if FNX is equal to F of FN -1 x then FX is equal to mx + C and it's given in the question also see the graph x represents a straight line is that
1:14:10 clear by it is very important and in the question also it's given directly so FX is MX plus C all right which also means that if I try to calculate FS2 x what do I get I'll get M of MX + C + C is the step clear to all of you is the Second Step clear to all of you [Music] second step is
1:14:56 clear H vishaka MX plus C straight line equation where m is the slope I hope all of you know this right m is slope and C is the Y intercept okay so FX is MX plus C f2x is this all I want to do is to find a pattern if I try to find f3x to M of this whole m² X Plus mc+ C is that understood plus C clear
1:15:42 clear expand what do I get mbx + m² C plus MC plus C do you find a pattern yet pattern except the first term these three are here these two or here only this this one C for MC plus ca for m² C plus m c plus c and so on so so if I go on like this F of NX X will be equal to m^
1:16:28 NX plus 3 2 2 one 1 Z so this will be m n - 1 * C plus MN - 2 * uh Power two power 2 * C + m ^ N - 3 * C and so on so on till the end we'll get C is that clear everyone is this clear all of you we got the value of FNX so simp to make it look prettier so I can take C common and then
1:17:16 write it like this just like binomial theorem all right is that fine once we have got that now I can calculate two things so let me do one thing what if I calculate fn0 basically putting x equal to 0 what do I get I'll get this hole all
1:18:02 right this is fn0 and fn0 is given as two this is equal to two it's given in the question okay all right and let me calculate fn1 also that's given in the question fn1 value 245 so let's do that calate fn1 to MN M Power n plus this whole this whole am I right is nothing but FN zero looks good this thing is equal
1:18:52 245 this is equal to m^ n so basically we get 243 is equal to m to the power n in the end we get this okay it's not that difficult it's pretty easy just that lot of values are written so it looks difficult clear M Power n is 243 want to find the value of M slope so what all condition what all things can be there we can have 3 to ^ 5
1:19:41 or we can have 243 to the^ 1 there are only two possibilities is that right is that right of course n cannot be one it's not just one term so we'll choose this possibility wherein m is equal to three that's the answer three is the answer interesting question isn't it for
1:21:11 PDF share okay one at one it won't be satisfying no see F1 you don't have the value for F0 h f of Z power me zero that's why okay now coming to the next one
1:22:10 answer basic questions a lot of you are doing it so you can find the shortcuts and whatever whatever comes to your mind but this in this question the main learning was that it represented the equation of a straight line c equal question exactly easy I prepared you for that in the live
1:22:57 classes moderate Zak for
1:23:56 do we have some IIT J parents also here F1 is four and F2 is five respectively all right H shun Shang your idj is Ben it so is pramesh I think
1:24:47 okay okay Chris way J Advanced Punit okay is it helping you for IID of course I'm not teaching entire thing but the questions are they helping you somehow are they relevant to you Rema VES all right here R elant
1:25:49 right har 10 maths drop k okay for okay so in this question what we need to
1:26:35 do polom it leaves the remainder of four and five so I think all of you know um remainder theorem isn't it that is something which is very common so here the value of F1 will be equal to 4 which means if I put one everywhere I'll get a minus b + C + D + 3 is equal to 4 all right is that clear which means a - b + C + D is equal to 1
1:27:22 right something missing no it's okay coming to F2 F2 F2 let's calculate okay here you can see that x² - 3x + 2 can be factorized as X - 1 * x - 2 isn't it so basically if I divide this GX by x - one or x -2 I'll get zero as the remainder they're completely divisible isn't it so the value of G1 is zero and the value of G2 is also zero is that right do you get
1:28:18 this yeah that is clear no take okay not just this now if you look at the uh Expressions FX and GX they look pretty much common uh similar isn't it problem the difference is that here we have D here we have two here we have three here we have e can I do something with FX in terms of GX can write FX is equal to GX can I manipulate this somehow D minus
1:29:06 2 * X GX me is that right plus 3 - e tell me everyone is this step clear can I write FX as this does that make sense everyone I'm looking at the comments v f says GX say huh GX is divisible by this and you factorize this as x -1 into xus
1:29:52 2 so if you divide GX by x - one you'll get Z as the remainder if you divide GX by x - you'll get 0 as the remainder that is why by remainder the I wrote this you just look at this FX minus GX FX G this now it makes sense now it makes sense everyone those who are not able to understand this if I subtract FX minus GX what do I get I'll get DX - 2x + 3 -
1:30:38 c okay now it looks good everyone got it great now listen me carefully to you know to avoid confusion and to make think simpler I will do was just one thing I will assume this as an other function let that be HX FX is now I can say that FX is equal to GX + HX how cool is that but HX is difference of FX and GX perfect now life will be very simple
1:31:26 at xal to 1 I'll get fub1 is equal to G1 + H1 F1 value 4 G1 value zero to H1 value four clear Floy exactly V here and similarly F2 F2 will be G2 plus H2 F2 value five G2 value zero H2 value
1:32:14 five yeah I'll wait for some a few seconds all of you if you're writing it down or absorbing the information just do that Ren H1 say that is nothing but d - 2 + 3 - e isn't it and that is equal to 4 which means the value of D minus E am I
1:33:04 right and H2 2D - 4 + 3 - e that is what 5 which also means 2D minus E will be equal to 6 now I don't think I need to say anything first equation second equation put you can calculate the value of d and e the value of D will be three value of e will be 0 2D + e 2 into 3 + 0 the answer is six is that clear okay AB arra it will be available
1:34:00 clever comment both English and Hindi okay is that clear everyone from next question I'll be using both the languages by the way tell me something how do you like this question okay Floyd I'm waiting for some time to you guys so that you can note it down noted mind bending huh McAn all right
1:34:59 yes exactly Dy here we have the next one this one is I think easy all of you can do it in 1 minute one of course it will be available on the the moment it gets ended 30 minutes May available pun given the
1:35:47 question F2 a value five it was given the live class when we doing there's many questions right Sid
1:36:38 one questions G x- or X divisible GX value question given it was given in the question PF
1:37:52 h for for
1:38:59 can answer 1.67 2.8 okay A + B is 60% that means 0.6 of A + B + C all right and ab is 320% more that means 4.2 * A + B + C clear if I divide a by A+ B I'll get
1:39:46 7 so far so good clear to everyone yes is that clear GM is greater than or equal to hm G so AB square root is greater than or equal to hm that means 2 a Upon A + B root a root a just for sake of easy easy let this be
1:40:32 equal to X to save some space so this will be root of 0.6 X is greater than equal 2 into 7 that is 14 clear it 4.2x not 0.6 I'm sorry squaring both sides what do I get 4.2x is greater than equal to 196 x is greater than or equal to this 196 into 10 upon 42 14 3 are 14 14 are
1:41:19 140 by 3 3 4 are 12 46 66 integral value the answer is 47 we use the inequality yes PR 3% mod is 4.2 so nice question isn't
1:42:09 it exactly here this first equation the second equation you know second divided by one when to use GMM whenever you have values like this this is the value hm just say when you two things are multiplying right GM when two things are when you divide okay divide the product by the sum you use hm is that clear
1:42:57 sui if in a question the product note everybody not it down somewhere you will use GMM inequality when you need to divide the product by the sum noted it down somewhere everybody LED tell me in the comments this is very important okay not
1:43:45 repeat when you have to divide the sum by the product of two numbers use GM hm inequality repeating use GM hm inequality when you need to divide the product by the sum of two numbers you can use amgm also yeah okay so let's go ahead the roots of the equation this are A and B where A and B is not real they're not
1:44:32 real if a cub is equal to this then what is the value of Q by P CH solve it real ual answer is 47 integral value integral value okay [Music]
1:45:38 CL come Cru way IIT aspirant I'm assuming that you will solve this question quickly quicker than gaspin let's see for e
1:47:06 naep hris say defined by B zero n
1:47:57 shagun two what is the value of p and Q for
1:49:04 roots of equation this are A and B where A and B is not real take okay this's one thing that we know um a + b will be equal to minus B and AB will be equal to Q okay we know this right don't we okay Sumit let's see is that right all of you a is equal to 16 < tk2 so can I say a minus 16 < tk2 equal 0
1:49:58 this is not the only case there more cases okay and I know a lot of you have done it the same way so this if I expand it I'll get a minus 2 < tk2 and a² + 2 < tk2 a + 8 is equal to Z how many of you did it did this well A and B is not real and a so of course 2k2 this is something which we do not have to consider this is what we need to consider isn't it so
1:50:43 let's consider that uh minus 2k2 + - b s 2un 4 2 are 8 - 32 upon two is what we get clear so the value of a will be if you solve it 22 minus < tk2 and is - 24 4 comes out right so plus minus 4 two plus minus square root of - 6 that's what we get isn't it and B are real is not real 2 root2 is real number irrational but
1:51:30 real is this clear so value of a is this a basically two Roots we got the two Roots so I can just say that a is this or conjugate is B so one is - 2+ Root 6 other is - 2 - root6 is that clear now let's calculate the value of P a plus b if you add these two imaginary part gets canceled out I have minus 2 < tk2 you calculate this AB to is square minus Square all right so a will be equal to uh 2 + 6 that is 8 a m q
1:52:29 clear not comp we have not used I yet isn't it so Q upon p first case a 2 root2 is real no Q by calculate that
1:53:38 yes I'm yeah bre them nice question isn't it do you like it of course you did let's go to next one
1:54:28 nil by using a cub minus B Cube identity a minus B into a s + 2 a + b sare option what happened uh
1:55:18 problem did I say that minus p H this will be what P Val plus 2un will be the answer notus 2 root2 A plus b is minus B okay yeah so right what is the symbol okay see what is Sigma all of you know that it's
1:56:04 summation and when you Pi Capital Pi this is product in series sum of series product of series series product Bic it is A1 multiplied by A2 multili by till a 620 this is the question this is what they are asking all
1:56:50 right nice let's take log everywhere so what do you have A1 will be log 6 by Log 5 A2 will be log log 7 by log 6 and so on till uh a 620 will be log 625 upon log 624
1:58:00 as you see that numerators basically 6 all be are left with is log 625 upon Log 5 now this can be written by change of bases log 625 base 5 that is four 5 4th power we get 625 of course let's go to next one
1:58:58 cuz we're right msan break is at 12:15 C multiplication RI Basics basic
1:59:45 playlist s crel
2:00:46 yes Punit for chrisy what's your name base two m not in M minus
2:01:48 n hey Ela how are you you late as usual CL oh kundan okay and I good I wish break how many of you not able to solve this it's log on M only Kosik not on M
2:02:34 minus n right okay Ela that was late anyways so log a minus log B is log a by B now I'm
2:03:21 assuming that you know these identities right can I do something can I factorize something the first thing that comes to all of our minds is factorizing um I can take certain things common here isn't it wow what if I take can I take something common let me take x² Comm say say so if I take x square common I'll get x square + x + 1 plus 2 but I'm taking it as x² +
2:04:08 x² is this clear 2 X+ x s all right is the is this clear everyone which also means what will I get x² - x + 1 x² plus x + power 4
2:04:57 looks good so as you can see here because the value of log can never be zero these guys can never be zero I can take x s X+ 1 common and cancel it out so I'll be left with log base 2 is this clear I have canceled out x² + x + 1 because this cannot be zero log log Z is not possible it can't be zero so it can be cancel out clear let us do one thing let Z is equal to this
2:05:54 that should be greater than zero all right this is more important whatever you feel like sending it there is that clear x² Comm zus one plus zus okay plus Z + x - 1 is equal to Z am I
2:06:40 right in now it looks like quadratic equation isn't it is that cool all right everyone comparing to quadratic equation here you can see B is 1 a is z minus 1 and C
2:07:26 is z minus one is that right this must be greater than or uh you know this this thing will be will always be positive isn't it imaginary real it must be real okay so this will be real how uh b s - 4 a c let's calculate that 1 - 4 * Z - 1 s must be greater than or equal to Z this has to be real is that right yes samak it must be
2:08:18 real which means 4 * - 1 whole Square must be less than or equal to 1 so value of Z lies between 1X 2 1X 2 1X 4 + 1 3x2 we got the value interval of uh Z 1x2 to 3x2 T this is Zed we want to calculate this hole and this whole was actually Y and question may given Y is equal to
2:09:06 this yeah do we have any queries here let me see 1X 4 Square b sare - 4 A C A and C are zus 1 each ran b² you discriminant by discriminant quadratic equation b² - 4 AC CH we have done that now what we got the value of M are you writing things
2:10:14 why take only okay now I'm going further noted Z 1 by 2 3x2 y just calculated let's calculate that so the value of y log log of 1x2 at base 2 log of 3x4 3x2
2:10:59 what was that 3x2 at base two okay so log to b 1 by isus one am I right 3 base two to Y Min is minus1 and Y Max is uh what log 3x 2 base 2 but log 3 base 2 minus log 2 base 2 can I write it like this
2:11:54 Ina sum sum of Maximum minimum values let's add them y Min plus y Max what do you get log 3 base 2 minus 2 is what you get compare with this m value Three N value 2 M plus n value five
2:12:40 n what happened Z last Z Cala inequality sign I told you that I said the pattern is what you have got you can calculate it in in case you want me to do that again I will do that now coming to numerator break that is what you guys want to know how you got minimum value of
2:13:27 Z taking discriminant what happened I'm erasing it the last step while getting Z if you remember we got this thing 4 * Z - 1 whole square is less than equal to 1 isn't it those who have problem those who have problem on this I'm showing that to you again okay all of you pay attention sending four there Z - 1 whole square is less than equal to 1 by 4 is that right taking Square root- 1 1 that is the value of zus
2:14:14 oneus let's add plus one plus one everywhere - 2 + 1 less than equal Z - 1 + 1 is less thanal 1X 2 + 1 now it's clear 1 by two Z 3x2 is this clear now those who are asking Z Val let me look at the comments got it H not for this question but for like how
2:14:59 we factorize this okay a k this is a b quadratic expression here the power is four factors isn't it break okay everybody pay
2:15:48 attention you write x s okay x² X or 1 positive we need to multiply it by something so that we get power x² x 4 but x² MTI x x cub x² is- xus X do you understand
2:16:35 that yeah all of you is that clear okay and that's not just it we like that that is why these two will be the factors is that clear
2:17:21 got it everyone that is how I factorized looks good plus one into plus one to plus one that is how I multiplied in my childhood so I still do
2:18:10 it I just shared that with you guys take let's go to next one so let's go ahead
2:19:18 break just one question more which at 12:30 there'll be break I said that
2:20:49 I for
2:21:34 okay musan very good 100 is the answer okay uh this function FX lies between 0 and one both including right it's given in the question itself am I clear I it's your question I will tell okay so now listen to me carefully everybody attention
2:22:26 what will we getra if we have a everybody pay attention all of us know that quadratic expression minim Maxim
2:23:17 quadratic and get a quadratic expression is that right is it understood everyone so we have x² T sending it here - x + t - A is equal to 0 we have this quadratic expression with us all right f b s - 4 a c 1 - 4 * T into t - a t ² - t a right this must be greater than or
2:24:05 equal to zero rest you can solve I think one minus minus inequality change we'll get t^2 minus t a is less than or equal to uh what 1x4 right T into T minus a is less than equal to 1 by 4 is that right is that right everyone
2:24:52 this is always positive this thing is positive FX is positive this is positive yeah t is positive positive we get real numbers take real Roots okay perfect we have to find the values of a in the interval 0 to 100 0 to 100 that's what we need to find out okay T into T minus a is what we have obtained
2:25:45 already okay let's look at this again so yeah x + a upon x² + 1 lies between 0 and 1 both included if I solve these two what do I get that X will always be this x will always be greater than or equal to minus a am I right say and also if I solve these two I'll get x + a is less than equal to 1 am I right one x sare + 1 all right these two things are
2:26:34 perfect which means a is less than equal to x² - x + 1 yeah combining both of them that a will be a will always be greater than 1X
2:27:24 4 which means from 1 to 100 get assume any values the answer is 100 by Alpha Trader by discriminant b square - 4 a all right
2:28:29 fresh air did this way 1X the range is from 0 to 100 1x4 integral values the 1 2 3 4 5 6 till 100 all right okay shortcut uh this is FX everybody pay
2:29:16 attention FX is x + a upon X square + 1 value of fub1 fub1 a + 1 upon 2 isn't it is that right see a lies Z and one a is a fraction it can be zero or at Max it can be one isn't it that means this is this thing is always greater than sorry less than or equal to zero can I say that F1 is always equal to
2:30:13 zero that's a the best way you can solve this question that's a shortcut shortcut okay isn't it in this how less than equal to zero a value it starts from Zer to one you put a as zero
2:30:59 you get 1 by two if you put a as 1 you get 2 by two Zer one sorry sorry sorry sorry one one one to F1 H at one at x equal to 1 it's not zero it's one it's one that was my typing mistake writing
2:31:48 mistake I'm got it chal so now it's a break and I will just do one thing new slide I'll just write here what's the time 12:38 12:40 break till 1255 p.m. so if someone visits for someone visiting in between okay they'll get to know there it's a break
2:32:34 time all right for
2:33:48 e
2:51:17 e e all right am I audible Great I Am
2:52:16 take did this we did this as well right let's do it once more there was some problem in place of P putting x equal to 1 we needed to put this entire FX equal to 1 this whole thing should be one that was the main problem over there okay so it's not F1 it will be FX equal to 1 so if I put X+ a that was the problem which we were solving is equal to 1 then we'll get the correct value yeah shortcut all right the problem I'm repeating the problem that we did here was that we
2:53:02 tried to put are you listening to me x equal one place of that just put this equal one exactly man yeah right once we do that it will be easy for us okay so what do we get now x + a is equal to x² + 1 which means a is equal to x² - x + 1 all right now it will be clear
2:53:53 yes now take out the discriminant try to calculate the discriminant now and things will be pretty simple moreover should be written as x² - x + 1 - A is equal to0 let's calculate discriminant now that will be -1 s - 4 * 1 - A all right is that clear is greater than equal to Z now we can solve it so what do we have this is one goes there - 4 to
2:54:41 1 - A is greater than equal to minus1 If I multiply whole by minus one the sign inequality it changes we'll have 1x4 all right or I can say a is greater than or equal to 3x4 so a is greater than 3x 4 that means a can assume any value from 1 to 100 the answer is 100 is that clear now
2:55:29 that cat is just 10 days away and people have done with the basic algebra okay still if you're asking that's that's rude right discriminant because the because the values of FX are real they lie between zero and one and when are the values real the values are real when discriminant is greater than or equal to zero isn't it CH let's go ahead so H it was given in the question
2:56:15 no FX contains the interval 0 to one that is why I took the maximum value that is one rash no need to take that we took the maximum value because
2:57:02 ultimately if I found a is greater than 3x4 automatically the range is from 0 to 100 so you got the answer z okay this question has yeah 2 minutes to you
2:58:17 e for for
2:59:42 rri answer 5 p + 4 q for [Music]
3:00:42 shagun what is 5x4 it's the value of p and Q what is it T to answer answer 5 p+ 4 q
3:01:46 for
3:02:39 [Music] e for [Music] the answer is
3:03:27 10 the value of P will be one and the value of Q will be 5x 4 just check that again [Music] all right let's see um by the way there's one thing you must understand let us look at the
3:04:14 graph GX graph would be looking like this am I right with this place is 1 comma 0 yes or no is that right okay and if a function quadratic function which is which will be Parabola if this cuts at four we have four distinct real Roots okay four common roots are there so graphically what is possible if the graph goes like this we'll have only two possibilities isn't it is that right if the graph goes like this let's say we have only one possible one one
3:05:00 common root right we can have four common Roots only if graph goes like this one 2 3 4 do you understand this or not do you guys understand this all of you okay so here in this question the main trick was to realize that this this function this will its discriminant will always be negative it has got imaginary Roots okay
3:05:49 all right take so if it got imaginary roots that means this discriminant must be greater less than 0 so - 2 square - 4 a must be less than 0 if we solve this what do you get a is greater than 1 so I got the value of p is this clear Perfect Isn't it one so we want to find the maximum value for that I can take x² - 2 x + a to be
3:06:36 equal to plus of x -1 or minus of x - 1 let's take any one of it let's take one one of it first of all take okay if I take positive value I'll get x² - 2x + a is = x - 1 which means x² - 3x + 8 + 1 is equal to tar it's open interval p is an open interval it doesn't say closed interval so one say one is not being included anything above one is included
3:07:21 got it's open interval all right tell me let's solve this one okay 9 minus 4 a - 4 all right so what do we get here 5 minus 4 a must be greater than zero here also must be greater than zero so value of a
3:08:08 will be you have minus sign changes automatically less than 5 by 4 if I take it on the right hand side will change and inequality will also change so a lies from 1 to 5x 4 5x4 will value of Q moment we got it let's put the values and get the answers 5 * p is 1 and 4 * Q we got 10 as the answer first part Tri Yes again graic graphically if you can can imagine that now though it will be understandable so here we use graphs
3:08:53 first of all yeah yeah yeah we can do that negative you will get one also second also will get I took positive if I took negative you'll get a is greater than one you'll get both the values okay man prashan tar clear let's go to next question um yes
3:09:52 there's no is less Z because it's imaginary is same same graphical there is no other way we then do realize that it has to be imaginary um roots
3:11:04 T answer last while is H for for
3:12:21 bat and Prasad when the roots are imaginary the graph is always positive it never touches the x-axis okay bat Prasad I'm repeating when the roots are imaginary it never touches or intersects the x-axis it lies somewhere in the air that's how for [Music]
3:13:20 all right let's solve it now let p and Q be the roots of this equation 3x s minus this plus okay equal to is that log Q is p is less than zero the range of a is M to Infinity then find the value of M by the way what can be said about Q or P it's less than Z that it's fractional it cannot be zero right right it can't ever be zero it has to be fractional so if p is greater than 1 then Q is less than 1 is that understood
3:14:09 everyone is that understood yes right Prasad no problem okay so can I say okay p and Q are the roots can I say these roots p and Q both the roots like lie on different sides
3:14:55 of one okay so we can say that opposite of one okay so the product of roots if I product of roots P * q q p * Q will be what this upon this this upon 3 isn't it a² - 3 a + 2 upon 3 and this will
3:15:46 always positive isn't it if you multiply them you will get positive only right which that's the basic thing no base P isn't it you'll get a positive which means this a squ if I factorize it a - 1 * a minus 2 must be greater than Z which also means the value of a it belongs to minus infinity 2 1 Union 2 2
3:16:32 Infinity all right no they're not less than zero all right and what can be said about the sign of F1 one F1 value negative the value of F1 would be negative or
3:17:42 positive it'll always be negative exactly F1 is always negative so let's put the value of F1 here fub1 let's put X in place of X we'll get 3 minus a + 1 + a² minus 3 a + 2 this will be less than 0o let's expand and try to find it out 3 - a - 1 + a s - 3 a + 2 less than 0 like terms
3:18:37 together there is a simple mistake here this is a² + 1 just make it correct yeah all right otherwise you won't able to solve there was a mathematical error to that question now it's perfect so a square - a square this get canceled out all we have left is 3 - 1 is 2 + 2 is 4 so - 3 a is less than - 4 which means a is greater than 4 by 3 a is greater than
3:19:27 4x3 okay if you look at this range that we have got and a great than 4x3 so where does it accommodate this value this interval right 4X 3 is 1.33 and 2 is greater than 4x3 that means M will be equal to 2 that will be our answer I think some people got that right cool
3:20:16 Krishna Q is not negative Q is just fractional Q is just fractional okay less than one less than zero let's go to next question STG we got the range of AO putting it there we got less than zero V how 4
3:21:02 by a is greater than 4 by3 that's why we took two moit Rana there no log of negative negative just see it once more log of something negative is not possible log itself can be negative moit isn't it for fractional values remember
3:21:47 that you got confused between L okay moit I hope it's clear now so this one can be done by using what amgm inequality a classic condition where we can use amgm inequality XY Z is given here we have x² 4X y 4 Y sare 2 Z sare okay well I can break something okay I can
3:22:33 break it as x² + 2xy + 2xy what I'm trying to do I want to have see here the powers of x y and Zed are one each so I'm trying to break this expression such a way that I get I get the powers in such a way that if I multiply them I'll get similar Powers isn't it okay goam
3:23:26 that's why I am paid right okay + 4 y s + z square plus Z Square so see what happened when I did that X power 2 one one so I got four X's isn't it power will be four If I multiply all I'll get X power 4 y power 4 and Z power 4 that is the main reason that I'm breaking it like this all right okay a yeah see the thing is
3:24:17 whenever the product is constant okay the concept product of some variables is constant then the sum is minimized when the variables are as close as possible all right that is the property that we use for amgm okay well so that is why what I did here I got here I can see that powers of this is XY are
3:25:06 equal so that when I try to you know take the nth root here if I take the fourth root four will be cancelled that is why I am doing it okay take so now let us try to find the we know this am is greater than or equal to GM all right how many uh terms do we have 1 2 3 4 5 6 let me consider as T just for sake of not writing it again T upon 6 must be greater than or equal to if you multiply all of it what do you get 2 2 are 4 4 are 16
3:25:52 16 and then we have x ^ 4 y^ 4 z^ 4 whole ra to^ 1X 4 is that right all right so now you realize that why I did that because now I have value of XY Z I can put it there moreover power fourth so TX 6 will be greater than or equal to 2x y z or t is greater than equal to 6 * 2x y z that is 32 96 is the correct answer minimum
3:26:40 value am GM hm inquality ohx is 1x six wait wait wait wait I'll just I just Rectify that you'll get the same answer don't worry 2 XY Z to the power 4X 6 or I can 2x3
3:27:25 now it looks good okay so XY Z is what 32 2 into 32 is 64 64 power 2x 3 6 to multiply so you will get 6 into 4 Cub 16 the answer is 96 the subsequent steps were right 1 by4 now it makes sense okay that was just manual writing error
3:28:12 nothing else okay amgm I'll just repeat see I considered this whole as T that is the only reason to am we have six terms number of sum upon number of terms T by 6 all right and here it geometric mean multiply all of this this into this into this into this into this you get this to the power 1X 6 because we have six elements over there sixth root that's all and XY Z is given as 32 so I put it
3:28:58 there uh is that clear here we have the next question X Char Bar y char
3:29:46 s multiply sub
3:30:42 no goik didn't get time um okay we have this and we have this if XY Z are positive real numbers as that this and this is there then the minimum value of XY is p by Q XY this is p by Q where p and Q are relatively prime find the value of p + Q okay x + y + z whole Square expand then what will I get x² + y² + z s + 2
3:31:29 * this whole all right which means the value of this you 7 2 into 4 8 I get 15 over here is the step clear answer eight answer seven and eight are the answers no that's a wrong answer okay now everyone listen to me very carefully
3:32:19 got it okay so here we have this value I can rewrite it as I can take uh y common left with X+ see I can take y common isn't it XY + y z okay XY is there I need to minimize this XY isn't it so let me have XY separately and then take Z common and write x + y over here okay now tell me everyone if I assume if I assume that this is four okay let us
3:33:07 have it here now if I assume two things if I assume that x + y to be one value and Z to be another value it arithmetic mean will be this isn't it x + y + z upon 2 now you will say y not 3 because I have assumed X+ y to be one single entity I have assumed X+ y to be one single entity all right and this arithmetic mean is greater than equal to geometric mean geometric mean root of x + y multiplied by Z does that look
3:33:57 good perfect now let us take the square of both the sides take the square of both the sides this is what I have already calculated it's 15 so 15 upon 2 CU Square must be greater than or equal to x + y * Z now you realize what is happening we wanted to minimize this so basically I'm maximizing this does that make sense okay I'll wait for a few seconds just see is that making some sense
3:34:52 all right so what is going to happen now x + y * Z must be less than or equal to what okay just pay attention to that 15 by 4 we have got
3:35:44 this see all right C cool question am then I realize that this is what I needed to this and this are two separate entities terms I'll I'll get geometric mean like this like assuming X Plus y to be one term Z to be second term
3:37:05 how many value of p+ Q minimum value of XY is okay so so tell me something if you solve this you will get what uh x + y MTI Z maximum value is 15x 4 isn't it this got maximized so this will get minimized yes or no is that clear let's put the value here so XY will be 4 minus this whole 4 minus 15 by 4 what do you
3:37:52 get yes minimum P by Q 1X 4 answer is five answer is five not 7 cool hi two is
3:38:38 common and X key that is why I did that kishna um am is greater than equal to GM that's it so let's go to next question now let's do this one Prime co- Prime co- Prime yes an it's co- Prime this question has some problem
3:39:23 don't do it it has a formation problem let's do this this is fine h sayak is co- Prime to every number
3:40:42 one or four copre Shi Chevy problem say x square huh it's not y Square [Laughter]
3:41:58 uh let the two numbers be let the numbers Be A and B all right so geometric mean the root a and we have two arithmetic means inser between y and z are two arithmetic means insert it in between okay so I can write Y is equal to 2 A + B by 3 and Z is equal to a + 2 B by 3 so 2 y- Z as the question asks 2 Yus Z minus and then 2 zus Y what do you get
3:42:45 A or B putting it here so a MTI B upon square root a square answer is one yeah simple simplest question yes or no let's go to next question
3:43:43 we yes that can also be done oh sorry what happened I'm so sorry I think ah now it's fine remote control okay number of real values there go this is a very important logarithmic identity this two can be sent here x squ and now we have plus 4x + 3 isal 0 this will be
3:44:29 x² if you solve you get value of x is -1 and Min -3 so a lot of people will you know forget and write it two but no this will be zero why because log of negative is not possible so the answer is zero there is no real value clear even if you get the value of x x and x are negative because X is negative log of negative is not possible so we'll get zero HTG now they realize it's it should not be zero
3:45:18 let's go to next question this one two minutes to you yeah man
3:46:13 is for for [Music]
3:47:24 how a n minus one n plus one obviously there office for
3:48:28 upe all right let's check this out
3:49:32 okay what is the answer you're getting 1 3x 4 4 okay loads of answers 140 by 81 wow anyways so I can do one thing S1 by A1 is equal to 1 can I rewrite this as look at this carefully A1 by A1 comma A1 + A2 by A2 comma A1 + A2 + A3 by A3 and so on yes or no
3:50:18 H yes okay it's also known that the common difference is 3x4 so basically this guy is 3x4 more than this A1 by1 question given which means A1 + A2 by A1 must be 1 + 3x 4 that is 7x4 right right so 4 A1 + 4 A2 will be equal to 7 A1 M 4 A2 is equal to 3
3:51:06 A1 or I can say A1 by A2 is equal to 4x3 so why I took the ratio the reason is simple because I want everything to be in the same format so I can write A1 is equal to 4K and A2 is equal to 3K all right a so this is 7 A2 4 A1 +
3:51:58 A2 3 A2 is equal to 4 A1 so A1 by A2 is 3x 4 now it looks good let me check it once 4 A1 + 4 A2 7 a A2 3 A2 4 A1 A1 by A2 is 4 3x 4 okay so 3K and 4K okay so basically I want to I want to have things in you know in terms of K that is why I'm doing it so I got A1 as 3K and A2 is 4K
3:52:46 is that [Music] right okay now this one is 3x4 more than this one isn't it so 7x4 + 3x 4 will be 10x 4 A1 + A2 A1 is 3K A2 is 4K A3 is we don't know what upon A3 is what this is what we get 5 by2 you can find the value of A3 as
3:53:34 well okay once it's given in the question see okay so now we have cross multiplying 14 k + 2 A3 is equal to 5 A3 that means that means 14 upon 3 K will be the value of A3 V the same thing you want to find the value of A4 by A1 see we are almost getting over
3:54:19 there Cho coming to 3x 4 More Than This 13 by 4 so 13x 4 will be the value of A1 + A2 + A3 + A4 isn't it whole upon A4 now you can solve it I think you you can solve it 3 + 4 7 21 35 K by 4 + A4
3:55:15 let's check the calculation once I got A4 as 35 K by 9 okay kamam we need to calculate A4 by A1 A4 is 35 K by 9 and A1 is what 3K so what do you get 35 by 27 is the answer's no F to solve it let's go to
3:56:01 next question now what's the problem 35 K by 3 13 minus 13 minus 4 to 90
3:57:28 next time 35 K by 4 can't I see get N9 four
3:58:38 my B to the a that's my mistake this will be three right this will be three okay everything will be changed now
3:59:37 some to again up take up four * this 13 * this okay take 435 140 K by 3 is equal to 9 A4 that means A4 will be 140k by 27 okay all right put and we get the answer 140 K by 27 take okay CH let's do this
4:00:48 take you got the approach solve see the thing is we not here to solve the exact correct solution we need the approach this is a very good
4:01:41 question okay yeah they are asking the sum of the roots of the above expression sum of roots it's always minus B by minus B by a isn't it this is what we need to find and B is the coefficient of the second expression second term and a is the coefficient of the first term isn't it so everybody look at this if I cross multiply of course you'll able to solve the question but that will take a lot of time isn't
4:02:27 it so we need to play SM over here everybody listen to me carefully a b quadratic equationa isn't it there will be a b quadratic expression so if you multiply it whole you'll end up wasting five minutes just have to get the the the coefficient of x power 4 and X power 3 if this is a this is B you'll find minus B by a this is what you need to find only this exactly goik that's it
4:03:15 okay so you multiply should I multiply this I think you can multiply you get 3x^ 4 minus X cub and something after that this is what you will get so minus of minus one get 1 by 3 as the correct answer question okay clear everyone let's go to next
4:04:07 one I just come in a while 2 minutes for
4:05:12 for e
4:06:33 I'm back uh okay have you got the answer so here we have okay what can be done can I take log on both sides can I take log on both sides Le log on both
4:07:24 sides okay is power yeah so I get the same thing again is that right is the first step clear this multiplied by this will be log a into B so I can write it as log of x + 1 base 10 is the first step clear to
4:08:10 everyone First Step should be clear the moment you do it you'll find a quadratic equation let T is equal to log of base 10 mtip and x + 1 log X + 1 base 10 put it there t² is equal to t t + 2 this will be two Now log 100 base 10 will be two am I right let's find the roots that's t² - t - 2 =
4:09:05 0 2us 1 and this is the value of log this is nothing but log of x + 1 base 10 this is 2 or minus one so I can say x + 1 is 10 Square 100 or 1 by 10 which means X is 99 or -1 - 9 by 10 all right
4:09:52 is that clear two values X1 X2 that's divide by 99 divided by - 9 by 10 inside a mod so you will get yeah 110 is the correct answer perfect very good next okay J his parents are there are they
4:10:39 still there this also applicable to you a h conent viewers that's really good
4:11:41 is there any mistake in the toe um let me see R what happened
4:12:30 question let us find the term first of all um T1 is what 1 Cub by 1 what is T2 1 Cub + 2 cub upon 1 + 3 what will be TN 1 Cub + 2 CU till n Cube whole upon 1 + 3 + 5 till 2 N - 1 am I right all of you can you can you identify the common term
4:13:19 all of you is that clear some of cubes of natural number starting from one we'll have n into n + 1 by 2 whole square square n okay odd numbers n odd numbers sum is n square right so if I write this TN n Square gets cancel out all we are left is with n + 1 square upon 4
4:14:13 all right okay summation summation you because they asking about sum right sum from till 12 they asking till 12 so n is equal to 1 2 12 that's what we need to find out this will be equal to 1x 4 * Sigma n² + 2 * Sigma n + Sigma 1 and I I'm assuming that all of you know how to find the sums right
4:15:04 n Sigma n s by two two cancel this will be n Only
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