 start and the first question on the board is so this question is based on the product of zeros of two polynomials so the question could be misleading so hence read the question properly and then only answer just a minute yeah I hope now you have got the correct answer okay just good so most of you have got the right answer that's good so let me explain the solution yes so this was the question it was very simple straightforward so if you what was the question about so it was nothing but x square minus zero is nothing but c by a so this is c so c by a is the product of zero so 12 upon 1 has any difficulty in this so here this was the catch so many people would have just summed this to our product you know I've multiplied these two so that was not the case because there's a plus sign something is getting added added over here so hence be careful so good so first question is done this is not allowed who's incognito name please who's this there incognito or shall I you know me hello just confirm yeah so anyone is there whose name is incognito yeah yeah mute yourself someone is not on mute please can you put your mic on mute please okay so to find out the maximum number of times the graph of fx and fx is given like that can cut the x-axis so this was very easy question I believe so it was fairly simple because how many roots are there in this you can see it is already factorized so hence this should not be a big deal so there are five you know the number of roots possible is five so and the idea was that all of them are real roots so which is very much evident yep so there are very much evident that all the roots are real so only when the roots are real it cuts the x-axis otherwise it will not cut the x-axis so you know this so this is x and this is fx so only when the roots are real all the roots are real here what are the roots of this polynomial for zeros is 0 1 2 3 and 4 these are the roots and all of them are real and hence they will it will cut the x-axis 5 times so hence answer was 5 okay so you have to just check whether the roots are real or not if they are not real then you can't say that they are cutting the x-axis let's go to the next question so the graph of the polynomial fx is symmetrical about which of the following lines so you know the polynomial is a quadratic polynomial I would be interested in knowing the result hmm most of you have not got it wrong how many if you could not solve it how many x yes x most of you could not solve it how many if you could solve it okay minus b by 2 a good main so your understanding is correct so the next question is this so let me take some space and explain it to you just give me one moment this was question number four give me one this was the question so fx is 2x square minus 9x plus 9 so all of you know that when you plot this curve so this is how it would look like okay so it's a parabola it will open up and it has two roots both are positive roots it's like that okay so it is always symmetrical about this line right this line and what is this line if this is alpha this is alpha root is alpha this is beta correct so this line this point is nothing but alpha plus beta by 2 and what is alpha plus beta some of the zeros in this case it is nothing but minus b by a so minus 9 by 2 right so this is 9 by 2 and there was this 2 also so whole by 2 so 9 by 4 right so this line this distance is x is equal to 9 by 4 right so hence this this is this entire curve will be symmetric about this line right always remember any parabola will be of this nature y is equal to 2x square so any parabola of y is equal to ax square plus vx plus c will be symmetric about line x is equal to minus b by 2a correct just mean of roots in the vertex very good understood now this is it and let's say if the question would have been like something this if this is the parabola x is a y square plus b y plus c what will be the line of symmetry can you tell me what will be the line of symmetry in this case now replace x by y yeah so yes in this case y would be it will be y is equal to minus b by 2a okay and hence the parabola would look like this something this is why this is x let's see your performance so okay at least all 10 of you have got it correct that's nice be very very careful with the powers which are there take your time don't rush through take your time don't rush through so there are two zero two zero minus x to the power two zero one nine plus x to the power two zero one eight so on and so forth still x most of you have already answered so a few few few only in the context of the problem in the context of the problem you have to answer that okay so mixed mixed response okay so this was a question effects is this sum of sum of two zero one eight zeros so right so this is the who's how is this true a to the power two zero one nine if you put will you get zero so this is not correct right so this is not correct yes this could have been this could have been one of the zeros of effects this is also not correct sir the power of x is 2020 so there are total of 2020 roots over here and the sum of 2018 is a so one of the zeros can be anything one of the zero is zero one of the zero is zero and two zero one eight roots is given as a right so nine this is 2019 through right so hence let's say the last root was alpha so alpha plus zero plus two zero one eight sorry what was that one is for a alpha plus zero plus a is what is sum of all roots here one why it is one coefficient of x power 0 doesn't have to be necessarily the 2019 through it can be one of the 2018 correct yes that could be so in the context hence I'm saying in the context of the problem so that's what maybe you know anyways so what is alpha one minus so this is a yes technically one also is correct but then as I as I'm telling you so in the context of the problem one minus a was the near and most correct answer it make now I'll explain just let this question be over okay let me explain it to you to her so what I'm saying is sum of roots sum of roots in any polynomial any polynomial is negative of coefficient of x to the power n minus one divided by coefficient of x to the power n right it can be one yeah but then I'm saying common in the context of the problem I told you right no how can I'll be how can a be zero if you put a a 2020 minus 2019 plus a 2018 will they add up to zero so hence some is right so hence they are not adding only in in case of a is equal to zero very specific case which is obviously I can't continuously telling context of the problem so then context of the problem this is the most appropriate answer okay so when when there are ambiguities in the options there could be many times this could be a situation then you have to pick up that which is most least ambiguous how is one root if you put one everywhere it will become zero put x equals to one it will become zero right they will cancel out in pair okay so hence minus coefficient of x so make now who whom was I was explaining to whom why does x to the 2019 have to be zero why will it be zero are in why will it be zero there is no I'm saying 1 to the power 2 0 1 20 minus 1 to the power 2019 will be zero this item is zero no didn't understand the question only so please elaborate if you have a mic please unmute and say meanwhile this was the thing so some of roots so there are 20 20 roots is it so alpha 1 plus alpha 2 dot dot dot alpha 8 2018 plus alpha 2 0 1 9 plus alpha 20 20 these are the roots and this will be equal to this will be equal to minus of 1 upon 1 sorry minus of minus 1 upon 1 so which is 1 okay so hence what will happen this is given this is how much 2018 zeros is a so a and since there is no constant term constant term is 0 that means x equals to 0 is a root right so hence a plus 1 a a plus 0 plus alpha 20 20 is 1 so a alpha 20 20 is 1 minus 8 okay so this was the explanation okay so this is the leaderboard next question please polynomial is a x to the power 5 minus b x to the power 4 plus c x cube plus dx square minus ex plus k how many maximas and minimas possible minimum value and maximum value of any polynomial it's called maximum and minimum how will be how many peaks and troughs are there in the graph in this case or how many are possible how many peaks and troughs are possible in this graph majority wrong anyway let me explain I think you need some for maximum and minimum you have to understand y is equal to fx y is equal to fx the curve how does you need to know how to plot the so y is equal to fx is the what is maximum and minimum in case of a quadratic curve what do you think how many minima maxima is possible in quadratic tell me if this is why this is x how many maxima or minimize possible in quadratic so this is how it looks like quadratic looks like this so this is there is one minima one minima lowest point correct for a three third degree polynomial how would maximum and minima look like how many will be possible so for a third degree this is how it will look like right three roots possible no so one one maxima and one minima just by the knowledge of degree you can say since there are three roots the curve has to cut the x-axis at least in three points or maximum in three points right so then only three roots will be there so when that is the situation then the curve will be something like that it will go up hit a maximum then start going down and then cut the axis again hit the minimum and then again go up like that so for three roots one maxima and one minima what about four if a degree four polynomial is there how many maxima and a minima okay so degree four that means it will have four cuts in the x-axis can you see that so either there will be two maxima one minima or the vice versa right either will be two two maxima so one maxima is here one maxima is here this is one minima either this or the other way around that is you understand just don't don't worry about this extra thingy so this is maxima this is minima this is minima right so for fifth degree what will happen how many maxima minima possible to each five times it will cut the x-axis one two three four five right so one maxima one maxima one minima one right so now you know in case of n degree n degrees generalizing n degree polynomial is there so how if n is odd n is odd how many maxima how many minima well in terms of n how will how many maxima and how many should be nothing but n minus one by two maxima and minima correct if n n is odd if n is even then n minus one by two and n minus two by two sorry n minus not n minus two by two n minus one by two minus one n minus three by two right so either these many maxima will be there and this many minima or vice versa depending upon how the curve is opening upwards or downwards is that clear to everyone generalized form here please remember this will be the next year when you're writing kvq y's and all that these are this will be low handing fruits okay next few people got it good so you could ask you could have now realize what kind of questions will be there where there are no numerals we are only talking about possibilities we are not talking about certainties so you have to be familiarized with this kind of questioning what could be the possibilities so far you have been habitual of tackling problems where there is a certain answer answers will be certain but then there will be lot no to reach to that lots of possibilities could be there so hence you have to eliminate by your knowledge what is possible what is not possible that's a quadratic polynomial with certain conditions you have to find out in which quadrant the vertex will lie if you want you can apply the formula of coordinates of vertex as well okay so it will be interesting to note how many if you could get the right answer oh my god equally distributed what happened real roots minima yes so fourth quadrant will be the right answer why let me explain this so this was a question they come how to tackle such such questions so what is given a is greater than a is just use this color now so a is greater than 0 correct and these also greater than a b is less than 0 and c is greater than 0 so and the discriminant is greater than 0 yep so under root b square minus 4 ac anyways this is greater than 0 actually under root is extra information given this is greater than 0 okay b square minus 4 ac that mean the roots are real real roots and which side the curve is opening upward or downward which side the curve is opening upwards right why because a is greater than 0 that means it will be off this form either the vertex would be lying here or it would be lying here so first and second is ruled out why because since discriminant is 0 there has to be two real roots so two real roots and opening upward means the vertex will be either in third or a fourth quadrant third or the fourth quadrant right but since b is negative since b is less and a is and a is a is greater than 0 b is less than 0 so some of roots let's say alpha plus beta will be greater than 0 you get the point alpha and beta are greater than 0 and alpha beta is also greater than 0 C by AC C by a is also greater than 0 or minus b by a is also greater than 0 so that means both alpha and beta are of same sign same sign right right so same sign but some is greater than 0 that means alpha and beta are positive so alpha is also greater than 0 beta is also greater than 0 right otherwise if alpha and beta are both of negative sign then it day the sum cannot be greater than zero right so hence we conclude that alpha is greater than zero beta is greater than zero that means both will have to cut on the positive side of the x-axis right hence the vertex will lie in fourth quadrant did you understand how to solve these kind of questions now what did we learn in this so you have to come out of the realm of formula you know so you have written down all the formula vertex axis this and that but then the question will not be straightforward that will ask you to think and that is how you eliminated right so first you eliminated the first and second quadrant then you eliminated the third quadrant by thought process that's what is the intention of such questions okay so good 17 people got the right answer so I hope they have not guessed it so good magna is magna and stress and yeah okay never mind let's go to the next question the question if you look at the options if you read the question carefully so let me see how many of your accurate how is your accuracy result right so far so tell me right upon attempted total number of right divided by total number of attempted fraction what is that for all of you those who have done this question finished five by six okay next manikets five by six anyone else magna five by six ananya four by six okay so we'll discuss this we'll come back to that okay so again not many people are able to solve this question okay what is the question by the way so this was very simple question guys why did you why did you you know get a wrong answer it was nine degraded problem actually so alpha beta gamma are the zeros so what do I know do I not know that alpha plus beta plus gamma is how much below alpha plus beta plus gamma is how much in this case alpha plus beta plus gamma is three by two right and what is alpha beta beta gamma gamma alpha this one three very good done what else is required in this case then you simply square the first one alpha plus gamma whole square will give you nine by four okay and so hence it is alpha square plus beta square plus gamma square plus twice alpha beta plus beta gamma plus gamma alpha is nine by four right and alpha plus beta plus gamma this one is given as three so alpha square plus beta square plus gamma square is equal to nine by four minus six that's it so that is nine minus 24 by what how much by four minus 15 by four any doubt is this isn't it ninth grade a problem with some information of the roots of so the only thing in you learned in 10th grade is this this was ninth grade problem in ninth grade this was given if alpha plus beta plus gamma is yes momita exactly that is what the catch of the question is since can it be possible that alpha square plus beta square plus gamma square all the squares adding together you're getting a negative number what does this mean good observation momita very good very nice so what is you know so you're adding three squares together and you're getting a negative number what does that mean that not all the roots are real right so this equation so the follow-up question could be will fx x axis twice yes or no twice yes or no twice will it cut twice maybe probably no it will not cut twice as well it will not cut twice why because come if all the if you see all the coefficients are integers in such cases in such cases the complex roots occur in pairs occur in pairs so there will be two complex roots right there will be two complex roots so hence it so the the follow-up question could be if this is given let's say if this is given and no other information is given only polynomial is given the question would be how many times will the curve fx equals to 2xq minus 3x square plus 6x plus 1 cut the x-axis how many times will it cut the x-axis can it be can can so happen can it can it so happen that it will not cut at all can it so happen that it it doesn't cut at all no that is not possible right it will cut once why because if it doesn't cut at all that means there are three complex roots which cannot be possible complex roots have to be in pairs but the point so it cannot have one complex or three right so hence if there are two complex one has to be real so hence it will cut the x-axis at least once not at least exactly at one point yes or no so this function so much of certainty is there that it cuts exactly at one point is that clear to everyone any doubt any doubt no let's move ahead then very good so this is a new Cali next zero of the polynomial x cube minus ax square plus bx minus c equals zero are three consecutive integers oh it should not be zero sorry so this is polynomial guys this is not an equation so there's a change slight modification the question but it will not hamper your approach so it's a polynomial or you what you can say is remove this polynomial word with equation whichever so it's a polynomial fx x cube minus ax square plus bx minus c okay okay okay at least majority are correct hmm let's go to the question was this difficult again then nothing beyond your scope okay zeros of the polynomial are three consecutive integers what does this mean the zeros could be alpha minus one alpha and alpha plus one correct this could be the zeros yes or no then what is the smallest smallest possible value of b so let's see so if you add all of them what do you what will you get you'll get three alpha and this is equal to what a three alpha is a so alpha is a by three that we know if you don't understand any point at any point now product of a sum of roots taken two at a time so this plus alpha into alpha plus one plus what is um this is alpha plus one alpha minus one right this is how much b right this is what they are saying so expand it alpha square minus alpha plus alpha square plus alpha plus alpha square minus one is equal to b am i right just just you know confirm if if you get any issues okay so this is three alpha square minus one is equal to b so here itself it is over you don't need to go for first also not required what is our minimum possible value of b minus one why minimum possible value of three alpha square will be how much zero isn't it so three alpha square is always greater than equal to zero right so three alpha square minus one will be greater than equal to minus one right so what is three alpha square minus one so b b is always greater than equal to minus one won't a is equal to zero what so where is it where is it written that a has to be non-zero or a has to be you know yeah so is there any is there any constraint on that and a will be zero or not is it written given not given so it could be zero let it be right so hence minus one yes once again i'm saying product if in a in a in a cubic equation if you take product of sum of product of pairs of roots together right so what are the roots roots are alpha minus one multiplied by alpha this is one then second product is alpha alpha plus one and then third one is alpha minus one and alpha plus one these are the sum of roots taken together sum of roots sum of product of roots to take two at a time so what is this this is nothing but the third coefficient or coefficient of x divided by the coefficient of x cube which is one so it is b and if you expand all of this you'll get three alpha square minus one is equal to b so the minimum possible value of b will be minus one because three alpha square is always greater than zero okay so this was the question here see again there are no there are none of the questions will be directly apply the formula and get the result types so you're able to think here is the oh so only Meghna well done very good Nyanesh Alipthia Tavishi Ananya very good so next question guys now I hope all of you have covered AP what is an AP you know that right all of you have covered this again this was very simple question if you know AP and the properties of our cubic polynomial so you can see all our mix type not only one type right so let's solve this but only only 14 are getting that means the concept of AP is not that established okay so zeros of FX is x they are in AP so whenever you have to choose three terms in AP what do you what do you take you take a minus d a and a plus d choose like this why because there are some of roots or some of the terms involved so what is the sum of the roots sum of root will be a minus d plus a plus a plus d and what is the sum some will be minus minus right so hence it is Surya what is the problem what what is happening to you why are you so bothered about question number and all that attention pay attention on this solution yeah so hence there will be lots of insights here so what do you see d and d gets cancelled so 3a is equal to 12 so what is a a is 4 so you got a s4 you have to find out common difference but so how to do it so you can do it either why product of root that will be easier because you don't need to do more calculation so let's do product of root so a minus d into a into a plus d is how much 28 right so i hope you know if alpha and beta and gamma are the roots then alpha plus beta plus gamma this is the fundamental minus b by a alpha alpha beta plus beta gamma plus gamma alpha is c by a and alpha beta gamma is equal to minus d by a correct these are the three things so i have applied this minus d by a so d was minus 28 and one and a was one so simply 28 now you can calculate a already you have got so 4 minus d into 4 4 plus b is 28 so this 4 we go 7 then 16 minus d square is 7 this implies 16 minus 7 is equal to d square so d square is 9 so hence d is plus minus 3 nothing is wrong nothing is wrong but the idea in a mcq and a time-based paper is to get the question correct so if you can get with that assumption solving high order equations then that's also fine but what did i say the idea is not to solve the problem only but also to solve within 100 seconds so these are the tricks you need to adopt and anyways we have discussed right selection of terms in an ap also when anyways we'll do ap then again similar type of problems will be there so whenever there are three terms to be chosen in ap whose sum is given always take a minus d a and a plus d okay good next question you're missing just a minute yeah here is the question graph of the polynomial fx x minus a x minus a minus one plus x minus a minus one cut the x axis add let me solve this for you okay the graph of polynomial it was again you need to just you know you don't need to expand at all you're right right now so you just you can assume this to be y that will help you y y y and y right x minus a as y so what will you get you'll get y times y minus one right then plus y minus one y minus two then plus right what is this y minus two times y this is the function or polynomial so you expand it what will you get you'll get y square minus y plus y square minus three y plus two and plus y square minus two y okay so this is three y square and minus y minus three y minus six y and okay this is the this is the so if x has to be real then y will also be real because y is nothing but x plus a so hence hence if you see that is the determinant of this will be v square that is 36 minus 4 into 3 into 2 that is 12 which is greater than 0 right so hence it is greater than 0 so discriminant of this that means y is y is real and distinct real and distinct right if y is real and distinct then obviously y was x minus a this means x is equal to y plus a so for two y's you'll get two x so hence exactly at two points yes all of you understood this question any doubt in this so the question was deliberately made longer so this is how they will set the paper they will make it deliberately longer so if you miss out on the fact that whether you take this as a one this as one entity or you expand and solve the answer will not be affected by it so knowledge was knowledge was in the same equation so let's say a y square or sorry a x square plus vx plus c let's say for this for this if let's say x is x is real and unequal let's say okay then for any x minus or plus alpha square plus beta x plus alpha plus c so if you add alpha also right then also you will get real values of x is that okay so if you add some constant to x the nature of roots that do not change so that's the learning so keep that in mind yes yes alisha degree of a polynomial will be equal to all possible roots all possible roots okay not only real next number of yes number of roots so the polynomial whose zeros are additive inverse which would be zeros yes here zeros are additive inverse of the zeros of the given polynomial is very straightforward actually I hope you know what additive inverse is what is the additive inverse of the number you know what an additive inverse is right you know or not shouldn't be plus five okay so let's see your only nine people got it hmm no so for additive inverse means negative values negative roots right so if alpha is the root of fx then the the new polynomial which you have to find out the root will be minus alpha correct so hence wherever you find x you simply replace it by minus x so you have to simply find out f of minus x okay so hence the constant term is not affected so hence three minus three x to the power five minus five x to the power four minus 11 x to the power three plus seven x square and minus five right so wherever there was x you simply put minus x and that will be the new polynomial right so hence minus a constant term doesn't get affected guys so hence it will not be plus is that clear clear to everyone simply replace x by minus x and you will get the new equation so hence answer was d okay so you could have eliminated this is straight up straight away eliminated this straight away right because plus three was there this is also eliminated plus three cannot be why it is minus five into four where oh because x to the power a slasher x to the power four it's an even power so hence if you take my take minus x to the power four you will get extra power four again so the fine sign of even powers will not change only odd powers will change is it so wherever there was odd power it changed it changed so wherever there was odd power so this was this will change this will change only but nothing else will change okay next question guys the last question of the evening we have just done some this type of problem sometime back 10 seconds hope you drew the map oh sorry the graph of it it was a graph based question surprising no one marked the d only a b and c and that too okay see fair enough so let us draw the plot in such questions immediately take the pen and paper and start drawing so fx is greater than zero when x is between minus infinity and minus two okay so let's say this is minus two these are the critical points of the question so minus two what else zero so this is another one zero and then what else two right right so hence hence from so if it is on the left hand side of minus two the values are positive so it is happening something like this okay and then and then from z minus two to zero it is negative and zero to two again positive and then from two to plus infinity again so hence how many times will it cut one so this is how you have to solve this question so fx is greater than zero in in this community here it is greater than zero here it is less than zero here between minus two and zero between zero and two again positive and between my plus two and infinity again negative so this would be the graphs something like that so it will cut at three places did you understand how to do it so whenever such kind of intervals are given immediately draw a rough diagram you do need not be accurate but you can get a sense of it correct I could have reversed the sign and the craft would have been something like this something like that then also it will cut in three it would have reversed these signs then okay so that's all for today so hence let's see the final leader board okay very good so Meghna leads the board good congrats Aditya Aranya Abhipsha Abhihi Ganesh Srijan Neo me who's this Srijan only and Neo me okay okay did you learn something new set of questions different type of questions yep what did you learn guys did you learn or nothing just like that yep so please keep an eye on such type of approach towards these questions these will be like you know you'll be tested on thought processes okay bye bye take care stay safe