 Hello all welcome to the YouTube live session on important problems for Jee main and Jee advanced so those who have joined in the session I would request you to type in your names in the chat box so that I know who all are attending the session guys can you all see the screen am I audible so alright so we'll begin our problem solving session with this very first problem now let me tell you there are 30 problems like this similar to the pattern that you'll be having in your Jee main exam and most of the problems are of Jee main level or lesser than that so I'm expecting you to solve 30 problems in the coming three hours alright so here's the first problem in a triangle A,B,C A is given to us for B is given to us 3 medians AD and B are mutually perpendicular then find the length of C that is the side AB so guys you have to be very fast today unlike previous sessions we were having around 30 problems in this session so please be fast today and type in your response in the chat box so the average turnaround time for every problem should not be more than three minutes max to max four minutes yes guys any response for the first question so good afternoon to all of you have joined the session so as I was telling this would be a quick round we'll have 30 questions to be solved in the coming three hours alright so the first one to answer this is Akshath he says option C what about other guys alright so Purvik also says C I need three more response before I start solving this so let's say this is X so this will be 2x and let's say this is Y so this will be 2y so since this is 90 degree over here I can say that 2x square plus y square is going to be 2 square okay this isn't from triangle B, G and D this is from triangle B, G and D right because BD is going to be the hypotenuse and BG and GD are going to be the sides similarly for triangle AEG from triangle AEG I can say that 2y square 2y whole square plus x square is going to be 9 by 4 that is 3 by 2 whole square right so let's add these two let's add these two I will get 5 times x square plus y square is going to be 4 plus 9 by 4 that's going to be 16 plus 9 by 4 so this implies x square plus y square is going to be 5 by 4 right now I also know that this is going to be 90 degree this is going to be 90 degree over here so my c square my c square will be equal to 2y square plus 2x square that's actually 4 times x square plus y square so c square is going to be 4 times 5 by 4 4 and 4 gets cancelled so c becomes under root of 5 units which means option number c is correct so well done Akshath was the first one to answer this guys let me tell you these are all very easy questions I am expecting the turnaround time for each question to be lesser than 3 minutes okay so please be fast any question with respect to this is there any question with respect to this so please feel free to stop me and ask questions if not I will move on to the next question all right so moving on to the next question yeah the next question is right there in front of you two distinct numbers two distinct numbers A and B are chosen randomly from the set to 2 to the power 2 2 to the power 3 all the way till 2 to the power 25 then then the probability that log B to the base A is an integer then the probability that log B to the base A is an integer is which of the following options please feel free to share in your response on the chat box all right so Pratik Kondinya has come up with the answer B let's see Pratik whether this is correct or not how about the others Akshath, Ramcharan, Shruti, Vashnavi, Urvayak guys back up remember not more than 3 to 4 minutes per question okay so Akshath says D Ramcharan also is of the opinion that it should be B okay guys so enough time was given for this question let's discuss this so of course when you pick up any number let's say if you pick up let's say 2 to the power of 1 okay or 2 then you can only have you can have 2 to the power 2 2 to the power 3 2 to the power 4 all the way till 2 to the power 25 on your as your B right so there are 24 ways you can pick up your B when your A is 2 to the power 1 similarly when your A is 2 to the power 2 then your B can only be 2 to the power 4 2 to the power 6 2 to the power 8 and so on till 2 to the power 24 right that is 11 ways that is 11 ways yes or no so these numbers will be 11 in number now what is happening is that your total number of ways in which you can actually choose B when you choose A as something is depending upon this power actually so if this is 1 then the total number is 25 by 1 gif minus 1 correct when this is 2 it is 25 by 2 gif minus 1 okay then you have gif of 25 by 3 minus 1 why I'm subtracting this one every time is because I have to have two distinct numbers I cannot choose the same 2 to the power once again as B correct so basically I'm trying to see how many multiples I can form except for the same number so I have to continue doing this till I reach till I reach the end so tell me how many numbers will I get I get 24 here I get 11 over here okay then if I choose 3 I will get 25 by 3 gif which is 8 8 minus 1 which is 7 then when you put 4 you get 6 25 by 4 you get gif as 6 6 minus 1 which is 5 okay when you put 5 you get 4 when you get when you put 6 when you get when you put 6 you get 3 when you put 7 you get 2 when you put 8 again you get a 2 when you put a 9 9 will give you 1 okay when you put 10 10 will again give you a 1 11 will also give you a 1 okay 12 will also give you a 1 but 13 onwards will start getting 0 so I will have 35 plus 12 plus let's say 4 7 11 11 plus 4 that would be around 47 plus 15 that is 62 such cases will be formed that will be your favorable events and total number of cases in the sample space will be picking up any two distinct numbers which can be done in 25 into 24 ways which is 600 in number so the probability that your log B to the base a would be an integer would be 62 by 600 that is nothing but 31 by 300 that's option number B is correct so the first one to answer this was Pratik Kondiniya okay well done Pratik good and of course Ramcharan and Kushal also gave the right answer great so we'll move on to the next one any question with respect to this please stop and ask me because I'm going to erase the screen and move on to the next one should I move on type yes if I have to if I should move on it okay great chalo so we'll move on to the next question that's your third question in a triangle XOY XOY is 90 degree let M and N be the midpoints of OX and OY respectively XN is 19 XY is 22 then XY equals 2 this is a very super simple question should not take you more than one minute to answer this just one minute so OXY so this is 90 degree M is the midpoint here and is the midpoint over here XN is 19 YM is 20 this is the midpoint yeah you have to find XY you have to find the length XY this is probably a CT level question it seems to be very similar to the first question we did today all right so Akshath has given one of the and one has given is a response for Rick also backs him up all right guys so let's discuss this let's say this is X this is X this is why this is why it's very simple to understand that X square plus 4 Y square is going to be 22 square and similarly 2 X square that is 4 X square plus Y square is going to be 19 square okay so we add them up we get 5 times X square plus Y square as 22 square plus 19 square okay and we actually need XY XY is nothing but 2 X square plus 2 Y square under root which is nothing but it's 2 times under root of X square plus Y square so we'll find X square plus Y square from here that's nothing but 22 square plus 19 square by 5 so so if I'm not wrong 22 square so 22 square is going to be 484 and 19 square is going to be 361 add them and divide by 5 and divide by 5 you get 169 as your answer okay so your XY would be 2 under root of 169 which is clearly 2 into 13 that is 26 units so your first option is going to be correct okay super simple question should have been done within one minute guys any question with respect to this can I proceed so moving on to the next question this question comes to you from the mathematical reasoning chapter a negation of p disjunction q disjunction negation of p p conjunction q is logically equivalent to which of the following guys a few things I would like to remind you when I say conjunction it is it means p and q so it follows the truth table of and this is called disjunction or alternation which is read as p or q okay this is p if p then q this is if and only if p then q okay is everybody aware of the truth table of these compound statements especially these two I'm interested in are you aware of the truth table for this who doesn't know the truth table for this please let me know now alright so I'm starting getting the responses mostly people are going for option C is there anybody who doesn't know that truth table for if p then q so just quick revision of truth table of if p then q and if and only if p then q please note the truth table let's say these are your inputs then the truth table of if p then q is true false true true okay good table of if and only if p then q will be true false false true so remember your if p then q is only false when p is true and q is false this is the only situation where it is going to be false else it'll be true everywhere okay and if and only if p then q will be true when both are true or when both are false okay so please be careful about this now for here you need not use the truth table it could be solved without that also if you see this I can apply de Morgan's law on this I can apply de Morgan's law on this which is negation p this junction conjunction negation q okay please note that your basic laws of sets will also be applicable because it is as good as saying union or intersection so this is like intersection this is like union okay yeah so I can take negation p common so I will have this operation correct and this is our tautology this is a tautology tautology means it will always be true it will always be true so this is a tautology so negation p conjunction with a tautology will always be negation p so option number c is correct option number c is correct guys please note that j e asks you questions mostly on tautology tautology is a statement which is always true which is always true okay then they'll ask question on fall see fall see is something which is always false you can solve these type of questions by using your truth table by using your truth table please note that statements which are neither tautology nor fall see are called contingency what is contingency contingency is something which is neither tautology nor a fall see neither tautology nor a fall see okay so they may ask you this statement is what tautology fall see can contingency none of these so you should be aware of these words as well is that fine can we move on to the next question so mostly everybody has answered this question correctly so that this shows that you guys are good with your mathematical reasoning at least can I move on guys shallow next question I'm sorry yeah next question this is your next question if the line 2x plus 9 y plus k equal to 0 is normal to this hyperbola then the value of k is super simple questions should not take you more than two minutes yes guys waiting for your response option a okay psi has given option a akshath has also gone for option a guys pretty simple we all have to be well aware of the equation of the normal so let's say we have this hyperbola okay we should be aware of the equation of the parametric form of the normal to the hyperbola which is nothing but a x cos theta plus by cot theta is equal to a square plus b square okay and now we can compare it with this equation given to us okay by the way the a square and b square role is being played by 23 by 3 and 23 respectively so we can write this hyperbola like this okay so your a square is 23 by 3 and your b square is 23 that I will substitute later on 23 by 3 and b square is 23 in this case okay so now compare the coefficients you can write this as a cos theta or a by 2 seek theta is equal to b by 9 tan theta is equal to a square plus b square minus k by minus k so I can say seek theta is minus k a by 2 tan theta sorry seek theta is minus k by 2 a square plus b square and tan theta is minus a kb by 9 a square plus b square okay now we can use the identity to get rid of theta which is secant square theta minus tan square theta equal to 1 which will give you the result as k square a square by 4 a square plus b square minus k square b square by 81 a square plus b square is equal to 1 so I can say k square a square by 4 minus b square by 81 is equal to a square plus b square whole square okay now let us put the value of let us put the value of a square as 23 by 12 minus 23 by 81 is equal to 23 by 3 plus 23 whole square okay so you can pull out a factor of 23 from here and get cancelled from here so 1 by 12 minus 1 by 81 is equal to one third plus 1 the whole square into 23 so this will give you k square 69 by 12 into 81 is equal to 4 by 3 whole square that will be 16 by 9 into 23 so this will go by a factor of 3 this will go by a factor of 27 again this will go by a factor of 3 okay so I will get I will get k square as a 4 sorry k square as 36 into 16 so k will be 6 into 4 that's going to be 24 that's option number a is correct is that fine guys so basically this is a problem which you could solve by comparison of equations you can solve it by comparison of the equation of a normal with the given equation right so psi was the first one to give this answer correctly followed by Akshath, Purvik and Lalitha of course guys any question please feel free to ask me comparison of equation is a big method where we use the method of comparison to eliminate the parameter very very commonly asked in locus questions can I move on guys simple alright so moving on to the sixth question so here we have a question where we have been given a function f of x which is x square minus x plus 1 when x is greater than equal to half then the solution of f of x equal to f inverse x guys just a 30 second question that's a 30 second question would not take more than 30 seconds alright so I only have the answer for Bashnavi 6a so please note any function and its inverse if they have to intersect any function and its inverse if they have to intersect they will always intersect on they will always intersect on y equal to x line so they'll always intersect on y equal to x line please note that so trying to find out where the f of x and f dash x meet which is indirectly saying trying to solve this you can directly relate this to x itself that means x square minus 2x plus 1 is 0 which means x minus 1 whole square is 0 which means x is 1 so that also satisfies the domain of your function so your option number a becomes correct well done Bashnavi she was the first one to answer this question guys any question can I move on so please note that instead of directly solving this equation you could actually solve this by relating it to x that would give you the same set of solution as if you are solving these two equations that is that was the trick in this question is that fine can I move on great so let's move on to the seventh question the minimum value of the function f of x given by sine x by under root 1 minus cos square x cos x by under root 1 minus sine square x tan x by under root secant square x minus 1 and cot x by under root cosecant square minus 1 as x varies over all numbers in the largest possible domain of f of x so you can say you know x can be anywhere from 0 to 2 pi all right so Akshath says a shares Bhaktram also says a so does Rohan kushal Vaishnavi Vaishnavi says D okay for a change for which says C so guys let's solve this question I don't think so it was a difficult question for you because you could write 1 minus cos square under root as mod of sine x so it is actually nothing but this mod of cos x this was nothing but tan x by mod of tan x this was cortex by mod of cortex okay now we have to be very careful we have to choose our quadrants over here let's talk about quadrant number one in quadrant number one all angles are positive so this this value will be 1 plus 1 plus 1 plus 1 which is 4 let's inquire in quadrant number 2 quadrant number 2 sine is going to be positive while cos is going to be negative tan is also going to be negative and cot is also going to be negative so that's going to give me minus of 2 right quadrant number 3 what will happen quadrant number 3 will have minus 1 again minus 1 plus 1 and again plus 1 that's going to give you 0 and in quadrant number 4 what will happen quadrant number 4 again minus 1 plus 1 minus 1 minus 1 that is going to give you I think minus of 2 again so I think the minimum value is minus 2 which is option number D is correct so the only one person who answered this correct question correctly was Vaishnavi Vaishnavi Deshpande very good Vaishnavi guys I think you you went complacent in this problem this was a super simple problem but you mess it so note that under root of 1 minus cos square is not just sine x it has to be written as mod of sine x and so does under root of 1 minus sine square and 6 square x minus 1 so mod is something which you should not ignore okay so these are the simple things Jay can trick you on is that fine are we good to go can you move on to the next problem alright so here is the next question in front of you the number of integral points on the parabola y square is equal to 4x from which or at which from which or at which exactly one normal can be drawn remember integral point is defined as a point both of whose coordinates are integers so how many integral points will lie on this parabola from which or at which exactly one normal can be drawn okay so Purvik says option C so does Rohan Rohan also says C okay Akshet also says C I need at least two more response okay Kushal also says C so guys alright let's discuss this we all know that the equation of the normal is given as y plus Tx is equal to 280 plus 80 cube right given the parabola is y square is equal to 4ax but here my a is one so equation of the normal I can write it as y plus Tx is equal to 2t plus Tq okay now let's say at the this parabola the normal add on to this parabola at a point t okay let's say it's on at t1 okay it goes and meets the parabola again at t2 right now what I'm trying to say is that if there is a unique normal that means first let me make this t2 satisfy this equation right so t2 will satisfy one so t2 will satisfy equation number one so it will become t2 point means and this is the short form of writing t2 square comma 2t2 okay so let me put y as y as 2t2 t1 x will be like t1 t2 square 2t1 plus t1 cube okay so let me bring everything to one side so I can say t1 cube minus t1 t2 square plus 2t1 minus t2 equal to 0 this equation will definitely have one of the factor as t1 minus t2 and you'll see how because you can actually factorize this like this okay so you can see now t1 minus t2 coming out from this okay so I can write it like this t1 minus t2 coming out from this so t1 minus t2 will come out and we'll have t1 square plus t1 t2 plus 2 as the other factor remaining now guys here if there is a unique normal like this that means the only possible solution from this if you consider this to be a cubic cubic in t1 then only possible solution should be t1 minus t2 equal to 0 okay there should not be any kind of solution coming from this equation okay because you can only draw one normal from that point right so it'll be a normal at t1 only it cannot be a normal at any other point this line cannot be a normal at any other point so t1 equal to t2 is the only possible solution right that means the quadratic equation t1 square plus t1 t2 plus 2 equal to 0 should have no real roots this should have no real roots correct if this has no real roots means its discriminant should be negative right so it's a discriminant will be b square minus 4 ac should be negative that means t2 square minus 8 should be less than 0 which clearly implies a t2 should lie between minus 2 root 2 and minus 2 root 2 and 2 root 2 correct now minus 2 root 2 root 2 is approximately 1.414 so 2 root 2 is minus 2.8 something so t2 should lie between t2 should lie between minus 2.8 to 2.8 so the only possible integers only possible integers is minus 2 minus 1 0 1 and 2 that means altogether there are five points from where you can actually draw or at which you can actually draw unique normals exactly one normal to this particular parabola so option number a is correct and nobody gave a right answer for this right mostly people were carried away with option number C but the right option here is option number a are you getting this point guys is any question is there any question related to this please do ask me can I move ahead can I move on to the next question anybody needs any explanation with this so moving on to the ninth question for the day so here we have a question f of x is given as x minus gif x is not equal to 0 and belonging to all real numbers then the number of solutions of f of x plus f of 1 by x equal to 1 then you have to find the number of solutions for f of x plus f of 1 by x equal to 1 is it 0 is it 1 is it infinite or is it 2 let's see who is able to answer this first and correctly okay poor Vick says see option for this infinitely many solutions guys just a hint try to think in terms of one concept that we had taken in binomial theorem see we know your f of x is going to be fractional part of x correct so this equation is trying to say this equation is trying to say a fractional part of x and fractional part of 1 by x that's actually equal to 1 so remember we had done something like f plus f dash equal to one type of scenario in binomial theorem I don't know how many of you recall that just take an example and I'll just take a very simple example for you let's say 2 minus root 3 if I take this as your x okay so we know that or you can say 2 plus root 3 let me take this as x so 1 by x will be what 1 by x will be minus 1 by root 3 correct so you would realize that the let's say this is i plus f and we know that this is always a fractional part okay so when you add them when you add them you get something for correct which means f plus f dash must also belong to some integer because this is an integer this is an integer so this also should be an integer right if this is an integer and you know that f lies between 0 and 1 and f dash also lies between 0 and 1 so the only possible integer that lies between 0 and 2 is 1 correct so you can recall that the fractional part of this plus the fractional part of this let me write it like this so the fractional part of this term and the fractional part of this term will actually add up to give you 1 and there will be numerous such examples like you can have 3 plus root 8 as well this can also be your x right you can have 4 plus root 15 also right okay you can also have 5 plus root 24 also so keep on going like this you'll realize that all these numbers are going to satisfy this functional equation this functional equation okay so your answer is there are infinitely many solutions like this they are in finitely many solutions like this okay so the first one to answer this correctly was Purvik well done Purvik good so guys can we move on to the next question any question so far great so the next question that comes before you is your question number 10 again it is less than a minute question please read this carefully and try to solve it okay so what should we say is option B greatest possible value psi says option D okay Kushal also says D one more response I need guys alright so let's discuss this so guys the word continuous and differentiable itself indicates that it's a question related to the mean value theorem okay so we know that for mean value theorem f dash x right where x belongs to the open interval a to b will always be equal to f of b minus f of a by b minus a so I can apply the same concept over here so there exists an x lying between minus 2 to 5 where the derivative of the function will be equal to the average slope or the slope of the secant which is actually f of 5 minus of f of minus 2 by 5 minus minus 2 which is 7 since f dash x happens to lie between minus 4 and 3 I can say f of 5 minus f of minus 2 by 7 will lie between minus 4 and 3 that means this expression f of 5 minus f of minus 2 will lie between minus 28 to 21 which means the greatest value is my is equal to 21 that's option number D is correct so the first one to answer this correctly was Psi Meher well done Psi very good simple question it was based on the Langrange's mean value theorem Langrange's mean value theorem any question is there any question with respect to this can I move on to the next slide all right next question is question number 11 there's a function from natural numbers to natural numbers where the function actually gives you the sum of the digits of n whatever is put into the function now is f of bijective function surjective function injective function it is not a function Psi says 11 b Purvik also says Ramcharan also says so Janta is going with option b all right of course it is not a bijective because it will not be a one-one function because if I put a number like 12 I will get answer as 3 and so would be the answer for something like 21 right so many to one mapping is happening so cannot be bijective cannot be injective okay has to be a function because whatever answer you put you will get a unique output for it okay for example if I put a 72 I will get always a 9 for this I cannot get any other answer for this because it is adding the digits of this number so it has to be a function so option number B is the only correct answer in other words you can always find a natural number which can be obtained by the sum of a digits of some natural number always okay so guys super simple question should not have taken you more than 30 seconds to solve it so option B is correct can I move on to the next one next if the roots of this equation z cube plus I z square plus 2 I equal to 0 represents the vertices of a triangle in the argon plane then find the area of the triangle then find the area of the triangle yes guys any response from anyone all right so Rohan says option A guys first we need to figure out what are the roots of this cubic equation I can clearly see that when you put z is equal to I it will satisfy this equation because I cube plus I cube plus 2 I will be actually minus I minus I plus 2 I that's equal to 0 so z minus I will definitely be a factor okay so let us see what will be the other factors of course I will have a z square term over here okay and I would have a minus 2 term over here now I just need to figure out what is the middle term over here what is the middle term over here for that it's very simple I would see what all terms are contributing to the z square term so there'll be something called kz over here so let's let's find out that k over here so z square term would be coming from kz into kz multiplication okay and it'll also come from this term minus I okay so this is going to be compared to I z square so k minus I is equal to I so k will be 2 I so you can finally factorize this as z minus I z square plus 2 I z minus 2 equal to 0 correct so from here I can get z equal to I no problem for this I have to use my Shridhar Acharya formula or the quadratic equation formula minus b plus minus under root b square minus 4 ac minus 4 ac will be plus 8 by 2 so that is going to give me minus 2 I this is going to be 4 under root of 4 is going to be 2 by 2 that's going to be minus I plus 1 in fact minus I plus minus 1 so we have two other roots coming out which is 1 minus I another is minus 1 minus I okay so these are the three roots let me write it properly yeah now it's like having three points it's like having three points I is like 0 comma 1 1 minus I is like 1 comma minus 1 and minus 1 minus I is like having this point so we can always use the determinant method of finding the area of the triangle which is x 1 y 1 1 x 2 y 2 1 x 3 y 3 1 okay so let us expand this so when we expand this I will have half times 0 will make everything 0 so they can take this one minus 1 which will give you 1 plus 1 which is 2 and plus 1 minus 1 minus 1 which is minus 2 that's going to give you half of minus 4 which is minus 2 and of course you have to take a mod of the answer so when you take the mod of the answer that is going to give you 2 square units as your final answer so option number a becomes correct in this case so option number a becomes correct in this case and I think the first one to answer this and the only one to answer this was actually Rohan right so well and one very good guys it was just a matter of identifying the points and using the determinant to get the area of the triangle okay so it was an easy question you should have all solved this so is there any question with respect to this any question any explanation please let me know can I move up can I move forward alright so let's next question is this the number of roots of this equation x square plus x plus 3 plus 2 sine x equal to 0 where x belongs to minus pi to pi again a very simple question you should all be able to answer this all right very simple question Purvik has already given his response which is option a what about others okay Kushal also says option a okay Ramcharan also guys very super simple question if you have to solve this question you basically have to see where these two graphs intersect okay now if you see this this expression if you complete this square you can actually see it's actually this and this expression is minus of 2 sine x now this will have the greatest value as this will have the greatest value of 2 whereas this itself has exceed 2.5 right it's approximately greater than 2.5 right so it cannot happen that the greatest value of this is not able to match the minimum value of this expression the minimum value of this expression is 11 by 4 so minimum value is 11 by 4 right so 11 by 4 is greater than 2 so we cannot have any possible values of x satisfying these two conditions that means there is no solution for this equation that means zero solutions so as I told you these all questions are very very simple they have to be solved well within three minutes in fact this question should not have taken you more than a minute or so is that fine guys can I move on to the next one next if x y z all are positive quantities then the minimum value of this is given as lambda x y z then lambda is okay. Rithvik says option D okay so starting getting the responses as D D D from almost everyone now Kushal Rohan Sai everybody's saying D alright guys again this is the question which indicates that you will be using your AMGM inequality over here right so we have the terms if you write these terms in an expanded fashion so these terms are actually this okay now all of these terms are positive right so these all terms see all these six terms are positive so I can say this will be greater than equal to the product of all of them okay so x y square into x z square into y z square into y x square into z x square into z y square hold to the power of 1 6th okay so let's say I call this term as s so s by 6 would be greater than now how many x's are there 1 2 3 4 5 6 so it's x to the power 6 similarly y will be also 6 z will be also 6 okay so 1 6th of this so s is greater than 6 times x y z which clearly implies the minimum value the minimum value of s is 6 times x y z that means lambda is going to be your 6 so option number D is correct super simple question again yes most of you have got this right so guys moving on to the next question which is question number 14 this comes to you from the properties of binomial coefficients 30 C 0 30 C 10 minus 30 C 1 30 C 11 plus 30 C 2 30 C 12 all the way till 30 C 20 times 30 C 30 is which of the following options just observe the pattern in our the difference of these two is always 10 difference of these two is always 10 that's a hint for you in this question okay so Kushal says 15 D okay how about others alright so Sai also says 15 D need three more response before I start solving this yes guys others please respond Akshath Ramcharan Kondanya Lalitha Sanjana okay guys let me start with the expansion of 1 plus x to the power of 30 so this is going to be 30 C 0 30 C 1 x I'll go till 30 C 10 x to the power 10 then 30 C 11 x to the power 11 and all the way till 30 C 30 x to the power of 30 okay now if I write x minus 1 to the power 30 expansion I start with 30 C 0 x to the power 30 then we'll have minus let me change the color minus 30 C 1 x to the power of 29 then plus 30 C 2 x to the power 28 da da da da all the way till 30 C 10 x to the power 20 30 C 11 x to the power 19 and by the way I have messed up the sign over here this should be negative if I'm not wrong and yeah finally I'll move on to 30 C 30 x to the power 0 guys what terms do I need I need the product of 30 C 0 and 30 C 10 right so 30 C 0 and 30 C 10 is actually associated with the coefficient T correct similarly 30 C 1 and 30 C 11 again are associated in fact minus 30 C 11 again are associated with x to the power 20 because when this x multiplies to x to the power 19 it will generate x to the power 20 so can I say indirectly indirectly I am looking out for the coefficient of x to the power 20 in the product of in the product of these two terms isn't it which is actually looking like which is actually like saying coefficient of x to the power 20 in this term right so if you expand this term it is going to give you 30 C 0 minus 30 C 1 x square plus 30 C 2 x 4 okay minus 30 C 3 x to the power 6 and so on so it is very clear that by the time I reach x to the power 20 I would have something like plus 30 C 10 over here right and it is clear that this will become your coefficient of x to the power 20 in this expression which is option number D is correct so the first one to answer this was Kushal Kushal was the first one to answer this well then Kushal it was though very simple question others should also have contributed so this question is based on the binomial coefficient series which we had done to a great depth last year is it fine guys any questions can I move on to the next question please feel free to stop me in case you have any doubt any concern anywhere alright so we'll move on to the next question which is question number 16 again a super simple question there's a relation R on set A such that if a comma B belongs to R implies B comma A belongs to R inverse and R is equal to R inverse R is equal to R inverse then R is choose the best option choose the best option okay super week says B why should we also say is B guys this is a sitter actually because it has to be option of movie so it's a property that all symmetric relations will satisfy this condition all symmetric relations will satisfy this condition so guys just a question I would like to ask you you know related to the relations if you have a set a containing let's say n elements a 1 a 2 a 3 till n elements okay how many reflexive relations are there how many reflexive relations are possible on this set a basically how many relations are possible from set a to set a which are reflexive which are reflexive I just try to type in your response in the chat box if you can think again think again it's very simple n is the number of n is the number of elements in a reflexive relation I'm asking you how many reflexive relations are possible there's a difference in the question this question may come in your no two marker or one marker all right so I I've already got the answer from Prateek Kondiniya Prateek you are absolutely correct see guys treat as if you have a set of n cross n okay where the elements are a 1 1 a 1 a 2 and so on till a 1 a n similarly a 2 a 1 a 2 a 2 and so on till a 2 a n okay so like that it goes now these elements must definitely be there in your symmetric relation right just assume that they have a common between assume there's a common between okay and let's make up pair from this so these elements which are there in the diagonal position must definitely be there right so these off these elements they don't have any option but to be there so they are they are having one option each correct but remaining elements they may or may not be there right so this option may or may not be there so two options for this similarly a 1 3 may or may not be there so two options for this so like that whatever elements are left off they will all have two options each left so please note that the elements in the diagonal position they don't have any other option but to be there but the remaining n square minus n they have two options each so it will be like 1 to the power of n into 2 to the power of n square minus 1 that is going to be 2 to the power n square minus n reflexive relations can be found so well done Pratik Pratik has given the right answer for this guys just to ask a follow-up question on this how many symmetric relations are possible can somebody tell me can somebody tell me how many symmetric relations will be possible on the set a number of symmetric relations Spurvik says 2 to the power n square minus 2n plus 1 okay anybody else okay so Pratik Kondiniya gives the answer as 2 to the power n square plus n plus 1 by 2 so Pratik let me tell you you are correct again guys how do I solve this particular scenario now see here try to imagine a situation where if you're trying to make a symmetric relation if you're trying to make a symmetric relation of course these elements they have two options each they may be there may be there may not be there right but let's say this is chosen then this will automatically be chosen so they are like tied up a kiss out a free types so if you choose a one comma a two as your pair you have to choose a two comma a one in your relation right so if you're choosing this whose probability is sorry whose options are to you this automatically comes along with it so if this is chosen this will be chosen if this is not chosen this will not be chosen are you getting this point similarly this has two options chosen or not chosen and with this is linked to the fate of a 3 1 also right so they're coming in pairs these are coming in pairs so you have to just decide whether you want to choose a 1 comma a 2 or a 1 comma a 3 or not the other pair will be automatically linked to it are you getting this so what is happening is that the number of ways of choosing a symmetric relation will be how many elements are you choosing from here right so you can choose any one of the elements on this pair or not choose any one of the elements from this triangle so all together there are 1 1 plus 2 plus 3 plus 4 till n that means n into n plus 1 by 2 numbers there in this upper triangular matrix okay so each one will have two options so your answer will be 2 to the power n n plus 1 by 2 so this will be your total number of symmetric relations so well done Pratik you was again correct please note that there is no direct method to find out the number of transitive relations is that fine guys so all the elements in the upper triangular matrix will have two options each whether to be chosen or not so there are total number of number the total number of elements in the upper triangular matrix is n n plus 1 by 2 so its answer will be 2 to the power n n plus 1 by 2 next question number 17 the number of permutations of the letters of the word Hindustan says that neither the pattern hint nor dust nor tan appears okay Pratik says 17 b first of all such heavy numbers you know they kill a lot of your time in calculation okay so noted down your answer Pratik what about others I can see a lot of piggybacking happening so one person says some option many people back it up any that is two more response alright so Rohan is the one with a different answer he says 17 d is the is the right answer okay let's see Rohan okay Apurva also says b alright so we have got five responses over here so we can start solving it so guys the meaning of nor nor means let's say a is the event that your letter has him let a be the event that your letter has him b be the event that your letter has dust and c be the event that your letter has or sorry the word has tan together so what I'm looking for I'm looking out for this and I now according to D moeville's sorry according to D Morgan's law I can say this is equal to this correct universal set minus the number of elements in a union b union c so we know that the universal set will be the total number of words we can form from this please be careful two n's are repeated over it so it will be 1 2 3 4 5 6 7 8 9 factorial by 2 factorial and when I'm writing n a union b union c I will use the inclusion exclusion principle that is n a plus n b plus n c minus n a intersection b and b intersection c and c intersection a plus n a intersection b intersection c okay now n a is like trying to ask how many words will have h i n together so if I tie up h i n together okay so I will have 1 2 3 4 5 6 7 words and none of them will be repeated so n a will be 7 factorial without any doubt right n b will be the where your dust is together so let's say I tied dust so we'll have hint and so we'll have 1 1 2 3 4 5 6 7 factorial by 2 factorial because 2 n's are repeated over here similarly n c n c will be where your tan is together so tan together will again be 7 factorial then n a intersection b that means you want in and dust both to be together so when you tie up hint and dust together and you have tan separate so 1 2 3 4 5 that is 5 factorial words can be there similarly b intersection c if I'm not wrong you're tying up our dust and tan dust and tan again will be 5 factorial then a intersection c you're tying up hint and tan hint and tan if you're tying up again 5 factorial and when you have hint dust and tan altogether a intersection b intersection c will be like 3 factorial so by the inclusion exclusion principle we can say our answer is going to be 9 factorial by 2 factorial minus 7 factorial by 2 factorial factorial minus 5 factorial minus 5 factorial minus 5 factorial plus 3 factorial well I have to pick up a calculator for this unfortunately just to save time for everybody so 9 factorial so 9 factorial will be 7 factorial into 8 into 9 divided by 2 181434 and I'm not going to subtract I'm not going to subtract 5040 minus 5040 by 2 minus 5040 again 360 sorry plus 360 plus 360 minus 6 minus 6 that gives me 1 sign 194 which is option number D is correct which is option number D is correct and the only one to answer this correctly was Rohan Arut well done Rohan good one guys these questions please do not mess it up they are so simple just you have to use the case of your addition theorem in an addition theorem you have to use your inclusion exclusion principle carefully that's always required so we'll take up one more question before we go for a small break next if two events A and B are such that PA complement is 0.3 PV is 0.4 PA intersection B complement is 0.5 then find out PB given a union B complement so not more than two minutes will be given for this question this this is a super simple question anyone okay so Rohan again is the first one he says 18 a okay Lalitha says 18 C I need three more response alright so this is not that difficult question so PA will be 0.7 undoubtedly okay now when you're finding this expression which means you are trying to find out P B intersection a intersection sorry a union is there right so my bad there has to be union over here yeah so BA okay divided by P a union B complement okay now you can use sets over here you can use the sets over here we know that B intersection a union B complement could be written by distributive property of intersection over union it could be written as P B intersection a union B intersection B complement B intersection B complement will be a null set so this will be union of null set with any set is the same set itself so we'll have PA intersection B here and we'll have PA union B complement over here correct now we know that PA intersection B complement is actually PA minus PA intersection B alright it's basically trying to address this area this area is a intersection B complement given that this is your set A and this is your set B right so PA minus PA intersection B will be PA intersection B complement so this is already given to us as 0.5 PA is 0.7 so I can easily find out PA intersection B so PA intersection B will be 0.2 will be 0.2 so this is 0.2 undoubtedly what about the denominator denominator I can write it as PA plus P B complement minus P a intersection B complement that's going to be 0.2 PA PA is going to be 0.7 P B complement is going to be 0.6 and this is already given to us as 0.5 so that's going to be 0.2 by 0.8 that's going to be 1 by 4 that means option number A is correct again Rohan has cracked it first well done Rohan guys I don't think so it was a difficult problem you just had to apply your concept of sets and addition theorems wherever applicable and of course the condition probability formula is it fine can I move ahead or before that I would like you to take a break just take a break okay and we'll resume at right now it's 5.57 let's resume at 5.08 p.m. let's resume at 5.08 p.m. alright so resuming back let's start with this question a super simple question on limits lift and pi by 4 plus ln x whole to the power 1 by ln x extending to 1 alright so Vaishnavi is saying 19 D so first of all the indeterminancy over here is 1 to the power infinity form whenever you have f of x raise to the power g of x 1 by g of x limit extending to a where f of x and g of x the limit as extends to a both of them is giving you zero okay then the result is e to the power f of x by g of x limit extending to it so here will be e to the power tan pi by 4 plus log x minus 1 by log x limit extending to 1 so this is clearly a case of 0 by 0 form so I can apply LH let's apply LH rule so by applying LH rule I will get limit extending to 1 e to the power the derivative of this will become secant square pi by 4 plus ln x times 1 by x and derivative of this is going to become 1 by x so 1 by x and 1 by x will get cancelled so putting x as 1 it will give you e to the power secant square pi by 4 plus 0 which is e to the power root 2 square root 2 square will be e to the power 2 so option number b becomes correct so Purvik is the only one to get the right answer for this option number b is the right answer is that fine guys I hope there's no question with respect to this yeah Vaishnavi I'm sure you must have done some silly mistake somewhere okay so can I move on to moving on to the 20th question again a super simple question f of x from r to r is a differentiable function and f of 1 is 4 then find the limit of this so we have done many more difficult questions than this so this should be an easy one for you very very simple questions guys should not take you more than 30 seconds all right so Purvik has already given the response as option 8 Kushal also says the same how about others Sanjana Ramcharan Kondiniya Lalita Shes Kushal Kushal also says a yeah so it's clearly a 0 by 0 form right because when you put x as 1 over here f of 1 also goes for till 4 so it's like integrating from 4 to 4 so again I can apply LH rule over here I can apply LH rule over here so to find the derivative of this term I will have to use the Leibnizun so it'll become 2 times f of x times f dash x and 4 I need not put because derivative of 4 will be 0 denominator will become 1 okay so now we have to find the limit of this expression as x tends to 1 so it'll be 2 times f of 1 f dash 1 which is going to be 2 times 4 into f dash 1 that's going to be 8 f dash 1 that's going to be option number is correct so again Purvik is the first one to answer this so I where are you okay so these questions are simple can I move on any question with respect to this please ask me moving to the next question if f of x is defined as this function and g of x is defined as integral of f of x from 0 to x x lies between 1 to 3 then which of the following option is correct they're mostly talking about the local extremars of g of x it should be a simple one okay so Purvik again says a you guys first find g dash x g dash x will be nothing but f of x correct now see where f of x is becoming 0 obviously this cannot become 0 only these two can become 0 so let's put them to 0 okay so this will give me e to the power x minus 1 is 2 so x is x is 1 plus l into guys be sure this happens to fall under the domain so this should fall in the domain of the function which very much it is falling because lawn 2 is approximately 0.6 something okay and this gives you x as e okay and this also belongs to 2 to 3 interval okay now how do I know whether this is the maximum or minimum very simple do the double derivative test so when you differentiate this once again when you differentiate this once again the derivative of this will become minus e to the power x minus 1 which is definitely a negative quantity that means this point is a point of local maximum okay and derivative of this expression is 1 which is definitely positive that means this point is a point of local minimum so option is correct which says g of x has local maximum at 1 plus l into and local minima at e so Purvik is again the first one and the only one to answer it correctly I don't think so this is a difficult question I don't see a point why people are not responding is there fine any question with respect to this can I move ahead with the next question so we have nine more questions to go so next one is an integral wow hat rick so Purvik has again given the answer first in fact he has given his response let's see what others have to say Purvik I'm assuming you are working equally hard in chemistry as well are you okay so guys I can clearly see that the derivative of this derivative of this okay since the answer is saying this term try to try to compare it with this form which we are done in integration by parts so for this case your answer is e to the power x f of x plus c right okay now what is happening over here is that since this is appearing in your answer this term is appearing in your answer it means derivative of this term which is 2x by x square plus 1 okay what we can do is yeah this to this b term we can do one thing this to be term should be equal to see actually okay because if I do that then automatically this term also becomes a b correct so to be equal to see is meeting the requirement and the only option which says to be equal to see is option number a okay or you can try to substitute also you can see when you put b as 1 c s 2 it perfectly fits into this expression and here also it becomes e to the power x ln x square plus 1 so option number a satisfies this required condition so a is your correct option you have to be equally good in chemistry as well chemistry improves the rank no matter what you have to become good in chemistry because maths could be difficult but chemistry there is less chance that chemistry will be difficult all right moving to the next question question number 23 find the value of integral of gif of x from 0 to n by fraction part integral from 0 to n again this is super simple question I am expecting everybody to answer this Sondarya says option d okay Vaishnavi says a kushal also says a all right guys so this gif function since then there's a natural number you have to break it from 0 to 1 which is 0 1 to 2 which is 1 2 to 3 which is 2 and so on till n minus 1 to n which is n minus 1 okay so end up getting 1 plus 2 plus 3 all the way till n minus 1 correct which is nothing but n into n minus 1 by 2 now what about integral of fraction part of x fraction part of x is a periodic function it's a periodic function so you can write it like this and in the interval 0 to 1 this function behaves like x it becomes n into x square by 2 it becomes n by 2 so your answer will be desired answer will be n into n minus 1 by 2 divided by n by 2 so n by 2 n by 2 gets cancelled your answer is n minus 1 which is option number d so Sondarya is the only one to answer this correctly guys again basics these are all basics right we have done many problems which are far more complicated than this the session was actually to boost your confidence can I move on now next area of the region bounded by the curve y equal to e to the power x in the lines x equal to 0 and y equal to e okay so Nanda says option a guys buck up not a very difficult question just make the right diagram so e to the power x is going to be an exponential function like this from x equal to 0 x equal to 0 is your y axis this is called x equal to 0 line and why is equal to e y is equal to e will be a line like this so we have to find out the area which is sandwiched between this okay so guys let me tell you this is 0 comma 1 point it's better to take horizontal strips for it because your options all has dy dy term appearing in it okay so if I take a horizontal strip I have to write this as x as a function of y rather than y as a function of x so x is the function of y will be x equal to ln y correct so it will become ln y minus 0 dy integration from 1 to e integration from 1 to e which is actually integration of 1 to e of ln y right now I can apply the extension of kinks property all of us know the kinks property that integral of f of x from a to b is integral of f of a plus b minus x from a to b right this is the extension of kinks property okay kinks property or kinks rule so I can write my answer as integral from 1 to e ln e plus 1 minus y dy so I think option number b is correct please note that derivative sorry integral of this will be this and when you put e and 1 when you put e it'll give you 0 and when you put 1 also it'll give you 0 sorry when you put 1 it'll give you 1 times 0 minus 1 yeah it'll give you 1 okay so option number b is correct in this case it cannot be a cannot be c and d either is that fine guys can I move on next question this is to be read as angle bisectors the angle bisectors bd and cf read it like that yes anybody you have to find the equation of the line bc or it is to look at this figure from here do you see that since this line is the angle bisector of angle b can I say image of image of a will lie somewhere on bc okay image let me call this as bd let me call this as cf guys is the screen visible to y'all so can I say image of a on bd will lie on bc similarly image of image of a on cf will also lie on bc because it's like angle bisector angle bisector means let's say this is a bisector we know that any point here its mirror image its mirror image will lie on this line itself is that fine everybody's convinced with that now what is the mirror image of a along the line y equal to x so this is your line bd so can I say mirror image will be nothing but 5 comma 3 so this will be your mirror image 5 comma 3 okay so this will lie on bc this will lie on bc similarly mirror image on x equal to 10 line would be the distance from here to here is 7 so you go 7 more on this side so it'll be 14 plus 3 which is 17 comma 5 this will also lie on bc so you just have to see which line satisfies these two points so 5 and 3 5 and 3 will not be satisfied by this 5 and 3 will be not be satisfied by this 5 and 3 will be satisfied by this similarly 17 and 17 and 5 check is it satisfied by this so 18 so 30 minus 17 that's going to be 13 yes it is also satisfied so option c is the right answer so option c will be your right answer is that fine so moving on to the next question question number 26 for the day so here we have a question tangents drawn from 1 comma 8 let's say this is 1 comma 8 to the circle x square plus y square minus 6x minus 4 y minus 11 equal to 0 touch the circle at a and b so let us say these are the tangents and this are the tangents touches the circle at a and b so we have to find out the equation of the circum circle of triangle p ab the circum circle of triangle p ab again just a 30 second question guys not more than that so it is obvious that this angle here would be 90 degree right and so would be this angle so if you make a circle which is passing through p a and b okay so basically it will be in such a way that it will be passing through c as well so the circle would be of such a nature that it's not very accurate so I'll just draw a rough one so it will be passing through this point this point like this okay that means pc will act as a diameter to this circle all right so all you need to do is find the equation of a circle whose diametrically opposite points are 1 comma 8 and this will be 3 comma 2 if I'm not wrong the center of this circle will be 3 comma 2 that will be x minus x 1 x minus x 2 plus y minus y 1 y minus y 2 equal to 0 which is x square plus y square minus 4x minus 10 y plus 19 equal to 0 that's option number b is correct that's option number b is correct correct I think we used to use this in our construction questions in class 10th right I don't know how many of you remember this in construction chapter in class 10th we used to use this concept chalo without much I do we'll move on to the 27th question now in a battle 72% of the combatants lost one eye 78% lost in here 75% an arm 80% a leg x% lost all the four what is the minimum value of x okay Purvix is a what about others okay so we have to find how many people have lost all the four so we know that a union b union c union d will be n a plus n b or I can take a shortcut so how many people have not lost an eye let's try to look at that situation how many people have not lost an eye not lost an eye will be 28 okay let's say total is 100 total is 100 not lost an year will be 22 not lost an arm will be 25 not lost a leg will be 80 will be 20 will be 20 okay so the minimum value would be let's say I'm assuming these people have also not lost everything so I'm assuming that these people who have not lost an eye an arm an ear or leg if you subtract it from total so that'll give you 50 plus 45 that is 95 so 100 minus 95 that is means 5% of the people is the minimum that you can expect have the lost all the four because these people have not at lost at least one of the things isn't it and of course there can be common as well right so to have a maximum difference I can assume that 28% 22% 25 and 20% have not lost at least one of the things so total minus have not lost at least one of the things will give you the number of people who have lost all the four so the minimum answer is going to be 5% which is option number d which is correct moving on to the next question question number 28 it's from 3d geometry so p is the point 326 and q be a point on this line on this line then the find the value of mu for which pq is parallel to the plane so pq is parallel to the plane in other words n vector is perpendicular to pq okay it's very easy you can assume q to be minus 3 mu plus 1i mu minus 1j and 2 plus 5 mu okay so this is the position vector of q position vector of p is already known 3i plus 2j plus 6k so pq vector will be the difference of these two that is position vector of q minus position vector of p that's going to give you minus 3 mu minus 2i cap mu minus 3j cap and 5 mu minus 4k cap so pq dot n n is i minus 4j plus 3k this should be equal to 0 so minus 3 mu minus 2 minus 4 mu minus 3 plus 3 times 5 mu minus 4 should be equal to 0 okay that means 8 mu that's going to be 12 12 gets get cancelled minus 2 equal to 0 so mu is going to be 1 fourth so option number a is correct guys yeah they are parallel parallel means line is parallel to the plane means like this this is parallel to this plane which means n is perpendicular to pq the normal to the plane will be perpendicular to pq so when line is parallel to the plane then this is perpendicular to the line n is perpendicular to the line right parallel to the plane is perpendicular to the normal is that fine so we can move on we can move on with the next question this is an extremely easy question, find the angle between the vectors a and b where a, b, c are unit vectors satisfying a plus b plus root 3c equal to a null vector. Guys, fast, these questions are super simple, should not take much time. All right, so size says option D, let's check. The first I wrote it like this and I take the dot product with a plus b on both the sides. In fact, I'll take with this with both the sides. Okay, so here I'll get mod a square mod b square to a dot b. And here I will get three times mod c square. Now, we know that this is one, this is one. This can be written as two mod a mod b into cos of the angle between them. And this is also one. So it's one plus one plus two cos theta is equal to three. So two cos theta is going to be one. So cos theta is going to be half. So theta is going to be 60 degrees. That's pi by three. That's option number D, which is correct. Is that fine, guys? No question with respect to this. All right, so moving on to the last question. This is from sets. I'm sorry, this is from statistics. So we have been given that the standard deviation of 10 observations is four, 10 observations, y1, y2 till y10 is three and summation of xi minus x bar, yi minus y bar from 1 to 10 is given to us as 80. Find the standard deviation of these observations. Find the standard deviation of these observations. That means you have to find this. Any response, guys? So if you see this expression, you can actually write it again as xi minus x bar minus yi minus y bar the whole square by 10, correct? Okay. Now, this could be written as summation xi minus x bar whole square plus summation yi minus y bar whole square one by 10, of course, outside minus two times summation xi minus x bar times yi minus y bar. Okay. Now, we have been given certain information that this in this expression that is one 10th summation xi minus x bar square is given to us as the square of the standard deviation that is equal to be four square. So this is four square. Similarly, one 10th summation yi minus y bar square is also given to me. That is going to be three square. Okay. And we have also been given minus one fifth. This summation is given to us as 80. So this is the required, let's say I call this as sigma square. I call this as sigma square. So this is the required variance of the data. So it will be one 10th 16 plus 9 minus 16. So, sorry, 16. We have 16, 9. Why am I getting a negative expression? One fifth is 16. Yeah. Oh, sorry, sorry, sorry, sorry, sorry. There's a mistake over here. One 10th is already known to us as 16. This is 9 minus 16. Okay. So that's going to be 9. And the standard deviation would be under root of that standard deviation would be under root of that expression, which is going to be under root of nine. That's going to be three as your answer. So your option number B will become correct in this case. Your option number B will become correct in this case. Is that fine, guys? So this formula was important over here. This formula was important. Any questions with respect to the last problem? Is it clear? Please note that this is the variance. And what you have been given was your standard deviation. So you have to square your standard deviation to get the variance. Similarly, this was the variance of the other data. And this is the data which we have been already given separately. This one. Okay. So we have, we got the variance of the, we got the variance of the, this data. So we took the under root of that, which is under root of nine, which is going to be three as your answer. All right, guys, thank you so much for coming live YouTube and all the best for your school exams. And of course, the J exam, which is going to happen on Sunday. So this is a full length paper. So it would be more or less of the same difficulty level. Please do not make any kind of silly mistakes, people who are writing it on nine. So all the best and do reach out to us in case of any doubt. Thank you for an out from my side. Bye bye. Have a good day.