IPMAT Indore 2026 Quant Marathon 3

Added: 2026-05-07

[00:00:48]Good evening everyone. How are you guys doing? I hope I'm properly audible and visible to you guys. Right, we already have some students here.

[00:00:59]Arush Gurv and uh Priti natural arr hello guys so last three days right Thursday Friday and Saturday then it's going to be the most important date of the year right IPMAT indor exam only 3 days away I hope all of you guys are positive right and uh working hard for it on daily basis. So in the first two sessions we were able to practice lots of questions from arithmetic from all the topics right we covered questions from percentages profit and loss ratio simple interest compound interest average mystery allegation time spread distance time and work all of these topics were done some questions more some questions less from a particular topic but we were able to do questions from all the topics today we are going to discuss questions on algebra Algebra and in algebra we are going to discuss questions from linear equations, quadratic equations, logarithm, indices, inserts, modulus, functions, series and sequence. These are the topics that we are going to discuss. So I hope all of you guys are ready with pen and page right.

[00:02:24]So let's start with our first question.

[00:02:28]Okay.

[00:02:31]12 hours marathon.

[00:02:37]Okay.

[00:02:44]Practice key, right?

[00:02:48]Okay. Okay guys, moving to our first question.

[00:02:55]This one.

[00:02:58]Everyone try this question.

[00:03:00]and let me know your answers. Okay.

[00:03:36]H I'm all ready guys. I'm ready. If you can arrange something like that then I'm ready to go with you guys. Okay.

[00:03:55]Which of the following cannot be a value of k if the given simultaneous system of equations has one unique solution 4x - 3 y =0 3x + 4 y - k y² is equal to 0. How do we solve this? Today you are going to see some of uh unique questions okay that you have never seen okay and uh all the concepts that you have learned on the basis of that those questions could be solved. So be alert and try every single question. Okay.

[00:05:21]Milan, you can write mocks. No problem with that.

[00:05:38]So how do we solve this question?

[00:05:40]Which of the following cannot be a value of k if the given simultaneous system of equations has one unique solution 4x - 3 y is equal to 0 and 3x² + 4xy - k y² is also zero. So if we think about only this equation there could be infinite solutions right. And if you talk about this one right for different values of k there will be infinite solutions. But if we check what is x from here. So we can say 4x is = 3 y or x is equal to 3 y upon 4. Now if I put this here what will happen? So all the values of x with respect to y we are putting here. Right? Then if if we take the condition here 3 * 3 y by 4² - 4 into 3 y by 4 into y - k y² is equal to 0 or this will become 27 y² upon 4 - 4 cancelled out 3 y² - k y² is =0. We can remove y from all sides.

[00:07:03]We can get 27 by 4 - 3 - k is equal to z or k is equal to k k.

[00:07:17]Something is wrong with that. We need to find this will be 16. My bad. This will be 16, right? This is 16.

[00:07:27]Okay. And 3x² + this is plus plus plus.

[00:07:32]So 16 16 + uh 48 this will be 75 upon 60.

[00:07:40]Now if you are taking if you are replacing x by y here that means you are using the solution of this into this.

[00:07:48]This can have infinite solution in that way it will have infinite solution. If k is equal to 75 by 16, these two equations will have infinite solutions.

[00:07:58]Right? So answer is option C. It says which of the following cannot be a value of K. So if you take K is equal to 75 by 16. That means both of these are giving you the same solution. Right? There could be infinite relation between X and Y. Infinite pairs of X and Y which will follow this. If K is 75 by 60.

[00:08:22]So only some of you were able to take solve it. Aayos got the answer. Ma got the answer. Son got the answer. Right. Okay.

[00:08:33]So let's move on to the next one.

[00:08:37]This one 80 natural both solutions are simultaneous equation means we need to find unique solution for both of them. Okay.

[00:10:14]And for which value of K that is not possible. That is only done when so we can take the ratio of X and Y from first equation and put it in the second one.

[00:10:49]a sh then try to solve it again.

[00:11:12]Some students were saying that uh questions were not difficult enough in first session.

[00:11:21]So that onwards I took some good level questions and now I don't see many responses. What is happening guys? Tell me.

[00:11:32]A local bakery is donating at most 18 identical cupcakes to three community centers X, Y and Z. Let the number of cupcakes they receive be X, Y, and Zed respectively. All non- negative integers means they can be zero as well. Center Y must receive more than center X but less than center zed. The combined share of X and Y must be greater than the share of Z. Each center must receive at least three cupcakes. How many distinct ways can the cupcakes be distributed? So at least three x will be smallest than y then zed. And what do we know that x + y should be greater than z. x + y + z should be less than equal to 18. These are the two conditions that we need to take care of.

[00:12:23]This must be greater than equal to three. We can start with three. If I take this three, this will be at least four, right? And what could be the value of zed? X + Y is 7. So zed could be 5 or 6. Then if I take this three, take this five. 5 + 3 8. Zed can be 6 and 7. Then if I take this three, this as 6. Then 6 + 3 9. It can be 7 and 8. Then if I take this three take this as seven then sum is 10 I can take this eight I cannot take this 9 because then it will be 19 right I cannot take this 9 because then it will be total 19 3 + 7 + 8 is 18 3 + 7 + 8 is equal to 18 that's why we cannot take it any more than that and then next we cannot take 3 and 8. 3 + 8 will be 11 and it must be minimum 9. That is not possible. So I take this as four. If I take this four, this must be five. 4 + 5 is 9. This I can take 6 7 and 8.

[00:13:43]Right? 4 + 5 9 + 8 9 + 8 is 17. And 4 + 5 must be greater than zed. Then if I take this four, this I take six. Then I can take this seven and eight. Not more than that. Then if I take this four and I take this seven then I can take this seven. Not possible right?

[00:14:08]So next if I take this five this will be six and this will be seven. That's it.

[00:14:14]That's the total number of solutions.

[00:14:18]So two solutions here.

[00:14:21]2 2 1 right then three here then 2 here then one here total 3 + 3 6 + 1 7 7 + 6 it will be a total of 13 so answer is option B there was nothing in this question just you need to take care of these three condition okay one condition x can be greater than equal to 3 less than y less than Z then X + Y is greater than Z X + Y + Z less than equal to 18.

[00:14:55]You need to take care of all of these and the question is done. Nothing is special right?

[00:15:01]Uh no it looks like arn that these type of questions can be solved by P and C but no it's not P and C it's simple linear equation question. Okay. So there is no other way you can solve it as per I know. No, combination permutation cannot be applied.

[00:15:22]Yes, Aos, you can you can uh subtract 3 into three and then make the cases.

[00:15:29]No problem.

[00:15:31]Okay, let's try the next one.

[00:15:34]An easy question, guys.

[00:16:49]Right. Answer. For the simultaneous system of equations x + 9 y = 12, k y - 4x= m to have infinite solutions.

[00:17:00]What should be the value of k and m? So a1x + b1 y is equal to c1. A2X + B2 Y is equal to C2 have infinite solutions.

[00:17:19]If A1 by A2 is equal to B1 upon B2 is equal to C1 upon C2. So if I write it x + 9 y = 12 - 4x + k y = m. So what we can say 1 upon -4 should be equal to 9 upon k that should be equal to 12 upon m or k will be equal to 9 into -4 that is equal to - 36 and m will be equal to 12 into -4 that is equal to - 48. So yes, option A is absolutely correct. You know, option A is the correct answer guys. So don't forget these three rules where we have infinite solutions, where we have unique solution, where we have no solution. Right? Don't forget this question can be asked uh on these concepts. Although these are very simple but definitely questions can be asked because these are very popular questions.

[00:18:32]Okay, let's try the next one.

[00:18:37]Ma cupcake wall already discussed there.

[00:18:42]We just need to take the values nothing else. X can be minimum three. If I take X3, Y will be minimum four. 3 + 4 X + Y is 7. Zed must be greater than Y.

[00:18:56]Greater than four it can be five. And six. It should be less than 7. Right?

[00:19:01]Then three. Then five. And rest. Same same thing. This will be the last case.

[00:19:07]And we just need to count count it.

[00:19:09]There is nothing special in this question. You just need to consider all the cases.

[00:19:32]Cat p.

[00:19:36]But uh there is a very important concept that you should know here right How many of you have seen this question before?

[00:21:27]Anyone 58 59 for the given pair X Y of positive integers such that 4x - 7 17 Y is equal to 1. X is less than 1,000. How many integer values of y satisfy the given condition? First of all, we need to find one integer solution and then rest can we can find out easily. So 17 if I put y is equal to 1, it will give us 17. There should be one more than 17. 18 is not a multiple of four. Take 34. 34 is uh even 34 + 1 35. 4x cannot be 35. If I take y is equal to 3. This will become 51. 4x will become 52. X will be 30.

[00:22:28]Now what's the pattern that it will follow? The value of X will increase with the positive coefficient of Y which is 30 47 uh 43 13 after 13 it will increase by 17 which is 30 then 47 then 64 and so on. This will this we can say 13 + n -1 * 17 this will be the last term it will become it will increase by the cofficient of x. So 3 + 4 is 7. Whenever the cofficient are of opposite sign both value either increase or decrease. So what we can say 13 n 13 + n -1 * 17 should be less than 1,000 or 17 * n -1 should be less than 987 less than 987 or n minus one should be less than 98.

[00:23:39]So 5 * 85 137 17 into 8 is 126. So 58.x or n will be less than 59.xx.

[00:23:55]So n will be 59.

[00:23:59]So total maximum value that we can take for n here will be 59. The number of terms which are possible number of values for X will be 59. Correspondingly Y will also take 59 values. So the answer is option D. Okay.

[00:24:19]Option D is the correct answer.

[00:24:25]Okay.

[00:24:33]Moving to the next question guys. This one.

[00:24:48]It's one way that you can find the first value is doing heat and trial. Take the different value of y. Right? 4x is equal to 17 y + 1. We should put the values of y so that it becomes multiple of four.

[00:25:05]y= 1 is not possible. 2 is not possible.

[00:25:08]When you put y= 3, you get x is equal to 30. That's the minimum possible value of y. Right? And then you can increase the values. Increase the value of x with the positive coefficient of y. Increase the value of y with the cofficient of x. And if you want to solve it properly, then I use this method. If you don't want to do heat and trial, 4x can be written as 16 y + y + 1. Now left hand side is a multiple of four. Right hand side must be a multiple of four. y + 1 must be a multiple of four. Minimum value I can take is four. So y will be equal to three and that will be minimum. This method is what I use in these type of situations where I need to find the first integral solution. Okay? Or you can do hit and trial by putting values.

[00:26:02]Okay.

[00:26:07]How did n minus one came uh come? So 13 next value will be 13 + 17 into 1. Then next will be 13 + 17 into 2. If there are n terms so nth term will be 13 + n -1 17 or 17 * n -1 this will be nth term right so and this value should be less than 1,000 so how do you find the value of n using this okay okay manish and uh aren Hat.

[00:27:18]Yes Achala. Uh we cannot take negative because it's given in the question that X and Y are positive integer. Okay.

[00:27:25]Achala read the question properly. It's given in the question. X and Y are positive integer. Okay.

[00:27:36]A not X is not less than,000.

[00:27:40]The number of terms 57 terms 57 is less than uh 1,000. Okay. There will be 57 values of x that can be taken. The 57th value will be 13 + 56 into 17. This is the last value of x.

[00:27:59]Last value of x or greatest value of x.

[00:28:03]That will be the last value of x. You keep adding 17. You can add 56 times before it crosses 1,000.

[00:28:52]is asking sir should I do this marathon or 250 most expected questions question 250 question solve experience you know learning 250 series you can do anytime right once I had been to post office to buy stamps of 5 rupees 2 rupees and 1 rupee I paid the clerk 20 rupees and since he did not have change he gave me three more stamps of 1 rupee if the number of stamps of each type that I had ordered initially was more than one. What was the total number of stamps that I bought?

[00:29:50]So number of one rupee stamp is equal to x. Okay. and uh 2 rupees stamp is equal to y and 5 rupees stamp as equal to set. These are initial value. These are initially so what we can say he bought three he had to get uh accept three one rupees time for change 20 - 3 is equal to 17. What will be the value? X + 2 Y + 5 Z should be equal to 17. Right? X + 2 Y + 5 Z is equal to 17.

[00:30:37]All of these values are greater than 2.

[00:30:39]X Y Z all of these are greater than equal to 2. Right? I had I had ordered initially was more than one. Right? Each of these values is more than one. So at least two.

[00:30:53]See if you put at least two what will happen? This will become 10. This will become four and this will become two.

[00:31:02]That will become 16. If x = y = z = 2.

[00:31:10]This will become 16. Now there is only one way possible when so what we can say one rupee less. X must be 3, y must be 2 and zed also must be 2. 10 + 4 + 3 17 total it will be 3 + 2 + 2 and three more three more 1 rupees that will be equal to 6 + 4 10 total 10 steps option A is the correct answers all of you who got option A is the correct answer guys option B is not correct So, Arian Gerve best clip option A is correct.

[00:32:00]Guys, don't spam the chat. Okay, short slayer. Don't spam the chat otherwise you will be kicked out. We are very strict about that.

[00:32:22]Yes, last three add because of this three rupees change. Okay, this three this three more stamp will be added here. These three is added here because it says how many what was the total number of stamps that I bought. So including this three this is x y z + three three more as change. Okay.

[00:32:54]Okay. So in question it says that out of 20 rupees he bought some stamps and in return he got all the stamps that he needed plus three more one rupee stamp.

[00:33:07]Right. So basically he had bought 17 rupees worth stamps initially.

[00:33:14]One rupee stamp is X 2 rupees stamp is Y 3 rupees stamp is Z. All of these values is greater than equal to two more than one. So X one rupee stamp X rupees 2 rupees Y stamps 2 Y 5 rupees Z stamps.

[00:33:30]So 5 Z total value this should be 17.

[00:33:33]Now if I take all of them two stamps it will become 16.

[00:33:38]So in 16 rupees you are getting two one rupee stamps, two two rupees stamps and two five rupees stamps. Now you are left with one rupee. For that one rupee you can get only one one rupee stamp. That means x is equal to three. Number of stamps that he went to purchase right is three. Number of two rupees stamp two. Number of five rupees stamp five two and three more stamp as change.

[00:34:07]So total will be 10. Okay Acha I hope that is clear now.

[00:34:13]Let's try this one.

[00:34:17]Okay. Three.

[00:34:20]I hope that is clear short slayer X1 Y7 Z1 is not possible because it's given in the question that each type of stamps is more than one.

[00:34:44]Okay.

[00:35:14]This is a very interesting question guys and uh you really need good hold on linear equations special equations to solve Yes.

[00:36:06]Heat. Heat. N.

[00:36:26]Do you know this axus b y is equal to k must have infinite integral solutions if hcf of ab is equal to 1.

[00:36:39]So randomly you can take any equations 7 x - 2 y = 30 9x - 5 y is = 70 3x - 17 y is = 3 all of these all of these will have infinite natural number solution. Okay, all of these will have infinite integer solutions that we need to find but not required. But but but but what we can say that uh if k if k is not a multiple of scf of a and b then there will be zero integral solution. there will be zero integral solution. See in this case 3 s minus nt 3s - nt is = 5 will have infinite integral solution. will have infinite integral solution if HCF of 3 and n is equal to 1. What does that tell you? It says no integral solution have no integer solution. So for no integer solution n must be multiple of 3. So n is less than 40 right? So n can be 3 6 till 39. So 13 values for n are possible.

[00:38:46]Answer is option B. See I have written everything here. All right. Everyone was getting 14 or option D. But the correct answer is option B. Guys you know Achala got this correct.

[00:39:03]A we can also take three. We can also take three as well. Okay.

[00:39:32]Okay, done. Remember this is an important concept, guys. Okay, let's go to the next one.

[00:39:49]Try this.

[00:41:50]Yes, unatted guys to different possibilities otherwise otherwise possible if unattempted questions are not there unatted zero marks that's it otherwise there will be only one possibility right when uh number of correct questions 25 wrong questions 50.

[00:42:20]It's not given but it's obvious. Okay.

[00:42:22]In these type of situations, if there are no if there are no unattempted question then correct question plus wrong question will be 75.

[00:42:33]Correct question wrong question will be 50.

[00:42:38]5 C 125. C will be 25 and wrong will be for 50. There will be only one solution possible, right?

[00:42:49]Okay.

[00:42:51]But that is not true. It says how many different combinations of correct and wrong are possible.

[00:42:58]So unattempted zero. That is understood. Okay. Unattempted zero marks. there will be unattempted otherwise it would have been given in the question that every question must be attempted. Okay.

[00:43:18]So what do we say number of correct questions number of wrong questions must be less than equal to 75. Right? Then correct question four marks. Wrong question minus one. This should be 50.

[00:43:35]What we can say from here that correct question must be less than equal to 50 by4.

[00:43:42]So correct question must be sorry greater than equal to so number of correct questions must be greater than equal to 50 by 4 correct question must be greater than 12.5 right till what values? Till what values? So if I want to maximize if I want to maximize the number of correct questions the W will also be maximized right. So what we can say let's put it here C we can say W is equal to 50 - 4 C. Let's put it here. C + 50 - 4 C should be less than equal to 75 or uh W is equal to 4 C - 50 by mistake 4 C - 50. So C + 4 C - 50 or 5 C should be less than equal to 125. C should be less than equal to 25. C should be less than equal to and greater than equal to.

[00:44:50]So C must be greater than equal to 12.5 but less than equal to 25. So the values that C can take will be 13 14 till 25.

[00:45:05]So number of combinations is equal to 25 - 13 + 1 that is equal to 13.

[00:45:19]So 13 different combinations are possible right? Who got 13? Got 13.

[00:45:27]Ishant part Aush push ship all of these got 13 which is absolutely correct.

[00:45:40]Vab also got 13 and he was also the first one guys.

[00:45:51]response.

[00:46:00]Hannah So let's try the next one. A question from quadratic.

[00:47:11]You already got the correct answer right?

[00:47:16]C + W total 75 questions. There could be some unattempted as well. So correct plus wrong will always be less than equal to 75. Four marks for correct minus one for wrong. So 4 C minus W should be 50. These two are given in the question. Now what we can say 4 C will always be equal to 50 + W. Right? W is what? Number of wrong answer wrong attempted questions. So basically non- negative value. So 4C will always be greater than 50. So C will always be greater than 50 by 4. C will always be greater than equal to 12.5.

[00:47:58]Then I can take W is as 4 C minus 50 from here and let's put it here. So we get this inequality where we get C less than equal to 25. C will always be less than equal to 25. So this is the range we get for C. For every C you can find W. Right from here for every C you can find W and that will be within the restrictions.

[00:48:23]This is how you Let gx= x² + px + q be a quadratic function with real roots alpha and beta.

[00:49:46]It is given that g0 is equal to g4 and g1 is less than equal to minus1. Another equation x² + fx + n a=0 has roots that are squares of roots of g is equal to0.

[00:49:59]Roots are alpha square and beta square.

[00:50:03]Find the value of m in terms of q. First of all, we are going to find what do we need? Right? What is m? m is nothing but minus alpha² + beta square. Right? And this m I can write alpha + beta² - 2 alpha beta.

[00:50:30]So this I can write minus alpha + beta is what? Minus p. So - p² - 2 alpha beta. So 2 q. So we can say m will be equal to minus of p² - 2 q or m is = 2 q - p². So what do we need to find? We need to find p² only. We don't need to do anything. We just need to find what is p. How do we find p? So let's check the condition. G 0 is equal to z4. Means if I put x = 0, what I'm going to get? 0 + 0 + q is equal to 4. So 16 + 4 p + q. So 16 16 cancelled out. P is equal to -4.

[00:51:28]This I can use here. So what we can say m will be equal to 2 q - 16.

[00:51:36]Now in this one, this one is totally irrelevant, right? This one is totally irrelevant. There is no use for that. We needed the value of N and M. We got 2Q - 16.

[00:51:51]Option A and know.

[00:51:55]So some irrelevant information as well.

[00:51:59]VV Sony Maverick Achala Gurv GV check your answer correct answers guys very good okay let's try the next question.

[00:52:39]I have done nothing out of the box.

[00:52:42]Question was regarding find M. So I tried to find M. I reached here. I got to know I can not find value of Q and still it will work. I need to find the value of P. What information is given?

[00:52:56]This one and this one. I use this information. I got the value of P.

[00:53:00]That's it.

[00:53:04]So whatever is said in the question I have done only those right nothing nothing excess if you can find the answer even even if you are using this you got some inequality later you will find that there is no use of that right.

[00:53:17]So yes, it's not easy to avoid that.

[00:53:22]But that's why you need to find what do what do you need? If you know what do you need then uh it's easier right M is asked find the value of M.

[00:53:41]So M question right.

[00:53:46]So you will find the value of M right click.

[00:53:56]Okay. So, let's try the next Okay. How many got the correct answer?

[00:58:43]Short slayer wrong answer. Gerve wrong answer.

[00:58:47]Vav correct answer. An aancha wrong answer. GV now got the correct answer.

[00:58:59]Turfalar. Thank you.

[00:59:06]Okay. A person has a total of 50 coins in three denominations rupees 1 2 and five. Let's take this as X. Let's take this as Y. There are exactly five five rupees coins. So this is five. Okay.

[00:59:23]Rupees five denomination. The total value of rupees 1 coin is at least what we can say total value total value will be x + 2 y + 5 into 5.

[00:59:38]So x + 2 y + 25 this will be the total value rupees 1 coin the total value of rupees 1 coin is at least 50%. So x is greater than equal to x + 2 y + 25 by 2. Now one thing that we know that a total of 50 coins x + y + 5 is equal to 50 or x + y = 45 or we can say x = 45 - y because we need to find the value of y.

[01:00:16]Right? The total value of rupees one coin is at least 50% of the total value of the coins in the back. The total value of rupees two coins is at least 15% of the total value of the coins in the bag.

[01:00:29]What is the number of rupees two coins?

[01:00:31]So we need to find the value of y. So I'm taking x in terms of y. Here we can say 2x is greater than equal to x + 2 y + 25 or this I can take left side it will become x - 2 y should be greater than equal to 25 or instead of x I can put 45 - y - 2 y is greater than equal to 25 or 3 y is less than equal to 20 or y is less than equal to 6.66.

[01:01:07]That's a say our first observation.

[01:01:11]Second, total value of two rupees coin.

[01:01:13]Two rupees coin is greater than equal to 15 by 100 of x + 2 y + 25.

[01:01:22]So this is 3 * 5 20 * 5. This will be 40 y is greater than equal to 3x + 6 y + 75 or this will become 40 - 6 34 y - 3x is greater than equal to 75.

[01:01:43]Now 34 y 34 y instead of 3x I can put 135 - 135 + 3 y is greater than equal to 75 or 37 y is greater than equal to if I add this it will become 210 or y is greater than equal to 210 220 5 something this is our second so what we can say that y is greater than equal to 5 something but less than equal to 6.66 66 y can take only one integer value that is option B six great work guys all of you who we are able to solve it that's really good work you know it was not an easy question okay is that clear let me know if you have any doubts clear. Okay, let's check the next one.

[01:03:22]a relatively easy question. Okay.

[01:04:17]But I don't encourage that guys. Don't do this. Okay?

[01:04:21]You don't need to send any kind of uh money.

[01:04:27]Okay? You guys are in 11th 12th. Use that for your purpose.

[01:04:34]We surely don't need that from you guys.

[01:04:37]Okay.

[01:04:39]Your presence is the ultimate help guys. I'm eating chili jutka.

[01:04:57]Oh, total sugar 15 g.

[01:05:16]Added sugar 12 g.

[01:05:22]Protein.

[01:05:25]It has protein as well. Can you believe that?

[01:05:29]Okay guys, P, Q and R are the roots of the equation Z Q - 3 Z² + Z + 1 is equal to 0. So what do we know? If PQR are the roots, we can say P + Q + R will be minus of minus 3 which is equal to 3.

[01:05:50]product of roots taking two at a time will be one and product of all the roots will be minus one. What we need to find 1x p + 1x q + 1x r if I take pqr as lcm that will be this will be qr + p r + pq. Now we know this value this value is one and this is minus one overall it will be minus one.

[01:06:21]Option D that is correct.

[01:06:28]Yes Akancha I am from Bihari.

[01:06:43]Yes. Congratulations Arush.

[01:07:01]Folks, see whenever you have a x² + bx + c =0 roots alpha and beta. So you can say alpha + beta is equal to minus b by a alpha beta is equal to c by if you have a x² + bx a x cq + b x² + cx + d is equal to 0 and the roots are alpha beta gamma. So we can say alpha + beta + gamma is minus b upon a alpha beta plus beta gamma plus gamma alpha is equal to c by a and alpha beta gamma is minus d by a this will follow even in higher degrees. Okay first single roots taken together two roots at a time three roots at a time four roots at a time and keep going like that. This is what we are using here. Okay. So this is the equation. PQR are the roots. So this is what we can observe.

[01:08:06]From there you need to use this. That's it.

[01:08:10]Okay.

[01:08:12]Fox.

[01:08:17]Let's try the next one on the similar concept.

[01:08:22]Very good question.

[01:09:45]Yes. You see question a 100 x ^ 100 a 99 x ^ 99 a 98 x ^ 98 and so on.

[01:10:01]A1 x to the power 1 and a 0.

[01:10:07]This is the equation and the roots. What will be the roots?

[01:10:12]Roots will be -1 2 - 3 4 and so on. Sum of roots is equal to minus a 999 upon a00. Right?

[01:10:29]minus a 999 upon a 100. A 100 will be equal to zero. A 100 will be equal to 1 because if you multiply the cofficient of x ^ 100 will be 1. So what we can say this will be -1 + 2 - 3 + 4 till 98 - 99 + 100 this should be a 99 or a 99 is equal to 50.

[01:10:59]Uh 9 this should be minus this is minus so minus 50 minus a 999 option B right option B is the correct Answer.

[01:11:28]Ashep G an correct answers guys. Sony correct answers.

[01:11:43]Okay.

[01:11:44]So let's move on to the next question.

[01:11:47]Let's try this one.

[01:12:44]When is in polomials this is the pattern that is followed.

[01:12:49]Sum of the roots will be minus of this.

[01:12:54]Then taking two roots at a time sum of that will be a98 upon a 100. You need to know this. If you know this then good.

[01:13:02]If you don't know this you can follow the pattern from here. Sum of the roots is always the negative of the cofficient of one less power upon a like this. This one this one same thing will be here. Sum of the roots will be minus a 999 upon a 100. A 100 will be one because if you multiply all this cofficient of X to the power 100 will be one. The cofficient of X to the power 100 will be one. So a 100 is one.

[01:13:39]That's it. Then we added all the roots.

[01:13:43]We got minus 50.

[01:13:57]That's the formula of polomials. Okay.

[01:16:14]Okay, lots of answers. If alpha, beta, and gamma are the roots of the equation xq - 4x² + 3x + 5 is equal to 0. Find the value of the product. Alpha + 1, beta + 1, gamma + 1. Usually, we think that we need to find the roots here, but that is not necessary. See if alpha, beta and gamma are the roots of this then this can be written as x - alpha x - beta x - gamma right if these are the roots. Now if I put x is = -1 what will happen?

[01:16:55]-1 - alpha -1 - beta -1 - gamma will be equal to put min -1 here. So min -1 this will become -4 this will become -3 + 5 or this will become minus 1 + alpha 1 + beta 1 + gamma and that will be equal to minus so 1 + alpha Uh 1 + uh 1 + alpha 1 + beta 1 + gamma will be equal to 3. That's your answer and that's a good question. You don't need to find the value of alpha, beta, gamma or you don't need to multiply all this and then find the answer. Right? How many of you have used this method to solve this question? Because from here I can surely see that alpha + beta plus gamma will be four. Alpha beta + beta gamma + alpha gamma will be three. Alpha beta gamma will be minus 5. And then I can use this here. But that will take so much time.

[01:18:22]If you understand this, then use this right. That's much easier.

[01:19:04]Okay. Same concept we are using for last three questions guys and because it's it's important.

[01:19:16]Okay, some of you are going doing good.

[01:21:16]This is a very simple question from linear equations. If you know how to write the correct equation from the question, it should not be difficult for you guys.

[01:22:20]Goldie Hawk, the gold digger whose height is 180 cm is standing in a ditch that she is digging.

[01:22:29]She is 1/4 done right now. When she had finished digging, her head will be twice as far below the ground as it is now above the ground. Then the depth of the hole when Goldie Hog is finished digging will be let's say this is the hole right and so far C is done by 1/4 this is D by 4 and C is standing like this okay so this is the surface what will be this height this height will be 180 D minus DX4 this part of her body is above the head of her body is above the ditch. Right?

[01:23:19]Then in the second situation when she is in the ditch h when C is in the ditch her head will be below. So total distance is D. We have taken total distance is D. So this distance will be d minus 180. That's it.

[01:23:43]If you are able to figure out this you are done with the question. What we can say according to the question that she's done 1/4 right now. She is done 1/4 right now. When C has finished digging her head will be twice as far below the ground as it is now above the ground.

[01:24:01]So dus 180 is equal to twice of 180 minus dx4 right. So what we can say d - 180 is 360 - dx2 or 3dx2 is equal to 540.

[01:24:24]This is 180 or d is equal to 360 cm.

[01:24:31]360 cm. Option C is the correct answer.

[01:24:34]Right? So yes, Aayush your answer is correct. Achala, an Sony or SE correct answers.

[01:25:01]Okay, I hope the diagram is clear, right? I'm not a very good uh not very good in drawing pictures, but I tried my best to make you understand, right? So, I hope this is useful.

[01:25:25]180 cm is long, right?

[01:25:30]Best clip it's given in the question that C is done 1/4. C has dug so far.

[01:25:36]This part she has dug so far. This part she has dug so far. This part is 1/4.

[01:25:43]She's done 1/4. Total distance is D. So her head is above the ground, right? By how much? Total C is 180 cm in length of which this is dx4. Her head is above the ground by this much 180 minus dx4. Then when she is done digging this will be the uh ditch. She will be standing on the base of the ditch. Total length is D. Her head is below this surface by how much length? Total is D.

[01:26:19]C is 180 m 180 cm. This length will be d minus 180. That's what it says. Okay.

[01:26:26]Okay. Best clips.

[01:26:35]Okay. Let's go to the next one.

[01:28:16]So this is a very popular type question right?

[01:28:21]Whenever you see that a to the power b is equal to 1 then there will be three different situation when a is equal to 1 b could be anything second b is equal to0 a could be anything and third a is = minus1 and b is equal to even now can you try this and don't think that the answer is always six, right?

[01:28:54]Answers could be anything, guys.

[01:29:29]So I'm getting answers 3 4 5 6 first of all case one x² - 4x + 3 is = 0. So x - 3 into x -1 is =0 or x = 3 1 two values we get from here. Second case when d is =0 so x² - 7 x + 10 =0 uh wait a second this should be one right this should be 1.

[01:30:11]This is equal to 1 or x² - 4x + 2 is equal to zero.

[01:30:23]Say solution right a one. So no integer solution.

[01:30:35]Right? No integer solution.

[01:30:42]Correct guys? Is that clear? x² - 4x + 2. No integer solutions. Right?

[01:30:50]I use distinct. Of course, we are talking about distinct.

[01:30:55]Okay. Then x² - 7 x + 10 is equal to 0.

[01:30:58]x - 5. x - 2 is =0. X = 5 and 2. Right.

[01:31:08]Right. Say real solution.

[01:31:25]Two real solutions then it okay then it's okay two real solutions okay integer it's not says it doesn't say integer two real solutions are possible okay complex b square - 4 is c 4² - 8 is greater than zero right so two real solutions Find x= 5 and 2. Then we take x third case x² - 4x + 3 = -1.

[01:32:07]x² - 4x + 4 is = 0. x - 2² is = 0. So we get only one value of x which is 2. If you put two here for x = 2 this will be even right. So it it's okay. So answer is five two a two to already answer will be four because this two and this two is common right? These two are common.

[01:32:44]So answer will be four. 2 here, 1 year, 1 year.

[01:32:50]Four is the correct answer.

[01:32:54]Correct.

[01:33:06]Okay. Understood.

[01:33:24]So remember the answer is not always six. It could be anything less than equal to six. Okay, some people had answered six without even solving it. So be careful. Not always the answer is six. Ch. Let's try the next one.

[01:36:24]Verde wrong answers guys. Check your answers Okay.

[01:37:26]Yes, 10 is the correct answers. 10 is the correct answer. DHS, correct answer, guys. For what minimal positive integral values of k are both roots of the equation x² + k - 3x + k is equal to 0.

[01:37:43]Real, negative, and unequal. Real, negative, and unequal. What does that tell you? First of all, if we talk about negative roots, so a x² + b x + c =0 has negative roots means c is greater than 0 because both the roots are negative. Negative into negative is positive. Product of the roots will be positive. Means c by a is greater than zero. Then uh end and end b² - 4 a c should be greater than zero. That's it.

[01:38:26]So what we can say here k - 3² - 4k should be greater than 0 or k² - 6 k + 9 - 4k should be greater than zero or k² - 10k + 9 should be greater than zero. k² - 9 k - k + 9 should be greater than zero.

[01:38:55]K - 9 K -1 should be greater than zero 1 9 Here you get positive positive negative meaning K should be less than 1 or K should be greater than 9 but here it says that K is greater than zero right here it says K is greater than zero k is greater than zero so what we can say k must be greater than Fine. So K minimum will be equal to 10.

[01:39:30]Minimum positive integer value.

[01:39:34]So 10 is the correct answer.

[01:39:42]Equally one me. Yes. Equal yoga.

[01:39:54]No sha same questions, same syllabus.

[01:40:00]Let's try the next one.

[01:40:19]So guys, K is positive, right?

[01:40:24]K, this expression will be positive when K is greater than 9 or K is less than one.

[01:40:30]But K less than one, for K less than one, there won't be any positive integer value. So K must be greater than 9, right? K must be greater than 9. Greater than 9 and positive and integer, what's the minimum value K can take? 10. Okay, got it. Spurge clear 10 to two or five are roots.

[01:41:02]Now 2 or 5 - 2 - 5 x² + 7 x + 10 = 0 x= -2 - 5 Eight match form.

[01:41:32]Okay, done.

[01:41:36]Try the next question, guys.

[01:45:56]Okay. Find the number of integral values of x that satisfy the following equation. x ^ 5 - 4xqub - 344x² + 165 16,513 is equal to zero. Now these values are here for a reason. What is that reason?

[01:46:17]48 into 344 is equal to 16,512 that you must have gotten it if you had tried it right. So what we can say we can write it is x ^ 5 - 48 xqub - 344x² + 16 512 + 1 is equal to 0 or if I take x cub common I get x² - 48 - 3 44 common we get x² - 48 is equal to -1 1 or x² - 48 into x cq - 3 44 is equal to -1 which can be -1 into 1 or it can be 1 into -1 right so let's check if I take x² = 48 x² - 48 is = 1 then x² will be 49 so no integer solution If I take x² - 48 is = -1 then x will be equal to 7. If x is equal to 7, we need to check this x cq - 344.

[01:47:40]Right? So in this case, see in this case there will be two situations that x will be x²= 49 or x = 7 and - 7. If I take x cq - 344 then this value should be 1 or this should be yeah this should be yes so if I put 7 so 343 - 344 344 is equal to -1 right that is only possible when x is equal to x= 7 So that's the only solution. So answer will be equal to 1. One positive integer solution possible.

[01:48:35]Okay.

[01:48:40]Clear.

[01:48:52]Okay guys.

[01:49:08]Is this clear to everyone? Why not 49 integer solution? Why GV? Why 49 integer solution? Tell me that. Why do you think 49 interior solution?

[01:49:20]X² we are getting 49. So how are you getting 49 integer solutions?

[01:49:28]X can be 7 or minus 7.

[01:49:32]This is only possible when X is equal to 7, right? Not minus 7.

[01:49:40]So there will be only one.

[01:49:44]So root of 49 is 7. So 49 solutions say okay all good let's go to the next one.

[01:50:16]Uh take minus one + one this is + one this is minus one to 47 correct now it's okay all good Sure.

[01:50:52]Let's go to the next one.

[01:51:28]Manish, we already did boat and stream problems. Okay, we already did that in in arithmetic session.

[01:52:46]Now this is not a regular question but still solvable.

[01:52:59]Maverick this is what we are doing right revising algebra quickly aren't we revising all the concepts along with all the questions we have another session tomorrow so what's better than this revision other than that you can write sectional tests you can try previous year questions as well option B is not correct. Wherever can you tell me the solution.

[01:53:46]See we can write this as y ^ 4 is = 20x - 12x y² or y ^ 4 is we can take common as 4x 5x - 3 y². Now x and y both are non- negative integers. x and y both are non- negative integers. Think about this.

[01:54:13]This is y to the power 4 right? This is y to the power 4. If this is positive, y to the power 4 is always positive. Right hand side should also be positive. So what we can say if I take y, y is equal to 0. So what will happen? This will become zero. This can never be zero. So 0 0 is one solution possible. Okay. Uh question is regarding number of solutions, right?

[01:54:42]Okay. Okay. number of solutions that is that is one that is correct. So when y is equal to0 x is equal to0 that is possible but when you increase the value of y when you take y is equal to 1 this will become one there are no solution possible. If you take y is equal to 2 no the value will keep keep increasing the value will keep increasing left hand side but not right hand side and that will be the only solution. So answer is option P. Yes, web correct answer right you four 4x to five huh take this will be five check so y 0 x0 is possible no other solutions are possible okay yes I have written it 5 - 3 y square thank you to mistake on Okay.

[01:55:46]So again this left hand side is positive. 4x will always be positive.

[01:55:53]Right? If not zero 4x will always be positive.

[01:55:58]But if you take the value of y anything greater than 2 it will become negative and that won't be possible. If you take y is equal to 1 that is not possible anyway. So only one solution possible.

[01:56:09]Okay, is that clear now? So, GV correct answer.

[01:56:20]UNS correct answer. VAV correct answer.

[01:56:27]So, it looks a very difficult question but it was not right.

[01:56:39]Yes, I use zero is a non- negative integer. Correct.

[01:56:43]Zero is definitely a non- negative integer.

[01:58:14]always cancel.

[01:59:23]This is a divided by b where a and b are a by b. Yes. Yes. A is divided by B. Correct. Correct.

[01:59:42]Small.

[02:00:16]Hello S.

[02:00:19]Okay.

[02:00:22]X divide. Divide this by X and then try to solve it. Okay. Then try to solve it. Divide this by X.

[02:01:04]Divide to 2x + 3 + 2 = + 2x or 2 * x + 1x = - 3 or x + 1x = -3 upon 2. Can you find x² + 1x²?

[02:01:37]Can you find this?

[02:01:42]What will be x² + 1x?

[02:01:45]That is nothing but x + 1x² - 2. So that will be equal to 9x 4 - 2 which is equal to 1x 4. Right? Fair x ^ 4 + 1x ^ 4. How what can we write that?

[02:02:06]That will be x² + 1x² - 2 which will be equal to 1x 4² - 2. So that is 1x 16 - 2. So that is - uh - 32 by 1 - - 31 by 16 say confirm. Then we can find x ^ 8 + 1x ^ 8 will be x ^ 4 + 1x ^ 4 square. This will be square minus 2. What will be this value?

[02:02:54]What will be this value?

[02:02:57]Tell me what will be x ^ 8 + 1 by x ^ 8.

[02:03:02]Everyone answer that.

[02:03:14]Folks, again if you have practiced algebra question then you will surely able to find it that here we are x² + 1x² x ^ 4 + 1x ^ 4 x ^ 8 + 1x ^ 4 8 and you need to know if you know x + 1x then you can find x² if you know one value you can find rest of the values right.

[02:03:37]So, Maverick, what's the final value? What's the final value? Tell me the final value.

[02:03:57]Okay, I want you guys to solve it. Tell me the final value.

[02:04:10]Yes s correct guys be is answering 449 upon 256 is that correct ma 449 by 256 is that correct or not?

[02:04:34]Yes. So 449 by 256. Now can you find a by b a by b will be equal to x² + 1x² 1x 4 - 31 by 16 and this is + 449 upon 256. Now tell me what's this sum?

[02:04:58]What is this sum? You can take the LCM as 256 and then calculate.

[02:05:14]What's the final sum? This is this will be you can take these two together then these two together then these two together calculation say if you are attending the class then try to do as I say and you will improve every single day after every single question what's this Are you 17 by 256?

[02:06:05]Yeah. R say 64 and uh 16 into 31.

[02:06:14]So now it's in irreducible form. So minimum value of a + b a + b minimum will be 256 + 17 which is equal to 273.

[02:06:29]Option a right.

[02:06:34]Now all of you guys who have ever solved a question where you need to find x² + 1x² when x + 1x is given. Can you tell me this question was solvable or not? Was this question solvable? Yes, a little bit calculative.

[02:06:57]But was this solvable?

[02:07:01]DH Praapati got this correct. 273. That is amazing.

[02:07:08]Very good.

[02:07:33]AMGM.

[02:07:50]There is no requirement for AMDM again you need to check what is required to find in the question right. Question and from there you can understand once you know x² + 1x² you can find x ^ 4 + 1x ^ 4 you can find x ^ 8 + 1x ^ 8. So you need to find this once you are able to find this. How do you find this? You can find that from here only. Right?

[02:08:42]This is the hint. So from here you can see 2x² + 2.

[02:08:49]So you can find x + 1 by x that's it.

[02:08:52]And of course we are doing this so that you can learn how to solve these type of questions right.

[02:09:04]Uh in case of alternative yes again you can take this right hand side. You can take minus 3 on right hand side then take x on the left hand side. Ultimately you need to know we need to find x² + 1x² and further and for that we need what we need x + 1x that's it and that you need to find from the given equation.

[02:09:30]Okay, let's try this one. An easy one to difficult questions.

[02:09:43]So after that an easy question guys.

[02:10:01]Hello Sham. Sorry for late uh 2 days ago. Yes, you can attempt mock. Okay. 2 days ago. Don't attempt the mock on uh Saturday. Okay.

[02:10:24]Try to relax. Try to revise basic concepts.

[02:10:30]Okay.

[02:11:13]If f(x)= xq + 3x² + 4x + k and it has one root as x = -2. So put x = -2. So this should satisfy it - 8 + 12 - 8 + k should be equal to 0. Right? If I put root in place of x, we should get zero as well. So from there we can say k will be equal to 4. k is equal to 4. Kus 3 will be equal to 1.

[02:11:50]K will be equal to 4. k + 3 will be equal to 7. So the equation that we need is x -1 into x -4 into x - 7. So one thing that sum of roots should be 12. So -2 you can discard these two. Product of the roots 7 into 4 28. So minus 28. So - 28 is here. Option A. We don't need to solve it. But if you solve it, you are going to get the same thing. This if you solve this should be equal to zero or it will be xq - 12x + 12 x² + 39 x - 28 is equal to 0. You don't even need to solve it. You can eliminate the options. Did you see that what I did there?

[02:12:44]I found the sum of the roots. Then I discarded two of the options. Then I found the product of the roots and then again I discarded two options uh one option and we get the correct one.

[02:12:57]So option A is correct.

[02:13:00]Uh deep vari Mik correct answers guys clear.

[02:13:20]Let's go to the next one.

[02:14:09]about management. Let's see if we can have a session on permutation combinations.

[02:14:42]A polomial f ofx upon division by x - 1 gives a remainder of 5 when fx is divid by x + 2 the remainder is - 7. Find the remainder when fx is divided by x² + x -2. So what do we know? remainder when f_sub_x is divided by x - a is equal to f a right I hope everyone is aware of that correct reminder when f_sub_x isid x - a is nothing but f a now what we can say reminder as f_sub_x is divided by a quadratic expression x² + x - 2.

[02:15:30]This will be a x + b. Means the remainder could be a linear expression or it could be a constant. Right? It could be anything like 15 10 divide. The remainder 10. If you divide 50 by 11 the remainder will be less than 11. Same thing is happening here. If you divide a polomial by quadratic expression, the remainder will be linear or constant. If it's constant, a will become zero. Now the interesting part x² + x - 2 is x² + 2x - x - 2 which is x + 2 into x -1.

[02:16:17]Meaning these two are factors. x -1 is a factor of this expression. x + 2 is a factor of this expression.

[02:16:26]Marlo, if I divide 27 by 15, remainder is 12. If I divide that remainder by 5, the remainder is 2.

[02:16:39]Which is same as when 27 is divided by 5, that is 2. Then 15 is 5 into 3. 27 when divided by 15 remainder is 12.

[02:16:50]If you divide 12 by 3 the remainder is zero. And if you divide 27 by 3 the remainder is zero. That means factor or remainder expression say. So if I try to divide this by this I will get the same remainder as it is when fx is. So what we can say remainder when f_sub_x is divided by x -1 will be same as the remainder when ax + b is / x - 1 and that will be equal to a + b right and similarly remainder when f_sub_x is divided by x + 2 will be same as the remainder when ax + b is divided by x + 2 meaning putting x is equal to -2 so -2 + b follow this right now according to question a + b is = 5 - 2 a + b is = -7 so subtract this so 3 a is = 12 a = 4 if a is = 4. What we can say that b will be equal to 1. So a x + b means 4x + 1. Option B is the correct answer.

[02:18:25]Right?

[02:18:32]Okay. So much a x + b we took from our side. Yes, Maverick as I said I said that when you divide a polomial by quadratic then remainder will be x + b right. Same remainder can be obtained when f(x) is divided by x - 1 or x + 2 because these two are factors of this. Okay, that's why manish check this first you need to check this.

[02:20:10]Whenever fx is divided by xus a the remainder is fa.

[02:20:15]After that second thing is this one f_sub_x when divided by x² + x - 2 that will be a x + 2 that we have let that we have taken right is that clear that we have let here a x + b because when you are dividing by quadratic so remainder will be linear or constant and what is x² + x -2 this is nothing but a multiple of these two expressions.

[02:20:47]So whenever you are dividing f_sub_x by this same as dividing by this this will be the remainder.

[02:20:56]When you divide a x + b by x - 1 remainder will be a + b.

[02:21:02]When you divide fx by x + 2 remainder will be -2 a + b. This is x - -2. So you put instead of x you put -2. And in question it's given that a + b is 5 and this is minus 7.

[02:21:26]Okay.

[02:21:31]Put put x is equal to x= 1. Put x= 1.

[02:21:37]Put x= minus2. Then you are going to get this. Okay.

[02:21:54]factors divide you will get the same remainder number remainder when 143 is divid by 100 is equal to 43.

[02:22:09]Now if I want to divide 143 by 25 this should be same as the remainder when 43 is divided by 25 and that is equal to 18. Okay.

[02:22:25]Yes.

[02:22:28]And in the same case remainder when 143 is divided by 4 will be same as the remainder when 14 when 43 is divided by 4 that is equal to 3. So it's remainder factors divide that will be same when the number is divided by the factors. I hope this is clear now. Okay.

[02:22:54]Yes, Maverick. You can solve all the questions like this. Okay, you can solve all of these type of questions like this.

[02:23:04]Okay.

[02:23:17]Okay. Let's try the next one.

[02:23:50]Manise that is true not only for IPMAT it's for all the exams okay for all the situations doesn't matter how much you prepare there will always be one question there will all there will always be one situation where you will feel like your preparation is not good enough and that is true and that is for everyone not for only you right so ignore that part okay try to be a learner whenever you are not able to solve a question just think that it's a part of the learning and try to learn it okay that's it can we do this by using options if you can then Hi.

[02:24:36]Sure.

[02:25:18]Question solve. For what minimum value of C will the equation 4x + 7 y is equal to c have exactly eight non negative integer solutions? Now can you tell me can you tell me if I say one of the solution of this is uh 1 and 8. Next solution I will decrease the value of x by 7. So it will become min - 6. I will increase the value of y with the coefficient of x. Do you understand this? Do you understand this part that the values of x will increase with the cofficient of y value of y will decrease with the cofficient of x and vice versa do you understand this part if I take next as minus3 this is not the solution these are not the solution just trying to make you understand something this is just letting it yes you understand that okay we say non- negative integer means zero can be taken zero can be taken right. So guys, if I want to find the least value, what is the value of x that I can start with? The least value of x that I can start with. Tell me least value of x that I can start with. Non- negative integer solution.

[02:26:50]Sub least non- negative integer say what's the least non- negative solution I can take x0 right then next value of x will be 7 then 14 then 21 and so on it will be what so 7 into 7 49 right I want to keep this minimum I want to keep C minimum. So what do I need to do? I need to take Y is 0. If I take Y0, what will be the next value of Y? Can you tell me that S values? Let's not skip anything. 21 28 35 42 49. These are the least eight values of X which are possible. Now what could be the least value of Y that can be taken? When x is maximum, y will be minimum. Minimum value of y that I can take is zero. If this is zero, next value will be increasing by four. So 4 8 12 16 20 25 29.

[02:28:08]Say increase five 24 and 28.

[02:28:21]Right? Now if you want C, you can use any of these. So 4X + 7 Y is equal to C.

[02:28:31]C will be equal to 4 into 0 + 7 into 28 which is equal to 196.

[02:28:40]That will be your answer guys. This is how you solve these type of questions.

[02:28:44]Okay.

[02:28:46]This is how see if you start with x being one you will end with 50.

[02:28:52]So the value will increase right? So you must start with x0 or y0 doesn't matter.

[02:28:58]In each of these cases you are going to get same value of c which is 196.

[02:29:04]Same value of c.

[02:29:08]Okay.

[02:29:10]24 28. Huh? See?

[02:29:14]All right.

[02:29:20]So, this is how you solve these type of questions. Okay guys, take care. So, that's it for today, guys. Okay. I hope you are happy with this session because uh we solved some really difficult questions and we were able to revise really good concepts right which are important concepts. So I hope this session was fruitful to you guys and you are feeling charged right and uh ready for the next session tomorrow 6 p.m. We will meet again. We will continue from here. We are going to solve questions from logarithm, modulus, functions and series and sequence. Right? How about that? It's going to be another fruitful session. Okay? So, thank you for joining us. Okay? Bye-bye. Good night. Take care.