Day 32 - Questions for the hardest JEE Advanced - 100 Days

Added: 2026-05-12

[00:00:00]Oh, hello people, and welcome back once again.

[00:00:03]So, today we are back with our 100-day JEE Advanced question series after a long time.

[00:00:09]And this question today is a really good question from uh Wait.

[00:00:16]Okay.

[00:00:17]So, this question is a really good question from Narayana GTA.

[00:00:22]And they have a lot of other good questions as well, but this one was pretty interesting. I will not say this is too hard, but this is just the level that JEE Advanced actually belongs to nowadays, right? Like this type is exactly the level that you can expect in your actual exam, okay? So, I guess I should pick this up.

[00:00:44]Anyways, so I hope you have read the question, and therefore we guess should start off.

[00:00:49]So, the question says that f(x), f'(x), and f''(x) are all continuous on 0 to ln 2.

[00:00:58]And they have given some conditions that f(0) is 0, f'(0) is 3, f'(ln 2) is 4, and f(ln 2) is equal to 6.

[00:01:14]And along with that, they have given two more things that integral 0 to ln 2 e^(-2x) f(x) dx = 3.

[00:01:21]And there is a continuous function g(x) which satisfies another equation that integral 0 to 1 g(x) whole multiplied by 4x^2 - g(x) dx = 4/5. So, on the basis of all this info, we have to find out the following. Okay.

[00:01:37]So, yeah, I guess f(x) and g(x) are pretty separate from each other because there is no relation given amongst them.

[00:01:45]So, okay.

[00:01:46]We'll consider the equation separately and then solve and find out the answers of the two subparts. That's the gist of the question, I hope.

[00:01:52]So, I hope you have seen the question, guys, okay?

[00:01:55]So, moving down.

[00:01:57]Starting off, first thing we have is your integral 0 to ln 2 e power of minus 2x into f of x dx is equals to 3.

[00:02:13]So, for this particular part, like Okay. So, for this particular part f of x, if you see the first question, we are supposed to find out this thing, right?

[00:02:26]The value of integral 0 to ln 2 e power minus 2x into f double dash x dx is equal to what?

[00:02:38]Yeah.

[00:02:38]So, in order to find this, I guess the best thing I can see right now is integration by parts.

[00:02:44]Moving on.

[00:02:46]Let's take this to be I. And uh okay, I can change the thing.

[00:02:53]Okay.

[00:02:55]So, I equals to 0 to ln e power minus 2x into f dash x So, for integration by parts, in this particular thing, if we take e power minus x as uh and if we take your v as thing So, applying by parts like that, we get I is equals to uh okay, I guess this will become power minus 2x and integral and then from ln 2 ln 2 plus of two times integral 0 to ln 2 e power minus 2x into f dash x something like that now So, now we have to like the find out this value, we need to compute the value.

[00:03:59]So, at x is equal to long two your e power minus two long two perfect square is So, your x dash long two That means the upper value will be one by four into four, right?

[00:04:22]Which is equal to one.

[00:04:25]Okay, this we have it downwards.

[00:04:27]Next one we have at This will become e power zero into f dash of zero. The values are given, right? I hope you remember. So, one into three is So, this particular part e power minus two x f dash x from zero to long two This becomes one minus three is equals to minus two.

[00:04:54]Okay.

[00:04:56]That's uh half of the job done.

[00:04:58]Now, moving on to the next part uh if we see this equation is now becoming I is equals to minus two plus two times integral zero to long two e power minus two x into f dash x Wait.

[00:05:24]Okay.

[00:05:25]So, I is equals to minus plus two times then let us take this particular integral as J.

[00:05:35]Okay. So, I is equal to minus two plus two J. Now, to further evaluate J, that is your integral zero to long two e power of minus two x into f dash x dx we again have to do by parts.

[00:05:50]Okay.

[00:05:52]So, here also we take this as u and this thing as v. So, once again applying applying integration by parts, we will get j is equals to first term will come out to be e power minus 2x and this will be f of x from zero to lawn plus of your two times integral zero to lawn e power minus 2x f of x dx.

[00:06:26]Okay.

[00:06:27]So, once again in this particular thing the same thing we are going to do we will evaluate the boundary terms once again.

[00:06:37]x is equals to lawn two e power minus two lawn two f of lawn two will be equals to one by four into six, right?

[00:06:50]This is becoming three by two.

[00:06:52]And at x is equals to zero, this thing is e power zero f of zero, which is one into zero, which is equal to zero.

[00:07:03]So, this thing your e power minus 2x into x from zero to lawn two, this becomes three by two minus zero, which is three by two. Okay.

[00:07:15]And if you see carefully, integral zero to lawn two e power of minus 2x into f of x dx is given to you as equal to three, right?

[00:07:27]It's given in the question.

[00:07:29]Very genuinely your j comes out to be three by two plus two into three, which is three by two plus six 15 by two.

[00:07:41]The value of j comes out 15 by 2.

[00:07:44]Which means I is equals to minus 2.

[00:07:49]Plus 2 multiplied by 15 by 2.

[00:07:54]Just 13.

[00:07:57]So, the first part's answer is 13.

[00:08:01]Okay.

[00:08:03]Cool.

[00:08:04]Going on to the second part, which is part number B.

[00:08:08]Where we were supposed to find out integral I mean we were actually given that integral 0 to 1 g of x 4x squared minus g of x dx is equals to 4 by 5.

[00:08:25]Okay.

[00:08:26]This was given to us.

[00:08:30]Okay.

[00:08:31]So, let's try to simplify the integral, right?

[00:08:34]So, if I'm going to do this part but g of x into 4x squared minus g of x If we break this up, it's this will become 4x squared. I'm writing this as g. I'm not writing g of x instead I'm writing g.

[00:08:52]4g minus g squared.

[00:08:55]Now, if you look carefully 4x squared minus g squared can be written as 4x to the power 4 minus of 2x squared minus g whole squared. This is a very crucial step, okay?

[00:09:11]Uh this actually means If we can write this like this, the integral becomes 0 to 1 4x power 4 minus 2x squared minus of g whole squared dx equals to Now, if we split this this becomes four integral zero to one x power four dx minus integral zero to one two x squared minus g of x squared dx.

[00:09:53]Uh okay. g of x whole squared dx which is equals to 4 by 5.

[00:10:00]So, computing this I guess is very easy.

[00:10:03]This part is obviously 4 by 5 without and integral zero to one two x squared minus of g of x whole squared dx is equals to 4 by That's the real deadlock, right?

[00:10:22]Nice deadlock we have appeared into because 4 by 5 is getting cancelled from both So, I guess because a square is always non-negative, right? So, the integral of a non-negative continuous function because given g of x continuous.

[00:10:42]So, this is a non- negative continuous function.

[00:10:50]The integral of a negative non-negative continuous function will be zero only when the function itself is zero everywhere.

[00:11:00]Okay? Let me say the line once again.

[00:11:03]Integral of a non- continuous function will be zero only when the function itself is zero everywhere. Okay?

[00:11:15]So, from here cuz we have found that integral zero to one two x squared minus g of x squared dx equal to 4 by 5 taking into consider it Sorry, not 4 by 5, zero.

[00:11:29]So, taking into consideration two factors, non-negative and continuous.

[00:11:33]We can conclude from here that 2x squared minus it is itself equal to zero. Okay.

[00:11:41]A very important step and like the second part of the question was actually more interesting.

[00:11:48]Okay.

[00:11:49]So, from here we directly get g of x as 2x squared.

[00:11:55]So, yeah.

[00:11:56]And this is for all x belonging to zero to one.

[00:12:02]So, okay.

[00:12:03]Job's done almost. I guess we have to compute something like this, right? To compute the arithmetic mean of g of half g of 1/2 squared dot dot dot till we have g of 1/2 power 10. Okay, maybe a GP.

[00:12:21]Let's see.

[00:12:23]So, g of x is 2x squared. So, I guess the numbers are becoming two will be there will be one squared and because of two one power will be reduced. So, I guess they are the odd powers. This will be 1/2 1/2 cubed 1/2 to the power five but dot dot dot dot dot till your 1/2 to the power 90.

[00:12:45]Okay.

[00:12:46]Finding out the arithmetic mean thing.

[00:12:48]What is the formula for this? E n will obviously be 1 by 2 to the power of 2k minus 1.

[00:12:58]We write down the sum till your Okay, let's write it out again once we I guess it was H or something like capital A given in the question. I don't know.

[00:13:06]I'm writing it as S, the sum. Okay.

[00:13:08]So, S is 1 by 10.

[00:13:11]Sigma k equals 2 1 to 10 1 by 2 power of 2k minus 1.

[00:13:21]Okay.

[00:13:23]So, I guess evaluating think is not a very difficult task because this is a GP with first term A is equal to half R is equal to 1 by 4 N equals 10.

[00:13:38]The sum in by 10 and this will be half into 1 minus 1 by 4 to the power 10 divided by 1 minus 1 by 4.

[00:13:52]So on some evaluation I guess this is coming out to be 1 by 10 and 3 by 4 will have calculations, right? So I guess this will be Wait.

[00:14:05]Uh let me check.

[00:14:08]Okay. So we can uh I am just skipping a few steps. Okay, there will be some sort of translation. I calculated this thing this is coming out to be 1 by 15 into 1 minus of 1 by the power 10.

[00:14:22]And they have given this to be equivalent to 1 by M 1 minus 1 by 2 to the power of N.

[00:14:31]So I guess N will be becoming N plus 10.

[00:14:36]So M plus N comes out to be 10.

[00:14:43]Okay.

[00:14:46]So we have found out both the answers for part A and part B was I guess more interesting than part A, right? I hope you guys will agree.

[00:14:55]Anyway, so guys that's it for this session today, I guess. This was a really interesting question. Although I solved it in 40 minutes something. Yeah, it will take less time in the exam.

[00:15:04]So yeah.

[00:15:06]That's it for today, guys. I hope you have a nice Have a nice Join my channel That's it.

[00:15:14]Any doubts you have comment section. I guess that's it. Thanks. And obviously best of luck for JEE advanced because advanced is already Hope you are confident enough. taken the ones that are already there.

[00:15:28]Hope you are confident enough.

[00:15:30]So, that's it for today, people.

[00:15:32]Best of the best.

[00:15:33]Best of luck. Thanks. Thanks for watching.