ISI UGB PYQs | Must solve | Most Important Questions | Part 05 | Anirudha sir

Hinzugefügt: 2026-05-12

[00:00:01]Hello everyone, welcome to this video. I hope you've been making the best use of these videos of ISI preparation.

[00:00:08]And this is probably the last recording video which I'll be posting and then all the best for your exams. On Saturday, try to relax as much as possible.

[00:00:18]And that's it.

[00:00:21]So guys, you must be knowing that there is a possibility that pattern is different.

[00:00:27]Right? The UGA questions there'll be certain number of questions which are having multiple correct options.

[00:00:35]So so that is a possibility. So we'll see how that happens. Yet the difficulty would be the same.

[00:00:44]So I hope you are mentally prepared for that.

[00:00:48]All right?

[00:00:49]We are solving UGB problems as much as possible, right? Because mainly these questions are tough and they require not tough exactly, but they require you to think and present your thinking.

[00:01:02]In UGA, the plus point or easy point is that you can kind of guess the options.

[00:01:09]Right? You can substitute and check or something like that.

[00:01:12]So I hope you are practicing UGB as much as possible.

[00:01:17]All right.

[00:01:18]So in today's session, we will be It is official sample paper. If you go to the website, sample papers are uploaded. They're actually PYQs. The latest one we have kept for the last so that you'll find it most helpful. I will solve more number of questions in this video so that you get the feeling of actually solving a paper.

[00:01:41]So here we go.

[00:01:44]This is the first question. Pause the video. Try it for yourself.

[00:01:48]I hope you have paused the video.

[00:01:50]I will be starting now.

[00:01:52]The question says this and we are asking we want a point such that f of x is equal to x or x equal to some x naught. We want to find that for some x naught this happens.

[00:02:11]So, almost always this is what you should be doing. You take g of x to be f of x minus x.

[00:02:20]And we know something about f dash.

[00:02:23]Right? So, what do we know about g dash?

[00:02:26]g dash of x is f dash of x minus 1.

[00:02:32]If f of f dash of x is between uh or rather mod f dash is less than 1.

[00:02:39]What can we say about g dash?

[00:02:41]So, let's see.

[00:02:44]Uh f dash of x is less than 1 by 2 mod.

[00:02:51]That would imply if I remove the mod, what happens? f dash of x will be less than 1 by 2, but greater than minus 1 by 2.

[00:02:59]Right?

[00:03:00]Now, that should imply that What was g dash of x?

[00:03:07]If you see, g dash of x is this. So, if I write f dash of x is equal to g of x g dash of x plus 1.

[00:03:17]All right? And we are talking about the whole R and everything.

[00:03:21]Okay?

[00:03:23]So, that implies that g dash of x plus 1 is less than 1 by 2 greater than minus 1 by 2.

[00:03:31]Now, that will imply that g dash of x is less than minus 1 by 2. I'm just subtracting on both sides.

[00:03:40]And greater than minus 3 by 2.

[00:03:43]So, you see that g dash of x is strictly less than 0. It is strictly negative, you know, not even close to zero.

[00:03:54]So, what can we say?

[00:03:57]We can say that uh g of x is a s- trictly strictly decreasing function.

[00:04:12]Right?

[00:04:14]So, if g dash x is a strictly decreasing function, what how does that help us?

[00:04:19]We know that uh Now, if g of x is strictly decreasing function, that means that uh it'll be like this.

[00:04:31]No, uh this is in- increasing function.

[00:04:33]I am so sorry.

[00:04:34]So, we want strictly decreasing function.

[00:04:39]So, if you see, it'll be like this. So, it will be intersecting the x axis at some point.

[00:04:44]Right?

[00:04:45]So, wherever it is intersecting the x axis, let's say that is x not.

[00:04:50]Right? That implies g of x has to be equal to zero for some x not.

[00:04:59]You can use various things for this. You can use uh Rolle's theorem.

[00:05:05]A- all the mean value theorems you can use.

[00:05:07]And you can come to this conclusion.

[00:05:09]This is very natural.

[00:05:11]All right?

[00:05:12]And uh yeah.

[00:05:14]So, this will imply that f of x minus x will be equal to zero for some x equal to x not.

[00:05:24]And that will imply f of x not is equal to x not.

[00:05:28]So, this question was fairly easy.

[00:05:31]Right? Quite intuitive.

[00:05:33]But you had to convert that first g of x part.

[00:05:37]All right. Let's move to the next one.

[00:05:40]If the interiors of angles of this uh uh of a triangle ABC satisfy this, what do we have?

[00:05:49]So, we will uh shift the cos to this side.

[00:05:56]So, we will use the fact that cos squared theta is equal to 1 minus sin squared theta.

[00:06:03]All right?

[00:06:05]Okay. So, that will imply that summation sin squared A I'll write it as summation sin squared A is equal to 2 into summation 1 minus sin squared A.

[00:06:23]Right? So, this will be 2 into summation uh if I am adding 1, how many times am I adding 1? Three times. So, 3 minus summation sin squared A.

[00:06:35]So, this will be 6 minus 2 times summation sin squared A.

[00:06:42]Now, if I bring this to the other side, I will be having 3 times summation sin squared A is equal to 6 actually. So, if I cancel out, I'll be left with 3. So, we will be having summation sin squared A is equal to 2.

[00:07:07]I'm canceling out 3, so I'll be left with 2 actually. What are we doing?

[00:07:13]Yeah?

[00:07:16]So, uh uh uh I had 6. I'll cancel out and I'll get 2. Yeah.

[00:07:24]So, this is what we have.

[00:07:27]Right? So, this is basically for in a triangle ABC sin squared A plus sin squared B plus sin squared C is equal to two.

[00:07:43]So, what does this mean to us?

[00:07:45]Let us see if we can find a relation in a in a triangle if A + B + C is equal to Um this thing, pi.

[00:07:57]What is this exactly? So, let's try to find that out.

[00:08:03]Uh Obviously, we see that mhm What is sin squared A + sin squared B?

[00:08:15]First of all.

[00:08:16]Yeah?

[00:08:19]If I have to write this, this can be if I multiply two and divide by two outside, this will be 1/2.

[00:08:27]1 minus cos 2A plus 1 minus cos 2B.

[00:08:36]Right?

[00:08:38]Let's see what happens.

[00:08:40]This is half of 2 minus cos 2A plus cos 2B.

[00:08:52]So, if you see cos 2A plus cos 2B, that will be Uh if I'm not wrong, that is two times uh cos A + B cos A - B if you add, yeah.

[00:09:06]This is A + B cos A - B.

[00:09:12]Right?

[00:09:14]Everyone is clear about that.

[00:09:16]Now, A + B is pi minus C.

[00:09:19]And cos pi minus C is minus cos C. So, two cos C cos A minus Okay.

[00:09:31]>> [clears throat] >> Okay.

[00:09:33]That is interesting.

[00:09:35]So, you will have basically 1 by 2 2 minus and that becomes plus 2 cos C cos A minus So, this will be 1 plus cos C cos A minus Now, what do I have to add? I have to add sin squared C to this to get our LHS over there.

[00:10:06]So, this sin squared C I'll write as 1 minus cos squared C because I have cos C here.

[00:10:14]If I'm able to take it common, I think that will be better.

[00:10:18]So, I have this to be 1 plus 1 2 plus cos C cos A minus B plus or rather minus, I should have minus.

[00:10:35]cos squared C So, this will be 2 plus cos C cos A minus B minus cos C Now, what is cos C?

[00:10:51]If I have to write it in terms of A plus B that is pi minus of A plus B.

[00:10:59]The second cos C, one cos C is outside, the second cos C is this. And that I'm writing it like this.

[00:11:04]So, what will this become? This will become plus because there is minus already.

[00:11:10]It'll become plus cos A plus B right?

[00:11:19]And now what is this whole thing? This is 2 cos A cos B.

[00:11:26]So, at the end, what do we have?

[00:11:29]We have that summation sin squared A is equal to 2 + 2 cos A cos B cos C.

[00:11:41]And we are given that this is equal to 2. This is given.

[00:11:47]So naturally, what does that mean? That means cos A cos B cos C is equal to 0. What do we want to prove?

[00:11:58]Prove that the triangle must be must have a right-angled triangle.

[00:12:03]Right?

[00:12:04]So that means obviously, since this is 0 and we are talking about angles of a triangle so since this implies that some cos cos of some angle should be 0.

[00:12:17]A could be one of the angles, let's say without loss of generality.

[00:12:22]Right? And since all the angles are between, you know 0 and pi only choice is A is pi by 2.

[00:12:34]Okay?

[00:12:35]So yeah, one of the triangle one of the angles has to be 90°.

[00:12:40]Right.

[00:12:41]Next, pause the video, try the question.

[00:12:44]It seems very similar to the question we saw in the first one. Right?

[00:12:50]It is kind of, yeah. It has to deal with differential equations and all that.

[00:12:56]We are giving we are giving initial thing and we want to play around. Okay.

[00:13:04]So immediately we have one thing. Since this is mod, that means this is greater than or equal to 0.

[00:13:13]Right?

[00:13:14]So first of all, we have that f of x is greater than or equal to 0.

[00:13:20]Note it down, all right?

[00:13:23]Okay. Now what does this imply?

[00:13:25]This implies f dash of x is less than or equal to f of x and greater than or equal to minus f of x.

[00:13:33]Okay.

[00:13:34]Now, what does that imply?

[00:13:37]What What can we say about this?

[00:13:39]So, if I have f dash of x less than or equal to f of x only if I consider only this.

[00:13:46]We want to show that f of x is zero. So, I would advise we don't do uh this thing.

[00:13:52]Division and all that. Let's not divide.

[00:13:55]Although it is greater than or equal to zero, let's not divide that.

[00:13:59]Um what can we do? We can subtract. So, f dash of x minus f of x is less than or equal to zero.

[00:14:08]If you have solved differential equations, you know this.

[00:14:13]If I multiply If I multiply e power minus x the inequality will remain the same, like that will be preserved.

[00:14:26]But this becomes d by dx of e power minus x f of x.

[00:14:37]If differential of this function is less than or equal to zero, that means either either e power minus x f of x is strictly decreasing. I mean, it cannot be Are we said that it is continuous? Yes, it is a differentiable function.

[00:14:59]So, either it is a constant function or it is strictly decreasing function.

[00:15:05]Right? You understand?

[00:15:07]So, that is the only possibility. We cannot have something like it is constant like this and suddenly it is going down like that.

[00:15:16]Right?

[00:15:17]Maybe it will smoothly decrease or something.

[00:15:21]So, all that is possible.

[00:15:23]Overall, it is a decreasing function.

[00:15:26]All right?

[00:15:28]So, what does that mean?

[00:15:29]This is uh for X between zero or greater than zero.

[00:15:38]e ^ - X f of X is less than e ^ - 0 f of 0.

[00:15:46]Right? At least less than or equal to because it's not strictly decreasing.

[00:15:50]Loosely decreasing, but we can say this much. This is for sure. You can take it for granted. And we know that f of 0 is 0.

[00:15:58]So, we have that.

[00:15:59]So, immediately we can say f of X is less than or equal to zero.

[00:16:06]But, what did we have initially? We had f of X was greater than or equal to zero because f of X was greater than or equal to mod f dash of X.

[00:16:15]Which is greater than or equal to zero.

[00:16:16]So, this is greater than or equal to zero, less than or equal to zero.

[00:16:19]Together, that should imply f of X is equal to Right?

[00:16:25]Key thing is we are using this X greater than zero. Why? You have to look at the domain carefully.

[00:16:31]Domain is one zero to one.

[00:16:35]So, why do we Why are we getting 10 points for this?

[00:16:38]Because in the exam pressure, you should be able to think this.

[00:16:42]This is very easy.

[00:16:43]Obvious kind of uh and it is there in the official paper.

[00:16:49]So, yeah. Think about it.

[00:16:52]Right?

[00:16:53]Now, this one.

[00:16:55]Pause the video and try it for yourself.

[00:17:00]>> [snorts] [clears throat] >> First of all, intuitively, just think about it.

[00:17:06]When should we be having this?

[00:17:09]And why is it specifically odd?

[00:17:12]What I would imagine is that one of them is getting canceled out. You see, if these two are getting cancelled out and these two are getting cancelled out, we're left with this.

[00:17:23]If that is happening, what does that mean? And K is odd specifically. Why does that uh you know, matter so much?

[00:17:31]Probably because A + B is zero.

[00:17:33]If A + B is zero, that means A is minus B and any power if you take, you'll get A power K + B power K is also zero.

[00:17:41]I hope you're following that. So, with that intuition, that is the first thought when I'm solving this. So, I'll keep that in mind. And I'll try to achieve that from this uh given thing.

[00:17:53]All right?

[00:17:55]So, I have this. So, let's just take uh all that multiplication and everything.

[00:18:00]So, the initial condition is AB + BC + CA by ABC is equal to 1 by A + B + C.

[00:18:15]That means A + B + C into AB + BC + CA is equal to ABC.

[00:18:26]So, let's see what happens.

[00:18:28]Here, if I multiply A, A squared B plus ABC plus uh CA squared plus AB squared plus um all the other things.

[00:18:48]Da da da da.

[00:18:50]B squared C plus ABC plus C ABC plus BC squared plus C squared A.

[00:19:04]This is equal to ABC.

[00:19:05]One of the ABCs will get cancelled, let's say.

[00:19:08]And we have this. And since I have this in my mind that a plus b or b plus c or c plus a these can be zero. I'll try to factor that out.

[00:19:18]All right, let's see what this becomes.

[00:19:20]What is this exactly?

[00:19:22]So, here I can see that I can take a plus b common.

[00:19:27]If I take ab, I'll get a plus b.

[00:19:31]And I'll write the rest of the things.

[00:19:33]Plus Mhm.

[00:19:36]c a squared plus b squared c plus I'm trying to take ab common, ab common, right?

[00:19:45]Plus uh Okay, one of the abc's canceled out, so abc plus abc plus c squared a plus Mhm.

[00:20:00]c squared a plus plus plus bc squared.

[00:20:04]Okay.

[00:20:06]Let's see. What can we do? Can I get a plus b out of this?

[00:20:10]Can I get a plus b out of this?

[00:20:12]So, if if I'm taking bc common in these two, right? I'll be getting a plus b.

[00:20:21]So, so I will write that here down.

[00:20:25]If I take bc common, that will be a plus b.

[00:20:30]Right?

[00:20:31]And what about what about this and this?

[00:20:39]No. What about this and this? If I take ac common, Mhm. No, that's not helping.

[00:20:49]What about these two, huh? If I take ac common, huh? So, I'll get ac into a plus b.

[00:20:58]Right? So, what are left? These two are left.

[00:21:02]I'll have c squared a plus bc squared.

[00:21:05]So, I if I take c squared common, this is a plus b.

[00:21:09]Yeah. So, that is successful.

[00:21:13]So, if you have an agenda in mind, this is probably helpful, very very helpful.

[00:21:17]Now, if I take a, I can take ab common.

[00:21:20]You see?

[00:21:22]So, now I can take ab common, that will be equal to a plus b, I mean, I'm so sorry. a plus b common, and I'll be having uh I'll be having c squared, bc, ac, ab.

[00:21:36]c squared plus bc plus ac plus ab.

[00:21:41]This is very clearly c plus a into c plus b.

[00:21:46]Wow.

[00:21:48]So, our this thing gave us, you know, and there is in the RHS we have zero equal to zero. This becomes zero.

[00:21:55]So, this becomes zero.

[00:21:57]So, this is equal to zero.

[00:21:59]Wow.

[00:22:00]So, we basically have a plus b, b plus c, c plus a equal to zero.

[00:22:09]And the moment we have let's say, without loss of generality, here is where you use without loss of generality.

[00:22:16]Any one of these terms, if you take zero, our purpose is solved, right? So, for the sake of argument, let's say a plus b equal to zero.

[00:22:24]That will imply that uh a equal to minus b. And for k odd, a power k equal to minus b whole power k, which is minus b power k. And that will imply a power k plus b power k is equal to zero.

[00:22:41]And all these things, you you can just continue from here.

[00:22:45]I hope you got the point.

[00:22:47]Yeah?

[00:22:48]So, yeah. That's it.

[00:22:51]Uh [sighs] another question.

[00:22:54]Try it out.

[00:22:56]Pause the video. Try this question.

[00:23:09]>> Mhm.