I
iTeachChem2mo ago
Opt

S&S

How? I have the solutions but i didn't understand that at all. I tried using the Bell number expansion and subtracting but it didn't work either.
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101 Replies
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SirLancelotDuLac
I have an idea but am not sure if it works.
SirLancelotDuLac
Ah I see what they did there. When you put x=omega and omega^2, the real part of the power wala portion is -1/2 which you get 2 times so you get 1/sqrt(e) and the imaginary parts are conjugates. So you get 2/3*cos(sqrt(3)/2) So answer would be $\frac{2e}{3}-\frac{2cos(\frac{\sqrt{3}}{2})}{3\sqrt{e}}$
TeXit
TeXit2mo ago
SirLancelotDuLac
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Opt
OptOP2mo ago
Oh, complex values for x....dammit i forgot that was valid. I did that for the binomial theorem stuff, but didn't do it here.
SirLancelotDuLac
This is fiitjee aits right? They're just built different 🗿 Like I would have surely left this in the paper.
Opt
OptOP2mo ago
Ye lol I got 180
SirLancelotDuLac
Nice.
Opt
OptOP2mo ago
Yeesh, this was a tough one. And, ik it's not S&S, but I do have another doubt
SirLancelotDuLac
Sure.
Opt
OptOP2mo ago
Q53
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SirLancelotDuLac
The (iii) wala toh is true because of I and stuff. And I have like, zero ideas how to go about proving the first 2. But the (i) need not be necessary ig.
Opt
OptOP2mo ago
What? I don't get it. Why does it have to be 2025?
SirLancelotDuLac
Oh yeah right.. It doesn't need be.
Opt
OptOP2mo ago
Answer is C btw. I have no clue why. There's no solution either. It's too simple for the FIITJEE people to give one
SirLancelotDuLac
https://en.wikipedia.org/wiki/Idempotent_matrix It does say that the trace=rank of the matrix So that does give 2 options.
Opt
OptOP2mo ago
The only non-singular idempotent matrix is the identity matrix;
Nvm
SirLancelotDuLac
Ah.
Opt
OptOP2mo ago
Trick question tha
SirLancelotDuLac
Fr. But how to prove this tho? Ah nvm the proof is just multiply by A inverse
iTeachChem
iTeachChem2mo ago
Taylor series hai kya syllabus me? Afaik college first year me hai.
iTeachChem
iTeachChem2mo ago
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iTeachChem
iTeachChem2mo ago
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No description
Opt
OptOP2mo ago
A² = A Means (Tr(A)-1)A = |A|I Now take det on both sides. Should be enough to show |A|= 0 or 1
SirLancelotDuLac
I mean, it doesnt say explicitly but a lot of questions have comes from it. And is very useful in cases also.
iTeachChem
iTeachChem2mo ago
Advanced me possible hai then. Main me bhi?
SirLancelotDuLac
Yes.
Dexter
Dexter2mo ago
idempotent is A²=A right? yes advanced mein to limits ka sawaal is either taylor series or pure hell
Opt
OptOP2mo ago
Usually both
Dexter
Dexter2mo ago
mains mein still L hopital works sometimes Fr
Opt
OptOP2mo ago
x²⁰²⁵ types in Adv
Dexter
Dexter2mo ago
😭😭😭I hope mains mein aisa kuch na aaye they like giving exponential on the year we're giving in the exams mostly binomial mein milta hai ye tho
SirLancelotDuLac
Ah right and 2025 tends to be a scary number on that part :sweaty:
Dexter
Dexter2mo ago
wait what 45² na crazy we're living in a square year last time it was 1936 next gonna be 2116
SirLancelotDuLac
Yeah that was the trick in the question. non-singular=inverse exists and A^2=A meaning A=I
Dexter
Dexter2mo ago
Goddamn matrices is Weird like that dude
SirLancelotDuLac
Kinda midly dislike questions like these where they trick ya by the language Fr.
Dexter
Dexter2mo ago
Fr Mujhe to poori algebra iske chalte naa pasand hai ngl
Opt
OptOP2mo ago
And it's not something you think of checking either, because every question mentions non singular.
SirLancelotDuLac
Exactly. Like unless you have seen this thing in writing somewhere your mind doesn't go to it.
Dexter
Dexter2mo ago
ngl i observed in mains 2024 ke papers the only chapters in algebra that they asked extensively is binomial and sns PnC Matrices were very less
Opt
OptOP2mo ago
S&S is one of my faves
Dexter
Dexter2mo ago
they asked 2D alot tho and fun too
Opt
OptOP2mo ago
But then you come across questions like this.
Dexter
Dexter2mo ago
2D is my favourite topic in maths
Opt
OptOP2mo ago
I'm screwed then. Conics is the bane of my existence
Dexter
Dexter2mo ago
fr
SirLancelotDuLac
Same 'ere Entire coordinate was destroyed in 11th ngl.
Dexter
Dexter2mo ago
I've seen that the people who like Permutations, Algebra jaise topic tend to hate 2D and 3D and vice versa too like i hate algebra with a passion
SirLancelotDuLac
But now that tangent normal is removed its slightly easier maybe?
Dexter
Dexter2mo ago
last year tha last year bhi tha they asked questions still just mention "line touching the curve at a single point"
SirLancelotDuLac
Bruh I skipped all that in revision and stuff. I'm cooked. (Trusted nta for once)
Dexter
Dexter2mo ago
it was pretty basic tho AoD se bhi hojaayega utna
Opt
OptOP2mo ago
Fun fact, you know those sums of the type $\Sigma_{n=0} \frac{n^{k}}{n!}$ ? Those have a general result.
TeXit
TeXit2mo ago
Opt
No description
Dexter
Dexter2mo ago
e^ something wala na?
SirLancelotDuLac
Well mocks mein I didn't see much tangent and normal wale stuff.
Dexter
Dexter2mo ago
bhai i thought diffraction was removed
Opt
OptOP2mo ago
Well, no. Hell nah.
Dexter
Dexter2mo ago
but 2024 mein wave optics se bas diffraction aaya hai lmao wahi
SirLancelotDuLac
Ye, you can differentiate taylor and multiply by x repeatedly maybe?
Dexter
Dexter2mo ago
i revised it again
Opt
OptOP2mo ago
It's not x^n. It's n^k K is the external variable here.
SirLancelotDuLac
Power comes infront of x repeatedly and then you just put x=1 Did this a lot in binomial ig.
Dexter
Dexter2mo ago
yeah integration based series
Opt
OptOP2mo ago
Yeah, you can do it there, but not here. Because terms end up cancelling up and down. The only legal method is actually partial fractions iirc. But if you have k=5, enjoy fifteen rounds of partial fractions. No wait More
SirLancelotDuLac
Wait terms cancel?
Dexter
Dexter2mo ago
Crazy and how would we do that?
Opt
OptOP2mo ago
I forgot the LaTex for ∞ What is it again?
SirLancelotDuLac
\infty
Opt
OptOP2mo ago
$\Sigma_{n=0}^{\infty} \frac{n^{k}}{n!} = B(k) e$
TeXit
TeXit2mo ago
Opt
No description
Dexter
Dexter2mo ago
What B whats
Opt
OptOP2mo ago
Where B(k) is the k-th Bell number.
Dexter
Dexter2mo ago
Bell number kya hota hai 😭😭😭
Opt
OptOP2mo ago
Equivalence relations nahi yaad hai kya? No. Of equivalence relations ke liye ek triangle banate hai na? Bell's triangle.
Dexter
Dexter2mo ago
wait what y'all studied relations in that depth? 😭😭😭 i have no clue I'll have to look it up
Opt
OptOP2mo ago
You want Relations notes of mine?
Dexter
Dexter2mo ago
🐴 I'll watch a lecture first uske baad if I'll need I'll ask you
SirLancelotDuLac
I didn't cover that in relations. Is it relevant for jee?
Opt
OptOP2mo ago
Adv more. Not Mains.
SirLancelotDuLac
For lower powers it works ig
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Opt
OptOP2mo ago
They might ask no.of symmetric and reflexive if anything.
SirLancelotDuLac
Ah I see.
Opt
OptOP2mo ago
Nice approach, but what about k=7?
SirLancelotDuLac
Exactly. Works for lower powers only.
Opt
OptOP2mo ago
One more not so relevant question.
No description
Opt
OptOP2mo ago
I'm going through the paper again now. I evaluated the integral (diff.under the integral) and got I = 1 - cos(1), but I have no clue how to proceed. Or is this some sandwich theorem based question?
Dexter
Dexter2mo ago
But that was in limits na?
Opt
OptOP2mo ago
An integral is a limit of a sum.
Dexter
Dexter2mo ago
yeah this correct ngl 0 ke around sinx/x tends to 1 and 1 pe cos1 hoga mhm allen aiot mein bhi to tha ek sin8x/x wala wo bhi nahi hua tha merese 🐴
Opt
OptOP2mo ago
Yeah i messed that one up Got it later Just expand and integrate
SirLancelotDuLac
Taylor expansion se approximation ig?
Opt
OptOP2mo ago
To how many terms? We have 1800 in the options. I need to go to at least 7! for that denominator
SirLancelotDuLac
Three terms no? x,x^3 and x^6 Painful neverthless.
Opt
OptOP2mo ago
Oh right it's sine. Yeah. The options all differ by 4/1800
SirLancelotDuLac
Ya three terms would do it
Opt
OptOP2mo ago
I think this is good. I'll close the thread +solved @SirLancelotDuLac @Dexter
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