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#help-23
shut gust
glossy dock
dude chill
glossy dock
oh good lord
- T (Temperature): 1 = Too high, 0 = Acceptable
- W (Wind speed): 1 = Too high, 0 = Acceptable
- R (Rain): 1 = Detected, 0 = Not detected
- M (Manual override): 1 = On, 0 = Off
Read the bullet points carefully. Look for keywords like AND, OR, and NOT (or
words that imply a NOT, like "off" or "acceptable" when the default '1' state is
"on" or "high").
- Part A: "the temperature is too high"
- Looking at the table, "too high" means T = 1. So we just use T.-
Keyword: "and" \rightarrow This will be an AND gate.
-
Part B: "the wind speed is acceptable"
- Looking at the table, "acceptable" means W = 0.
- Because logic circuits trigger on 1s (True), if we want a condition to
be met when the input is 0, we must invert it. We use a NOT gate. - This becomes NOT W.
-
Keyword: "and" (The second bullet point connects the first line to the third
line with an AND). -> This will be a final AND gate combining everything. -
Part C: "rain is not detected"
- Table says "not detected" is R = 0. We need to invert this to make it a
1 for our logic. - This becomes NOT R.
- Table says "not detected" is R = 0. We need to invert this to make it a
-
Keyword: "or" -> This will be an OR gate.
-
Part D: "the manual override is off"
- Table says "off" is M = 0. We need to invert this.
- This becomes NOT M.
-
etc
glossy dock
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daring sonnet
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daring sonnet
In part b
Why is theta, in fact, equal to 3/4 pie
Answer key
Nvm got it
.close
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vagrant brook
The answer is B. Why do we consider the cases where alpha = 1 or -1?
vagrant brook
I got the bounds by making the function either strictly increasing or decreasing
median sand
What is sin (-1) and sin (1)
You are trying to make them a linear functions that is monotonic.
median sand
it is.
The reason why you include alpha = +-1 is because it still make f a strictly monotonic function.
Which satisfies conditions to be an injective function.
noble mango
I got the bounds by making the function either strictly increasing or decreasing
Out of curiosity, how did you make the function "strictly increasing"?
vagrant brook
set the derivative > 0
noble mango
I see I see.
Let me ask you a question
Do you think y = x^3 is strictly increasing?
golden nebula
set the derivative > 0
for strictly incerasing, the derivative is greater than or equal to zero right
noble mango
for strictly incerasing, the derivative is greater than or equal to zero right
Ehhh also not true
Take the function y = 0 from x <= 2 and y = (x - 2)^2 from x > 2
This is not strictly increasing.
vagrant brook
oh
golden nebula
Take the function y = 0 from x <= 2 and y = (x - 2)^2 from x > 2
but its derivative is negative
noble mango
This is sort of the point I was leading up to; showing the derivative is non-negative or non-positive only shows that it is increasing or decreasing (not strictly).
golden nebula
for x greater than or equal to 2, the function is strictly increasing
noble mango
Sure, but for all x the function is increasing.
vagrant brook
how do we identify if it would be one-to-one for case where f'(x) = 0
noble mango
And the derivative is greater than or equal to 0.
The point is, f' >= 0 does not imply f is strictly increasing.
golden nebula
The point is, f' >= 0 does not imply f is strictly increasing.
can you give an example
noble mango
I literally just did
The function f(x) = 0 from x < 2 and f(x) = x^2 from x >= 2 is an increasing function with a non-negative derivative.
Or even a simpler example than that
The constant function
The derivative is 0 but it isn't strictly increasing
noble mango
how do we identify if it would be one-to-one for case where f'(x) = 0
You need to show that f'(x) = 0 has at most one solution
Well
golden nebula
your right, i got confused between increasing and strictly increasing
noble mango
Actually you need to show that for if f'(a) = 0 and f'(b) = 0, you need to show that f(a) is not f(b).
vagrant brook
why is that the case?
noble mango
Because if a, b are different, you want to show f(a) is not f(b)
Otherwise if f(a) = f(b), that automatically means it's not strictly increasing
There's probably a nicer result to show but this is the one that comes to mind immediately
vagrant brook
ok, thank you
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runic gale
or have i done something wrong
outer pollen
1/arccosec is NOT arcsin.
the relationship does not hold that way.
what IS correct is that $arccosec(x) = \arcsin(\frac1{x})$.
flat frigateBOT
go maximax or go home (Chiaki)
outer pollen
please don't say that. anything else though?
runic gale
nono
outer pollen
ah, alright then. you can close the channel when done.
proven smelt
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nimble wyvern
when u substitued (2/3,1) you put -2(2/3)^2 instead of -2(1)^2
nimble wyvern
when u substitued (2/3,1) you put -2(2/3)^2 instead of -2(1)^2
at the -2y^2 term*
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nimble wyvern
ok wait is this in the right track?
yep perfect
proven smelt
but these answers could be wrong
like this company has hella errrors in their solns so i can never rely on them
proven smelt
💀 true
nimble wyvern
fahhh
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hoary seal
Im a bit confused, do we use series or the pascals triangle
safe radishBOT
Im a bit confused, do we use series or the pascals triangle
desert juniper
pascal's triangle is just the graphical representation of n over k
broken forum
Woah woah, why does it say you use binomial theorem for small powers...
hollow pelican
Woah woah, why does it say you use binomial theorem for small powers...
Yeah I was confused about that too, the point of using it is to not have to expand big powers.
broken forum
For small powers we use Pascal's triangle
Because to see how the n'th power looks I have to calculate how every power from 1..n-1 would look
brisk shard
For small powers we use Pascal's triangle
Why? You can use nCk for all
broken forum
I mean yeah
hollow pelican
Why? You can use nCk for all
Yeah but that is annoying sometimes, I hate having to compute choose stuff
broken forum
hoary seal
so how do I go about this
broken forum
Well you look at your theorem
And look at the question
from that we can see n=3
So k needs to loop from 0,1,2,3
Here me out
To make it very clear substitute every value except for k
Into the binomial theorem
solar hazel
broken forum
$\sum_{k=0}^{3} \binom{3}{k} a^{3-k}b^k$
flat frigateBOT
Nyxzore
broken forum
Agreed?
broken forum
It's a dummy variable for the sum
broken forum
Why we using permutations and combinations
Ooooh
Exciting question
You could do this by hand
Let's say (a+b)^1
Then you could use your foil or whatever to do
hoary seal
It would get messy after 2
broken forum
Indeed
The relationship is kind of interesting
If you try your tedious algebra
And look at the coefficients that you get
And then draw Pascal's triangle next to it
You'll see a striking resemblance
Then the question is is there a way that I can get the n'th row and the k'th item in Pascal's triangle
And that's where our nCk comes from
hoary seal
broken forum
Well not really there's nothing mathematically wrong from how you're doing it
It's just messy and easy to F up
hoary seal
Fair
broken forum
But it's bad to say we use the binomial theorem because this is easy to mess up
That's a consequence of discovering the binomial theorem
I should ask myself is there a way
To generalise this process
For the n'th power?
Well I can kind of see a pattern if I look at the first 3 or 4 powers
You should take note of:
How do my powers of a change as I look through the terms
How do my powers of b change
And how do my coefficients change
The last part is the hardest
And where I will cheat and tell you to draw Pascal's triangle next to your list of coefficients
hoary seal
How do my powers of a change as I look through the terms
Are you taking about this two similar algebraic identity, where the powers of a sand b increase/decrease consecutively
hoary seal
Isn’t it a^n-1
broken forum
.
If you'd be willing I'd be more than happy to give a more visual explanation in the obsidian vcshea later today
plush ore
I have a question regarding binomial theorm
For example
You have (1+2i)^-2
Upto what terms do we expansion?
hoary seal
If you'd be willing I'd be more than happy to give a more visual explanation in ...
Sure im down
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pine wind
<:meowdy:1211089139550920895>
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coarse tendon
Hi, I am starting with limits and i am js wondering whether this is correct.
Show that lim x->1 (5x-3) = 2
Heres my answer :
We can set x0 as 1 since the value of x is approaching 1. Hence 0<mod(x-1)<delta. Epsilon>mod((f(x)-L)) Therefore Epsilon>mod(5x-5) So : mod(x-1)<Epsilon/5 0<mod(x-1)<Delta Hence delta = epsilon/5.
I get the premise with the delta and epsilon stuff, but i dont rlly know how to draw a conclusion. This is kinda the best i can do and am wondering how to clearly write a proper conclusion, i mean i think the rest is pretty decent. js starting out tho
warm spindle
do you understand why and how we are trying to use the epsilon and delta? like what do these variables intuitionally mean
verbal cloud
Hi, I am starting with limits and i am js wondering whether this is correct.
Sh...
I think you have the right idea, however delta epsilon proofs are usually written in the other direction: you pick epsilon , then set the appropriate delta and you do the calculations after. What you did is kinda like draft work, you only need to write the steps in reverse I think
Then it will be easier to see the conclusion
coarse tendon
do you understand why and how we are trying to use the epsilon and delta? like w...
i know that delta is kinda like a domain, that strays from x0, and that epsilon is kind of like the range, at least thats what i understood.
warm spindle
isnt it like we have the delta
and we're trying to obtain an epsilon?
coarse tendon
I think you have the right idea, however delta epsilon proofs are usually writte...
alr lemme try it out
verbal cloud
isnt it like we have the delta
and we're trying to obtain an epsilon?
usually no or maybe I'm tripping
warm spindle
i know that delta is kinda like a domain, that strays from x0, and that epsilon ...
yeah so when we say that lim (x --> 1), to use this information, we introduce delta as a variable which says how much we are absolutely straying from x_0
warm spindle
$\delta = |x-x_0|$
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coarse tendon
i mean if you use the values i got you get mod(x-1) for both, so i thought that was it
i inputted the value for delta, which is epsilon / 5
warm spindle
now with this information, we are trying to prove that (after assuming that L = 2)
$| f(x) - L | < \epsilon $
coarse tendon
mhm, then we get | 5x-5 |
warm spindle
where f(x) is 5x-3
so we can say g(x) = f(x) - L
and we have to prove | g(x) | < epsilon
where epsilon is something we dont really know about
we just know it is some positive quantity
the way we say that the limit holds, is that
if for ANY delta, the g(x) is less than SOMETHING
then even when delta approaches a minuscule number (almost zero)
the inequality holds
if we can show that epsilon is a function of delta
then the epsilon also approaches zero in that case
so we can say that the limit of g(x) has to equal 0
basically we have proved that no matter how close x is to x0, the value of f(x) keeps getting closer to L and never strays off
in other words, the limit of f(x) has to equal L
does the logical flow make sense?
coarse tendon
yeah tysm
i think i get it now, tho i was kinda lost when you were using that g(x) terminology
So if i wanna get full marks for example, i js put in my value for delta, which is epsilon / 5, then multiply both sides by 5, hence getting the original equation for epsilon. is that fully right?
warm spindle
you would find the function g(x)
which is 5x - 5 in this case
then from the x - inequality, you try to prove the g(x) inequality
coarse tendon
oh, so we are js substituting for practicalitys sake?
warm spindle
substituting what?
warm spindle
So if i wanna get full marks for example, i js put in my value for delta, which ...
the thing is, youre technically supposeds to arrive at the epsilon at the end
coarse tendon
substituting what?
like we are putting g(x) as 5x-5
warm spindle
(although my uni profs give marks your way too, i wouldnt rely on it)
coarse tendon
the thing is, youre technically supposeds to arrive at the epsilon at the end
mhm, order matters here?
warm spindle
like we are putting g(x) as 5x-5
yeah because g(x) = f(x) - L
and we need to show that limit of g(x) is zero
so we need to obtain g(x) to start with the logic
coarse tendon
oh okk
warm spindle
mhm, order matters here?
yeah thats the beauty tbh
coarse tendon
i kinda js learned from this textbook, mind if i send you a picture? i was trynna reverse engineer the way to solve it since its kinda old
warm spindle
can you try to show how the logic flows for your specific limit here?
it should take about 3-4 steps
yeah sure
ill really have to learn latex lol
coarse tendon
see, i kinda js copied the logical flow here. find the a kinda common thing between the values of epsilon and delta, then equate to one another to find the value of delta in terms of epsilon
then you kind js substitute epsilon/5
did i really js mess it up that badly or is this method js kinda different?
coarse tendon
uhhhh
idk lwk
this textbooks pretty old
maybe this specific example is wrong? i mean this did make sense and yours also made sense
warm spindle
but you can do it both ways
you basically have to show that one inequality implies the other
(and derive a suitable value from the other in the process)
quiet plume
I'll leave this here just since it can help frame what you need to do in general.
warm spindle
I'll leave this here just since it can help frame what you need to do in general...
could you tell me if and where i went wrong ehehe
quiet plume
I think the only issue from what I read is that it seemed you were explaining that epsilon ought to be a function of delta whereas it should be the opposite. The idea of the limit is that you can make f(x) as close to L as you want (epsilon-close to be exact) provided you choose a suitably small delta for x to range about x_0. In other words, delta is chosen in response to epsilon, not the other way around.
warm spindle
yeah that makes sense
thanks a lot
coarse tendon
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supple basin
Help
molten ibex
I've been looking at this for like 20 minutes straight and actually dont understand how im wrong its got to be something dumb someone please help me spot the mistake
I was trying to verify the red circled part but i keep getting something different
like why do i end up with divided by 1-a instead of a-1 bruuuh
gleaming coral
like why do i end up with divided by 1-a instead of a-1 bruuuh
Whaat
So the answer spreadsheet has a-1?
gleaming coral
Ehat if
molten ibex
I was like going crazy cause i couldnt get why it went from the red circle answer to the thing below it by subbing in Bsin^2(x) for a
gleaming coral
-1-a = ay - y
gleaming coral
-1-a = ay - y
Exactly
This gives us
-1-a = (a-1)y
y=(-1-a)/a-1
Which is allegedly the same
gleaming coral
Thats the answer right?
molten ibex
took y out and left 1-a
molten ibex
Thats the answer right?
yeh
it basically subbed in what a was which was the right side of this
molten ibex
but i was confused to why he got 1+a/1-a when subbing in when the step before he got something else
gleaming coral
Now manipulate him into putting you a higher grade
5% chance to get very difficult exam because he’s mad
molten ibex
oh i really should since Im bound to fail next test
gleaming coral
oh i really should since Im bound to fail next test
You got it
molten ibex
I need 30% on this next one to pass the year yet I only started revising this module today 4 days left
ezzz
aight thx for telling me im not the only one who sees his mistake
makes it a 2v1 at least
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unique spruce
Can you shift the frequency of an arbitrary wave f(x) by multiplying x? e.g f(2x) has twice the frequency of f(x)? Seems to work at least for a sin wave but is that correct for all waves?
And if so, can frequency shifting a wave be described as a form of delay? like you're delaying the wave in time by a multiple of its time axis?
I made this graph and if d is the amount of delay you're applying to the wave my conclusion seems to check out but im a little unconfident in my math skills having graduated a while ago :p
safe radishBOT
Can you shift the frequency of an arbitrary wave f(x) by multiplying x? e.g f(2x...
tardy mango
Can you shift the frequency of an arbitrary wave f(x) by multiplying x? e.g f(2x...
Seems you're thinking of horizontal compressions, so yeah.
For $a>1$, $y=f(ax)$ is the result of horizontally compressing the graph of $y=f(x)$ by a factor of $a$.
flat frigateBOT
Civil Service Pigeon
tardy mango
This is not the same thing as a delay though
iirc the term in English is time scaling, which can roughly be thought of as altering how fast the wave cycles
this is all pertaining to this
for this, you're doing a horizontal translation (so shifting left or right without changing the step)
that is a delay (or time shift), which can be thought of as "postponing" the wave in a rough sense
unique spruce
maybe I can phrase it this way, lets say I had a signal going into speakers, and I can delay the time it takes the wave to reach the speakers by an amount in seconds, if I continually increased that delay linearly with time, could that be modelled as f(x*d)
tardy mango
maybe I can phrase it this way, lets say I had a signal going into speakers, and...
mhm
unique spruce
and that f(x*d) would result in a change in frequency like f(2x) would have twice the frequency of f(x) did I get that right?
tardy mango
and that f(x*d) would result in a change in frequency like f(2x) would have twic...
yes, that's what I was referring to above
unique spruce
so couldn't you then say a change in frequency is a delay that increases/decreases linearly with time?
tardy mango
so couldn't you then say a change in frequency is a delay that increases/decreas...
constantly changing delays result in frequency shifts if that's what you're trying to get at
this is basically what the Doppler effect is tbf
If a time delay increases linearly with $x$, then the delay can written as $cx$ for some constant $c$
flat frigateBOT
Civil Service Pigeon
tardy mango
The delayed signal would thus be
$$f(x-cx)=f\left(x(1-c)\right)$$
but $1-c$ is just a constant, so let $d=1-c$ and you just get $f(x \cdot d)$
flat frigateBOT
Civil Service Pigeon
unique spruce
ok yeah so I figured I had it right but I wasn't entirely sure, this is all as you may have guessed related to audio engineering, theres a common sayign that every effect on a wave can be described as either gain or delay, and if I understand it right thats because any change in y axis can be described as an arbitrarily complicated gain and same for x and delay, seems to check out
anyways thanks, you helped a lot
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full ridge
So Im tryna understand derivatives and stuff (just out of curiousity) im wondering why the derivative of this function (just a random function i thought of) crosses through at that soecific point. I have a firm grasp on graphing and functions but i cant seem to grasp derivatives quite yet, all the YouTube videos i watch dont really explain it that clearly.
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So Im tryna understand derivatives and stuff (just out of curiousity) im wonderi...
wind cypress
i think you need to encase the function entirely when applying the d/dx on desmos
i think it reads it as just d/dx (5x^2) and then adds the rest of the terms, because the real derivative is 10x + 5 meaning it should cross the y-axis at x=0 and should be 5 , not 2
whole hill
So Im tryna understand derivatives and stuff (just out of curiousity) im wonderi...
Yeah, you need parentheses around the function
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ionic blaze
progress:
nothing I forgot the formula
safe radishBOT
progress:
nothing I forgot the formula
ionic blaze
do I get the cross product of the normal vectors and insert that to like R(t)=<normal vector>t+<point>
dw guys this is for my finals I know I'm definitely not cooking
wait
ionic blaze
do I get the cross product of the normal vectors and insert that to like R(t)=<n...
does this work wtf
stoic torrent
does this work wtf
yeah first cross the two normals
ionic blaze
yeah first cross the two normals
well now I assume the cross product is the direction vector
and the point insert yadda yadda
stoic torrent
you would get lets say uh vector c and then yeah
yeah thats the method
if the determinant comes out to be zero then do you know what it implies?
altho idts it comes as 0 here
ionic blaze
I assume <1, -10, 6>t+<-1, 4, 2>=0 or something
stoic torrent
I assume <1, -10, 6>t+<-1, 4, 2>=0 or something
-1,10,6 was what you got after corssing?
ionic blaze
am I correct or am I gonna have to restudy this for
ionic blaze
-1,10,6 was what you got after corssing?
1, -10, 6
stoic torrent
aight thats correctthen
stoic torrent
am I correct or am I gonna have to restudy this for
nah you good
ionic blaze
I have some atrocious notes
stoic torrent
OH WAIT
ionic blaze
nah you good
I'm prolly restudying this
stoic torrent
it asks the plane
ionic blaze
it asks the plane
yeah
stoic torrent
not the line
ionic blaze
isn't
oh shi
wait hold on
no wonder I felt something was wrong
😭
(x+1)-10(y-4)+6(z-2)=0
stoic torrent
I assume <1, -10, 6>t+<-1, 4, 2>=0 or something
you cant do -1,4,2 here in that way
stoic torrent
(x+1)-10(y-4)+6(z-2)=0
yeahh
ionic blaze
you cant do -1,4,2 here in that way
had a feeling 🥀
if I ever get linalg I'm failing
stoic torrent
if I ever get linalg I'm failing
same🥀🥀
stoic torrent
if I'm remembering this right
?
ionic blaze
we can get the normal vector and the vector containing both points and then figure out the plane containing both vectors from there
nope it doesn't work
brute force 🥀
stoic torrent
brute force 🥀
yeah i think so
ionic blaze
ok we have a plane and two points
stoic torrent
we can get the normal vector and the vector containing both points and then figu...
wait this works
i dont see why it shouldnt
ionic blaze
we can get a normal line out of the plane and we can get a line from -2 1 4 to 1 0 3
now the issue is how do I find the plane containing those vectors
stoic torrent
we can get a normal line out of the plane and we can get a line from -2 1 4 to 1...
so thats two lines in the plane reqd
ionic blaze
just cross the the two line?
well that'd be THE NORMAL VECTOR OF THE PLANE
OH MY GOD YOU'RE A GENSIU
wow I'm stupid
and washed
alr that's the last of my question
.close
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stoic torrent
OH MY GOD YOU'RE A GENSIU
-# i get called a disgrace every morning😭
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barren fjord
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split kayak
I’ve done part a bit in a bit stuck on part b
Consider what the condition (i) states.
barren fjord
Consider what the condition (i) states.
Wdym
barren fjord
If F + T > mgsin30
barren fjord
F> mgsin30 -lamdax mg?
split kayak
Im not precisely looking for an equation here.
What happens if B is too heavy? > say, 3,4 or 5 times as heavy as A
barren fjord
A moves up the plane
split kayak
Okay, so our problem is that we have to show that B, can, at most, be half the mass of A (in reality just below that)
barren fjord
Ohh
split kayak
Notably, we will show that if massB = m/2
Then the system remains static. > which is usually thought as the frontier condition for (i) to not be satisfied
barren fjord
I see
split kayak
Ill have to hint you at it, consider that the coefficient μ here is for the dynamic friction.
barren fjord
Mu x R = mgsin30 - lamdamg
split kayak
lambda here = 1/2
barren fjord
Oki
barren fjord
friction equals 0 when there’s no movement ?
split kayak
friction equals 0 when there’s no movement ?
If its the dynamic friction, yes
Theres the static one that responds entirely to the forces being applied to the object, but youll see that all forces cancel here.
split kayak
The Dynamic Friction is a constant force that resists movement
The Static Friction is a force that is applied opposite to another force that wants to move the object.
Dynamic one is always present in the case of movement
Static matches the force that is being applied (up to a certain limit)
barren fjord
How do we know this is dynamic
split kayak
(i) A is moving down
barren fjord
Oki
split kayak
You could assume that the dynamic and statics factor are equal, but even so, since your forces cancel out, then it doesnt really matter
there wont be any friction force.
split kayak
You should probably set it up as
Σ F = 0.5mg - (sin30)mg
To denote that the sum of forces will be 0
split kayak
1/2mg =mgsin30
since this alreayd assumes that the sum of forces is 0 which isnt necesarilly true
(it is the case anyways)
barren fjord
I see
safe radishBOT
@barren fjord Has your question been resolved?
mossy ridge
Prove that eqn a²+b²= 3c² has no solutions in positive integers a,b,c.
elder lagoon
Prove that eqn a²+b²= 3c² has no solutions in positive integers a,b,c.
well
idk if im right but if you see
a^2 + b^2 = c^2
whats the formula that pops up in your head?
ionic blaze
Prove that eqn a²+b²= 3c² has no solutions in positive integers a,b,c.
I'm thinking you could consider squares mod3
mossy ridge
pythagoreas yeah but how do i connect that here
elder lagoon
pythagoreas yeah but how do i connect that here
consider a,b,3c to be part of a triangle
an imaginary one of course.
atomic gorge
consider a,b,3c to be part of a triangle
sqrt(3)
mossy ridge
I'm thinking you could consider squares mod3
Same from there i get a has to be 0 mod 3, same as b
mossy ridge
consider a,b,3c to be part of a triangle
a,b,c are integers 😭
ionic blaze
Same from there i get a has to be 0 mod 3, same as b
yeah both have to be divisible by 3
so what can we say about their sum
elder lagoon
a,b,c are integers 😭
.. and?
mossy ridge
c² also divisible by 3
elder lagoon
Ok i shall leave this to nautilius.
@ionic blaze but do try saying if my method is appropriate.
mossy ridge
I assume triangle of a,b, c sqrt 3
ionic blaze
Ok i shall leave this to nautilius.
<@910426153217581057> but do try saying if ...
don't think it is
ionic blaze
c² also divisible by 3
well
let a=3n, and b=3m for arbitrary integers m and n
square those what do you get
mossy ridge
9(n²+m²)
mossy ridge
3c²
ionic blaze
bare with what I have to say I don't know contradictions
ionic blaze
Same thing no
same thing
ionic blaze
This is where i got stuck
well intuitively you can say that n^2+m^2 must be divisible by an odd power of 3 since c is an integer squared
which isn't possible
mossy ridge
well intuitively you can say that n^2+m^2 must be divisible by an odd power of 3...
Wait how did we get to this
ionic blaze
Wait how did we get to this
well
elder lagoon
(a + b)/c = root (3)
(a^2 + b^2)/c^2 = (sqrt(3))^2
(a^2 + b^2)/c^2 = (a^2 + b^2 + 2ab)/c^2
2ab = 0, we considered both integers
a = 0 or b = 0 or both 0.
however since part of a triangle cant be 0..
Ok i feel im going with bs logic, dk why but it clicked lol.
elder lagoon
(a + b)/c = root (3)
(a^2 + b^2)/c^2 = (sqrt(3))^2
(a^2 + b^2)/c^2 = (a^2 + b^2 ...
feels right but doesnt feel right.
mossy ridge
(a+b)² is not a²+b²
ionic blaze
c^2=3(m^2+n^2)
c must be divisible by 3 hence 9 divides 3(m^2+n^2) which means m^2+n^2 must be divisible by an odd power of 3
ionic blaze
(a+b)² is not a²+b²
^
ionic blaze
(a + b)/c = root (3)
(a^2 + b^2)/c^2 = (sqrt(3))^2
(a^2 + b^2)/c^2 = (a^2 + b^2 ...
well for one, where'd you get (a+b)/c=sqrt3
mossy ridge
ionic blaze
it's not going to be a triangle via triangle inequality
elder lagoon
well for one, where'd you get (a+b)/c=sqrt3
I said consider it to be a right angled triangle.
ionic blaze
c^2=3(m^2+n^2)
c must be divisible by 3 hence 9 divides 3(m^2+n^2) which means m...
^ so anyways going back to this, mod would yield you that both m and n have to be divisible by 3
mossy ridge
i may know how to proceed
elder lagoon
I said consider it to be a right angled triangle.
as part of pythagorean formula.
(as a^2 + b^2 = c^2)
ok fine someone ping me after yall finish, mb nautilis.
ionic blaze
which would only lead to even powers of 3 in m^2+n^2 which would prove the thing as false yadda yadda
mossy ridge
which would only lead to even powers of 3 in m^2+n^2 which would prove the thing...
Oh contradiction is neat
But can u hear me out
ionic blaze
I said consider it to be a right angled triangle.
well, if a, b, and csqrt3 are sides of a triangle
a+b>csqrt3 in ALL cases.
mossy ridge
I think we can do this by infinite descent
atomic gorge
$$
(a^2 - c^2) + (b^2 - c^2) = c^2
$$
$$
a^2 - 2b^2 = -(b^2 - c^2)
$$
$$
3(a^2 - c^2) +- (a^2 - 2b)^2 = 3c^2
$$
$$
3a^2 - 3c^2 + 2b^2 = 3c^2
$$
$$
3a^2 + 2b^2 = 9c^2
$$
ionic blaze
well, if a, b, and csqrt3 are sides of a triangle
a+b>csqrt3 in ALL cases.
of a euler triangle
elder lagoon
well, if a, b, and csqrt3 are sides of a triangle
a+b>csqrt3 in ALL cases.
yes..
mossy ridge
What is this 😭
elder lagoon
damn i was dumb
elder lagoon
so how did you equate a+b=csqrt3?
xD my bad.
honest perch
i haven't read everything so far but consider both sides mod 4. it's not that long this way
ionic blaze
ain't the 2nd line just kinda
saying that -c^2=c^2?
mossy ridge
Okay but hear me out
mossy ridge
i think this i can do by well ordering principle
honest perch
it's easy to see that a,b,c must all be even
ionic blaze
even is 0 mod4 while odd is 1mod4
oh wow smart
atomic gorge
ain't the 2nd line just kinda
saying that -c^2=c^2?
I got problem in my latex
honest perch
so dividing a,b,c by 2 we get the same equation
and again, they are all even
this cannot continue forever, contradiction
mossy ridge
So a²+b²= 3c²,
Let there be a smallest value of c such that this is true
We concluded a,b divisible must be divisible by 3 using modulo 3,
So a= 3a1
b= 3b1
And also c= 3c1 since c divisible by 3 as well
So (3a1)²+ (3b1)²= 3(3c1)²
9a1²+ 9b1²= 27c1²
a1²+ b1²=3c1²
We got same eqn but here c1= c/3 which is contradiction cuz we chose smallest c
Is this valid or am i making a mistake
ionic blaze
So a²+b²= 3c²,
Let there be a smallest value of c such that this is true
We conc...
well
valid actually
mossy ridge
im so tuff
ionic blaze
ur gonna be in the big leagues one day
mossy ridge
poggers
honest perch
So a²+b²= 3c²,
Let there be a smallest value of c such that this is true
We conc...
looks good to me
elder lagoon
Ignore this for now and come later (ans could be stupid):
||b^2 = 3c^2 - a^2. I consider all to be integers. b = sqrt((sqrt(3)c- a)(sqrt(3)c + a))||
||I believe RHS is defo irrational and b must be an integer so hence proved?||
honest perch
same idea as mine
mossy ridge
I have my math subjective writing exam tomorrow and im so cooked
mossy ridge
looks good to me
Ey thanks
u js used modulo 4 instead of 3
But anyways man the geometry problems are the real headache
ionic blaze
Ignore this for now and come later (ans could be stupid):
||b^2 = 3c^2 - a^2. I ...
sqrt2 is irrational, I multiply it to itself, wtf it's rational
mossy ridge
Ignore this for now and come later (ans could be stupid):
||b^2 = 3c^2 - a^2. I ...
RHS is NOT irrational
I mean
elder lagoon
sqrt2 is irrational, I multiply it to itself, wtf it's rational
+- there tho so thought of it .-.
ionic blaze
+- there tho so thought of it .-.
being able to factor into irrational numbers does not prove that they're not integers
elder lagoon
sigh i guess i suck at thinking
whts mod 3 etc?
mossy ridge
Where i could go by intuition
elder lagoon
@mossy ridge r u in iit?
mossy ridge
sigh i guess i suck at thinking
whts mod 3 etc?
Remainder of a number when divided by 3
mossy ridge
<@761453215836798976> r u in iit?
yeh
mossy ridge
Remainder of a number when divided by 3
Except no uniqueness
mossy ridge
is he 1st year too
mossy ridge
I see
mossy ridge
Good luck to you
How did bro find a soln with 3 variables 1 eqn
Truly albert einstein
atomic gorge
Truly albert einstein
Lol
safe radishBOT
@mossy ridge Has your question been resolved?
safe radishBOT
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Available help channel!
dense lark
hi ik this is a math channel but can I possibly ask regarding a rlly simple phy question here?
safe radishBOT
hi ik this is a math channel but can I possibly ask regarding a rlly simple phy ...
dense lark
<@&286206848099549185>
mild plaza
vagrant hill
stone shale
Hlo hlo
buoyant shadow
i won't help if you never ask
warm warren
<@&286206848099549185>
!15m
safe radishBOT
<@&286206848099549185>
Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.
warm warren
hi ik this is a math channel but can I possibly ask regarding a rlly simple phy ...
yes.
Please post the question
dense lark
<#1364708617072148591>
pfft
vagrant hill
this looks like science
warm warren
buoyant shadow
warm warren
It's actually $n^0 \to e^- + p^+ + \bar{\nu}_e$
the last one is an electron neutrino
dammit latex
i dont care
flat frigateBOT
Annie Maqionde
warm warren
the last one is an electron neutrino
warm warren
they have equal and opposite charges
its called free neutron decay i think
what is your question?
dense lark
this looks like science
its phy, comes under electrostatics
safe radishBOT
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Available help channel!
potent oar
where did I go wrong?
safe radishBOT
where did I go wrong?
potent oar
median sand
Which one?
potent oar
my result is x=0 and x=-1 but it's not there...
median sand
Oh it's a multiple choice question?
crude bear
where did I go wrong?
So you got t(t+1) = 2
potent oar
yeah
crude bear
You figured out that t= 1 is a solution
crude bear
What about the other one
potent oar
t = 2
crude bear
2(2+1)=2?
crude bear
Coincidence
crude bear
Show work
crude bear
👍
turbid ore
huh
potent oar
log_2(-2) doesnt exist no?
crude bear
No
potent oar
so it's just "ignore the second solution"?
cause i was making -log_2(2) but that's not how equations work...
potent oar
ok so solution is D, right?
turbid ore
believe in yourself
potent oar
believe in yourself
brother i'm crying
i'm so cooked
the exam is the 17th of june
i'm absolutely burnt to a crisp
thanks tho
.cloe
.close
safe radishBOT
turbid ore
the exam is the 17th of june
which standard?
potent oar
calc1
potent oar
thanks 🥲
damn
if someone actually clicks on this link in 2026, they deserve to get hacked, that's on them
upper forge
<@&268886789983436800> just mass delete it bruh
honest perch
@subtle falcon gng the invite is invalid
potent oar
<@1373610459378487330> gng the invite is invalid
that's CRAZY 😭 💀
junior smelt
<@&268886789983436800> just mass delete it bruh
You do realise you need a senior mod to be able to do that, right 
upper forge
You do realise you need a _senior_ mod to be able to do that, right <:catlove:92...
i don't
😭
dry sierra
<@&268886789983436800> just mass delete it bruh
Sorry we're just human and not machines
proud carbon
<@1373610459378487330> gng the invite is invalid
lol
potent oar
You do realise you need a _senior_ mod to be able to do that, right <:catlove:92...
senior mod? does that mean you're really old 👴 ?
dry sierra
Back to the topic at hand folks please
upper forge
Sorry we're just human and not machines
hmm. we should def use an AI
junior smelt
senior mod? does that mean you're really old 👴 ?
No comment, mostly to not derail this any more 
potent oar
i already closed this ticket, i dont know why it's not going away
honest perch
hmm. we should def use an AI
people in this server don't like ai
potent oar
can i still open another one?
turbid ore
can i still open another one?
as many as your doubts
proud carbon
can i still open another one?
.reopen to open it
potent oar
it's another question so i'll just create a new one
proud carbon
sure
safe radishBOT
Available help channel!
patent citrus
safe radishBOT
acoustic wyvern
simplify the sin term using lim(x-->0)sinkx/kx =1 , and then use the taylor expansion series for sin and cos terms to simplify the equation and to get the value as (1+(k+1)x)^2/x
now use the property
lim(x-->0) (1+x)^1/x = e
in this case,
(1+(k+1)x)^1/(k+1)x = e
(1+(k+1)x)^6/(k+1)x = e^6
so, you have 6/(k+1) = 2
k+1 = 3
k = 2
earnest nacelle
<@&268886789983436800> one more here
golden nebula
<@&268886789983436800>
proud carbon
simplify the sin term using lim(x-->0)sinkx/kx =1 , and then use the taylor expa...
id say theres an easier way
if the x goes in the bracket
then using algebra it can be split into 3
right
and then just small angle approximation because x tends to 0
wont that work
nvm
nvm
denominator
mb
high garden
Can we actually just use angle approx
proud carbon
standard limits
high garden
or we really need sin x / x
proud carbon
Can we actually just use angle approx
nope because x is in the denominator too
so the better way is
standard identities
prob
patent citrus
standard identities
but then how do i get sinkx in the Dr
proud carbon
but then how do i get sinkx in the Dr
multiply and divide
by sin kx
so you can get sin(sinkx)/sinkx and sinkx/kx both
write x as kx
btw
safe radishBOT
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Available help channel!
wind cypress
I already did #1-#4 and got that in #3, the eigenvalues were $1, -\frac{1}{2} + \frac{\sqrt{3}}{2}i, -\frac{1}{2} - \frac{\sqrt{3}}{2}i$ using the characteristic polynomial, so then it follows that the 3x3 matrix has 3 distinct eigenvalues implies T is diagonalizable over C. My question is: is there an easier way to do #3?
safe radishBOT
I already did #1-#4 and got that in #3, the eigenvalues were $1, -\frac{1}{2} + ...
flat frigateBOT
TestTickler
I already did #1-#4 and got that in #3, the eigenvalues were $1, -\frac{1}{2} + \frac{\sqrt{3}}{2}i, -\frac{1}{2} - \frac{\sqrt{3}}{2}i$ using the characteristic polynomial, so then it follows that the 3x3 matrix has 3 distinct eigenvalues implies T is diagonalizable over C. My question is: is there an easier way to do #3?
```Compilation error:```! You can't use `macro parameter character #' in horizontal mode.
l.49 I already did #
1-#4 and got that in #3, the eigenvalues were $1, -\frac...
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.```
wind cypress
we didnt really cover much about rotation matrices and only did the 2x2 case, so i had to look up the 3x3 case around a given axis
stoic dune
You shouldn't look up the matrix. What does T do to the standard basis? Take the vectors you get from that, slap them together, that's the matrix for T
Actually, you probably needed to do that for 2
Anyway, looking at 3:
since rotating three times returns the vector back, you have:
T³ = I
Is that what you did? I imagine that's the easiest way.
@wind cypress
If you happen to be skilled with complex numbers, you'll recognize those as:
cos(0pi/3) + isin(0pi/3)
cos(2pi/3) + isin(2pi/3)
cos(4pi/3) + isin(4pi/3)
Which represent the complex numbers that cube to 1
safe radishBOT
@wind cypress Has your question been resolved?
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Available help channel!
spark leaf
how do i do this?
safe radishBOT
how do i do this?
spark leaf
please @ me
median sand
how do i do this?
Sine theorem?
Or first, recall, what is the total angle of a triangle?
spark leaf
180
spark leaf
180-58-48?
median sand
Start with a.
median sand
180-58-48?
Great.
spark leaf
and then i find side c?
median sand
Now, let's roll with sine theorem?
spark leaf
bet
median sand
Well firstly you got the ratio of sin48 / 7.
For example, if you want to find side AB, then what is the angle opposing it?
median sand
Ah, question c?
spark leaf
no im answer the question opposite of AB is C
median sand
Correct.
spark leaf
u answered at the worse time possible, i need to go have breakfast can i get 30min?
spark leaf
ok cya
median sand
Ping me when you are done. Might go for 30 or more,
austere goblet
looks good
spark leaf
thank u
@austere goblet ive got 1 more question
<@&286206848099549185>
im finding hard to find which angle to use
median sand
<@1500328705116082346>
Great.
median sand
im finding hard to find which angle to use
Well if you know sine theorem, just pick an angle that is opposing a side with a given length.
glossy dock
im finding hard to find which angle to use
send the problem
spark leaf
Used c for multiplication
median sand
Correct.
flat frigateBOT
Moey
median sand
$\frac{\sin{C}}{x} =\frac{\sin{B}}{7}$
flat frigateBOT
Mercury (ヤフォダ)
median sand
Then would it be: $x= \frac{7\sin{C}}{\sin{B}}$?
flat frigateBOT
Mercury (ヤフォダ)
spark leaf
**Mercury (ヤフォダ)**
Yes
Now it makes sense
But how are u so sure we use sin B and not sin A
median sand
Because you are trying to find AB, not BC.
spark leaf
So is there a formula?
spark leaf
Yes pls
median sand
$\frac{\sin{A}}{a} =\frac{\sin{B}}{b}=\frac{\sin{C}}{c}$
flat frigateBOT
Mercury (ヤフォダ)
median sand
Like this.
spark leaf
Idk the answer of each of them are equal?
median sand
Like this triangle.
spark leaf
Yes I see
spark leaf
Yes
I have the same formula u sent me
But it don't explain it
Ohh I get it now
I think
median sand
I think it is because of how sine function is defined and the geometric properties of triangles...
spark leaf
I get it sin C/c =sin B/b
median sand
Correct.
spark leaf
Now it makes lots of sense appreciate the help brotha/sister
And that's all
Have an awesome morning/day/night
.close
safe radishBOT
median sand
Have an awesome morning/day/night
No worries!
safe radishBOT
Available help channel!
regal igloo
Hey
proud carbon
please post your question
versed wave
hi
proud carbon
also hi
turbid ore
Good morning
blazing swallow
hi x4
regal igloo
I need fomula like to a2+b2+c2-ab-bc-ca=0 then a=b=c. Or. if a2+b2+c2+ab+bc+ca=0 then a=b=c=0
I need some stuff of formula which indirectly reduce the questions
For competitive exam
regal igloo
theres alot of algebraic identities
I know them
But i didn't knew has one can u help me prove it
Proved it
Need help
,roate
versed wave
,rct
regal igloo
,rct
regal igloo
,rccw
regal igloo
Is p³=q³=0
glossy dock
this is C right
regal igloo
this is C right
How
glossy dock
not sure based on instinct
proud carbon
this is C right
yeah
glossy dock
alright cool i know how to do it then
regal igloo
Bro in jee if there is 0 then answer us 90% 0 i also know iy
regal igloo
I wrote 90%
desert pasture
Instead, I thinkg it's be useful to compuet det ((P+Q)^3)
regal igloo
Instead, I thinkg it's be useful to compuet det ((P+Q)^3)
I also thought same
regal igloo
that AND det (P^3+Q^3)
It's it 2p³/q³
proud carbon
Instead, I thinkg it's be useful to compuet det ((P+Q)^3)
subtract the 2 given conditions and take determinant of both sides
desert pasture
oh that's better
crisp mountain
Hey I have a question in statistics, where do I go for that?
proud carbon
oh that's better
you have given jee?
proud carbon
Hey I have a question in statistics, where do I go for that?
this one's occupied you can try other available 1s
desert pasture
you have given jee?
unfortunately
flat frigateBOT
desert pasture
unfortunately
I say "unfortunately" as the only reason I signed up for JEE was for the physics questions 
proud carbon
damn
dense lark
wsup
proud carbon
Same as me
cool
dense lark
who pinged da helpers
desert pasture
and then they decided to give me the easiest questions ever which scrwed me over(for obvious reasons)
dense lark
<@&286206848099549185>
hy bbg who's troublin u
proud carbon
hy bbg who's troublin u
bro theres another channel he has opened
dense lark
bro theres another channel he has opened
🤦🏼♂️
regal igloo
proud carbon
and then they decided to give me the easiest questions ever which scrwed me over...
easiest 😭?
dense lark
🥹
regal igloo
Nah
proud carbon
oh lord
desert pasture
||compute by hand||
desert pasture
||compute by hand||
there's no other way
regal igloo
||compute by hand||
What u mean
desert pasture
find the product
regal igloo
there's no other way
It will take million
dense lark
It will take million
yea ignore him
which question?
regal igloo
See bit up
proud carbon
2nd
regal igloo
2
desert pasture
It will take million
the only other way is to break it down into block matrices and do that
dense lark
alr so you need the determinant of the product of A nd B
computing each one by hand is a long process
regal igloo
the only other way is to break it down into block matrices and do that
There is a way to directly get answer 0 by sir told us
desert pasture
the det of each one isn't defined
dense lark
why dont you split the determinant
regal igloo
But i am vonfused
proud carbon
the only other way is to break it down into block matrices and do that
you cant even say that the AB would give us 0 because theres a chance it couldnt
too
regal igloo
you cant even say that the AB would give us 0 because theres a chance it couldnt
Answer 0
desert pasture
you cant even say that the AB would give us 0 because theres a chance it couldnt
wdym "split"
wrong reply
regal igloo
jst split the determinant?
U can give it try it would make 4 detements
dense lark
wrong reply
yea, splitting as in using the properties
desert pasture
the det function isn't defined for rectangular matrice....