One could just say symmetry
#help-39
crystal lion
ebon cloak
i cant see this
unique summit
@chrome niche u got it?
unique summit
i cant see this
me neither icl
ebon cloak
but how to prove it?
crystal lion
Symmetry is proof if you write it formally
unique summit
yeah i give up imma stick with integrals and vectors 😭
crystal lion
yeah i give up imma stick with integrals and vectors 😭
Kinda ironic. As that is considered harder than normal euclidean geometry
ebon cloak
i really cant see how you got that it was a trapezoid
crystal lion
i really cant see how you got that it was a trapezoid
KX and GI are both perpendicular to the line i drew
crystal lion
Well GI is perpendicular because i drew it to be perpendicular. KX is perpendicular because X is reflection of K around that line.
ebon cloak
x is the intersection of HJ and omega isnt? but why its perpendicular too?
unique summit
Kinda ironic. As that is considered harder than normal euclidean geometry
when you study something more it becomes easier my focus is alot less on geometry and alot more on integration
crystal lion
when you study something more it becomes easier my focus is alot less on geometr...
Fair
crystal lion
Consider a map that takes a point and reflects it around the line i drew. It maps K to a point on a cirle , and it maps K to a point on line HJ. Aka it maps K to X
crystal lion
But if you ask me you dont really need to prove that.
ebon cloak
that was a beautiful solution
crystal lion
Just stating it is a trapezoid due to symmetry woule probably get all marks on most exams i know.
Also triangles HGX and HKI are congruent. So that also 1 way to prove it
But the trapezoid sol is easier due to it being a common configuration with known properties
feral sedge
It's literally just angle chasing
pearl pondBOT
@chrome niche Has your question been resolved?
unique summit
It's literally just angle chasing
do it
its harder than it seems
unless ur like really really good with geometry
feral sedge
I did it already
unique summit
I did it already
whats K
feral sedge
80
unique summit
.
feral sedge
I didn't even know lines could be supplementary
I thought that was only a thing that angles can be
unique summit
form a 180 degree angel
feral sedge
You can prove that it does even if you're not given that fact actually
unique summit
how
if u look really close at the end of K u will notice K is alittle shifted
so they are infanct not the same line
unique summit
yep
feral sedge
So KHA is 80
feral sedge
Ah maybe I used collinear at some point let me look
unique summit
Ah maybe I used collinear at some point let me look
KHA is indeed 80
i just checked
pearl pondBOT
feral sedge
Ok maybe I used collinearity on accident, I can't rederive
pearl pondBOT
Available help channel!
trail forum
pearl pondBOT
pearl pondBOT
trail forum
.reopen
pearl pondBOT
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trail forum
is this one correct?
feral leaf
is this one correct?
Looks correct
orchid token
shouhld I be expanding on the concept of the reiman's sum here? If so what should i try to expand on or explain better?
cobalt hinge
shouhld I be expanding on the concept of the reiman's sum here? If so what shoul...
I guess it is sufficient enough for an informal proof? Really here, it's just an intuitive understanding. Though, you should (probably) show the actual integration over the bounds.
orchid token
I guess it is sufficient enough for an informal proof? Really here, it's just an...
what about on the part where I say "Using the integral... assembled."
i felt like that was not well explained but i js didnt know where to take the explanation from there
pearl pondBOT
@orchid token Has your question been resolved?
lilac siren
for this question I had a helper help me out and point in me in the direction to further my proof but I am once again stuck on what to do. I realize that I am supposed to use the hints but I don't really see a blatant connection to be made.
lilac siren
here is my work
I am stuck on proving 2G(x) > f^2(x) I had an idea of making it 2f(x) and saying that 2 > x from the domain and G(x) is bigger than f(x) or eequal but I couldn't find anything helpful
pearl pondBOT
@lilac siren Has your question been resolved?
pearl pondBOT
@lilac siren Has your question been resolved?
pearl pondBOT
@lilac siren Has your question been resolved?
pearl pondBOT
@lilac siren Has your question been resolved?
acoustic path
for this question I had a helper help me out and point in me in the direction to...
differentiate again
this part $2\int_0^x f(t), \mathrm{d}t - f^2(x)$
jolly parrotBOT
Mqnic_
acoustic path
and you will be done
lilac siren
I am at a lost for words how I didn't see that
I think that f(x) on the outside scared me
welp thanks for the help.
.close
pearl pondBOT
lilac siren
.reopen
pearl pondBOT
✅
lilac siren
@acoustic path Ok so I was able to prove that it was greater than zero but I still have to deal with the f(x) on the outside don't I. and If I differntiate again using product rule I get back to an even messier formula. Am i doing something wrong or is there a rule that lets me differentiate only within the paranthesis.
acoustic path
<@392866251149410305> Ok so I was able to prove that it was greater than zero bu...
you should simply be using the property that the product of two positive numbers should again be positive
(in particular 0 < f'(x) < 1 for 0 < x < 1 and f(0) = 0 allows you to conclude that f is positive from 0 to 1)
lilac siren
Ok so going back I have f(x)(2G(x)-f^2(x)) i differrentiate this and get f'(x)(2f(x)-2f(x)(f'(x))) + f'(x)2(G(x)-f^2(x)) but the left side is basically back where we started so i don't understand
acoustic path
you don't differentiate that
acoustic path
this part $2\int_0^x f(t)\, \mathrm{d}t - f^2(x)$
you differentiate this
lilac siren
but the problem is F'(x) = f(x)[2G(x)-f^2(x)] I can just skip over the f(x) on the outside?
unless you're saying by seperrating it and proving that the derivative is positive thus the regular is also positive?
acoustic path
the point of this is to check the sign
the end result you will end up getting is that [thing on left] and [thing on right] are both positive numbers
so their product is positive, from which you conclude F'(x) is positive and the problem is done
lilac siren
ok I'm a bit confused but let me get this straight. we know f(x) is positive bc 0 < f(x) < x on 0 < x < 1. and we know that right is positive because by differentiating it we have proven it is also positive. Thus bc both are positive f'(x) is positive
lilac siren
ok
final question and then i'll stop bothering you what justification could I use to say that the derivative being positive justifies the regular function being positive?
acoustic path
so F'(x) > 0 means it is strictly increasing and you can manually verify by plugging in zero that F(0) = 0, this suggests it is positive
lilac siren
sorry I was not clear I meant that why is it that the derivative of [2G(x)-f^2(x)] justify that it is positive
acoustic path
this is for the same reason as above, since if you plug in x = 0 you get zero
lilac siren
oh ur so righht
ok i release you
thank you for helping me ur the goat
.closee
.close
midnight haven
Can someone check this for me
pearl pondBOT
Can someone check this for me
midnight haven
I am not sure as to how this is taught but the topic is DE of first order and first degree
midnight haven
Can someone check this for me
I checked if the given DE was exact, which it was not.
I saw that the DE was homogeneous, so I used the "rule 1" to find the Integrating Factor which came out to be $\frac {-1}{2x^2y^2}$
Which I multiplied to the given DE and proceeded to integrate for the solution
However the "lecture" I am referring used "rule 2" where the IF is $\frac {1}{Mx -Ny}$
Which I multiplied to the given DE and proceeded to integrate for the solution
However the "lecture" I am referring used "rule 2" where the IF is $\frac {1}{Mx -Ny}$
Tldr: I am not sure why the method I used to find the IF wasn't used and whether what I did is correct
jolly parrotBOT
backtick
pearl pondBOT
@midnight haven Has your question been resolved?
pearl pondBOT
midnight haven
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midnight haven
<@&286206848099549185>
drowsy adder
I think there’s a super clean trick that can work here
You can multiply divide by 2 so you get terms like -(2x^2ydy + 2y^2xdx)
Which is just -d(x^2y^2)
So you get a exact differential
midnight haven
Tldr: I am not sure why the method I used to find the IF wasn't used and whether...
^
compact ridge
Can someone check this for me
it isn't exact so why are you proceeding if it were exact
I mean why would you do it that way if it isn't exact
midnight haven
it isn't exact so why are you proceeding if it were exact
so what i have done is wrong?
i am sorry but this was taught to me in a VERY superficial manner
literally writing down "formulas" and then solving questions so i do not the understanding i would want
drowsy adder
I don’t think it’s wrong just a shorter and way quicker version is available
drowsy adder
literally writing down "formulas" and then solving questions so i do not the und...
Oh
midnight haven
I don’t think it’s wrong just a shorter and way quicker version is available
ah, the reference video i was watching had solved it using "rule 2"
which meant his final answer was pretty different from mine
this is what i mean
midnight haven
https://users.math.msu.edu/users/newhous7/Math_235/Lectures/lec_5.pdf
i lost you here
compact ridge
i lost you here
it doesn't matter
you're working in R^2 so you can just let D = R^2
midnight haven
regardless thank you for your help, i hate learning maths this way
midnight haven
you're working in R^2 so you can just let D = R^2
do not know what that means
compact ridge
unfortunately for this topic you have to go with the substitution and see for yourself
compact ridge
do not know what that means
R^2, the Cartesian plane in 2 dimensions
$\mathbb R^2$
jolly parrotBOT
south
midnight haven
i see
i will have to make do with superficial understanding because time
thank you guys.
.close
pearl pondBOT
compact ridge
,w (d/dx (-(1 + xy)x) - d/dy ((1 - xy)y) ) /((1-xy)y)
jolly parrotBOT
pearl pondBOT
Available help channel!
simple eagle
we learnt complex numbers but i dont exactly know how to do the matrics part
simple eagle
like i dont get the 6i+4 0 0 step
light helm
they performed the row operation indicated
R_1 → R_1 + R_2
row 1 becomes the sum of rows 1 and 2
gusty heron
I need to state if the expression
$\frac{3 \cdot 7 \cdot 11 \cdot 15 \cdot ... \cdot 599}{5 \cdot 9 \cdot 13 \cdot 17 \cdot ... \cdot 601}$
Is greater than or less than 1/13
jolly parrotBOT
Oak
gusty heron
This is what I did so far, I am not sure how to continue
I can expand the series and add ln(13) to both sides, then, the only positive term I have is ln(11) I need to prove that the entire series is still negative, correct?
A few disclaimers:
This is a problem I randomly encounted online.
I didn't really do anything of this sorts, I am not used to proving things like that.
English is not my native language, I wrote it in English to show my work and ask for help
pearl pondBOT
@gusty heron Has your question been resolved?
errant cedar
i wrote it like that but idk what to do next, maybe expand term by term and pray for a pattern to show up
pearl pondBOT
@gusty heron Has your question been resolved?
stuck crypt
rigid lava
an even intger would have 0,4,8 at the end
stuck crypt
yes
stuck crypt
i might as well tell u the answer
do u have to split them
rigid lava
yes
make 3 cases
first one of 4 digit numbers
second one of 5
and third one of 6
stuck crypt
and third one of 6
what is wrong
rigid lava
now add them all up
stuck crypt
now add them all up
3600
rigid lava
done
rigid lava
wot
stuck crypt
the answer is supposed to be 3020
rigid lava
7
pearl pondBOT
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pearl pondBOT
pearl pondBOT
Available help channel!
pearl pondBOT
calm wing
<@&268886789983436800>
high furnace
too tired to think for sparx can someone help me finish this question 😭 it should be easy for me im grade 9 further maths gcse but i just cant thkn
high furnace
think
autumn fossil
in one third of all cases, he does the first part incorrectly
in four fifths of that, he also does the second part incorrectly
so what's four fifths of one third?
autumn fossil
hmm
that's weird
oh wait
given that he had to retry
there is nothing saying that the 2 tries are related
autumn fossil
isnt there at least sth that tells us how many options are there for each question?
high furnace
isnt there at least sth that tells us how many options are there for each questi...
sometimes but this one its just leaving it toyou to decide
high furnace
i already have
it both incorrect/retry t oget that
4/15 / 9/15
i can try again but itll likely change the question a bit as you get 3 attempts before it changes numbers
high furnace
they got 4/11
and now its right
😭
jusst didnt agree the first time ig
mb
thank you for the help
autumn fossil
mb
not your bad lol
its an incredibly stupid question
not enough info to solve it
you have to guess what they're asking for
high furnace
you have to guess what they're asking for
its very common thing for sparx
autumn fossil
Also this is quite literally just wrong
the probability that aiza answers both parts incorrectly definitely isnt 2 / 15
its 4 / 15
high furnace
😭
i tried using 3 ai before asking here they all got it wrong or said they need more info
.close
magic olive
pearl pondBOT
magic olive
Where does the k not equal 8 come from?
shrewd mist
determinant can't be zero, otherwise they represent the same line and your system becomes underdefined
cobalt hinge
Where does the k not equal 8 come from?
You get simulatenous equations (constant multiples) if you have k=8
Observe the final matrix produced
shrewd mist
You can check by plugging into the matrix, youll get the same eqtn twice^
magic olive
Huh
cobalt hinge
Which one😭
$\begin{pmatrix}2x+y\kx+4y\end{pmatrix}$
magic olive
Ohhh
cobalt hinge
Is it pmatrix
jolly parrotBOT
;(
cobalt hinge
It is pmatrix
magic olive
You get simulatenous equations (constant multiples) if you have k=8
I don’t see it
😭
Ohh yaaaa
cobalt hinge
I don’t see it
4(2x+y)=kx+4y for what value of k?
cobalt hinge
Does that mean I’ll just get the same invariant line twice?
Yep
sharp fractal
why is this true
pearl pondBOT
why is this true
tropic saddle
L(...) can either be 0 or 1 and why should it prefer one over the other
sharp fractal
L(...) can either be 0 or 1 and why should it prefer one over the other
im not sure if the exclusion of 0^w has any effect
i mean if it affects the probability
tropic saddle
well 0 always gets mapped to 0
but we are talking about a nonzero y. that can be mapped to 0 or 1
sharp fractal
and there are as many y's that map to 0 as ones that map to 1?
sharp fractal
right
tropic saddle
more formally, we can extend y to a basis (y, b2,...,bn) and then we can describe every possible L by saying what it equals on those basis elements
we can either set L(y)=0 or L(y)=1
and the choices for L(bk) are independent of that
so we get the same number of Ls with L(y)=0 as with L(y)=1
pearl pondBOT
@sharp fractal Has your question been resolved?
sharp fractal
ty @tropic saddle
pearl pondBOT
pearl pondBOT
Available help channel!
proud frost
im a bit lost on hwo to even start consrutrdting such a sequence
wary bane
might help to draw it out first to at least see the claim
proud frost
im a bit lost on hwo to even start consrutrdting such a sequence
hmm wait yeah i think I frogot about def of infimum
wary bane
largest lowerbound, but yh check defn
proud frost
largest lowerbound, but yh check defn
yeah wait, so there exists f in F, k in K such that d(f,k) < D(F, K) + 1/n
for all n in N
also we know that D(F, K) <= d(f,k)
proud frost
yeah wait, so there exists f in F, k in K such that d(f,k) < D(F, K) + 1/n
if this is true for all n in N, this implies d(f,k) <= D(F, K)
so d(f, k) = D(F,K)?
problem is i need to shwo ti converges to apply sequential comapctness
tiny lintel
yeah wait, so there exists f in F, k in K such that d(f,k) < D(F, K) + 1/n
Name that fn
proud frost
Name that fn
ah that is where i use sequential compactness
hwoever, i think i need to show that lim d(f_n, k_n) = d(x,y)
where x, y in K, F repseciivly
tiny lintel
lim fn = f
d(f,k) = 1/n
Since F is compact, f belongs to F
proud frost
lim fn = f
d(f,k) = 1/n
Since F is compact, f belongs to F
do you know where we need a "subsub sequenece"?
im only needing a subsequecne
tiny lintel
Yup my bad
So if you take fn, I assumed it converges
But nothing guaratees that
That's where subsequences are useful.
How can you prove that fn, as you defined it, has a subsequence that converges ?
pearl pondBOT
@proud frost Has your question been resolved?
proud frost
That's where subsequences are useful.
ye i mean by sequential compactness we know a subsequecne of it converges
however
im tryign to shwo ti converges to d(a,b)
given that f_n -> a
tiny lintel
It has to
proud frost
and k_n -> b
tiny lintel
Ok
proud frost
im thinking of maybe using limit defitinon?
tiny lintel
so fn is defined like this : d(fn,k) < D(F, K) + 1/n
let tn be a subsequence of fn that converges (given by compacity)
tn = fq with q >n
meaning
d(tn,k) <= D(F,K) + 1/q <= D(F,K) + 1/n
tn converges towards t, and t is in F (given by compacity)
that means d(t,k) <= lim (D(F,K) + 1/n) = D(F,K)
and since t is in F then we also have d(t,k) >= D(F,K)
We conclude d(t,k) = d(F,K)
proud frost
ig im saying for of how we know that d(t_n,b_n) -> d(t, k)
tiny lintel
d(tn,k) <= D(F,K) + 1/q <= D(F,K) + 1/n
This
Compacity tells you that tn converges towards a t
and by definition you have : d(tn,k) <= D(F,K) + 1/q <= D(F,K) + 1/n
If you tanke the limit : d(t,k) <= lim (D(F,K) + 1/n) = D(F,K)
proud frost
d(tn,k) <= D(F,K) + 1/q <= D(F,K) + 1/n
This
ye okay, i mean d(t,k) not D(F,K)
since -> D(F, K) follows by Arhcimdean pirnciple
pearl pondBOT
@proud frost Has your question been resolved?
outer hare
For this question, I proved it is not surjective/ a homomorphic image by saying that 2^x will always be even and therefor you cant get R. Is that also accurate?
lilac jackal
For this question, I proved it is not surjective/ a homomorphic image by saying ...
wdym 2^x is even?
outer hare
2^(Z) is always even @lilac jackal
by def its gonna be a multiple of 2
so it doesn't map to any odd numbers in R
lilac jackal
x is real not an integer
pearl pondBOT
lilac jackal
np
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Available help channel!
boreal frigate
Can someone clearly explain how i should approach this problem and what methods / formulas i should use
feral leaf
Can someone clearly explain how i should approach this problem and what methods ...
You can apply the concept about axis of symmetry
jolly flicker
Just a real quick question, can someone explain how ‘a’ wouldn’t affect the range???
cobalt hinge
Just a real quick question, can someone explain how ‘a’ **wouldn’t** affect the ...
Because it stretches/shrinks in the horizontal direction.
pearl pondBOT
pearl pondBOT
Available help channel!
cerulean prairie
Help me pls
pearl pondBOT
Help me pls
cerulean prairie
How do you get r^2 = 4
cobalt hinge
,rotate 180
jolly parrotBOT
cerulean prairie
How do you get r^2 = 4
cobalt hinge
How do you get r^2 = 4
Look at what it says
cerulean prairie
im confused
cobalt hinge
$\frac{a_6}{a_4}=\frac{ar^5}{ar^3}=r^2=\frac{-96}{-24}$
jolly parrotBOT
;(
cerulean prairie
oh shoot I didn't even think of that, thanks
cobalt hinge
oh shoot I didn't even think of that, thanks
Welcome
pearl pondBOT
@cerulean prairie Has your question been resolved?
tender vapor
how do i find the projection of $f(x) = e^x$ on the subspace $\mathbb{W} = { g \in C[-1, 1] : g \text{ is a odd function} }$ where $C[-1, -1]$ is the vector space of all real valued continuous functions defined over $[-1, 1]$
jolly parrotBOT
carburetor
pearl pondBOT
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toxic lichen
how do i find the projection of $f(x) = e^x$ on the subspace $\mathbb{W} = \{ g ...
what inner product does W come with
pearl pondBOT
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Available help channel!
junior vapor
am i allowed to do this?
pearl pondBOT
am i allowed to do this?
junior vapor
the solutions did something different
pearl pondBOT
@junior vapor Has your question been resolved?
junior vapor
<@&286206848099549185>
pearl pondBOT
@junior vapor Has your question been resolved?
junior vapor
<@&286206848099549185>
green dew
am i allowed to do this?
seems correct
junior vapor
seems correct
do i hv to write why p^2 divisible by 6 means p is divisible by 6?
green dew
do i hv to write why p^2 divisible by 6 means p is divisible by 6?
do u mean 2
junior vapor
do u mean 2
oh yes sorry i was doing another question
green dew
oh yes sorry i was doing another question
preferrably i would show why
junior vapor
preferrably i would show why
how do i show it?
marsh wharf
Im not really sure where to start, what is that symbol between the f and the g
green dew
Im not really sure where to start, what is that symbol between the f and the g
composition symbol
(f○g)(x) is the same as f[g(x)]
green dew
How should I plug the numbers in properly
f[g(x)] means f(x) but instead of x, replace it with g(x)
means that f[g(x)] = -4g(x) - 3
green dew
means that f[g(x)] = -4g(x) - 3
and youre given g(x) = -7x - 2, so js substitute it in here
green dew
yep
radiant coyote
what is C
5
9
(
F
−
32
)
pearl pondBOT
what is C
=
5
9
(
F
−
32
)
pearl pondBOT
reef marlin
?
slow grove
i dont know how to get it to help me
you just need to wait for someone to come along
reef marlin
i dont know how to get it to help me
What you just sent is the formula for converting Celsius to Fahrenheit. Is there anything you don't understand about it?
merry stirrup
what is C
=
5
9
(
F
−
32
)
C is celcius
pure rapids
what is C
=
5
9
(
F
−
32
)
!nogpt
radiant coyote
Wait didn't I close this?
pearl pondBOT
what is C
=
5
9
(
F
−
32
)
Please do not trust ChatGPT or similar AI tools for mathematical tasks, as they often generate output which "sounds correct" but has numerous factual or logical errors. Use of these AI tools to answer other people's help questions is strictly against server rules (see #rules).
pure rapids
If theres an original question then we might help
reef marlin
Wait didn't I close this?
You already did, but you can still type here.
pure rapids
Wait didn't I close this?
Takes some time to close
radiant coyote
Oh
pearl pondBOT
Available help channel!
marsh wharf
What do I do?
pearl pondBOT
What do I do?
marsh wharf
I know what to do when its formatted like
is it different when its in that orientation
toxic lichen
no, the lack of brackets does not make a difference here
it's more of an oversight on the question writer's part
they're using $f \circ g(x)$ to mean $(f \circ g)(x)$ here, and likewise for the other 3 parts
jolly parrotBOT
ann.in.a.teacup
marsh wharf
So its just the same thing?
so for the first one it would be
(x-7)^2 - 9(x-7)
and then simplified
toxic lichen
yes that is correct
fathom light
I need some help with proving that laplace transform of sin(at)=a/s^2+a^2
fathom light
using the defintion
I did partial integration twice and ended up with this expression
What do I do next? The proof I'm following simply writes this as (1+a^2/s^2)*the integral. But where does the 1+ come from?
this part is where I'm lost.
pearl pondBOT
@fathom light Has your question been resolved?
dire parrot
remember you have the same integral on the left hand side
as well as in the right side
terse tartan
You could use e^iat = isin(at)+cos(at)
Afterwards use the linearity of the Laplace transform.
Which results in L(e^iat) = 1/(s-ia) . Multiply by the complex conjugate of the denominator and extract real and imaginary values.
This method is FAR easier.
pearl pondBOT
@fathom light Has your question been resolved?
fathom light
You could use e^iat = isin(at)+cos(at)
I was unsure whether or not this method uses the "definition of laplace" but yeah, maybe that method is accepted, which would be easier for sure 🙂
candid jacinth
pearl pondBOT
vagrant briar
beautifully drawn 😍
candid jacinth
any hints? <@&286206848099549185>
chilly sand
any hints? <@&286206848099549185>
I think there's actually a theorem
vagrant briar
focus on triangles XYB and XBP
vagrant briar
I think there's actually a theorem
yeah
candid jacinth
how is it called?
vagrant briar
the angles subtening the same arc lengths or eaual length of arc lengths are always equal
subtending*
chilly sand
Angles subtended by equal chords at the center of the circle are equal,
chilly sand
how is it called?
Just gimme a min that's in my book I suppose
chilly sand
how is it called?
It's called angles in the same segment of a circle are equal
limpid heath
I think use properties of cyclic quadrilaterals and angle chasing
And also tangent secant theorem
Cuz like if u observe it, X and Y lies on both circles which it states that the X and Y are common points of intersection
And A and B are points of tangency which youll have right angles at A and B
angle AXO
and
angle BYO
While X Y P B lie on the same circle w_2 which is quadrilateral XYPB is cyclic
That's what I observe
pearl pondBOT
@candid jacinth Has your question been resolved?
pearl pondBOT
@candid jacinth Has your question been resolved?
candid jacinth
.close
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midnight haven
do i just send my question here?
pearl pondBOT
do i just send my question here?
ivory wasp
yes
hasty edge
the answer shows x= -5/2 , x= 3
x equals? This is an inequality.
midnight haven
x equals? This is an inequality.
yes yes but idk how to write it on a keyboard
<=
midnight haven
it actually is
yeah gow
hasty edge
the answer shows x= -5/2 , x= 3
It should just be $x\le -\frac52$, $x=3$. Is this what you mean?
midnight haven
no its x<= -5/2
hasty edge
$f(x)<0$ and $f(x)=0$ is what that notaion is essentially saying.
hasty edge
i dont get why x= 3 is there
Where is f(x)=0
midnight haven
oh
i didnt even look at the graph tbh
no need to look at the graph lol
what
(x-3)² = 0 doesnt need a graph
to figure where it equals
hasty edge
i didnt even look at the graph tbh
Just use algebra
midnight haven
Just use **algebra**
thats a bad word we dont say that where am from
anyway ty ill send further questions
if i need to
hopefully not
pearl pondBOT
@midnight haven Has your question been resolved?
scarlet trellis
I found $x=\frac{12}{\frac{3}{y}+\frac{6}{z}}\x=\frac{12y}{3}+\frac{12z}{6}\x=\frac{4y+2z}{1}$
I don't see how I could possibly get a y and z on top and bottom of that fraction. Multiplying top and bottom would result in squares.
Also, I don't see how I could flip the fraction without effecting x to get the addition on top.
Any ideas?
jolly parrotBOT
UCYT5040
jolly parrotBOT
mathisfun
hasty edge
$\frac{3}{y}+\frac{6}{z}=\frac{3z+6y}{yz}$
jolly parrotBOT
mathisfun
hasty edge
So $\frac{12yz}{3z+6y}$
jolly parrotBOT
mathisfun
scarlet trellis
ohh, you just added the fractions together
hasty edge
Yep
scarlet trellis
it can be further simplified though right?
jolly parrotBOT
mathisfun
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proven sable
i have no idea atm what to do for 11ii), 11iii) and 11iv)
pearl pondBOT
i have no idea atm what to do for 11ii), 11iii) and 11iv)
proven sable
feral sedge
For 2, try cubing them
pearl pondBOT
@proven sable Has your question been resolved?
proven sable
For 2, try cubing them
so i brute forced it
i can get w^2 and w^3
wait
no im dumb 💀
my exponents are 1 up
ok so thats 2 done
proven sable
For 2, try cubing them
that feels tedious as well though
feral sedge
Do you know the binomial theorem?
proven sable
yea
so id get 1 + 3w + 3w^2 + w^3
but then w^3 is just 1
so id have 2 + 3w + 3w^2
and then from there im not sure where to go
then for 11c and 11d) it gets worse
hasty edge
i have no idea atm what to do for 11ii), 11iii) and 11iv)
If $\omega^3=-1$ then $(\omega+1)(\omega^2-\omega +1)=0$
jolly parrotBOT
mathisfun
proven sable
im stil confused for this
why does omega^3 = -1
cuz it says if omega is a root of unity
so its a solution to z^3 = 1
then by demoivers we get
cos 3theta + i sin 3theta = 1
so 3theta = 2k pi
theta = 2kpi/3 for k = 0,1,2
so our roots will be in the form
roots : cos (2kpi/3) + i sin (2kpi/3)
k = 0 gives w^0 = 1
cos(0) + i sin(0) = 1
w^1 = cos(2pi/3) + i sin(2pi/3)
w^2 = cos(4pi/3) + i sin(4pi/3)
proven sable
?
distant basin
?
proven sable
😭
theres no mark scheme at the back for me to even look at either
since this is a prove that question
floral prism
and then from there im not sure where to go
add and subtract 1, use 1+w+w^2 = 0
proven sable
ohh thats proper
alr
its 11iii) im confused about
maybe a similar trick is gonna be involved
proven sable
hm
floral prism
theres another identity for iv
proven sable
so ive expanded the 3 brackets
floral prism
u can use
1+w+w^2=0
floral prism
so ive expanded the 3 brackets
thats more work but it should still work
proven sable
although something feels a bit off
floral prism
can you show it?
proven sable
yea hold on rq
proven sable
take a^2b+ab^2 common
cant do that
i think ill have to split it up in 2 of those though
proven sable
yea but then what can i do with that
floral prism
and use 1+w+w^2=0
^
floral prism
ahhhh
also even if u left one ab^2 and a^2b out for now, u can just put w+w^2 = -1 and simplify later to cancel them
proven sable
also even if u left one ab^2 and a^2b out for now, u can just put w+w^2 = -1 and...
alr now we left with the nightmare question
11d) 💀
definitely not expanding that out
so there has to be some other method of doing it nicely
floral prism
1+w+w^2 = 0
that and (a + b + c) (a^2 + b^2 + c^2 - ab - bc - ca) = rhs
proven sable
thats a big identity
proven sable
that feels like
too much 😭
only the first 3
all of that
is without multiplying the 1st bracket
floral prism
yes thats correct
floral prism
youre only 1 step away
proven sable
i feel like 10 a
floral prism
use the w identity i said before
floral prism
yeah
floral prism
that and (a + b + c) (a^2 + b^2 + c^2 - ab - bc - ca) = rhs
then its just this
proven sable
proven sable
then its just this
tysm
floral prism
yep np
proven sable
this question came out of left field iwl
the rest of the questions didnt use much identities like this with factorising and stuff like that
but its definitely good
cuz atleast i got bamboozled now
then later on 💀
floral prism
this question came out of left field iwl
idk what that is but nice
proven sable
the question came out of nowhere
pearl pondBOT
proven sable
a
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tacit crown
Used method my proffessor used and checked over it 5 ish times and still ends up wrong, I did it previously by using the (density) (depth of liquid) (area of surface)
pearl pondBOT
Used method my proffessor used and checked over it 5 ish times and still ends up...
@tacit crown Has your question been resolved?
tacit crown
<@&286206848099549185>
pearl pondBOT
@tacit crown Has your question been resolved?
pearl pondBOT
@tacit crown Has your question been resolved?
glossy root
pearl pondBOT
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sharp smelt
pearl pondBOT
sharp smelt
Trying to figure out a parametrization of the path
rough forge
it's a closed path
sharp smelt
yea
rough forge
could be anything
rough forge
hmm kinda hard to imagine to find a specific general parameterization
maybe just defining alpha = (x(t), y(t)) suffices and then somewhere you will have to use that alpha(a) = alpha(b)
ember cloak
Have you learned Stokes?
sharp smelt
No
jolly parrotBOT
sharp smelt
hmm
sharp smelt
I don't see how this helps
rough forge
where are you at
sharp smelt
Still stuck
rough forge
yea what step, any workings you can show?
sharp smelt
I haven't been able to make any progress
rough forge
yet you started somewhere
wet osprey
Still stuck
What do you normally do
sharp smelt
spend a few hours on it
sharp smelt
$\int_{a}^{b} ( y(t),x(t)) \cdot ( x'(t), y'(t))dt$
rough forge
dont forget dt
wet osprey
Space at the end
jolly parrotBOT
What a wonderful world !
wet osprey
Ok and that gives you what
sharp smelt
$\int_{a}^{b} [y(t) x'(t) + x(t) y'(t) ]dt$
jolly parrotBOT
What a wonderful world !
wet osprey
Keep going you don’t need me to bug you for every line
glacial sequoia
won't this be faster with stokes' theorem
sharp smelt
$\left[ y(t)x(t) \right]{a}^{b} - \int{a}^{b} y'(t) x(t)dt + \int_{a}^{b} x(t) y'(t) dt$
wet osprey
What the
jolly parrotBOT
What a wonderful world !
wet osprey
What the beans
wet osprey
Sir please look at this
sharp smelt
**What a wonderful world !**
This works too though, does it not
wet osprey
Yeah but it’s so much more to write and I also just don’t know where you’re pulling shit from
rough forge
prob ibp
rough forge
because we live in a wonderful world
wet osprey
I mean it’s the same I suppose
But one is definitely more clear than the other with what you’re doing
glacial sequoia
grown ups don't IBP
rough forge
ok but dont close the channel finish it
sharp smelt
so I have $\left[ y(t)x(t) \right]_{a}^{b}$
rough forge
i wanna see how you apply alpha(a) = alpha(b)
jolly parrotBOT
What a wonderful world !
rough forge
well done
rough forge
you're welcome
wet osprey
I haven't been able to make any progress
This is sometimes a problem I see with people
They aren’t willing to yolo if they don’t immediately see the full solution
sharp smelt
Yeah, that's me. I'm sorry
wet osprey
Don’t be sorry be better
sharp smelt
.close
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short orchid
guys, does the formula of arc perimeter (theta*r) only works when the theta is in radian?
toxic lichen
yes
if you want to find it with theta in degrees, it's L = pi/180 * theta * r instead
short orchid
if you want to find it with theta in degrees, it's L = pi/180 * theta * r instea...
ohhhh i seeee
short orchid
if you want to find it with theta in degrees, it's L = pi/180 * theta * r instea...
thank you soo muchh
.close
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lucid moth
pearl pondBOT
lucid moth
I dont understand the question, it says g_p intersects the parabola, what parabola?
timber tartan
gl hf
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lucid moth
pearl pondBOT
lucid moth
I solved all the questions, im just wondering about the 'Extra Item'
What are they asking me to do?
Like yeah sub in p = 4 and p = 6 then what?
coarse harbor
Like yeah sub in p = 4 and p = 6 then what?
Find the roots of the polynomial in the numerator
lucid moth
(x-2)(x-3)
lucid moth
(x-2)(x-3)/2(x-2)
So x = 2 is removable
coarse harbor
Yeah you can cancel out one factor
lucid moth
And x = 3 is removable for p = 6
coarse harbor
Yep
lucid moth
Is that all they were asking?
coarse harbor
I suppose yes
lucid moth
Is it just me or is that question really really vague?????
coarse harbor
Because those are the only p values when g_p becomes just a linear function
coarse harbor
Is it just me or is that question really really vague?????
It's indeed vague
It's more of a "research this" question
pearl pondBOT
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lunar osprey
pearl pondBOT
ivory wasp
there no root for x/y ?
compact ridge
sqrt(x/y) + sqrt(y/z) + sqrt(z/x) > 3 by AM-GM though
so you just need to prove that x/y - sqrt(x/y) > -1/2
well let u = x/y and hence u > 0
lunar osprey
Wait a sec
lunar osprey
And to solve these types of inequalities
Should I create perfect squares out of them or what?
It gives me $u^2-u+0.5>0$
jolly parrotBOT
Heisenberg
lunar osprey
@compact ridge
compact ridge
It gives me $u^2-u+0.5>0$
yeah, well you just need the vertex of that
u = -b/(2a)
to find the minimum value of the quadratic
compact ridge
And if it works it works for every u?
yes
you want this inequality to be true for every u
it's true for all real u but actually we just need it to be true for all real u > 0
compact ridge
to find the minimum value of the quadratic
yeah and you should get u = 1/2, min value = 1/4
pearl pondBOT
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keen jolt
can someone please help me define "literal equations"
pearl pondBOT
can someone please help me define "literal equations"
@keen jolt Has your question been resolved?
mystic wraith
guys
can someone explain the last question
pearl pondBOT
guys
can someone explain the last question
mystic wraith
first question gradient is 7
2nd questione quation is y=7x-15
can someone tell me why is my 3rd question wrong
my working out is
(y+5/2) = -1/7 (x-5/2)
7y+ 35/2 = -x + 5/2
14y + 35 = -2x + 5
14y = -2x - 30
how is that wrong
feral leaf
my working out is
(y+5/2) = -1/7 (x-5/2)
7y+ 35/2 = -x + 5/2
14y + 35 = -2x + 5...
What's the midpoint of AB?
mystic wraith
What's the midpoint of AB?
uh
idk
why is that needed
ohhh
i alreayd slved that
ye midpoint is (5/2 , -5/2)
light helm
your midpoint is incorrect
mystic wraith
its incorrect?
light helm
how did you get that
mystic wraith
2+3/2 and -1+6/2
oh shit
no its not negative
ohhhhhhh
cuz my midpoint was wrong thats how my equation was wrongg
okay that makes sense it shouldve been 40 instead of 30 then
feral leaf
cuz my midpoint was wrong thats how my equation was wrongg
Yeah that's what I was pointing out. Sorry for the delay
mystic wraith
and where do i start for this
ive found that the equation of the line AB is y=10x+1
i think
do i do the line equation again
cuz i know gradient is 10/3
so (y+14)=10/3(x+2)
light helm
how are you getting 10 for the gradient
light helm
howw are you getting 10/3
mystic wraith
howw are you getting 10/3
wait but my thinking is right right
how do i even state that they are collinear
light helm
finding the equation passing through two,
and showing that the third lies on that line shows that they're collinear
as stated in the question
mystic wraith
okay so if i found the equation of AB
light helm
plug in point C and show that it satisfies the equation / ( you get something true)