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#help-49
safe onyx
dawn dagger
Haha no problem!
safe onyx
.close Thank you!
midnight plankBOT
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safe onyx
midnight plankBOT
safe onyx
I said that the surface is
$z\le3-x-y\left{0\le y\le3-x\right}\left{0\le x\le3\right}\left{z\ge0\right}$
grand pondBOT
smeagol
safe onyx
oh I think I forgot to curl F
brb
$F\left(x,y,z\right)=(x+y^{2})i+(y+z^{2})j+(z+x^{2})k$
grand pondBOT
smeagol
grand pondBOT
smeagol
dawn dagger
,w curl((x+y^2),(y+z^2),(z+x^2))
grand pondBOT
safe onyx
got it now yay
I had messed up and forgot that I needed the curl
thank you!
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neat silo
Can someone explain why is the inductive step like that
midnight plankBOT
Can someone explain why is the inductive step like that
neat silo
especially when the problem turns into (x^7 + 1)/(x + 1)
proven slate
especially when the problem turns into (x^7 + 1)/(x + 1)
find the relation between $7^{7^n}$ and $7^{7^{n+1}}$ first i think
grand pondBOT
looklikesomethingulike
neat silo
$7^{7^{n+1}} = \left(7^{7^{n}}\right)^{7}$
wait
$7^{7^{n+1}} + 1 = \left(7^{7^{n}} + 1\right)^{7} + 1$
proven slate
the main idea is just factorizing the remaining components so that it will have at least two new prime divisors, am i saying wrong? (nice factorization btw)
the only thing i was wondering about the solution is if these two components are just a power of the same prime number
(so there will be only one new prime divisor)
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blissful rock
,how would I go about solving this true or false? Sorry if it’s a stupid question, it’s grade 11 level work where I live
midnight plankBOT
,how would I go about solving this true or false? Sorry if it’s a stupid questio...
lethal path
,how would I go about solving this true or false? Sorry if it’s a stupid questio...
do you know what the distance formula is?
blissful rock
Yes
lethal path
cool, so what would be the distance between (2, 4) and (-3, 2), using the formula?
I'll put it up here to help you
blissful rock
Okay wait I see now lol, I read the question kinda wrong
lethal path
ahhh was it a reading issue? I see
blissful rock
Yes, thank you
lethal path
yeah okay, so basically the problem-solving approach is:
- find the 3 distances using the distance formula
- check if their sum is sqrt(33)
blissful rock
Alrighty thank you!!!
lethal path
cool if you're done type .close
blissful rock
.close
normal furnace
how to solve this using cosine rule?
midnight plankBOT
how to solve this using cosine rule?
slender walrus
have you made a sketch of the rhombus
normal furnace
no not yet im a bit confused on visuallising and sketching it
slender walrus
try start with drawing a rhombus
slender walrus
not ideal but its enough to do the problem
normal furnace
yes sorry im drawing this using a mouse
slender walrus
draw in that diagonal (that doesn't cut those 128° angles) to get triangles
normal furnace
normal furnace
like the two formulas?
oh wait i think i get it maybe
like is it 8^2 + 8^2 - 2* 8 * 8 * cos(128)
like to find the diagonal
slender walrus
looks good
now apply properties of a rhombus to get the other angles
and then same idea for the other diagonal
slender walrus
not quite
normal furnace
wait how since the sides are the same?
slender walrus
i mean yes, but not really what you want
normal furnace
oh ok
slender walrus
you want the other angles of the rhombus
normal furnace
do i need another diagonal cutting the 128 degree triangle
slender walrus
yes, they want the length of both diagonals
normal furnace
will it look like this
so the formula to find the other diagnal is 8^2+8^2 - 2* 8* 8* cos(52)=c^2
c=7.01
slender walrus
yeh. that works
slender walrus
no
slender walrus
there's
Area = ab * sin(C)
for a parallelogram (which is applicable here)
normal furnace
so 14.38*7.01 is ab right?
slender walrus
no
normal furnace
wait what so 8*8?
grand pondBOT
Result:
50.4019
,w 88sin(128 deg)
grand pondBOT
slender walrus
rounding issues
slender walrus
multiple methods can be used
slender walrus
since they ask for the diagonals first, they probs intended for you to use that formula
normal furnace
they probably wanted us to use the generic rhombus area formula
slender walrus
but use the unrounded values from your calc if possible
normal furnace
because it was only covering cosine law
normal furnace
but use the unrounded values from your calc if possible
yes
thank you for your help
slender walrus
np
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tribal quiver
midnight plankBOT
grand pondBOT
k
gaunt nimbus
@tribal quiver
tribal quiver
hi
gaunt nimbus
it is from the theorem
tribal quiver
huh
gaunt nimbus
From FTC2,
[ F(b) - F(a) = \int_a^b f(x) \dd x ]
Since, ${\dv{t} P(t) = P'(t)}$, we can sub them as the ${F}$ and ${f}$ pair. Also, let ${b = t}$ and ${a = 0}$.
[ P(t) - P(0) = \int_0^t P'(x) \dd x]
grand pondBOT
k
tribal quiver
huhh
tribal quiver
y ur adding p0
tribal quiver
y r ur letting b = t and a=0
gaunt nimbus
yes
gaunt nimbus
y r ur letting b = t and a=0
so that we can substitute t
to get P(t) at any time
for example t = 4 here
tribal quiver
mmm
gaunt nimbus
so we will sub 4 in
tribal quiver
ok then
gaunt nimbus
[ P(4) = P(0) + \int_0^4 P'(x) \dd x]
grand pondBOT
k
tribal quiver
i thought p4 is just equal to the integral part
gaunt nimbus
trapezoid rule is just one part
tribal quiver
ok i think i got it ty
also i have another question
for this one
i get up to theta is = 32
like up to this basically
(i used degrees not radians formula tho)
and how do u find the value of k from that
tribal quiver
r u sure
gaunt nimbus
there we go
grand pondBOT
k
tribal quiver
all g lol
tribal quiver
yeah so new is 25
gaunt nimbus
so ${r' = (1.25r)}$ and ${\theta' = (1-k)\theta}$
grand pondBOT
k
gaunt nimbus
plugging it in
[ \frac{\pi(\theta)}{360} \cdot r^2= \frac{\pi}{360}(1-k)\theta(1.25r)^2]
some canceling
grand pondBOT
k
gaunt nimbus
[ 1 = (1-k)(1.25)^2]
grand pondBOT
k
gaunt nimbus
,w 1 = (1-k)(1.25)^2
grand pondBOT
tribal quiver
oop back
gaunt nimbus
?
tribal quiver
can u explain it this way
grand pondBOT
k
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
midnight plankBOT
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chilly cobalt
hi im back with more horrible fes
midnight plankBOT
hi im back with more horrible fes
chilly cobalt
Find all functions $f : \mathbb{N}\rightarrow\mathbb{N}$ satisfying $\ f^{abc-a}(abc) + f^{abc-b}(abc) + f^{abc-c}(abc) = a + b + c \ \forall a,b,c \geq 2$
grand pondBOT
Copter
chilly cobalt
also i dunno if f(1),f(2) are findable are not
i will crash out if my sol is completely wrong 🙏
chilly cobalt
can someone check if this is correct? ty in advance
woeful turret
bro what is this question
chilly cobalt
can someone check if this is correct? ty in advance
<@&286206848099549185>
hidden thorn
can someone check if this is correct? ty in advance
It is correct
hidden thorn
also i dunno if f(1),f(2) are findable are not
f(1),f(2),f(3) are never used, so they can be anything in N.
midnight plankBOT
@chilly cobalt Has your question been resolved?
young jolt
Give me a method without using (dy/dt)/(dx/dt)
midnight plankBOT
Give me a method without using (dy/dt)/(dx/dt)
young jolt
I can't do that long
Is there any short way ?
This is a parametric form of something?
steel crest
solve one variable for sqrt(cos(2t))?
young jolt
What do u want to find
Derivative
steel crest
Give me a method without using (dy/dt)/(dx/dt)
probably lookingfor dy/dx
young jolt
Find the derivative find dy)dx
silent zodiac
Uhhh
desert siren
just write y=cot^3 t * x and do it
young jolt
log differentiation
Ahh I think it will make it more uglier , because it's already very ugly
young jolt
just write y=cot^3 t * x and do it
How did you reach that conclusion?
desert siren
How did you reach that conclusion?
eliminate sqrt cos2t
steel crest
are you evaluating at a point or you need a general answer
silent zodiac
Is this like a jee question?
young jolt
eliminate sqrt cos2t
Oh yes I can do that , but I have to eliminate t 🥲
steel crest
eliminate sqrt cos2t
i gave that suggestion , student didn't like it i think
desert siren
sadge
steel crest
Oh yes I can do that , but I have to eliminate t 🥲
what, why
young jolt
I already did it but I can't eliminate dt easily
silent zodiac
Oh yes I can do that , but I have to eliminate t 🥲
U should be afraid of everything
It’s math
steel crest
lol
young jolt
U should be afraid of everything
Yeah
silent zodiac
Like u should either practice more
desert siren
I already did it but I can't eliminate dt easily
u can
steel crest
The question asks you to eliminate t
were you planning on telling us the rest of the question or is it a secret
steel crest
so why differentiate at all?
young jolt
desert siren
bruh
young jolt
Yeah 🥲
desert siren
it says without eliminating the parameter
steel crest
WITHOUT
desert siren
why did u not say that
gaunt nimbus
LMAO
desert siren
gaunt nimbus
[ \frac{\dd y}{\dd x} = \frac{\frac{\dd y}{\dd t}}{\frac{\dd x}{\dd t}}]
grand pondBOT
k
desert siren
u cant eliminate the parameter and u want a shorter method 
silent zodiac
u cant eliminate the parameter and u want a shorter method <:Prayge:844257517529...
😱😱😱
steel crest
you have 3 ways
the original problem
eliminate sqrt cos 2t
and log differentiate
choose
young jolt
Actually these all are the parametric form of something like the first question is the parametric form of parabola so I thought this is a parametric form of something that I don't know
silent zodiac
I think he wants a way so u can directly imagine the derivative 
young jolt
eliminate sqrt cos 2t
Yeah okay
steel crest
not all parameteric systems can be written without the parameter
young jolt
I think he wants a way so u can directly imagine the derivative <:monkey:6518168...
I always look for approaches , I can easily do it by that long method but I thought that this could be a parametric form of something which will make it shorter
young jolt
not all parameteric systems can be written without the parameter
Yeah
steel crest
I always look for approaches , I can easily do it by that long method but I thou...
but the instructions said not to do so 😭
silent zodiac
but the instructions said not to do so 😭
It’s just a normal exercise
Doesn’t matter how u do it
young jolt
but the instructions said not to do so 😭
Yeah 🥲
silent zodiac
it’s the basics book 
young jolt
It’s just a normal exercise
I was doing cengage then my teacher told me to think about this question now I'm overthinking it 😂
young jolt
it’s the basics book <:ded:1042959877519200359>
Ha
silent zodiac
I was doing cengage then my teacher told me to think about this question now I'm...
Think abt it
well
lemme see
young jolt
Yes I've log and cot^3x I've already tried it , and to eliminate dt I have to go through that hard labour
And I think that is the shortest way to do it because there aren't many ways to do it
silent zodiac
Ur preparing for jee
young jolt
Yep
silent zodiac
U should be
Enjoying the labour
Not run away
Not all questions have a short way
young jolt
🥲 I don't enjoy labour I enjoy smart work but I already did the labour , now I'm finding different approaches
young jolt
Yes
I think this question doesn't have any short way
But thanks for the approaches
.close
chilly cobalt
$A$ and $B$ play the following game with a polynomial of degree atleast 4:$$\ x^{2n} + a_{2n-1}x^{2n-1} + ... + a_1x + 1 = 0 \$$ $A$ and $B$ take turns choosing a real number for one of $a_i$ until all are chosen. If the resulting polynomial has no real roots, then $A$ wins. Otherwise $B$ wins. If $A$ begins first, who has the winning strategy?
midnight plankBOT
$A$ and $B$ play the following game with a polynomial of degree atleast 4:$$\\ x...
grand pondBOT
Copter
chilly cobalt
not an fe this time woo!!
anyways can someone check this solution
thanks in advance!
edgy plaza
anyways can someone check this solution
✅
midnight plankBOT
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twilit field
A sequnce is called zero-havy if there exists $M \in |N$ such that for all $N \in \N$, there exists $n$ satisfying $N≤n≤N+M$ satisfying $x_n=0$
\
is $(0,1,0,1,\dots)$ zero -heavy
midnight plankBOT
A sequnce is called zero-havy if there exists $M \in |N$ such that for all $N \i...
grand pondBOT
wai
twilit field
so a couple of questions
one if I understand it right, M is fixed, right
- so for every natural number there exists a fixed neigbourhood consisting of atleast one zero term
nova yoke
yea M is fixed
basically if you take any segment of length M+1, there's a zero in that segment
twilit field
actually even M=1 works since the inequality has <= on both sides
hmm, yea
This feels like denseness of some kind
like essentially a zero-heavy can only converge if it's ultimately zero
nova yoke
no i don't see why that's true
for example: 1, 0, 1/2, 0, 1/3, 0, 1/4, 0, ... is zero-heavy (with M=1 like your sequence) and converges to 0
twilit field
oops, I should have said if the non-zero terms converge to 0
that's what I meant my abd
&bad
*bad
ugh typos
nova yoke
ah yea that should be correct
twilit field
Lemme try proving that now
midnight plankBOT
@twilit field Has your question been resolved?
twilit field
Theorem : If a sequence is zero -heavy, then it converges to 0.
\
We prove that if a sequence doesn't converge to $0$, then it's not zero -heavy
\
Let the sequence be $(a_n)$, let its limit be $a$.
\
We then have $\forall \varepsilon>0 \exists N >0 : n≥N \implies \abs{a_n-a}<\varepsilon$
\
However, as $(a_n)$ is zero-heavy $\exists M \in \N \forall N \in \N$ , there is a $n$ in $N≤n≤N+M$
\
Let the sequence be zero-heavy
\
we let $\varepsilon = \abs{a}{2}$
we then have that for all $n$ greater than some $N$ , $\abs{a_n-a}<\frac{\abs{a}}{2}$. But as it's zero heavy, there is some $N'≥N$ such that $a_n=0$
\
We thus have $\abs{a}<\abs{a}{2}$, a contradiction.
\
Thus if a sequence doesn't converge to zero it's not heavy and equivalently if a sequence is zero-heavy it converges to $0$
grand pondBOT
wai
lavish venture
**wai**
do you mean if a convergent sequence is zero heavy then it converges to zero?
twilit field
do you mean if a *convergent* sequence is zero heavy then it converges to zero?
yes
my bad
lavish venture
yea it needs to be a convergent sequence
fallow scarab
Oh I missed convergent sequence
twilit field
Doesn't the example (1,0,1,0,...) contradict your theorem
yea, so convergent, my bad
lavish venture
like you said "let the sequence be zero heavy" after saying it’s zero heavy
and it should be a_{N’} on the fourth line from the bottom
and \varepsilon = \frac{|a|}{2}
also you said you’re proving by contrapositive but then assumed it’s zero heavy as if you were doing a direct proof
lavish venture
note that if you do it directly you can’t take epsilon = |a|/2 since this may be zero and we need epsilon > 0
twilit field
I suppose making the contrapositive a proof by contradiction can save it
lavish venture
just do contradiction. assume the sequence is convergent and zero heavy but it doesn’t converge to zero that’s all you need
then you have |a|/2 > 0 so you’re fine and can do what you did
twilit field
why not what I did
lavish venture
because it just makes it confusing and is inconsistent
nova yoke
@twilit field do you know the theorem that if a sequence converges to some limit L, then every subsequence must also converge to L?
twilit field
<@796939110815629322> do you know the theorem that if a sequence converges to so...
Iyea
*yea
nova yoke
if so, then your desired result is almost immediate from that
lavish venture
true
fallow scarab
Balls on yo wires traus
midnight plankBOT
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Available help channel!
buoyant linden
just got a q that asked me to complete the left side of the graph, knowing it’s an odd function
midnight plankBOT
just got a q that asked me to complete the left side of the graph, knowing it’s ...
buoyant linden
it’s contradictory isn’t it
buoyant linden
what is 1 and 2
intercepts
grand pondBOT
k
gaunt nimbus
use this
buoyant linden
yeah but u cant just apply that
gaunt nimbus
why not
buoyant linden
because then you’d get
(0,2) and (0,-2)
and see how that makes it not a function
it becomes a relation
u get what i mean, right?
lavish venture
yep
lavish venture
odd functions need to go through the origin f(0) =-f(0) implies f(0) =0
buoyant linden
odd functions need to go through the origin f(0) =-f(0) implies f(0) =0
or
arent defined at x=0
lavish venture
sure
buoyant linden
alr
cuz it was an entry test and i js wrote that the sol can’t be found
and then i wrote “bad question”
thanks
.close
quaint shoal
Is this the correct graph?? I don’t see how I’d graph this, how do I appplt the H and K values?
midnight plankBOT
Is this the correct graph?? I don’t see how I’d graph this, how do I appplt the ...
quaint shoal
horizontal shift
they didn’t do it in the photo did they?
they just applied the stretch of 2?
black is parent graph I believe
gaunt nimbus
they just applied the stretch of 2?
yes
vocal inlet
that's what the table on the right suggests
quaint shoal
yes
yea that’s what I thought
so then after I find the stretch I can just do the vertical and horizontal translations?
gaunt nimbus
the stretching is somehow exaggerated tho
gaunt nimbus
so then after I find the stretch I can just do the vertical and horizontal trans...
ye
quaint shoal
ye
okay that’s what I thought but I just wanted to make sure I wasn’t missing anything
thanks
.close
stark gale
can any one do my sparx maths
midnight plankBOT
can any one do my sparx maths
stark gale
for me
stark gale
hellp
midnight plankBOT
The purpose of this server is to help you learn, not to hand out answers. Do not ask someone to give you the answer directly.
midnight plankBOT
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molten bay
R+---R+ how can this function is commutative binary operation?
midnight plankBOT
R+---R+ how can this function is commutative binary operation?
azure oracle
i don't think I understand you question
midnight plankBOT
@molten bay Has your question been resolved?
midnight plankBOT
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Available help channel!
viral dagger
midnight plankBOT
viral dagger
can someone help me on how i would wtite this
viral dagger
tldr, if AB=1 then the equation holds, we can scale the diagram as we wish so we can scale it to where AB=1, since the equatiom is homeogenous then AB/BP must be constant
also in retrospect i shouldve said something with the fact that the min and max is just a reflection on the perp bisector of AB
cause AB=-golden ratio doesent make sense
midnight plankBOT
@viral dagger Has your question been resolved?
vocal moon
@viral daggerproduct of the length of segments?
viral dagger
<@750992393502720010>product of the length of segments?
yea
tawdry laurel
nice question
vocal moon
Yu
fossil moth
tldr, if AB=1 then the equation holds, we can scale the diagram as we wish so we...
WLOG, we can assume AB = 1 as we only care about the ratio, which is preserved under scaling
viral dagger
and the ratio of AP×BP=AB^2 wont change after scaling?
viral dagger
this method of assuming something is 1 can actually work right
fossil moth
yes
viral dagger
do you eben need to actually
fossil moth
well it makes the problem easier
midnight plankBOT
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molten bay
Z91 is isomorpic to?? In modulo multiplication operation
midnight plankBOT
Z91 is isomorpic to?? In modulo multiplication operation
midnight plankBOT
Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.
lyric charm
molten bay
Is it not isomorpic to z6×z12?
molten bay
Yes. I know you told me early this morning. Actually I was explaining the question to someone else and he is not agree with line of isomorphic
lyric charm
Z_91 and Z_7×Z_13 are isomorphic as rings and therefore their unit groups are isomorphic also.
molten bay
Is it not isomorpic to z6×z12?
He suggested
lyric charm
and with the question of if $\bZ_{91}^\times$ and $\bZ_6^\times \times \bZ_{12}^\times$ are isomorphic, it's not immediately obvious to me.
grand pondBOT
Ann
lyric charm
can i see your convo with your friend
molten bay
He only suggested about it
molten bay
I told him in spoken
lyric charm
ok so no record of it
molten bay
And he suggested this thing to me
lyric charm
Z_6^* = {1, 5}, it only has two elements
but Z_91^* has no element of order 2
so idk what your friend was cooking
molten bay
and with the question of if $\bZ_{91}^\times$ and $\bZ_6^\times \times \bZ_{12}^...
If they are isomorpic then why it doesn't work for us
molten bay
He is pointing out to me that operation is modulo multiplication
Let's see what he says. I have pointed out your point to him
Z6-->{1,5}
lyric charm
i don't want us 3 to play broken telephone
molten bay
Yoo he is confused now. And he will call me this evening
Last message was...is it not that group where we take GCD 1
woeful turret
Dude is always asking qs for hus friends
midnight plankBOT
@molten bay Has your question been resolved?
chilly cobalt
Given any set $S$ of positive integers, show that at least one of the following two assertations holds: $\$ 1) There exists distinct finite subsets of $F$ and $G$ such that $$\ \sum_{x \in F}\frac{1}{x} = \sum_{x\in G}\frac{1}{x} \$$ 2) There exists a positive rational number $r <1$ such that $$\ \sum_{x\in F}\frac{1}{x} \neq r \$$ For all finite subsets $F$ of $S$
midnight plankBOT
Given any set $S$ of positive integers, show that at least one of the following ...
grand pondBOT
Copter
chilly cobalt
no idea where to start ;-;
fresh sparrow
Given any set $S$ of positive integers, show that at least one of the following ...
the second assertion is weird
isn't that always true
midnight plankBOT
Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.
chilly cobalt
you sure it's \neq?
yep
runic hamlet
well if S = Z, then every number can be written as a sum like that
chilly cobalt
one sec ill go through my thingies
runic hamlet
and presumably thats also true for other sets S
austere nova
Hi, sorry for this simple problem but I really am confused if the domain of this rational function (2x+1)/(x²+1) is (+-1) or all real numbers
austere nova
oh sorry
chilly cobalt
**Copter**
yep this is worded correctly
OH
well like
if S is finite then 2) holds
fresh sparrow
maybe they mean "for every"?
yeah it's that
jaunty canopy
the second assertion is weird
this is the original
chilly cobalt
else then S is infinite
chilly cobalt
this is the original
woah its from imosl??
jaunty canopy
yeah 2018 ISL
haven't done the problem, i can have a look later today when i'm done with work
chilly cobalt
yeah it's that
sadly no💔
fresh sparrow
i see it now woops
chilly cobalt
well like whats the idea behind this
jaunty canopy
no idea where to start ;-;
ummm as to where to start, maybe try creating a subset S such that (1) is not true
and then see why that forces (2) to be true
fresh sparrow
(you've already basically done the other way around)
chilly cobalt
aaaa
sps 1), 2) false, then let there be 2 numbers b < a and there exists a finite subset F of S such that sum(x in F) 1/x = 1/b - 1/a
If 1/b - 1/a < 1/a, it implies a is not in F so we can pick another distinct subset FU{a} and both sums are equal
if we want 1) to be false too then 1/b - 1/a >= 1/a => a >=2b
blehh
what do i do with this ;-;
oh well then
it follows that S = {a_1,a_2,...}where a_(i+1) >= 2a_i
crashes out
what if equality
yawn
chilly cobalt
what if equality<:cat_thonk:586238610957926430>
assume FTSOC that a(i+1) > 2ai for every i, then there exists some rational number r in the interval (2/a(i+1), 1/ai) such that it satisfies 2).
also a(i+k) > 2^k ai
since r < 1/ai, we have that r< 1/ai + 1/a(i+1) + ...
r < 1/ai (1 + 1/2 + 1/4 + ...)
r < 2/ai contradiction since we chose r in the open interval
then a(i+1) = 2ai for all i
i MIGHT be on something here
THEN S = {k,2k,4k,8k,...}for any integer k
no
which implies
any sum of x in F is of the form (p/2^q)(1/k) for some integer q,p
F is finite, so we pick r = 1/3k not in S, contradicting the fact that we chose 1,2) as false
so then atleast one of 1),2) are true
scratches head
<@&286206848099549185>
absolu cinema
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jaunty canopy
sorry i have an internship over the summer, this question does look pretty cool and it is a question that i wanna try, i'll have a look when i'm done working for today
chilly cobalt
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vernal escarp
vernal escarp
Question 8th pls
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dawn pine
not sure how to approach these problems, ive tried using law of cosines to get angle E but still dont know where to go from there
midnight plankBOT
not sure how to approach these problems, ive tried using law of cosines to get a...
@dawn pine Has your question been resolved?
heady orchid
you just find the ratio since they similar
then use law of cosines
this should help
now use law of cosines to find <D and <F
then u can find the length EG
by using sin
sinF = x/26
or sinD = x/24
i got an ss of what i got when you need to verify (i think it’s right, i hope so)
i mean it is an answer choice
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dawn pine
i got an ss of what i got when you need to verify (i think it’s right, i hope so...
ok lemme try
dawn pine
by using sin
wait we dont know if its a right triangle so i dont think we can use trig
oh nvm i got it by just using law of cosines a bunch
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heady orchid
oh nvm i got it by just using law of cosines a bunch
this also works
what did u get btw
i js realized i made a huge assumption 😭
dawn pine
20.73
rounded to 20.7
i basically used law of cosines to find angle E and F
G is midpoint of DF so FG is 14
EF is 26
then just used law of cos again to find EG
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narrow ivy
Can someone help me
midnight plankBOT
Can someone help me
marble epoch
You know that the answer will be of the form $y=ab^x$ for some constants $a$ and $b$ which you need to find. You have two pairs of points that you can substitute in to get $12=ab^2$ and $6=ab^3$. You can now solve this system of questions to find $a$ and $b$.
grand pondBOT
@marble epoch
marble epoch
Hint: ||make a the subject of each equation and then equate them||
narrow ivy
Could you just tell me the answer please
marble epoch
Could you just tell me the answer please
But then you don't learn anything
Also, I just worked through it and got the same answer as you
fallow scarab
!nosols
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As a helper, please do not give out answers that could be copied as a homework solution. Have the student work through the problem themselves and guide them along the way.
fallow scarab
Could you just tell me the answer please
!noans
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The purpose of this server is to help you learn, not to hand out answers. Do not ask someone to give you the answer directly.
marble epoch
Sorry
narrow ivy
Also, I just worked through it and got the same answer as you
Yeah, I don’t know why it keeps saying it’s Wrong
marble epoch
Maybe there's a mistake in the question
The second point looks a bit too low on the graph
narrow ivy
Ok thank you for trying
west iron
The second point looks a bit too low on the graph
yeah thats my thought lol
that point is at (3,4) not (3,6)
Or maybe the table displayed is for their function and not the actual point
It does say "for your use"
So
You know that the answer will be of the form $y=ab^x$ for some constants $a$ and $b$ which you need to find. You have two pairs of points that you can substitute in to get $12=ab^2$ and $4=ab^3$. You can now solve this system of questions to find $a$ and $b$.
grand pondBOT
Dreyuk
narrow ivy
that point is at (3,4) not (3,6)
Yes, this helped. Thank you very much.
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quaint shoal
how would I get a hole here?
midnight plankBOT
how would I get a hole here?
quaint shoal
I tried to factor it again and got it to this point
still don’t get a hole from here though 
quaint shoal
how would I get a hole here?
Here’s all the work if that helps
surreal moon
how would I get a hole here?
x/x would still have a hole at x=0
quaint shoal
x/x would still have a hole at x=0
Does that come from dividing the factored out x’s?
quaint shoal
I tried to factor it again and got it to this point
so that would mean this is wrong?
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quaint shoal
how would I get a hole here?
<@&286206848099549185>
surreal moon
Does that come from dividing the factored out x’s?
What
surreal moon
I tried to factor it again and got it to this point
Yeah this looks wrong
civic lynx
yes - because the denominator becomes 0 at x=0, but you can "remove" it by cancelling, then it is a hole
if the denominator is 0 at, say, x=c, and you cannot remove it by cancelling, then it is a VA
so x = -4 is still a VA
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quaint shoal
so x = -4 is still a VA
ohh I see
I think I’ll be good with these for now
I forgot how to do all these stuff from 2-3 weeks ago already🥀🥀🥀
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quaint shoal
Guys is there an easier way to do this 😭
midnight plankBOT
Guys is there an easier way to do this 😭
quaint shoal
9^16 can’t be right lol
civic lynx
\begin{enumerate}
\item would be best to write $\log_9\big((3p+9)^4\big)$ to avoid confusing it with $[\log_9(3q+9)]^4$
\item you can do something else with the 4 instead of bringing it into the exponent
\end{enumerate}
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haseeb
quaint shoal
\begin{enumerate}
\item would be best to write $\log_9\big((3p+9)^4\big)$ to avo...
ohh I can just divide it?
that’s A LOT easier 
civic lynx
alternately, when you get to $(3p+9)^4 = 9^{16}$, you can try to "get rid of" the exponent instead of expanding it out
grand pondBOT
haseeb
civic lynx
i.e. raise each side to the power of $\frac{1}{4},$ or $\sqrt[4],$ and get $3p+9 = 9^4$. but here, you need to show that $3p+9 \geq 0,$ because it's an even root
grand pondBOT
haseeb
civic lynx
but the first way is easier ;)
quaint shoal
i.e. raise each side to the power of $\frac{1}{4},$ or $\sqrt[4],$ and get $3p+9...
thats what I kinda wanted to do but got stuck mid way 😅
quaint shoal
but the first way is easier ;)
yeah I’ll stick to that
civic lynx
yeah, if you find yourself expanding $(a+b)^4$ for a 5 point question, stop and recognize your teacher is not sadistic
grand pondBOT
haseeb
civic lynx
sounds good, lmk if it works out
quaint shoal
yeah, if you find yourself expanding $(a+b)^4$ for a 5 point question, stop and ...
🤣
got it
quaint shoal
sounds good, lmk if it works out
=2184. easy with diving it out thankss for the help haseeb!!
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chilly cobalt
Let $n$ be a positive integer, and consider a sequence $a_1,a_2,...,a_n$ of positive integers. Extend it periodically to an infinite sequence $a_1,a_2,...$ by defining $a_{n+i} = a_i$ for all $i\geq1$. If $$\ a_1 \leq a_2 \leq ... \leq a_n \leq a_1 + n\$$ And $$\a_{a_i}\leq n+i-1 \$$ for $i = 1,2,...,n$ Prove that$$\ a_1 + ... + a_n \leq n^2$$
midnight plankBOT
Let $n$ be a positive integer, and consider a sequence $a_1,a_2,...,a_n$ of posi...
grand pondBOT
Copter
chilly cobalt
no idea where to start
and also the a_{ai} is kinda trippy for me
<@&286206848099549185>
well the bottom proof is equal to proving $a_j \leq n + j -1$ for all $i \leq n$
grand pondBOT
Copter
chilly cobalt
and its equivalent to proving that each of a_i is a permutation of {1,2,..n}? or something
pine wave
no idea where to start
Maybe try small n and see what sequences you can find
chilly cobalt
🤔
chilly cobalt
well the bottom proof is equal to proving $a_j \leq n + j -1$ for all $i \leq n$
wait is it possible to contradiction this
zealous heart
Maybe starting to define for each integer ( j \in {1, 2, \dots, n} ), define:
[
c_j := # \left{ i \in {1, \dots, n} \mid a_i = j \right}
]
In other words, ( c_j ) is the number of times the value ( j ) appears in the sequence ( (a_1, a_2, \dots, a_n) ).
grand pondBOT
Clint
chilly cobalt
proof by contradiction
zealous heart
Then
$\sum_{i=1}^n a_{a_i} = \sum_{j=1}^n c_j \cdot a_j$
grand pondBOT
Clint
zealous heart
By summing the inequality ( a_{a_i} \leq n + i - 1 ) over all ( i = 1, 2, \dots, n ), we get:
[
\sum_{i=1}^n a_{a_i} \leq \sum_{i=1}^n (n + i - 1)
]
grand pondBOT
Clint
pine wave
next time you can try the forum if you need it to stay between sessions
proud violet
I have something but it's not a complete solution and might be a red herring. Firstly, observe by the division algorithm that for each $a_i \in {a_1, \dots, a_n}$, $a_i \equiv r_i (mod n)$ for some unique $0 \leq r_i < n$ and by the condition that $a_{n+i} = a_i$ for all $i \geq 1$, $a_{a_i} = a_{r_i}$ for each $i=1,2,\dots,n$. Hence, as $a_{a_i} \leq n + i - 1$ for each $a_i$, we have $a_{r_i} \leq n + i - 1$ for each $r_i$ (this one could be useful). Moreover, by the inequality, $a_1 \leq a_2 \leq \cdots \leq a_n \leq a_1 + n$, each $a_j \in {a_1, a_2, \dots, a_n}$ satisfies $a_j \leq a_1 + j$ (this one might also be workable).
grand pondBOT
Ellie
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molten bay
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molten bay
What is proper fraction less than 100?
I didn't get it properly
Are they asking me to sum
1/100+1/99+1/98+....1
golden comet
a proper fraction is a fraction that is less than 1
1/100, 2/100, 3/100, 4/100, 5/100..........99/100, 1/99, 2/99............1/2
molten bay
So they are asking such sum?
molten bay
1/100+2/100+3/100+....99/100+
1/99+2/99+.....98/99
1/98+2/98+......97/98
99/2+98/2+97/2+..
Ohh that's easy one
lyric charm
the sum of ALL proper fractions with a denominator of 100 or less
including duplicates like 4/100 vs. 2/50 vs. 1/25?
golden comet
I think those count
lyric charm
like does this question mean $\sum_{j=1}^{100} \sum_{i=1}^{j-1} \frac{i}{j}$ or what
grand pondBOT
Ann
lyric charm
it is impossible to tell if this was meant or not
molten bay
golden comet
ok I searched up, the definition of a proper fraction is ANY fraction where the numerator is less than the denominator (both positive)
molten bay
lyric charm
black band : image ratio = 3 : 1 😭
molten bay
black band : image ratio = 3 : 1 😭
Left one was in hindi so i cropped
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gusty sky
Hey can anyone help me with physics i have some doubts regarding friction
midnight plankBOT
Hey can anyone help me with physics i have some doubts regarding friction
gusty sky
grand pondBOT
gusty sky
If the fbd's look like this then why doesn't the A block start moving first
And then B block and then C block
This is the next page of the illustration in case you need it
olive matrix
well in reality they'll all move at once
but yes you're pulling on C, which is pulling on B, which is pulling on A
gusty sky
but yes you're pulling on C, which is pulling on B, which is pulling on A
So why wont A move first then ?
hushed mauve
because friction can't act on something that's stationary
friction can't oppose motion where there's no motion to oppose
gusty sky
because friction can't act on something that's stationary
Static friction ?
hushed mauve
static friction is the force you need to overcome to get something to start moving in the first place
static friction does not cause an object to move
otherwise, everything will remain in motion
gusty sky
So till C moves no friction will begin acting on it from B then ?
hushed mauve
yes, because the primary force acts on C
once C starts moving all of them will begin to move
gusty sky
So order would be first C begins to move then B begins to apply friction to C then that causes B to move then A applies friction on B which then causes A to move right ?
hushed mauve
C moving means ALL three will start to move at once
but in what order they will start slipping is a different story
gusty sky
but in what order they will start **slipping** is a different story
Please elaborate 🙏
hushed mauve
if there is an infinite amount of static friction, none of them would slip, agreed?
gusty sky
Yes
hushed mauve
but for a limited amount of static friction, whichever block experiences a force that can overcome its static friction first will start slipping first
for example, if B is super dense and heavy compared to A, A will slip first even if B is touching C
also i need to correct a few statements that i made earlier, apologies
hushed mauve
friction can't oppose motion where there's no motion to oppose
static friction does oppose impending motion, but static friction itself cannot cause an object to move
hushed mauve
because friction can't act on something that's stationary
so this is also invalid
gusty sky
static friction does oppose impending motion, but static friction itself cannot ...
But then how is B moving doesn't static friction from C to B gets B moving ?
Ohh nvm thats kinetic friction
hushed mauve
But then how is B moving doesn't static friction from C to B gets B moving ?
ok maybe i should add frames of reference
static friction cannot cause an object to move from the frame of reference of the object it is resting on
but yes, you are right that from an inertial frame of reference, static friction between C and B causes B to get pulled along with C
gusty sky
static friction cannot cause an object to move **from the frame of reference of ...
Okk I think I got it
Thank you very much for your help
gusty sky
i can clarify if you don't
Please do so 😅
hushed mauve
ok
if you were standing by the side, watching C get pulled
to you, C and B moved together, and B's motion is very accurately justified by static friction
but let's say I am sitting on C
to me, B doesn't move
at least, not until F gets large enough
so static friction prevents B from moving against C until opposed
but you saw it moving
this is the difference the frame of reference makes
my frame of reference is non-inertial (another way to say that i'm moving)
gusty sky
at least, not until F gets large enough
So until the acceleration of both the blocks are same till then from C frame im not moving but when F gets sufficiently large enough then due to there being a maximum value of friction for B block B looks like it's deaccelerating from C frame
hushed mauve
So until the acceleration of both the blocks are same till then from C frame im ...
from C's pov, B will not be moving until F is large enough, then B will start to move backwards
but from an inertial pov, B will always be moving. when it slips, B will start to slow down and slip further back along C
gusty sky
Okk tysm I finally got it
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viral dagger
can someone help me with this i dont really know much about vectors
midnight plankBOT
can someone help me with this i dont really know much about vectors
viral dagger
im guessing O would be a good origin? considering P,Q,R,S (the ones we want to do with) have something with O in them
im stuck on finding the position vectors of P and Q
wait is it like somehow getting the midpoint of AD using position vectors, then position of P is 2/3 of the position of midpoint of AD?
call the midpoint E
how are you supposed to get position of E wsp O 😭
vivid yoke
wait is it like somehow getting the midpoint of AD using position vectors, then ...
Yeah it's a way
civic gazelle
vec(OE) = 1/2 (OC+OD)?
viral dagger
oh is e=(a+d)/2
viral dagger
vec(OE) = 1/2 (OC+OD)?
C??
civic gazelle
oh E is the midpoint of AD
My bad
but it does mean getting the vectors of OS and OR would be very complex
so there must be a smarter way
viral dagger
is the reasoning cause something something diagonals get bisected in a parallelogram?
civic gazelle
ye
viral dagger
p=(a+d)/3
q=(b+c)/3
im guessing the relation between r and a-b is that they are perpendicular?
ok how is PQ perp RS <=> s(a-b)=r(d-c) what the hell
which theorem do i not know
vivid yoke
<@1265924904969043979> Can you help him with this
Nahh, I'm watching cartoon man
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civic gazelle
coming back to this
chilly cobalt
ok how is PQ perp RS <=> s(a-b)=r(d-c) what the hell
is this not just dot product
viral dagger
is this not just dot product
teach me
chilly cobalt
hol up i havent read the qn😭
civic gazelle
ok how is PQ perp RS <=> s(a-b)=r(d-c) what the hell
because p-q is basically ((a-b) + (d-c))/3
rare maple
civic gazelle
we need (p-q)(r-s)=0
<=> ((a-b)+(d-c))(r-s) = 0
Expand and you'll get s(a-b) = r(d-c)
viral dagger
you can just manipulate vectors as if it were regular variables??
civic gazelle
uhh somewhat
there have been moments I did very weird things with vectors and got wrong results
chilly cobalt
whats your definition of vector
viral dagger
ok how did they get s(a-b)=ad-bc
sa-sb=...?
viral dagger
whats your definition of vector
arrow
viral dagger
(s-c)(b-d)=0
sb-sd-bc+bd=0
s(b-d)=bd-bc
(s-d)(a-c)=0
sa-sc-ad+cd=0
s(a-c)=ad-cd
we want sa-sb+bc-ad=0
sa-sc-ad+cd-sb+sd+bc-bd=0
(sa-sb+bc-ad)+(-sc+cd+sd-bd)=0
so cd+sd=bd+sc?
chilly cobalt
arrow
also this
which is mostly the same identities you see with regular variables
chilly cobalt
fuck cursive man i aint (cant) read that
just ignore the text gng
viral dagger
just ignore the text gng
as if i know what the symbols mean
isnt |p> used for quantum mech?
vivid yoke
can someone help me with this i dont really know much about vectors
Well, I solved this using similar triangle and dot product:) ||s.a=r.d and sb=rc||
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wind oxide
@viral dagger u still need help?
viral dagger
Well, I solved this using similar triangle and dot product:) ||s.a=r.d and sb=rc...
hows u get that?
ROA=SOAD
vivid yoke
||So OA.OS=OA.(OH+HS)=OA.OH + OA.HS=OA.OH and the same thing for OR.OD=OD.OK since OHD and OKA are similar triangles we have OA.OH=OD.OK||
vivid yoke
ngộ nhận à
Ko a
civic gazelle
vcl vector
wind oxide
can someone help me with this i dont really know much about vectors
this is much easier via vectors
viral dagger
||So OA.OS=OA.(OH+HS)=OA.OH + OA.HS=OA.OH and the same thing for OR.OD=OD.OK sin...
how does them being simmilar triangles imply they are the same
vivid yoke
how does them being simmilar triangles imply they are the same
uh..I don't know what you want to say when u said "imply they are the same"
viral dagger
im asking why they would be in the first place
does simmilar triangles work with vectors aswell?
OA/OD=OK/OH
OA×OH=OK×OD
does this really work?
civic gazelle
does simmilar triangles work with vectors aswell?
only in a few cases
vivid yoke
only in a few cases
Yeah
civic gazelle
in this case cos(AOH) = cos(KOD)
viral dagger
oh if your purely working with lengths its true add the fact that the cos are equal you get the cross product is equal?
civic gazelle
yeah
viral dagger
ohh
viral dagger
sa-sb=rd-rc
s(a-b)=r(d-c) which is supposed to equal to ad-bc?
im guessing we want ad=rd somehow? but is that really true
vivid yoke
im guessing we want ad=rd somehow? but is that really true
uhh, I don't think so but why we need ad=rd
viral dagger
guessinh
vivid yoke
We have S and R is orthocenters but we only make us of OS and OR and any point lie on OS and OR also have the same properties we have to some how make use of CS,SO and BR,AR. that's what I thought of while solving this problem
viral dagger
hm
CS×SO=CS×(SH1+H1O)=CS×SH1
wait BR and AR?
BR×AR=AR×RK
if you do the same as before dont you need OSH1 sim to ORK1
or something like that
vivid yoke
I label H,H1,K,K1 so I don't have to work with R and S but still have its properties
igtg sorry
viral dagger
damn alr ty
honestly why does this look so complicated for a single hint
or am i just that terrible (yes) at vectors so this is long and tedious
worldly pine
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pulsar dagger
What in the name of god is that
lyric charm
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6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
2
lyric charm
9x - 5x = 9
this comes from?
worldly pine
ratio.. (5 from 7:5 and 9 from 7:9)
lyric charm
ok but x represents?
lyric charm
i will be honest i don't see what meaning this x has at all
it's not clear to me and maybe not clear to you
i can share my method of doing it if you want
worldly pine
yes you can..this type of questions are only 2 in the sheet..
I can take this pill
worldly pine
it's not clear to me and maybe not clear to you
7 and 7 are same in liquid A because liquid A is not changed..but liquid B
lyric charm
ok here is my strategy:
- we take 9 L of mixture from the vessel. the portion we take has the same ratio of A to B as the original mixture, i.e. 7:5. How much of A and B have we taken away?
- we replace 9 L of mixture with 9 L of pure B. this means that however much A was in the removed portion, we gained that much pure B.
- the fraction of B in the vessel has gone from 5/12 to 9/16. by how much did it increase?
- you know how much B was gained and what fraction of the vessel this gain is. find the volume of the vessel.
- find the initial volume of A.