how do you know that all the triangles are congruent
#help-26
forest bloom
vivid spire
Diagonals of a rectangle bisect each other and are not orthogonal unless it is a square.
Not all triangles are congruent
forest bloom
Yeah but it is in the question right?
vivid spire
Only the pairs
shut belfry
@forest bloom
you could prove that
you know
you have equal sides
and the opposite sides are parrallel
forest bloom
i don't mind about the diagonal part anymore
vivid spire
It just says that the area of the triangles is equal
shut belfry
think how can you prove them congruent now
forest bloom
oh
shut belfry
i don't mind about the diagonal part anymore
ik
just prove the triangles congruent
you got 4 triangles
think which pair can you prove congruent
you have sides
you have angles
as opposite sides are parallel
forest bloom
as opposite sides are parallel
i don't see it
one is an obtuse angle
and one is an acute angle
the diagonals are bisected
so at least the side side part is okay
i just need to show an angle or a side is equal as well to show congruence
but neither is
shut belfry
i don't see it
abef is rectangle
forget thi big triangle
focus on 4 small triangles
shut belfry
i just need to show an angle or a side is equal as well to show congruence
you have opposite sides equal
alternate interior angles will be qual
find these in the fig
you are done
thorn elk
hi, can i have a quick chat with you @shut belfry if you're free?
forest bloom
you have opposite sides equal
okay but you're showing that the two opposite triangles are congruent in that sense right (but didn't we already know that?)
unless i'm missing something
sorry for dumb drawing but i just sketched
the other two angles are obtuse
there is not a single angle that is obtuse on the two green triangles
knotty ledge
Hes saying e.g. that ABO and OFE are the same
Or wait what are you even trying to show anymore
knotty ledge
okay but you're showing that the two opposite triangles are congruent in that se...
If you already know this then what are ya doin
topaz sinewBOT
@forest bloom Has your question been resolved?
forest bloom
Or wait what are you even trying to show anymore
i'm trying to show all those 4 triangles in the rectangle are congruent
ocean terrace
they are not congruent, but they have the same area
forest bloom
yeah but i'm showing BOE is congruent to AOF and also AOB and EOF are congruent
forest bloom
they are not congruent, but they have the same area
oh lol
ocean terrace
was that all lol
knotty ledge
Yeah the opposite ones are congruent
ocean terrace
um, just take a 2 by 1 rectangle for instance and you can calculate their lengths
of each triangle when you draw a line through the diagonals
forest bloom
um, just take a 2 by 1 rectangle for instance and you can calculate their length...
okay then
i'm guessing this area bisecting thing
holds for all parallelograms and rhombuses?
shut belfry
holds for all parallelograms and rhombuses?
you are not clear witht he property of quads are ya?
ocean terrace
a rhombus is a parallelogram
forest bloom
you are not clear witht he property of quads are ya?
maybe
i don't see why i should've known the area bisecting thing
i don't find it on google even lol
forest bloom
a rhombus is a parallelogram
okay the area thing holds?
forest bloom
ocean terrace
wait ill b back i worked it out wrong haha
so the opposite triangles are congruent in a parallelogram
due to that we can tell that the main diagonals intersect each other at the mid point
using the fact they intersect at the mid point, the areas of two adjacent triangles using the same main diagonal as the base, can be observed to have the same height. and the same base due to the diagonal intersection being the mid point. thus all 4 triangles have the same area
and rhombus is a parallelogram with the additional property that all lengths are equal,
rectangle is a parallelogram with the additional property that all angles are right angles,
so as the property holds for parallelograms, it automatically holds for rhombuses and rectangles.
shut belfry
due to that we can tell that the main diagonals intersect each other at the mid ...
YES
but they should have found that on their own
ocean terrace
eh, they didnt so give them a hand
forest bloom
using the fact they intersect at the mid point, the areas of two adjacent triang...
owhh yeah i get it now
thanks
forest bloom
but they should have found that on their own
i just asked about this?
i don't even have an assignment on this
forest bloom
how do you know that all the triangles are congruent
I initially asked this too, but you were trying to prove to me something that wasn't even the case
Anyway thanks regardless i guess
.close
ocean terrace
I initially asked this too, but you were trying to prove to me something that wa...
. haha
trail lake
Every vector space will have null space
topaz sinewBOT
Every vector space will have null space
trail lake
Addition elements are (0,0,0) so
ocean terrace
whats the question?
shut belfry
I initially asked this too, but you were trying to prove to me something that wa...
I had told to think of two triangles which you can prove congruent
sweet shard
I had told to think of two triangles which you can prove congruent
!occupied
topaz sinewBOT
I had told to think of two triangles which you can prove congruent
Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #βhow-to-get-help for instructions).
trail lake
whats the question?
Am I right?
sweet shard
Addition elements are (0,0,0) so
Do you know the definition of vector spaces?
ocean terrace
Am I right?
what about?
drifting swift
"every vector space will have null space" is either poorly worded or just nonsensical
trail lake
Vector spaces needed null space to fulfill addition property
So addition identity is 0
So null space will be subspace of every vector space
Am I right?
<@&286206848099549185>
Entire vector space and null space will be surely subspaces of every vector space
.close
topaz sinewBOT
topaz sinewBOT
β
drifting swift
!1c
topaz sinewBOT
Please stick to your channel.
drifting swift
@trail lake reopening the channel still makes it have your name on it
trail lake
.close
topaz sinewBOT
pure estuary
The null space is based on the linear mapping. Suppose I have a finite dimensional vector space with a basis $v_1, \ldots, v_n$. If T is a linear mapping from V to V, I can define $Tv_k = v_k$ for $k = 1, \ldots, n$. In this case, what would nullT be?
trail lake
I am not familiar with the basis yet
I have read eigen values eigen vector and started vector spaces
pure estuary
Okay, either way, any linear map $T$ will map $0$ to $0$. So at minimum, the null space of $T$ is ${0}$, which is a subspace of every vector space.
trail lake
Yes that's what i was talking for
pure estuary
Great
topaz sinewBOT
Available help channel!
slow flower
yo
slow flower
barren salmon
??
barren salmon
are they all correct?
yup
topaz sinewBOT
@slow flower Has your question been resolved?
restive inlet
a bit lazy with notation
you should aim to write those thetas
and also use appropriate () to indicate numerators and denominators of fractions as needed if you don't use horizontal fraction lines
terse trellis
Need help with this one π§
topaz sinewBOT
Need help with this one π§
terse trellis
I was thinking to work with ln(β¦)
ionic oar
ln also works
neon iron
[
\lim_{n \to \infty} \p{1 + \f xn}^n = e^x
]
thorny flameBOT
ionic oar
but the limit of e is probably sleeker
neon iron
well it's something you could use here instead of l'hopital which is what neon is going for
terse trellis
well it's something you could use here instead of l'hopital which is what neon i...
Im not too sure on how to apply this to my lim tho
neon iron
Need help with this one π§
[
\f x{2+x} = 1+ {???}
]
thorny flameBOT
forest bloom
nice
terse trellis
Okay that seems to work indeed
But its another rule id have to remember
What about the ln way
thorny flameBOT
jan Nejon
ionic oar
try using this
topaz sinewBOT
@terse trellis Has your question been resolved?
barren salmon
as x tends to infinity
2/x will tend to
0
1^infinity
apply that formula get your answer :))
this is your answer
enjoy
terse trellis
Ok lets see
barren salmon
which?.
terse trellis
Oh you did that to show me that it tends to inf?
barren salmon
Oh you did that to show me that it tends to inf?
tends to 1
(tending to 1)^infinity is an indeterminant form
not 1^infinity
terse trellis
.
barren salmon
yeah
terse trellis
Is .
barren salmon
?
terse trellis
Not exponent
barren salmon
no lol
terse trellis
Lol
barren salmon
Does that look like an exponent
terse trellis
Got me confused for a sec
barren salmon
my bad then
terse trellis
topaz sinewBOT
terse trellis
How did this timeout
barren salmon
Lol
terse trellis
.reopen
topaz sinewBOT
β
barren salmon
it is
terse trellis
Im confused now
barren salmon
terse trellis
π
barren salmon
is it good now
terse trellis
No, still solving that lim
barren salmon
shouldn't take that long
terse trellis
Got it
barren salmon
Yeah
terse trellis
But why take the lcm
Like
^0 is not intermediate form
So i wouldve never even thought about taking the lcm
terse trellis
barren salmon
:)
terse trellis
Zerome limit god
barren salmon
nah these are basic questions lol π
they don't even ask these types of question because its too easy
terse trellis
terse trellis
you'll soon realize these questions don't even require pen
If you have time, give me a limit you think is hard
barren salmon
If you have time, give me a limit you think is hard
I'll give one right now
split cloak
If you have time, give me a limit you think is hard
You won't be able to do it... Hard means it's hard
you can do medium i guess
terse trellis
I just wanna see what one looks like lol
vale furnace
||C||
barren salmon
But i will be 25% right
Ahah
vale furnace
barren salmon
its a good conceptual question though
barren salmon
is it C <@758008621299793970> ?
let me check
its a multiple choice
answer
i think
ivory sorrel
where is this from? Looks interesting
barren salmon
AY YO
vale furnace
barren salmon
<a:FeelsMeowTastyBusiness:1191996109065830451>
what did you do
ivory sorrel
a previous year paper
jee adv?
barren salmon
i took me a whole page
vale furnace
you get some integrals shenanigans
barren salmon
but ultimately, f is inc in (0,1) and dec otherwise
how
barren salmon
I didn't write anything down
yes please
vale furnace
Dont wanna
It's just riemann sum stuff
[\rb{\4{f'(x)}{\map fx} = \ln(\4{x+1}{x^2+1})}]
thorny flameBOT
Snow Leopold von Winterbranch
barren salmon
It's just riemann sum stuff
what is riemann
wtf
it took you 5min
wow
it took me way too long
vale furnace
barren salmon
<a:FeelsMeowTastyBusiness:1191996109065830451>
i had to use summation of series
using integrals
and then
it took a whole ass page
damn this looks helpful
what's the concept name?
barren salmon
have a nice day
vale furnace
You too
merry mason
I'm confused about the range of a quadratic x^2+2x+2
topaz sinewBOT
I'm confused about the range of a quadratic x^2+2x+2
merry mason
Is it [0, infty) or [-2, infty)
coarse tusk
Neither
merry mason
Ohh
neon iron
minimum value will always be the vertex
merry mason
I tried completing perfect square
neon iron
so, find the vertex
merry mason
(x+2)^2-2
neon iron
or just use the fact that [
x_{\t{ver}} = -\f b{2a}
]
thorny flameBOT
merry mason
or just use the fact that \[
x_{\t{ver}} = -\f b{2a}
\]
I have not studied that yet sorry
merry mason
Ohh
coarse tusk
Yes
topaz sinewBOT
topaz sinewBOT
Available help channel!
trail lake
topaz sinewBOT
trail lake
Hints please
drifting swift
this looks like a modified version of the Fresnel integral
trail lake
How to solve it?
neon iron
How to solve it?
like u would for any other integral
trail lake
What next?
trail lake
this looks like a modified version of the Fresnel integral
How to solve it? Or i have to remember it?
topaz sinewBOT
@trail lake Has your question been resolved?
topaz sinewBOT
@trail lake Has your question been resolved?
trail lake
.close
thick rune
topaz sinewBOT
thick rune
Hey what am I supposed to be doing to find the asymptotic distribution
I had tried
Setting the expression that leads to central limit theorem equal to
F(x) and making sn the subject
But tbh I have no idea what the process is or what Iβm actually trying to find
topaz sinewBOT
@thick rune Has your question been resolved?
topaz sinewBOT
thick rune
.reopen
topaz sinewBOT
β
topaz sinewBOT
@thick rune Has your question been resolved?
terse pier
The sum of the iid variables itself diverges; and after normalization taking the sum minus the mean and divided by "variance", you should just have the standard normal distribution with mean 0 and variance 1; i suppose they are asking you to express the normalization formula in terms of p and n?
pastel salmon
.reopen
$\text{you have to reach such a form: }\\\sum_{i=1}^{n}X_{i}\sim \mathcal{N}\left( n\mu,\sigma\sqrt{n} \right)\text{ , where}\\\mu=E\left( X_{i} \right)\text{ }\text{, and }\text{ }\sigma^{2}=Var\left( X_{i} \right)$
thorny flameBOT
Joanna Angel
pastel salmon
and to obtain it, it is enough to apply this:
In probability theory we call this the de MoivreβLaplace theorem, which is a special case of the central limit theorem, stating that under certain conditions the normal distribution can be used as an approximation of the binomial distribution.
thick rune
**Joanna Angel**
what do i have to equate my expression to get the expression for SN
Sn
since i kinda just have the expression chilling there not equal to anything
pastel salmon
do you agree on this:
$\sqrt{n}\left( \overline{X_{n}}-\mu \right)=\frac{\sqrt{n}}{n}\cdot \left( \sum_{i=1}^{n}X_{i} -n\mu\right)=\\=\frac{\sum_{i=1}^{n}X_{i}-n\mu}{\sqrt{n}}\overset{d}{\longrightarrow }\mathcal{N}\left( 0,\sigma^{2} \right)$
thorny flameBOT
Joanna Angel
pastel salmon
no, it is absolutely mandatory then
now you have to think how to transform this random variable into such one, you get N(0,1)
thick rune
um when square root n by p equal 1 i think
based on dividing the second term by square root n
pastel salmon
$\sqrt{n}\left( \overline{X_{n}}-\mu \right)=\frac{\sqrt{n}}{n}\cdot \left( \sum_{i=1}^{n}X_{i} -n\mu\right)=\\=\frac{\sum_{i=1}^{n}X_{i}-n\mu}{\sqrt{n}}\overset{d}{\longrightarrow }\mathcal{N}\left( 0,\sigma^{2} \right)\text{ }\text{ therefore}\\\frac{\sum_{i=1}^{n}X_{i}-n\mu}{\sigma\sqrt{n}}\overset{d}{\longrightarrow }\mathcal{N}\left( 0,1 \right)$
thorny flameBOT
Joanna Angel
thick rune
why is it 1 when it is divided y standard deviation @pastel salmon
pastel salmon
why is it 1 when it is divided y standard deviation <@1035542975503138836>
that is known theorem , I could name as : linear transformation of continuous random variable
thick rune
that is known theorem , I could name as : linear transformation of continuous ra...
noted, thank you
pastel salmon
$\text{ }\text{ let: }\text{ }Y=aX+b,a\neq 0\text{ }\text{ then:}\\f_{Y}\left( y \right)=\frac{1}{\left| a \right|}\cdot f_{X}\left( \frac{y-b}{a}\right)$
thorny flameBOT
Joanna Angel
thick rune
So for part b
for the uncorrelated sequence part
i had submitted this cause i coulfnt. figure how to actually prove the rest to be 0
or finish b all together
i was told about some proof by recursion thing
but i didnt rlly understand the proof
would u be able to help me? @pastel salmon
topaz sinewBOT
@thick rune Has your question been resolved?
thick rune
.close
leaden sparrow
topaz sinewBOT
leaden sparrow
In the diagram, the blocks are stacked onto one another
I got two values for F1, (f1 is labelled as the friction force)
One was from miu*R (124N) and one was from resolving mg (100N)
The answer says that 100N<124N so 100N is the right answer but I dont understand why
neon iron
whats the mass of m1 and m2?
neon iron
block A and B are placed on an inclined plane?
They are lil tilted so..
sterile river
One was from miu*R (124N) and one was from resolving mg (100N)
Miu.R is the maximum static friction...after which block starts sliding....the one you got from equation is the real friction when real friction is greater than max static...block starts sliding
leaden sparrow
block A and B are placed on an inclined plane?
They are lil tilted so..
Yeah, inclined at 20 degrees
leaden sparrow
Miu.R is the maximum static friction...after which block starts sliding....the o...
I got real friction is less than max static, so does that mean the block is not sliding?
sterile river
Yes
leaden sparrow
So if it asked for the static friction, I'd give 124?
sterile river
So if it asked for the static friction, I'd give 124?
Maximum static that is...cuz 100N here is also static friction
leaden sparrow
Ohh Okay,
This is the markscheme
I'm confused why it says if 100<124, the actual value is just 100N for friction
sterile river
Because 124 is the required theoritical friction which when applied as an external force...will result in sliding of block..but here in this case external force is 100N only so it cant cause sliding
leaden sparrow
Ohh, I get it now
If F1 was larger, would you still put F1 as the actual frictional force?
sterile river
Yes, actual frictional force is always resolved by equations from FBD..while max static is by formula miu. Times normal
topaz sinewBOT
topaz sinewBOT
Available help channel!
burnt widget
how does an object deform due to stress when it's in equilibrium?
vocal willow
topaz sinewBOT
vocal willow
can someone explain
on the last line
why they subtracted the inverese laplace transforms?
for refeerence we are solving ODE's using the laplace transform
i think its an error but im not sure
topaz sinewBOT
@vocal willow Has your question been resolved?
vocal willow
<@&286206848099549185>
topaz sinewBOT
@vocal willow Has your question been resolved?
vocal willow
.close
lofty cloud
I have a question i dont understand
topaz sinewBOT
I have a question i dont understand
lofty cloud
"Prove that if the roots of ax^2 + bx + c = 0 (a dont equal 0) are reciprocals then a=c"
i just dont know where to start
sweet shard
"Prove that if the roots of ax^2 + bx + c = 0 (a dont equal 0) are reciprocals ...
Use the definition of roots
And meaning of reciprocals
topaz sinewBOT
@lofty cloud Has your question been resolved?
topaz sinewBOT
topaz sinewBOT
Available help channel!
neon iron
we want to count all possible ways to create a 5 digit number using the digits 1,2,3,4 and 5 only and the numbers 1,2 and 3 occur in their natural order. I have found two methods to approach the problem. Method 1: consider a five digit number. The permutations of the numbers 1,2,3 do not have to be counted(since they occur in their natural order), so the answer would be (5!)/(3!) =20. Method 2: since 1,2 and 3 occur in their natural order, digits 4 and 5 have 4 choices to go to, before 1 or between 1 and 2 or between 2 and 3 or after 3. Hence, the answer would be 4Γ4=16. I don't understand why the answer is different in both methods.
topaz sinewBOT
we want to count all possible ways to create a 5 digit number using the digits 1...
violet sand
so
123XX
for XX u have 5.5=25
X123X
again 25
XX123
@neon iron do u see it?
neon iron
one more thing, repetition is not allowed
neon iron
yes
lucid bridge
we want to count all possible ways to create a 5 digit number using the digits 1...
Got it
Okay so
You have this right
1_2_3
Okay but when u choose one number let's say 4
1_2_4_3 there's now more options
Which is +1
So it should actually be 4*5
Not 4*4
U got it?
@neon iron
neon iron
yea
topaz sinewBOT
topaz sinewBOT
Available help channel!
toxic aspen
topaz sinewBOT
toxic aspen
At the end they say it's +74
but according to my calculation it should be -74
$x = x_0 + \frac{b}{gcd} * n\
y = y_0 - \frac{a}{gcd} * n$
Like this right?
chilly walrus
sgd=gcd?
toxic aspen
yeah sorry made it in Swedish lol
thorny flameBOT
Merineth
toxic aspen
But
y0 = 29
a = 74
so why on gods green earth do they make a be +74 and not -74
.close
grave patrol
I have a question regarding variable substitution.
(sin (x^0,5)) / x^0,5
If I substitute x^0,5 for u, do we not have to substitute both (x^0,5) but instead we can substitute one and use the replacement to cancel out the other x?
That would be the first part of the question. As I remember it I can substitute all (x^0,5) for u, but it seems like I do not have to as long as I remove all (x^0,5)?
grave patrol
I hope this question is readable, please ask me to clarify if it is confusing
visual oracle
Integrals?
grave patrol
Yeah
topaz sinewBOT
@grave patrol Has your question been resolved?
gray ridge
I have a question regarding variable substitution.
(sin (x^0,5)) / x^0,5
If I s...
i think that's the point of u-sub
it is to remove all excess info so that you can have a better form to apply integrating methods
see if you can understand what i mean π
grave patrol
it is to remove all excess info so that you can have a better form to apply inte...
I have always understood it like that I have to replace all (x^0,5) with u, but in the example I saw only one was replaced with the other one removed after the substitution
gray ridge
true, actually both way works as long as everything is clear
grave patrol
true, actually both way works as long as everything is clear
So I can do either ? anywhere I look this is so poorly explained π©
gray ridge
can you show your examples?
do you have a screenshot or picture?
or you can even write your steps down and circle your doubts, i can try to explain
@grave patrol ?
gray ridge
I have to figure out a second example tho : D
it's ok you can write it down first π
take your time
grave patrol
it's ok you can write it down first π
take your time
I hope you can read this lol
gray ridge
reading
grave patrol
So the left side is how I have learned it
But I saw the right side as an example
I understood it as U has to replace all variables that U is replacing
If that makes sense
gray ridge
very clear! I'll try to explain
In my opinion,
what you have learned (left side):
- replace all x with u accordingly inside the integral
pros: - standardized workout
- looks tidy for "easy" substitutions
cons: - hard to see what to be crossed out, you'll have to know what you're trying to eliminate to see what's happening
- sometimes you cannot relate the u's with the remaining x's
What you have seen (right side):
- partially replace varible x with u accordingly (mainly with the "hard to integrate" ones first)
pros: - Making it clearer what is left to be eliminated
- looks tidy for "hard" substitutions
cons: - non-standardized approach
- you need to have clear mind to see which variable is which
I personally prefer the second one because it gives more information and ways to see what's happening and it looks clearer to me
grave patrol
I personally prefer the second one because it gives more information and ways to...
Alright great, I was mostly curious if I was even allowed to do like in the second case, only replacing one variable
This makes my life much easier : D
grave patrol
and i might even do this for the left side one for clearer understanding:
Honestly I don't even understand .. du = 1/2x^0,5 dx
ooh nevermind, I do understand it
Thanks this was really useful for me
placid flicker
anyone please help me with this question-Consider that a, b, c, d are positive real numbers satisfying (a + c)(b + d) = ac + bd.
Find the smallest possible value of S=a/b+b/c+c/d+d/a
topaz sinewBOT
anyone please help me with this question-Consider that a, b, c, d are positive r...
placid flicker
is anyone trying this question
whole geode
Hi
I closed your other channel. Please slow down for a second.
So what have you tried and where are you at with this problem?
!status
topaz sinewBOT
What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
@placid flicker have you heard of Lagrange multipliers? Used in constraint optimization problems? It's a multivariate calculus topic, and probably necessary for this problem.
topaz sinewBOT
@placid flicker Has your question been resolved?
whole geode
In summary, let f(a, b, c, d) be your objective function, the thing equal to S. Let g(a, b, c, d) be your constraint rewritten such that the constraint is satisfied when g(a, b, c, d) = 0, in this case it is g(a, b, c, d) = (a + c)(b + d) - ac - bd. Then we construct a function h(a, b, c, d, Ξ») = f + Ξ»g
Then we make a system of 5 equations:
0 = dh/da
0 = dh/db
0 = dh/dc
0 = dh/dd
0 = dh/dΞ»
Solve the system, and then for each critical point figure out if it's a minimum or maximum. And find the global minimum if it exists.
@placid flicker ^ hopefully you get this message.
rocky stream
What is this symbol called/what is it used for in calculus?
topaz sinewBOT
What is this symbol called/what is it used for in calculus?
rocky stream
Thanks prime Minister.
neon iron
it is a sexy thing though its the bane of existence past hs algebra
rocky stream
It is a nice symbol. Is there anything I need to do to confirm my question has been answered for the system?
neon iron
i guess .close if u are done with ur question
grave pine
topaz sinewBOT
grave pine
can anyone help w this laplace transform im stuck
this is what the answer should be
vernal matrix
,rccw
thorny flameBOT
neon iron
can you not use like
convolution and make your life easier
oh wait you're finding the laplace transform not the inverse
grave pine
oh wait you're finding the laplace transform not the inverse
yes :c
topaz sinewBOT
@grave pine Has your question been resolved?
topaz sinewBOT
grave pine
.reopen
topaz sinewBOT
β
grave pine
grave pine
i tried using this formula
vernal matrix
i tried using this formula
But you have sin(t-1)cos(t), so need the sin(a)cos(b) one, no? 
That one is then one of these ones, which explain their sin(-1) they have
topaz sinewBOT
@grave pine Has your question been resolved?
grave pine
But you have sin(t-1)cos(t), so need the sin(a)cos(b) one, no? <:RooThink:532310...
oh woah
i need sleep
m so sorry
vernal matrix
m so sorry
No need to apologise, it happens 
vernal matrix
i need sleep
Sleep is very important for sure 
grave pine
.close
violet sand
Let ( n \geq 2 ) be a natural number. Determine how many functions ( f: \binom{[n]}{2} \to \binom{[n]}{2} ) exist such that ( f(A) \cap A = \emptyset ) for every ( A \in \binom{[n]}{2} ). (Reminder: the symbol ( \binom{[n]}{2} ) denotes the set of 2-element subsets of ( {1, \ldots, n} ), and if ( J ) is a set, then the symbol ( \binom{J}{2} ) denotes the set of 2-element subsets of ( J ).)
topaz sinewBOT
Let \( n \geq 2 \) be a natural number. Determine how many functions \( f: \bino...
thorny flameBOT
Fate
neon iron
!status
topaz sinewBOT
What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
$\binom{[n]}{2}$ is the set of all unordered pairs of numbers from 1 to $n$
thorny flameBOT
Ann
drifting swift
you want to count the functions from this set to itself such that for any pair {x,y} the pair f({x,y}) doesnt have any elements in common with {x,y}
neon iron
Im pretty sure she knows how to solve it, but the point of this is to solve it yourself, not us.
drifting swift
look at one specific pair
think about how many choices there are for what pair to send it to
thats all i can say bc it's past midnight and i need to sleep
violet sand
What is unordered pair
For example 1,2 us ordered pair, right?
@violet sand @neon iron
deft dew
What are the values of x
in which xβ¬]-β,-3]United with[3,+β[ And p(x)β€xΒ²-x
topaz sinewBOT
What are the values of x
in which xβ¬]-β,-3]United with[3,+β[ And p(x)β€xΒ²-x
deft dew
!status
topaz sinewBOT
What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
1
vernal vale
munzer the question doesnt make sense
are you sure you are interpreting it correctly
can you share the original
vernal vale
i mean what is p
deft dew
Oh
deft dew
i mean what is p
p(x)=|x-3|*|x+3|
deft dew
Yes
vernal vale
Where is $|x-3||x+3| \geq x^2-x$?
thorny flameBOT
jan Niku
deft dew
**jan Niku**
Yes
vernal vale
you should use your answer to the other question
$|x-3||x+3| = $\begin{cases} x^2-9 & |x| > 3 \ -x^2+9 & |x| \leq 3 \end{cases}
thorny flameBOT
jan Niku
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
thorny flameBOT
deft dew
,w graph x-3|*|x+3| >= x^2-x
thorny flameBOT
vernal vale
,w |x-3||x+3| == x^2-x
thorny flameBOT
topaz sinewBOT
@deft dew Has your question been resolved?
topaz sinewBOT
topaz sinewBOT
Available help channel!
robust flame
how do i use minimum and maximum points to determine the equation of a polynomial?
topaz sinewBOT
how do i use minimum and maximum points to determine the equation of a polynomia...
robust flame
2 minimum points of (2,5) and (-3,-7) and maximum point (0,10)
drifting swift
you have six equations that costrain your polynomial:
f(2) = 5
f'(2) = 0
f(-3) = -7
f'(-3) = 0
f(0) = 10
f'(0) = 0
robust flame
how do i then use this to form an equation
drifting swift
well this tells you that you're looking at 5th degree at least
bc six equations would call for six unknowns
the last two allow you to pin down the constant and x^1 terms tho.
robust flame
this is where i dont understand
why the 5th degree, and what does that mean f^5(x)?
drifting swift
no, it does not mean f^5(x).
it means deg(f) β₯ 5.
well
actually hm
let's say deg(f) = 5
robust flame
what does deg mean?
drifting swift
degree
the degree of a polynomial is the highest power of x that appears in it
for example deg(x^3 - x^2 + 11x + 1) = 3
$f(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + a_4x^4 + a_5x^5$ \ a 5th degree polynomial has 6 coefficients: $a_0, a_1, a_2, a_3, a_4, a_5$
thorny flameBOT
Ann
robust flame
ok
drifting swift
again, 6 equations => probably 6 unknowns.
robust flame
what do we do next?
drifting swift
f(2) = 5
f'(2) = 0
f(-3) = -7
f'(-3) = 0
f(0) = 10
f'(0) = 0
write out these in terms of the a_i
you know how to take derivatives, yes?
robust flame
yes
drifting swift
ok right
then find f'(x) in general
and then for three of the equations plug x=2, x=-3 and x=0 into f(x) and for the other three plug them into f'(x)
robust flame
then find f'(x) in general
how do i do this?
robust flame
i dont know how to find it in general though
@drifting swift does f(-3) give -27a+9b-3c+d?
robust flame
isnt f(x)= ax^3+bx^2+cx+d?
robust flame
so -3-3x-3x^2-3x^3-3x^4-3x^5?
drifting swift
so -3-3x-3x^2-3x^3-3x^4-3x^5?
why did you make all of the coefficients -3???
i didn't say to set the a_i to -3; these are unknown and you'll be solving for these
it is x that should have been set to -3...
robust flame
a-3a+9a-27a+81a-243a
topaz sinewBOT
@robust flame Has your question been resolved?
topaz sinewBOT
@robust flame Has your question been resolved?
wintry wraith
@robust flame do you know how to integrate? If so, I believe you should be able to use that f'(x) factorised by a(x-2)(x+3)(x)
topaz sinewBOT
@robust flame Has your question been resolved?
candid plinth
what is the cartesian form of a force?
topaz sinewBOT
what is the cartesian form of a force?
candid plinth
So for example, in a vector question, is that just splitting the force on X and Y values?
I have a vector question that has 3 forces acting on a point.
not sure how to answer this
pseudo bear
Have you done vectors before?
candid plinth
yeah , i know how to split the vector into its X and Y components and calculate thje total force in X and in Y
pseudo bear
OK, that's what they're asking for.
pseudo bear
Yes.
candid plinth
oh okay , thanks so much?
for part b? what does it mean by polar force then?
is that just saying, the direction as well as the force?
pseudo bear
By polar form, they mean the way it's written above, like 305N<35 degrees.
So, yes, you're right.
candid plinth
i see! so itll be like, 200N at like so and so degrees
pseudo bear
Right.
candid plinth
and for cartesian itll be like
150 N on X axis and like
70 on Y or smth
got it
pseudo bear
Yeah, like (150N, 70N) or something.
candid plinth
your amazing tysm
pseudo bear
You're welcome.
candid plinth
how do i close this
pseudo bear
Say .close
candid plinth
.close
neon iron
topaz sinewBOT
neon iron
How would I do this?
molten vine
How would I do this?
(x+2) is a factor of g(x) <==> g(-2) = 0
neon iron
Yes but how will I show it
They want the entire process
Yk the bus stop method for division
molten vine
just calculate g(-2) ??
neon iron
That but for this long algebraic stuff
molten vine
(x+2) is a factor of g(x) <==> g(-2) = 0
or do u need to prove this ?
neon iron
just calculate g(-2) ??
Oh
molten vine
Factor Theorem states that if Ζ(a) = 0, then the binomial (x - a) is a factor of polynomial Ζ(x).
neon iron
Factor Theorem states that if Ζ(a) = 0, then the binomial (x - a) is a factor of...
Itβs this way tho
Bus stop method
molten vine
you can do long division ( u have to find remainder = 0)
tired ice
Itβs this way tho
Χ©ΧΧΧ ΧΧΧ
neon iron
you can do long division ( u have to find remainder = 0)
Yes
I forgot how to do it tho
molten vine
but calculating g(-2) is way faster and simpler
neon iron
Χ©ΧΧΧ ΧΧΧ
ΧΧ¨ΧΧΧͺ
neon iron
but calculating g(-2) is way faster and simpler
They want to use that exact method so
I mean
π€·π»
Do yk how to do it?
molten vine
i do, but i think u're solving the question the hard way
it says, using the factor theorem and not using long division
u'll use long division in the 2nd question
neon iron
Oh
molten vine
Factor Theorem states that if Ζ(a) = 0, then the binomial (x - a) is a factor of polynomial Ζ(x).
so if g(-2) = 0 then (x+2) is a factor of g(x)
neon iron
And solve
molten vine
So I replace x with x+2
you replace x with -2
molten vine
$g(-2)=-32-48+30+50=0 \implies (x+2)$ is a factor of $g(x)$
thorny flameBOT
Adam Chebil
neon iron
$g(-2)=-32-48+30+50=0 \implies (x+2)$ is a factor of $g(x)$
Yeah I did that
But if I want to do long div
For part C
neon iron
**Adam Chebil**
I wrote that for A
When it asked to use factor theorem
Now itβs asking me to show it
How would I?
molten vine
question b ?
molten vine
yeah do long division g(x)/(x+2)
if u forgot how to do it, search yt to refresh ur memory
it's not so complicated
,rotate
thorny flameBOT
topaz sinewBOT
@neon iron Has your question been resolved?
bleak swallow
topaz sinewBOT
arctic heath
hi
bleak swallow
BF and FC are tangents
arctic heath
im lazy to sleep
bleak swallow
that BD / DC = area of triangle ABF / area of triangle ACF
arctic heath
uh, just write what i type isnt it
neon iron
use that S(ABD)/S(ADC)=BD/DC
neon iron
what can you say about S(ABD)/S(ABF)?
bleak swallow
im not sure
arctic heath
area of tri ABF = area of tri ABD + area of tri BDF
area of tri ACF = area of tri ACD + area of tri CDF
area of tri ABD / area of tri ACD = BD / DC
area of tri BDF / area of tri CDF = BD / DC
let BD = x and DC = y
so area of tri ABD = xn
area of tri ACD = yn
area of tri BDF = xm
area of tri CDF = ym
then area of tri ABF / area of tri ACF = (xn+xm) / (yn+ym) = x(n+m) / y(n+m) = x / y = BD / DC
bleak swallow
they have same angle
arctic heath
like this?
oh, sorry, maria, u explain
; - ;
(continue, maria.
(my proof is not good, it is low level proof
arctic heath
im not sure
bruh you are sure
bleak swallow
i understand that proof
arctic heath
its just BD/DC
bleak swallow
but maybe maria has something better
bleak swallow
arctic heath
use that S(ABD)/S(ADC)=BD/DC
maria said this, see
actually, it's the same as mine (?)
but maybe maria have better form to write the prove
neon iron
no, i think your proof is the same as mine
spice fractal
,w integrate 1/(x^2-4)
topaz sinewBOT
,w integrate 1/(x^2-4)
thorny flameBOT
spice fractal
,w integrate 1/(16-x^2)
thorny flameBOT
stuck forum
You already did it with a^2. Why does it matter if you change it?
spice fractal
You already did it with a^2. Why does it matter if you change it?
Just making sure I understood it right lol
stuck forum
Ah, just replace a
If you can do it for some arbitrary letter, you can do it for any number
wet frigate
M & M
spice fractal
sup pop
thorny flameBOT
spice fractal
,w integrate sqrt(36-x^2)
thorny flameBOT
spice fractal
,w integrate sqrt(x^2-36)
thorny flameBOT
spice fractal
.close
stiff bear
Best crash courses for learning higher-math then current level
topaz sinewBOT
Best crash courses for learning higher-math then current level
@stiff bear Has your question been resolved?
stiff bear
<@&286206848099549185>
gusty bane
topaz sinewBOT
@stiff bear Has your question been resolved?
topaz sinewBOT
topaz sinewBOT
Available help channel!
woeful trout
if anyone knows parabola
topaz sinewBOT
if anyone knows parabola
woeful trout
in tangent equation
parametric form
why is t = 1 / m ?
t is parametric form
m is slope
topaz sinewBOT
@woeful trout Has your question been resolved?
robust jasper
topaz sinewBOT
robust jasper
Math is sometimes a joke
Part B)
So change in displacement in the third second is t=2 and t=3
Because of part A
Or just in general --> (This option I think)
Because t = 0 and t =5
So its in general,
Change in dispalcement in third second is difference of t = 2 and t =3
yeah its in gernal
.close
topaz sinewBOT
topaz sinewBOT
Available help channel!
flint star
$tex help
flint star
.close
topaz sinewBOT
pseudo bear
$tex help
You can check #latex-help.
topaz sinewBOT
Available help channel!
rigid osprey
topaz sinewBOT
rigid osprey
Hello, may I get some help? I don't understand why x=3/2, y=1/2 like where do you see that?!
burnt quartz
Hello, may I get some help? I don't understand why x=3/2, y=1/2 like where do yo...
You don't, since they didn't gave you the equations of the lines. Don't worry about the x=3/2, y=1/2 thing. It's only trying to tell you that there's one solution and one solution only.
What do you need help with on the second screenshot?
rigid osprey
How do I know its that tho
How do I see the solution as the answer?
I still don't get it ;-;
burnt quartz
does the second screenshot come after the first one
rigid osprey
Haha yeah khan academy doesn't tell me
burnt quartz
so to graph the two lines, for example the one with "y = -x + 2", you first need to find two points that are on the line.
for example if x = 1, y = 1.
If x = 2, y = 0.
Once you have these two points, you draw a line between them and that's going to be your line.
rigid osprey
yess
and then..
i think ik how to graph it
I dont know what the fraction doing there..
burnt quartz
I dont know what the fraction doing there..
whcih fraction?
burnt quartz
oh, for that you need to solve the system of equations, so you have two lines
y = -x + 2
y = 3x - 4
you have to find the point where these two intersect, the point they're talking about (3/2, 1/2) is where they intersect
opaque gyro
hi
rigid osprey
oh, for that you need to *solve* the system of equations, so you have two lines
...
Why is it 3/2...
So confusing lol
burnt quartz
Why is it 3/2...
so to get the answer you put the two equations right next to each other, so if you had
y = -x + 2
y = 3x - 4
you put it together like this: 3x - 4 = -x + 2
and then you solve for x
you will get 3/2
opaque gyro
if you figured out x = 1.5, then you can plug x into one equation y=3x-4 -> y=3(1.5)-4 , and solve for y
rigid osprey
if you figured out x = 1.5, then you can plug x into one equation y=3x-4 -> y=3(...
Tyyyy
opaque gyro
np