The correct answer is 64.
I have tried to use b^2 = 4ac
#help-13
rotund vine
cedar kilnBOT
The correct answer is 64.
I have tried to use b^2 = 4ac
cedar kilnBOT
@rotund vine Has your question been resolved?
rotund vine
<@&286206848099549185>
Thanks
rotund vine
<@&286206848099549185>
fervent mason
Let $a = k^l$, $b = k^m$, and $c = k^n$
wraith daggerBOT
Si Arya
fervent mason
The equation $ax^2 - bx + c = 0$ have a solution like this
$$x =\frac{b\pm \sqrt{b^2 - 4ac}}{2a}$$
wraith daggerBOT
Si Arya
worldly iron
The equation $ax^2 - bx + c = 0$ have a solution like this
$$x =\frac{b\pm \sqrt...
no
it has only one solution
so the discriminant is zero
the thing inside the square root is zero
fervent mason
You mean discriminant
fervent mason
Np. Then because the discriminant is equal to 0, then
b² - 4ac = 0 → b² = 4ac
fervent mason
**Si Arya**
Then substitute this
cedar kilnBOT
@rotund vine Has your question been resolved?
teal topaz
How would part 4 be done?
cedar kilnBOT
How would part 4 be done?
tropic oxide
well there's two ways you could go about it
either read off the value at t=2.5 from the graph
or calculate the number of atoms after 2.5 days using the halflife
probably most conveniently done with the formula $N(t) = N_0 \cdot 2^{-t/h}$ (where $N_0$ is the initial amount and $h$ is the halflife)
wraith daggerBOT
Ann
teal topaz
What would the number of atoms be tho?
Because the half life is approximately 3.8 days
teal topaz
by atoms do they mean the number of nuclei?
tropic oxide
atom = nucleus in this context yes
violet flume
use inverse?
like
its not gonna like degrees
lol
but sin^-1 (x)
or arcsin
cas im not sure
wraith daggerBOT
crimson sedge
Uh never heard of this either
violet flume
ive heard arc seconds
but no one uses that here
there are probably inverse versions of these?
tropic oxide
lmao
i mean arc-seconds still find use in navigation dont they
but also seriously? treating the acronym "cas" as standing for a function?
obviously CAS's are magic boxes which you throw your problem into and the answer magically comes out of 
rocky basin
hi im not sure how to do this
cedar kilnBOT
hi im not sure how to do this
teal vault
For a use inscribed angles
Since angle DAB is 79 degrees, what the measure of arc DB?
Angle ADE is 115 degrees, so what is the measure of major arc ABD?
Using those arc measures, you can find the measure of arc AB, and then angle ACB
rocky basin
im not sure how to find the measures of arcs tho
teal vault
Have you learned inscribed angles?
Angle DAB is inscribed in arc DB, so arc DB is double the measure of angle DAB
rocky basin
Have you learned inscribed angles?
i dont think so
teal vault
Hm
I don’t think there’s another way to do it
What have you learned?
That you might think you could use to solve this
rocky basin
well this chapter is about tangents
there's tangent is perpendicular to line from center
there's two tangents are equal provided they're on the same side
and there's angle in alt segment
teal vault
I see
Then probably it has to do something with the angle in alt segment theorem
Although I can’t quite figure it out
rocky basin
-oh wait actually i might have learnt inscribed angles after all
under a different term
teal vault
Oh
teal vault
Since angle DAB is 79 degrees, what the measure of arc DB?
Then could you answer these questions?
rocky basin
not really becos i dont know what the measure of an arc is
teal vault
Measure of an arc is the same as the measure of its central angle
When you use inscribed angles the arc measure is double the measure of the inscribed angle
teal vault
Since angle DAB is 79 degrees, what the measure of arc DB?
So here arc DB would be 79*2
teal vault
No it’s arc measure
rocky basin
uhhhhhhhh ok
teal vault
Arc length is a different thing
So then measure of arc DB is 158
Then using angle ADE can you find the arc measure of arc ABD?
rocky basin
230?
teal vault
Yep
rocky basin
but that angle isnt inscribed
rocky basin
oh ok
teal vault
Here’s an example from geogebra
So then measure of arc ABD - measure of arc DB = measure of arc AB
So what’s the measure of arc AB?
rocky basin
72?
teal vault
Yep
teal vault
You got it
cinder brook
Im quite stuck on question 9. I dont really know how to solve it. If i can get some guidance on it . It would be very appreciated
toxic flame
the two dashes on the lines indicate that they are the same length I believe - this should help
cinder brook
But how do I exactly find out what length they are?
toxic flame
try forming an equation with a variable
gloomy turret
how do i set this up in an equation
cedar kilnBOT
how do i set this up in an equation
gloomy turret
the y co-ordinate is meant to be (0,1)
the second graph is my additions to it (markings and simple labelling)
not sure if its right tho
<@&286206848099549185> pls
alrighty all good
.close
frail scarab
Claim
wraith daggerBOT
frail scarab
How to solve this question?
cedar kilnBOT
@frail scarab Has your question been resolved?
sonic kestrel
I got this question but don't know what to set for the Particular Integral
I tried y = Axe^3x but that didn't work
I think it's because the complementary function is (A+Bx)e^3x
astral crown
for this case you have to use Ax^2 e^(3x)
sonic kestrel
ahh gotcha
astral crown
I think it's because the complementary function is (A+Bx)e^3x
yep
its because of that
granite drum
cedar kilnBOT
fervent mason
Connect O with P and find the relation between ∠POR and ∠PQR
digital oak
I got the ans as 31 kek
granite drum
Uhm
granite drum
Connect O with P and find the relation between ∠POR and ∠PQR
Wuts next
obsidian coral
Apply the concept of central and inscribed angles
granite drum
Dont get it.
granite drum
Nope dont have lmao
granite drum
Ok..
obsidian coral
No
crimson sedge
u can get psr
obsidian coral
You would use inscribe and central angles, to find arc PSR and arc SR
crimson sedge
and u can get SPR too
obsidian coral
What about arc SR?
granite drum
Not sure abt arc sr
obsidian coral
Remember the hints I gave, central and inscribed angles
obsidian coral
i never learned what arc means
Then you don't need to help
granite drum
Remember the hints I gave, central and inscribed angles
Okay what's next tho..
Idk whats arx sr
c*
crimson sedge
i do know how to solve it i just dont know the term
obsidian coral
Apply the concept of central and inscribed angles to find those arcs
You used inscribed angles to find arc PSR
granite drum
Yes
obsidian coral
Can you use central or inscribed angles to find SR?
granite drum
Like which part is sr
obsidian coral
The arc that is SR
granite drum
Teacher never taught us something like arc sr before
obsidian coral
Use your pencil and draw it
obsidian coral
Where would arc SR be, if you drew it?
obsidian coral
lemme draw
Didn't ask you to do the work
obsidian coral
Because I'm trying to teach the person, how arcs are labeled, and they should know how to determine what arc it is
So you don't need to draw it out for them
granite drum
I dont understand, we never need to draw doing these kind of question
obsidian coral
You are confused on where arc SR
granite drum
Yes
obsidian coral
What is an arc?
crimson sedge
ok ill just let u help
granite drum
Cause never learned b4
obsidian coral
What is an arc, like I asked?
obsidian coral
Yes, that's arc SR
granite drum
Ya we never learned that b4
obsidian coral
Can you determine the value of that arc using central or inscribed angles?
granite drum
Calculating the arc
granite drum
Can you determine the value of that arc using central or inscribed angles?
As in "°"?
obsidian coral
No
The key hint is central and inscribed angles
Did you look up what central angles are?
granite drum
Did you look up what central angles are?
Yes i did
obsidian coral
What is a central angle?
granite drum
ROS?
obsidian coral
No, I'm asking conceptually, what is a central angle?
halcyon swan
any help needed?
halcyon swan
No, I'm asking conceptually, what is a central angle?
central angle in a what?
circumcentre?
where's the question
halcyon swan
this is easy
obsidian coral
Yes i did
The relation between central angles and the arc is the arc has the same value length of the angle, correct?
halcyon swan
angle prs is an angle outside the arc sr, so it's an inscribed angle
hence, the answer is 62/2 = 31
huh
no options?
halcyon swan
Because it's not
A translation would be nice to be honest
I think O is actually not the centre anymore
obsidian coral
this is easy
It may be easy for you, but not to the OP, because people are at different learning levels
obsidian coral
A translation would be nice to be honest
It has a translation
granite drum
I still dont get it
Proof that my calculation is correct:
QPR and QRS= 91
PSR=124
PQR=54
halcyon swan
It may be easy for you, but not to the OP, because people are at different learn...
I thought O was the centre and hence answer was 31
granite drum
Add all up, 91+91+124+54=360
halcyon swan
but let me try
granite drum
That's for PQRS
halcyon swan
cyclic quadrilaterals
granite drum
Help plz!
obsidian coral
I still dont get it
The cental angle, that oversees the arc, both has the same measurement, correct?
obsidian coral
cyclic quadrilaterals
Definitely not needed
obsidian coral
Translate the question please
It's on the paper
halcyon swan
Definitely not needed
I just saw the problem right now, that's something I thought of
halcyon swan
oh nevermind sorry
obsidian coral
hence, the answer is 62/2 = 31
Also, FYI, don't give out answers
obsidian coral
The cental angle, that oversees the arc, both has the same measurement, correct?
@granite drum
Knowing that definition, what is the measure of arc SR?
halcyon swan
23 is the answer
obsidian coral
23 is the answer
Don't give out answers
Not the purpose of the server
I'm trying to teach the OP
halcyon swan
Uh what
You are right, giving out answers is not correct but giving out solutions???
what's wrong with that
???
obsidian coral
I'm helping them through the problem
As you can tell
By hinting them in the right direction
halcyon swan
By hinting them in the right direction
Ah, to give them an idea on how to solve it?
great
obsidian coral
By telling them, they need to apply inscribed and central angles
They are confused on how to find arc SR using central angles
halcyon swan
hmmm I see
by the way, is the OP even here?
@granite drum
yeah, they're offline 😂
obsidian coral
Once that obstacle is passed, all they need to see is, PS is the difference between PSR and SR
Then applying inscribed angles, they can find angle PRS
halcyon swan
It's a status
and 80% of the time it's correct
obsidian coral
My status is offline
obsidian coral
Probably got distracted
halcyon swan
Yeah
obsidian coral
The process you need to apply, inscribed angles to find arc PSR, then central angles to find arc SR, then visually seeing, you can see that arc PS is the difference between arc PSR and arc SR. So you can use that to find arc PS, then using inscribed angles, you can find angle PSR
halcyon swan
The process you need to apply, inscribed angles to find arc PSR, then central an...
I think drawing a solution might give the click for him/her to get it
let me draw a rough diagram
obsidian coral
I think drawing a solution might give the click for him/her to get it
If you're going to draw it, I suggest just drawing the arcs and overlaps, and not including values, let the OP figure out the final answer, using your hint and mine
halcyon swan
If you're going to draw it, I suggest just drawing the arcs and overlaps, and no...
Sure!
@granite drum See this
You have angle SOR. and SPR is an inscribed angle of angle SOR
Try to figure out what the angle marked as ? is, in terms of x
cedar kilnBOT
@granite drum Has your question been resolved?
obsidian coral
i got the ans already
Close the channel if you are done
granite drum
.close
echo trench
i'm stuck on this problem
cedar kilnBOT
i'm stuck on this problem
echo trench
I have shown that $H_1\cap H_2$ is a subgroup of $H_1,$ so $(G:H_1\cap H_2)=(G:H_1)(H_1:H_1\cap H_2).$ I don't know how to show that $(H_1:H_1\cap H_2)$ is finite though.
wraith daggerBOT
monkeman
vernal palm
perhaps it is best for this question to go in #groups-rings-fields
you'll probably get a quicker and better response there
echo trench
perhaps it is best for this question to go in <#496784958430380033>
ok thank you
@vernal palm should i close the channel?
vernal palm
sure
echo trench
.close
languid robin
A small island on the Pacific is inhabited by a tribe of logicians, which is made up of 1000 people of different eye colours. Their religion forbids them from knowing thei rown eye colour, and any conversation on the topic. Thus follows that none of the inhabitants know their own eye colour (There are no mirrors on the island). Each inhabitant has perfect sight, therefore they know everyone’s eye colour but their own.
If one of the inhabitants were to learn their own eye colour, the following day they would have to leave the island, be banished forever. No caveats.
Out of the 1000 inhabitants, 100 have blue eyes and the remaining 900 have brown eyes.
Once upon a time, a blue-eyed traveler came to the island which made friends with the inhabitants. One evening, with every inhabitant as a witness he raised a toast saying „I wouldn’t have guessed that in this part of the world I would meet someone who, like me, would have blue eyes!”
What will be the consequences of the toast?
Hint: This is a mathematical induction puzzle
How do I solve this?
cosmic steppe
He gets shot and killed
winged owl
cedar kilnBOT
Available help channel!
brittle field
cedar kilnBOT
cosmic tusk
$-y = -1 \cdot y$
wraith daggerBOT
anamono
cosmic tusk
so if you rearrange the original expression,
$(-y^9 z^6)(9x^2y^2z^4) = (-1 \cdot 9)(x^2)(y^9\cdot y^2)(z^6 \cdot z^4) = -9x^2y^{11}z^{10}$
wraith daggerBOT
anamono
silver hollow
why is ln e 1?
cedar kilnBOT
why is ln e 1?
silver hollow
omg no I get it now
im basically saying
e^x=e
it has to be 1 right?
okokok tysm
.close
rustic ridge
How do I most compactly linearly interpolate between A and A*B with a single additional variable?
I can do something like A(1-t) + AB(t) for t=[0..1] — but I’m wondering if there are cleaner or alternate formulas (eg not requiring the two terms with A?)
crimson delta
well you could factor out the A. so A(1-t+Bt). and then maybe also factor out the t if you want to A(1+t(B-1))
but all of that already seems pretty compact. doubt it's possible do to much better
frigid coral
Derivate x^3-2x
cedar kilnBOT
Derivate x^3-2x
cosmic steppe
Chunkin moment
frigid coral
aight, need to find 2 extremepoints. derivated it and got the result that x = +- 1.22.... but putting it in a graph got it to x=+-1
frigid coral
no idea what they called in english
cosmic steppe
Like local maxima and minima?
bitter reef
critical point or smth?
cosmic steppe
^
bitter reef
f'(x) = 0?
frigid coral
yes
frigid coral
its zero
bitter reef
I guess wasn't that his original question?
cosmic steppe
Yeah
frigid coral
havent done it yet cuz i need the corrext X for it
fair hare
Use the power rule
frigid coral
its 2x^2-3
cosmic steppe
No
fair hare
its 2x^2-3
😳
cosmic steppe
Check your coefficients and constant
bitter reef
y'all spending too much time with power rules and limit definitions and chain rules and whatever whatever
I introduce, Mathematica theorem
cosmic steppe
Useless
bitter reef
anyways $$\frac{d}{dx} (x^n) = nx^{n - 1}$$
fair hare
y'all spending too much time with power rules and limit definitions and chain ru...
The terms are arranged the wrong way
bitter reef
keep this in mind
cosmic steppe
My face when computer
wraith daggerBOT
Doggo
bitter reef
hm interesting the d's looked better this way, I usually use \mathrm
frigid coral
aight so y=-3x + x^3 aint y'= -3 + 3x^2?
cosmic steppe
No
frigid coral
nvm
cosmic steppe
You derived x^3 incorrectly
cosmic steppe
Yeah
frigid coral
.close
cedar kilnBOT
fair hare
You derived x^3 incorrectly
🤨
cosmic steppe
He edited it
dusty hazel
Exactly. Edits are dangerous.
cosmic steppe
It was originally 2x^2 before he changed it to ^3.
cedar kilnBOT
Available help channel!
charred oxide
Hi I need help
cedar kilnBOT
Hi I need help
fair hare
Great
charred oxide
Uh
fair hare
🗿
fair hare
What’s ur question 💀
charred oxide
I'm stuck on 4
No.4 b
I have to find x and y for the matrix
And I wanted to attempt by multiplying then doing simultaneously equation but it sidnt make sense
bitter reef
you either multiply (13; 1) by the inverse you found in a), or you can write down the equation as a matrix and reduce it
crimson sedge
cedar kilnBOT
crimson sedge
Ok
I know how to get the reference angle
but
I dont know what sign to put
because its a large number
and I dont know what quadrant it is in
What do I do?
crystal raptor
Keep taking away 2π until its between 0 and 2π
crystal raptor
Then you'll know what quadrant it's in
crimson sedge
Keep taking away 2π until its between 0 and 2π
I can take out like 18pi?
crystal raptor
What do you mean
crimson sedge
I meant do I only take out 2pi? can I take 3pi or 4pi or more?
nvm my question
I got the point
thanks
.close
tidal hatch
can someone help me find x
cedar kilnBOT
can someone help me find x
crystal raptor
Use ya circle theorems
tidal hatch
.close
cedar kilnBOT
cedar kilnBOT
Available help channel!
wanton bison
I don’t get what do I have to do first? To simplify it
cedar kilnBOT
I don’t get what do I have to do first? To simplify it
upbeat meadow
do you know exponent rules?
vestal bear
I don’t get what do I have to do first? To simplify it
you know how to do fractional exponents right?
cedar kilnBOT
@wanton bison Has your question been resolved?
azure fable
I don't understand how to get to the average time formula
crimson delta
which one? the $T(n) \approx cn^2$? or the $\frac{x\times n^2}{100^2}$?
wraith daggerBOT
Denascite
crimson delta
calculating T(n) depends on the specific algorithm and can be hard
azure fable
I got the part, if it takes x seconds to do 100 then each one takes x/100
where did the squareds come from?
slate token
help
slate token
ancient talon
Just count the boxes lol
slate token
oker
how bout this
count again?
how do i count like
faster
i hav no time
what volum
obsidian coral
You just count
obsidian coral
There's not really that many shortcuts in counting besides counting by twos or threes, etc
slate token
i so dumb in math i dont even know multiplication table
idk if i can still graduate
😂
obsidian coral
i so dumb in math i dont even know multiplication table
Then learn it
obsidian coral
Just count the blocks
obsidian coral
Google it
If you don't know what that is
Google exists
We don't have to Google things for you
slate token
okr
cedar kilnBOT
Available help channel!
midnight zealot
How is this equal to 3^2?
cedar kilnBOT
How is this equal to 3^2?
midnight zealot
For context ^
crimson sedge
Like I am expecting it to be x^3/3
Coz that looks the limit definition of the derivative
So when u differniate a at 3
U will get 3^2
midnight zealot
I'm sorry I don't understand any of that. This is a intro to calc course
crimson sedge
Oh ok
midnight zealot
I did the calculation and I got 1
crimson sedge
Have you learned derivatives?
crimson sedge
Oh
midnight zealot
Yeah sorry
crimson sedge
I don't think there is any way to justify this without derivatives
midnight zealot
Oh, alright fair enough
crimson sedge
,w ((3.001)^3-3^3) / (3*0.001)
midnight zealot
I'll just keep watching and see if I get it later on
wraith daggerBOT
crimson sedge
I'll just keep watching and see if I get it later on
Ya u will require derivatives
midnight zealot
Okay thank you
cedar kilnBOT
@midnight zealot Has your question been resolved?
crimson sedge
cedar kilnBOT
modern compass
what hav eyou tried?
crimson sedge
what hav eyou tried?
so i pluged in 1
for x
so
√1-9
=
√-8
then i plugged in -1
wait
im dumb
wrong question
.
i meant this one
😅
so
1/1^2-1
which is
1/1-1
1/0
1/-1^2-1
oh
this is odd
i just needed to type it out
.close
cedar kilnBOT
cedar kilnBOT
Available help channel!
midnight basin
What does the square mean?
cedar kilnBOT
What does the square mean?
leaden otter
That square generally means Cartesian product of a set with itself
Here it means set of all ordered pairs (a,b) where both a,b are from the set {0,1}
x belongs to the set {0,1}² which means x is one of those pairs I suppose
But then 0x doesn't really make sense for me
cedar kilnBOT
@midnight basin Has your question been resolved?
cedar kilnBOT
@midnight basin Has your question been resolved?
modest storm
Can someone verify that this goes clockwise?
cedar kilnBOT
Can someone verify that this goes clockwise?
@modest storm Has your question been resolved?
modest storm
<@&286206848099549185>
dense hornet
yes it does trace the path out in clockwise direction
dense hornet
.close
modest storm
.close
kind lagoon
cedar kilnBOT
tropic oxide
,rccw
wraith daggerBOT
kind lagoon
Wondering if anyone can check my work there and help me find domain of d(h(i(x)))
Not sure how to go about domain of log base 2 cubed while it’s also a radicand
: (
I hope my writing is legible enough
reef venture
what you've written looks correct
now for g(h(i(x)))'s domain
we know for g(x), x>=0
and h(i(x)) is 27-(log_2x)^3
so 27-(log_2(x))^3 >= 0
solve for x here
reef venture
that's correct
kind lagoon
do i just square?
reef venture
no
reef venture
square would have removed a sqrt
kind lagoon
right, mb
reef venture
so you'd get
x <= 2^3
kind lagoon
im confused at the movement of the logarithm
do you know what log law that is so i can look it up aha
log_2(x) <= 3 becomes
2^(log_2(x)) = x?
reef venture
kind lagoon
oh
reef venture
you understand this right? its just this
kind lagoon
yeah that makes sense
kind lagoon
so 2^3 >=x
reef venture
yes
kind lagoon
8>= x
reef venture
now you can find domain of f by looking at the common domains of g and h and i
kind lagoon
ok, let me think real quick
D of f = (0, 8]
?
😅
seems too simple to be correct
reef venture
Its correct
reef venture
kind lagoon
wonderful
do you do python as well?
I'm taking computing science in my university, and we learned a bit of python
learning*
kind lagoon
I don't know how to say it in english 💀
your english is great dw
reef venture
do you do python as well?
mhm
reef venture
kind lagoon
.close
swift axle
cedar kilnBOT
tropic oxide
you can have radius 6 and opposite 1, or radius 1 and opposite 1/6
it doesn't matter
cedar kilnBOT
@swift axle Has your question been resolved?
charred yacht
i have a 3x5 matrix, the dimension of the col(A) is 2, the dimension of the Nul(A) is 3, do i have enough info to find the dim(Nul(A^T))
cedar kilnBOT
i have a 3x5 matrix, the dimension of the col(A) is 2, the dimension of the Nul(...
@charred yacht Has your question been resolved?
flint plinth
i have a 3x5 matrix, the dimension of the col(A) is 2, the dimension of the Nul(...
yes, dim(col(A)) = 2 tells you that the column rank of A is 2, but row rank = column rank, hence the row rank of A is also 2, and hence the column rank of A^T is 2. From this you can work out the dimension of Nul(A^T)
charred yacht
yes, dim(col(A)) = 2 tells you that the column rank of A is 2, but row rank = co...
ok so whats the rule about this
i feel like i just dont know it
flint plinth
ok so whats the rule about this
which rule?
charred yacht
how does row rank help me find nullity
flint plinth
row rank = column rank?
that's the key theorem here, along with rank-nullity
if you know that dim(Col(A^T)) = 2, you can find dim(Nul(A^T)) using rank-nullity applied to A^T
charred yacht
ok so is dim(col(A)) = dim(col(A^t))
flint plinth
yes, because dim(col(A)) = column rank of A = row rank of A = column rank of A^T = dim(Col(A^T))
where the second equality is the only hard one
charred yacht
ok nullity theorem is what im missing then
ok so quick google search rank(A) + nul(A) = # columns
how does that help me with A^T
flint plinth
and rank(A^T) + nullity(A^T) = # of columns of A^T = # of rows of A
cedar kilnBOT
flint plinth
pleasure
cedar kilnBOT
Available help channel!
rocky wave
Would like some guidance on this question. have some parts but don't know how to fully finish
rocky wave
ik that AD is 6.4, angle BCD is 96 and angles at B and D are 84 but need to find diagonal
tropic oxide
well you know that the two diagonals of a rhombus intersect each other at a right angle and also bisect each other, right?
rocky wave
sounds familiar
thats my working for the bottom question
how do u rotate?
also don't know the next step @tropic oxide
tropic oxide
,rccw
wraith daggerBOT
rocky wave
thanks mate
tropic oxide
yes, the diagonals bisect the angles of the rhombus
rocky wave
ok cool
so now I just find a side
and work the rest out, right?
also, do u mind helping with another question?
tropic oxide
you're overthinking it with this one
rocky wave
probably
rocky wave
fair
here is the other question
am I right in saying that the two triangles are congruent?
which means that the middle line is √2?
fervent mason
Nope
rocky wave
i said that the middle line is common in my congruence check
which is a lie
oh its just sin rule right?
fervent mason
Yes
rocky wave
ok cool
tropic oxide
they are not congruent but what they are is similar
rocky wave
ok ... which means?
fervent mason
Same shape but different size
tropic oxide
you applied the law of sines correctly by the looks of it
rocky wave
ok cool
rocky wave
Same shape but different size
if AB is √2 and according to my maths, BD is 2√3, what does similarity mean here?
@tropic oxide @fervent mason
tropic oxide
similarity means the same thing no matter what values two particular sides may take
but these values let you calculate the scaling factor
which in this case is 2sqrt(3)/sqrt(2), or sqrt(6).
rocky wave
option v is square root btw
so now I just multiply 2√3 by the scaling factor to get x right
if so, ur a legend
if not, im dumb
which should be 6√2 if im doing it right
bc u would get 2√18 which is 2√2x√9 which is 6√2
ah crap its divide
tropic oxide
yes, 6 sqrt(2) is correct
tropic oxide
well then either i fucked up or they fucked up
or you fucked up but i didn't notice, which i'd still count as me having fucked up
tropic oxide
as if i am not
rugged tusk
How many numbers at most can be selected from {2,3,. . . , 25} such that every two number are coprime ?
tight bolt
i suck at this xd
but
from what i think
- you cannot choose any other even number except 2
- choose all the prime numbers in the set
rugged tusk
its meant for pigeonhole principle
tight bolt
ahh
rugged tusk
yes, i have chosen exactly prime numbers
tight bolt
that dirichlet thingy
i have already forgotten how you do these kinds of problems 🤣
the only dirichlet related thing i remember
is how you use it to solve inequalities
sorry
languid imp
cedar kilnBOT
sacred stone
i dont even know where to start tbh
you can calculate each part separately
$\tan(90^\circ - \theta) = \cot \theta$
wraith daggerBOT
秋水
sacred stone
$\sin^2 x+ \cos ^2 x=1$ and $\sin(90^ \circ -x) = \cos x$
wraith daggerBOT
秋水
sacred stone
what ab the sinˆ2 25+sin^2 65
so it's 1
languid imp
so it's 1
so whats gonna happen if i multiply the tan together in the end?
bc addition makes it 1, what ab multiplication
sacred stone
so whats gonna happen if i multiply the tan together in the end?
wdym?
sacred stone
$\tan(90^\circ - \theta) = \cot \theta$ and $\cot x \times \tan x=1$
wraith daggerBOT
秋水
sacred stone
so $\tan5^\circ \times \tan 85^\circ =1$
wraith daggerBOT
秋水
languid imp
OHHHH OKAY
im so dumb
thank u sm
one last thing, how do i equate with the root 3?
sacred stone
one last thing, how do i equate with the root 3?
what?
languid imp
after im done w that
sacred stone
$\tan15^\circ \times \tan 75^\circ =1$
wraith daggerBOT
秋水
languid imp
nvm mb
sacred stone
$\tan 30^\circ = \frac{\sqrt{3}}{3}$
wraith daggerBOT
秋水
smoky geyser
Can someone help me with this question please
cedar kilnBOT
Can someone help me with this question please
crystal raptor
What have you tried
cedar kilnBOT
@smoky geyser Has your question been resolved?
frigid canyon
let him tell what he/she has done
halcyon swan
let him tell what he/she has done
Oh, sure
smoky geyser
sorry
halcyon swan
no problem
frigid canyon
man helping people is so fun :o
ik ima helper
halcyon swan
What did you try so far?
halcyon swan
ik ima helper
yeah! :D
lilac birch
cedar kilnBOT
digital junco
$\tan 30^\circ = \frac{\sqrt{3}}{3}$
@sacred stone
lilac birch
huh
sacred stone
P is on a line, when PN is the shortest distance, we have NP is perpendicular to the line
The distances from P to M and N are equal, so P is on the vertical bisector of MN.
the shortest distance is MN/2
smoky geyser
What did you try so far?
not sure how to start
cedar kilnBOT
@lilac birch Has your question been resolved?
mighty kestrel
I can’t seem to do this. I can do it without the square root but it seems impossible with the square root.
sacred stone
I can’t seem to do this. I can do it without the square root but it seems imposs...
it's Complete elliptic integral of the second kind
mighty kestrel
it's Complete elliptic integral of the second kind
Thanks i’ll look up some stuff before trying again.
.close
lime kelp
Hiya, I have this question, I was just wondering if someone could help me conceptualise the quotient ring given by $\mathbb{Z}/2 [x]/(x^3+x+\bar{1})$ please. I don't quite understand how the double quotient works. Thank you!
wraith daggerBOT
proazy
crystal raptor
So the top quotient is just the polynomial ring with coefficients in Z_2 right?
So just imagine that it's own ring maybe call it R
And so then they're just quotienting R by that polynomial
(Or rather the ideal generated by that polynomial)
lime kelp
so it's just a double quotient basically
crystal raptor
Well its just a quotient, but the ring you're quotienting happens to already be a quotient
wraith daggerBOT
proazy
lime kelp
@crystal raptor
crystal raptor
No its gonna be an element that looks like $a + \left<x^3+x+1\right>$ where a is some element in R, the top ring
wraith daggerBOT
iCaird
crystal raptor
Quotient rings are collections of cosets
lime kelp
so it's essentially the addition of any element in the top ring + any element generated by the ideal
ahh algebra is so confusing and abstract! am i correct in saying quotient rings R/I have that the ideal is just a subset of the larger set with a zero, closure and the 'black hole' property so just special subsets. I am just having a hard time conceptualising it.
@crystal raptor
crystal raptor
Yeah, the point is R\I only has the structure of a ring when I is an ideal
So the way it was most likely discovered is they made the set R\I for a random subring I, and then asked what conditions do we have to put on I for R\I to be a ring?
And if you follow that through you'll get the requirements of an ideal
lime kelp
And if you follow that through you'll get the requirements of an ideal
okay thank you, i'm still a little confused though. how would i go about solving this question?
crystal raptor
So they way quotienting by polynomials work is you can say the polynomial is zero for all x
So in this case we can say x^3+x+1 =0 for all x
Or rather x^3 = x+1
So we can just start listing elements in the original ring, but replacing x^3 with x+1
So in this particular case we don't have to keep talking about cosets, we can just identify the quotient with the original ring but two elements are equivalent if you can change one into the other by repeated applications of x^3=x+1
lime kelp
So in this particular case we don't have to keep talking about cosets, we can ju...
okay and am i correct in saying that in Z/2[x] any x with deg(x)>2 is just 0? just making sure
crystal raptor
In Z/2[x]?? No you're given one with degree bigger than two in the question
In the quotient by the polynomial, any of degree bigger than two is equivalent to one of degree less than or equal to 2
lime kelp
In the quotient by the polynomial, any of degree bigger than two is equivalent t...
ohh okay so it's Z/2[x] is the group generated by <1,x,x^2> so once I have those elements, since none are of the x^3 term nothing is replaced?
thus all the elements of R are just the elements generated by $<1,x,x^2>$
wraith daggerBOT
proazy
lime kelp
wait no
halcyon swan
hello there!
lime kelp
In the quotient by the polynomial, any of degree bigger than two is equivalent t...
okay just processed it lol i get how it works better now, but i'm still confused as to how you would generate all the elements. given the quotient ring do you just assume that since we can write x^3 = x + 1 that all the elements in the overall ring are of deg<3?
lime kelp
hello there!
hello!
crystal raptor
okay just processed it lol i get how it works better now, but i'm still confused...
Yes the quotient is just going to have degree 2 or less polynomials
But its going to have a different multiplication structure! Bc we could multiply two of them together and get something with degree higher than 2, in which case we use x^3=x+1 to see which element the product is actually equal to
It may be a good exercise to write out the multiplication table for the quotient ring, it may be long but gives you a bit of understanding
Or atleast compute a few product
Note that addition is the same as we can't add two of our guys and get a poly with degree bigger than 2
lime kelp
Note that addition is the same as we can't add two of our guys and get a poly wi...
ooo i think i get it
so i did this for another question
with the ring R in the top left
would you agree what i've written makes sense and that R is the set at the bottom? :)
cedar kilnBOT
@lime kelp Has your question been resolved?
lime kelp
Looks like you get it!
thank you so much! i don't have a mark scheme so do you mind checking if i answer the question from before
:)
lime kelp
Sure just ping me and if I'm around I'll have a look
thank you so much for your help i really appreciate it :) I did the following!
crystal raptor
thank you so much for your help i really appreciate it :) I did the following!
Looks good to me!
Very clever to use g(x) in part b because you know you're gonna need to use it in part c!
cedar kilnBOT
@lime kelp Has your question been resolved?
lime kelp
Very clever to use g(x) in part b because you know you're gonna need to use it i...
thank you so much :)) really appreciate the help and hope you have a good rest of your day!
wanton vapor
cedar kilnBOT
wanton vapor
how
crystal raptor
t substitution I imagine?
wanton vapor
how?
wanton vapor
Oh didn’t know that
wanton vapor
No, that tangent identity
verbal sphinx
Yes