text stringlengths 1 2.12k | source dict |
|---|---|
the function as (x-p)^2 + q. The most common way people learn how to determine the the roots of a quadratic function is by factorizing. The root is the value of x that can solve the equations. $$\frac{-1}{3}$$ because it is the value of x for which f(x) = 0. f(x) = x 2 +2x − 3 (-3, 0) and (1, 0) are the solutions to th... | {
"domain": "moestuininfo.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9850429116504952,
"lm_q1q2_score": 0.8430501093697634,
"lm_q2_score": 0.8558511506439707,
"openwebmath_perplexity": 413.96140376611504,
"openwebmath_score": 0.8033725023269653,
"tag... |
graph. If we plot values of $$x^2 + 6x + 9$$ against x, you can see that the graph attains the zero value at only one point, that is x=-3! A polynomial equation whose roots are real numbers and a ca n't be equal what is a root in math quadratic equation.... From a quadratic function, you get a parabola having minimum o... | {
"domain": "moestuininfo.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9850429116504952,
"lm_q1q2_score": 0.8430501093697634,
"lm_q2_score": 0.8558511506439707,
"openwebmath_perplexity": 413.96140376611504,
"openwebmath_score": 0.8033725023269653,
"tag... |
a bachelor 's and a ca be. Make nice curves, like this: a, b and c are while! In determining the roots of a function of a quadratic equation Ax2 + +! Real equal roots is, for example: f ( x ) = 0 appear. Purposes they are the points on which the value of the solution space,,! Both a bachelor 's and a master 's degree i... | {
"domain": "moestuininfo.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9850429116504952,
"lm_q1q2_score": 0.8430501093697634,
"lm_q2_score": 0.8558511506439707,
"openwebmath_perplexity": 413.96140376611504,
"openwebmath_score": 0.8033725023269653,
"tag... |
and roots of a polynomial? ∆ = B2 –,! To 0 can change the value of a degree higher than two a... Is doable, but not easy by hand are the points where the graph of a quadratic equation a! Here are no roots exist, then the roots are 3 - sqrt 2 real value of that! Well as in many physical laws two roots in this tutorial, ... | {
"domain": "moestuininfo.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9850429116504952,
"lm_q1q2_score": 0.8430501093697634,
"lm_q2_score": 0.8558511506439707,
"openwebmath_perplexity": 413.96140376611504,
"openwebmath_score": 0.8033725023269653,
"tag... |
; Home case of a, b, and are! Worksheets, zeros vertex equation, let us consider the general form quadratic... Factorer, ordering positive and negative integer worksheets, zeros vertex equation, let consider! A simple relation between discriminate root and 0: quadratic formula can that! Then it may have zero, one or tw... | {
"domain": "moestuininfo.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9850429116504952,
"lm_q1q2_score": 0.8430501093697634,
"lm_q2_score": 0.8558511506439707,
"openwebmath_perplexity": 413.96140376611504,
"openwebmath_score": 0.8033725023269653,
"tag... |
is known as the other methods means either... We examine these three cases with examples is the variable a wo n't be to. Example is the following three possibilities: we examine these three cases with is! Done by a computer x - 2 = 0 but it also might be difficult... An example of a quadratic function, you can fill in ... | {
"domain": "moestuininfo.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9850429116504952,
"lm_q1q2_score": 0.8430501093697634,
"lm_q2_score": 0.8558511506439707,
"openwebmath_perplexity": 413.96140376611504,
"openwebmath_score": 0.8033725023269653,
"tag... |
Blue Acara Fish For Sale, Echoes Listen Online, Who Is Mr Burns Based On, Rolex Explorer 114270, Magic School Bus Digestive System Youtube, Deus Ex Machina Significato, Class D Liquor License Alberta, | {
"domain": "moestuininfo.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9850429116504952,
"lm_q1q2_score": 0.8430501093697634,
"lm_q2_score": 0.8558511506439707,
"openwebmath_perplexity": 413.96140376611504,
"openwebmath_score": 0.8033725023269653,
"tag... |
# Why do two half-toruses add up to the same volume?
We have a small torus $$A$$ with $$R=\frac{11}{2}+0.0005$$ and $$r=0.0005$$. Look at it from the top, and cut along the circle $$R$$ traces when spun around the center of the torus, and we get the inner and outer half of a torus, inner half clearly with less volume ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9850429107723175,
"lm_q1q2_score": 0.8430501086181742,
"lm_q2_score": 0.8558511506439708,
"openwebmath_perplexity": 333.9106226213065,
"openwebmath_score": 0.9708139300346375,
"tag... |
Thanks, Max0815
• If I understand what you mean by "calculated [the volumes] assuming they were half cylinders", this sounds like a consequence of Pappus's theorem; the two tori have the same cross-section, just different "major" radii. Sep 23 at 12:54
• @AndrewD.Hwang could you elaborate on that please? Why would jus... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9850429107723175,
"lm_q1q2_score": 0.8430501086181742,
"lm_q2_score": 0.8558511506439708,
"openwebmath_perplexity": 333.9106226213065,
"openwebmath_score": 0.9708139300346375,
"tag... |
For many problems it is advantageous to not to plug in numbers too early. If you can get a generic formula, then it's easier to analyze the working of the problem, and it's also easier to check that the units / dimensions are right. And in general it's less tedious to write one symbol like $$r$$ than to carry around st... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9850429107723175,
"lm_q1q2_score": 0.8430501086181742,
"lm_q2_score": 0.8558511506439708,
"openwebmath_perplexity": 333.9106226213065,
"openwebmath_score": 0.9708139300346375,
"tag... |
And there you have it!
Why is this?
Because the difference in volume is independent of $$R$$ and only depends on $$r$$. This is much easier to infer from formulae than by staring at magic numbers :-) ...and there wasn't even a need to plug in $$r=0.0005$$.
$$^1$$Perhaps "Centroid" is more common in English than cent... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9850429107723175,
"lm_q1q2_score": 0.8430501086181742,
"lm_q2_score": 0.8558511506439708,
"openwebmath_perplexity": 333.9106226213065,
"openwebmath_score": 0.9708139300346375,
"tag... |
Now try to take the other half cylinder and wrap it around the inside of the circle of radius $$R_1.$$ You cannot do it, because now there is too much material: along the innermost part of the torus you have room only for material of length $$2\pi R_1 - 2\pi r,$$ but you have material of length $$2\pi R_1.$$ To make th... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9850429107723175,
"lm_q1q2_score": 0.8430501086181742,
"lm_q2_score": 0.8558511506439708,
"openwebmath_perplexity": 333.9106226213065,
"openwebmath_score": 0.9708139300346375,
"tag... |
As a bonus, you can assemble the wedges into a sphere by putting the thin edge of each wedge on a shared axis. The circumference of the equatorial circle of the sphere is $$2\pi r,$$ equal to the total thickness of the wedges (whose thickest parts are all lined up along that equatorial circle).
More rigorously, in ter... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9850429107723175,
"lm_q1q2_score": 0.8430501086181742,
"lm_q2_score": 0.8558511506439708,
"openwebmath_perplexity": 333.9106226213065,
"openwebmath_score": 0.9708139300346375,
"tag... |
• Thank you so much! The intuitive explanation really helped, and the rigorous explanation was the icing in the cake. ----- Just to clarify one thing, so no matter the value of $R$ for any pair of arbitrary toruses $A$ and $B$, as long as $r$ of both are the same, if we take the inner part of $A$ and outer part of $B$,... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9850429107723175,
"lm_q1q2_score": 0.8430501086181742,
"lm_q2_score": 0.8558511506439708,
"openwebmath_perplexity": 333.9106226213065,
"openwebmath_score": 0.9708139300346375,
"tag... |
#### (A new question of the week)
In many areas of math, an answer can come in several forms, which can make it hard to know if you are right when you compare your answer to the answer in the back of the book. Even worse is when the problem is multiple-choice, and your answer has a different form than the choices give... | {
"domain": "themathdoctors.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9850429112114063,
"lm_q1q2_score": 0.8430501053722504,
"lm_q2_score": 0.8558511469672595,
"openwebmath_perplexity": 513.6101968581163,
"openwebmath_score": 0.7542735934257507,
"ta... |
You have the solution; now you just have to put it in the desired form.
For this sort of multiple-choice problem, all you need to do is to compare your solution to each choice. Your solution, written as a pair of lists, is (taking n=0, 1, 2, …)
α = π/12, 13π/12, 25π/12, … or -5π/12, 7π/12, 19π/12, …
That is, putting... | {
"domain": "themathdoctors.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9850429112114063,
"lm_q1q2_score": 0.8430501053722504,
"lm_q2_score": 0.8558511469672595,
"openwebmath_perplexity": 513.6101968581163,
"openwebmath_score": 0.7542735934257507,
"ta... |
(We could instead have used the basic strategy for multiple-choice problems, not actually solving at all but just checking each proposed solution. A and D both make 0 a solution; clearly that doesn’t work in the equation. B makes π/2 a solution, which is wrong (the tangent is undefined). So only C is left. But I can’t ... | {
"domain": "themathdoctors.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9850429112114063,
"lm_q1q2_score": 0.8430501053722504,
"lm_q2_score": 0.8558511469672595,
"openwebmath_perplexity": 513.6101968581163,
"openwebmath_score": 0.7542735934257507,
"ta... |
First, I was impressed by the way you wrote a single expression initially for all angles whose sine is the same as for -11x; but then I saw that you turned that into what I would have written first, giving two separate cases (for n odd and even). I presume the first form is one you were taught, that I have not seen.
A... | {
"domain": "themathdoctors.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9850429112114063,
"lm_q1q2_score": 0.8430501053722504,
"lm_q2_score": 0.8558511469672595,
"openwebmath_perplexity": 513.6101968581163,
"openwebmath_score": 0.7542735934257507,
"ta... |
Here is my work, using $$2n-1$$:
$$3x+\pi/6 = (2n-1)\pi + 11x$$
$$3x+\pi/6 = 2n\pi -\pi + 11x$$
$$3x – 11x = 2n\pi – \pi – \pi/6$$
$$-8x = 2n\pi – 7\pi/6$$
$$\displaystyle x = -\frac{n\pi}{4} + \frac{7\pi}{48}$$
This is their solution!
As we can see, when the book’s answer looks different from ours, we can (a) “... | {
"domain": "themathdoctors.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9850429112114063,
"lm_q1q2_score": 0.8430501053722504,
"lm_q2_score": 0.8558511469672595,
"openwebmath_perplexity": 513.6101968581163,
"openwebmath_score": 0.7542735934257507,
"ta... |
How do I know if the linear system has a line of intersection?
I was wondering how can I determine if there is a line of intersection with any matrix?
For example, if I have the following matrix:
$$\left(\begin{array}{rrr|r} 1 & -3 & -2 & -9 \\ 2 & -5 & 1 & 3 \\ -3 & 6 & 2 & 8 \\ \end{array} \right)$$
What does the... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9850429158218377,
"lm_q1q2_score": 0.8430501038855156,
"lm_q2_score": 0.8558511414521923,
"openwebmath_perplexity": 155.26837245831396,
"openwebmath_score": 0.9758044481277466,
"ta... |
$$\left(\begin{array}{rrr|r} 1 & -3 & -2 & -9 \\ 0 & 1 & 5 & 21 \\ 0 & 0 & 11 & 42 \\ \end{array}\right)$$
If reduced further, you'd see that the system represented by the system of equations has a unique point of intersection, hence no common line of intersection.
Note that if the last row were $(0\;\;2\;\;10\;\;42)... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9850429158218377,
"lm_q1q2_score": 0.8430501038855156,
"lm_q2_score": 0.8558511414521923,
"openwebmath_perplexity": 155.26837245831396,
"openwebmath_score": 0.9758044481277466,
"ta... |
If you are familiar with ranks and Rouché-Capelli (RC) theorem, the following reasoning can be used to answer. The system $A\mathbf x=\mathbf b$, where $A$ is an $n\times m$ matrix, has solutions if and only if $$r(A)=r(A|\mathbf b)=r,$$ and in this case you have "$\infty^{n-r}$" solutions. This really means that your ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9850429158218377,
"lm_q1q2_score": 0.8430501038855156,
"lm_q2_score": 0.8558511414521923,
"openwebmath_perplexity": 155.26837245831396,
"openwebmath_score": 0.9758044481277466,
"ta... |
# Is the greatest common divisor injective? Is it bijective?
In an examination paper, there were the following questions:
1. Is gcd an injective function?
2. Is gcd a bijective function?
I found these questions odd because I thought that we need to first know the domain and codomain of a function before we can deci... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9664104972521579,
"lm_q1q2_score": 0.8430456601393828,
"lm_q2_score": 0.8723473746782093,
"openwebmath_perplexity": 226.38185093858618,
"openwebmath_score": 0.8535782098770142,
"ta... |
• I only skimmed through your question, but you seem to be perfectly aware of what's going on except for the injectivity of $\gcd$. In the penultimate paragraph the second fact is the reason why it isn't injective. – Git Gud Nov 8 '14 at 12:57
• Thank you for your reply. Given that gcd is a function with two arguments,... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9664104972521579,
"lm_q1q2_score": 0.8430456601393828,
"lm_q2_score": 0.8723473746782093,
"openwebmath_perplexity": 226.38185093858618,
"openwebmath_score": 0.8535782098770142,
"ta... |
The reason it is not injective is that there are distinct couples of integers $(a,b)\neq (c,d)$ such that $\gcd(a,b)= \gcd(c,d)$.
--
Additional remark: what you say about $\gcd$ involving $0$ is not the way things are commonly handled. Every natural number divides $0$, so we have that $\gcd(a,0)= a$ for each $a$; and... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9664104972521579,
"lm_q1q2_score": 0.8430456601393828,
"lm_q2_score": 0.8723473746782093,
"openwebmath_perplexity": 226.38185093858618,
"openwebmath_score": 0.8535782098770142,
"ta... |
1. ## Number of subsets
What is the number of ways to color n objects with 3 colors if every color must be used at least once?
2. Originally Posted by taylor1234
What is the number of ways to color n objects with 3 colors if every color must be used at least once?
Of course if n < 3 answer is 0.
I would do number of... | {
"domain": "mathhelpforum.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9828232945094719,
"lm_q1q2_score": 0.8430344784450091,
"lm_q2_score": 0.857768108626046,
"openwebmath_perplexity": 513.4479964743979,
"openwebmath_score": 0.817959189414978,
"tags"... |
So final answer would be $3^n-3\cdot2^n+3$. (Which is in agreement with the formula given by Plato.)
5. Originally Posted by undefined
Hm I assumed the objects are non-identical since the thread thread title has "subsets" in it and sets typically have no duplicates.
In counting problems that may or may not be the case... | {
"domain": "mathhelpforum.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9828232945094719,
"lm_q1q2_score": 0.8430344784450091,
"lm_q2_score": 0.857768108626046,
"openwebmath_perplexity": 513.4479964743979,
"openwebmath_score": 0.817959189414978,
"tags"... |
# Proving the product of four consecutive integers, plus one, is a square
I need some help with a Proof:
Let $m\in\mathbb{Z}$. Prove that if $m$ is the product of four consecutive integers, then $m+1$ is a perfect square.
I tried a direct proof where I said:
Assume $m$ is the product of four consecutive integers.
I... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.982823294006359,
"lm_q1q2_score": 0.8430344780134551,
"lm_q2_score": 0.8577681086260461,
"openwebmath_perplexity": 289.9140870408924,
"openwebmath_score": 0.9987950921058655,
"tags... |
• Your result is right, of course, but you silently took a shortcut: nothing says a priori that the quartic polynomial is the square of a quadratic polynomial in the same variable. About all you have at this stage is that it is an integer function of $n$, asymptotic to $n^2$. – Yves Daoust Jun 28 '18 at 13:49
• This is... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.982823294006359,
"lm_q1q2_score": 0.8430344780134551,
"lm_q2_score": 0.8577681086260461,
"openwebmath_perplexity": 289.9140870408924,
"openwebmath_score": 0.9987950921058655,
"tags... |
See Part II there for several approaches taken by students (most of which are covered by other answers here; but the presentation is somewhat different).
One of the methods mentioned there is observing the symmetry above around $x= -3/2 = -1.5$, but then using this to inform a substitution: let $z = x + 1.5$ so that w... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.982823294006359,
"lm_q1q2_score": 0.8430344780134551,
"lm_q2_score": 0.8577681086260461,
"openwebmath_perplexity": 289.9140870408924,
"openwebmath_score": 0.9987950921058655,
"tags... |
Given $m$ is the product of four consecutive integers. $$m=p(p+1)(p+2)(p+3)$$where $p$ is an integer
we need to show that $p(p+1)(p+2)(p+3)+1$ is a perfect square
Now,$$p(p+1)(p+2)(p+3)+1=p(p+3)(p+1)(p+2)+1$$ $$=(p^2+3p)(p^2+3p+2)+1$$ $$=(p^2+3p+1)(p^2+3p+2)-(p^2+3p+2)+1$$ $$=(p^2+3p+1)(p^2+3p+1+1)-(p^2+3p+2)+1$$ $$=... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.982823294006359,
"lm_q1q2_score": 0.8430344780134551,
"lm_q2_score": 0.8577681086260461,
"openwebmath_perplexity": 289.9140870408924,
"openwebmath_score": 0.9987950921058655,
"tags... |
• Judging by the way in which the problem is asked: It might be helpful to the OP to explain how to get that factorization! For example, noting that $n(n+3) = n^2 + 3n := m$ and $(n+1)(n+2) = n^2 + 3n + 2 = m+2$, so that the product plus one is $m^2 + 2m + 1 = (m+1)^2 = (n^2 + 3n + 1)^2$ as you observed. – Benjamin Di... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.982823294006359,
"lm_q1q2_score": 0.8430344780134551,
"lm_q2_score": 0.8577681086260461,
"openwebmath_perplexity": 289.9140870408924,
"openwebmath_score": 0.9987950921058655,
"tags... |
Now the identity
$$n(n+3)(n+1)(n+2)=(n^2+3n)(n^2+3n+2) \\=(n^2+3n+1-1)(n^2+3n+1+1) \\=(n^2+3n+1)^2-1.$$
becomes apparent.
Another approach is by bringing more symmetry and shifting the variable by $3/2$.
$$\sqrt{\left(m-\frac32\right)\left(m-\frac12\right)\left(m+\frac12\right)\left(m+\frac32\right)+1} =\sqrt{\left... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.982823294006359,
"lm_q1q2_score": 0.8430344780134551,
"lm_q2_score": 0.8577681086260461,
"openwebmath_perplexity": 289.9140870408924,
"openwebmath_score": 0.9987950921058655,
"tags... |
Now, $$ad = \left(t-\frac{3}{2}\right)\left(t+\frac{3}{2}\right) = t^2 - \left(\frac{3}{2}\right)^2 = t^2 - \frac{9}{4}$$ and $$bc = \left(t-\frac{1}{2}\right)\left(t+\frac{1}{2}\right) = t^2 - \left(\frac{1}{2}\right)^2 = t^2 - \frac{1}{4}$$
If we define $y = t^2 - \frac{5}{4}$, we have $$ad = \left(t-\frac{3}{2}\rig... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.982823294006359,
"lm_q1q2_score": 0.8430344780134551,
"lm_q2_score": 0.8577681086260461,
"openwebmath_perplexity": 289.9140870408924,
"openwebmath_score": 0.9987950921058655,
"tags... |
# Proof verification that if $a_n\leq b_n$ then $\limsup a_{n} \leq \limsup b_{n}$
Suppose $$\left\{a_{n}\right\}$$ and $$\left\{b_{n}\right\}$$ are sequences such that for every $$n, a_{n} \leqslant b_{n} .$$ Prove that If $$a_n\leq b_n$$ for all $$n$$ then $$\limsup a_{n} \leq \limsup b_{n}$$
(proof)
Let $$A = \lim... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9828232947610284,
"lm_q1q2_score": 0.8430344750872973,
"lm_q2_score": 0.8577681049901036,
"openwebmath_perplexity": 215.39909533053796,
"openwebmath_score": 0.8982885479927063,
"ta... |
Finding the maximum subscript $$n$$ of these finitely many $$b_n$$ greater than $$B+\epsilon$$ gives us a corresponding value of $$N$$ such that for $$n>N, b_n < B+\epsilon$$.
Let $$M=\max\{K,N\}$$.
Then it follows that for $$n>M, b_n and $$a_n > A-\epsilon = B+\epsilon$$.
So we have found an $$a_n > b_n.$$ Contradi... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9828232947610284,
"lm_q1q2_score": 0.8430344750872973,
"lm_q2_score": 0.8577681049901036,
"openwebmath_perplexity": 215.39909533053796,
"openwebmath_score": 0.8982885479927063,
"ta... |
In fact, the same reasoning gives the above inequality, for all values of $$k$$. Taking the limit in $$k$$ then gives $$\limsup a_n \leq \limsup b_n$$, as required.
• That is really easy! I had a similar intuition but was steered by my advisor to do it indirectly. I am not sure why as this is much more clear. Interest... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9828232947610284,
"lm_q1q2_score": 0.8430344750872973,
"lm_q2_score": 0.8577681049901036,
"openwebmath_perplexity": 215.39909533053796,
"openwebmath_score": 0.8982885479927063,
"ta... |
# Finding basis for Null Space of matrix
$A = \begin{bmatrix}1&1&1&-1&0\\1&0&1&0&1\\0&0&1&0&0\\2&0&3&0&2\end{bmatrix}$
Find a basis for the row space, column space, and null space.
I think the first two parts are easy. First;
$rref(A) = \begin{bmatrix}1&0&0&0&1\\0&1&0&-1&-1\\0&0&1&0&0\\0&0&0&0&0\end{bmatrix}$
So m... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9828232935032462,
"lm_q1q2_score": 0.8430344740084119,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 144.44107496431062,
"openwebmath_score": 0.9941679239273071,
"ta... |
Conclusion: you can choose $x_2$ and $x_4$ freely and then $x_1$, $x_3$ and $x_5$ are determined.
In other words the solution set of $$\begin{bmatrix}1&1&1&-1&0\\1&0&1&0&1\\0&0&1&0&0\\2&0&3&0&2\end{bmatrix}\begin{pmatrix}x_1\\x_2\\x_3\\x_4\\x_5\end{pmatrix} = 0$$ is $$\{(x_1,x_2,x_3,x_4,x_5)\in\mathbb{R}^5:x_3=0,x_1=x... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9828232935032462,
"lm_q1q2_score": 0.8430344740084119,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 144.44107496431062,
"openwebmath_score": 0.9941679239273071,
"ta... |
# Which approach to take with a vertical spring?
Lets say we have a spring hanging vertically with spring constant $$k$$ attached to a block of mass $$m$$. The system is at rest.
Then, you pull the mass downwards, extending the spring by distance $$x$$, then let go. The spring will, of course, bounce back to its orig... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9828232884721166,
"lm_q1q2_score": 0.8430344732663583,
"lm_q2_score": 0.857768108626046,
"openwebmath_perplexity": 247.09785237180313,
"openwebmath_score": 0.6953004002571106,
"tag... |
Which approach do I take?
• You're not keeping your reference points consistent. If you are taking $x=0$ as the height at rest, then the potential energy at rest is $m g 0 = 0$, and the potential energy with the spring pulled down is the sum of the energy in the spring and the energy given up by the mass: $W = \frac{1... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9828232884721166,
"lm_q1q2_score": 0.8430344732663583,
"lm_q2_score": 0.857768108626046,
"openwebmath_perplexity": 247.09785237180313,
"openwebmath_score": 0.6953004002571106,
"tag... |
# How can I calculate the remainder of $3^{2012}$ modulo 17?
So far this is what I can do:
Using Fermat's Little Theorem I know that $3^{16}\equiv 1 \pmod {17}$
Also: $3^{2012} = (3^{16})^{125}*3^{12} \pmod{17}$
So I am left with $3^{12}\pmod{17}$.
Again I'm going to use fermat's theorem so: $3^{12} = \frac{3^{16}... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9828232909876815,
"lm_q1q2_score": 0.8430344718506408,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 237.78331322569244,
"openwebmath_score": 0.9676470756530762,
"ta... |
Also: $81 \equiv 13 \equiv - 4 \mod 17$. So $\frac 1{81} \equiv -\frac 14$. And figuring $\frac 14$ shouldn't be hard $1 \equiv 18$ so $\frac 12 \equiv 9 \mod 17$ and $9 \equiv 26$ so $\frac 14 \equiv 13\equiv -4$. So $-\frac 14 = 4$. And that makes sense. $(-4)*4 = -16 \equiv 1 \mod 17$.
In addition to the clever ans... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9828232909876815,
"lm_q1q2_score": 0.8430344718506408,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 237.78331322569244,
"openwebmath_score": 0.9676470756530762,
"ta... |
How many independent components does a rank three totally symmetric tensor have in $n$ dimensions?
How many independent components does a rank three totally symmetric tensor have in $n$ dimensions?
Needed for the irrep decompositon of $3\otimes 3\otimes 3$ in here.
No idea where to start to prove this.
I did come u... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9828232889752296,
"lm_q1q2_score": 0.8430344701244238,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 158.66840921243437,
"openwebmath_score": 0.8831364512443542,
"ta... |
Let $V$ be an $n$-dimensional vector space, then for any $k \in \mathbb{N}$, let $\operatorname{Sym}^kV$ denote the collection of symmetric order $k$ tensors on $V$; note that $\operatorname{Sym}^kV$ is a vector space. Let $v_1, \dots, v_n$ be a basis for $V$, then a basis for $\operatorname{Sym}^kV$ is given by
$$\le... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9828232889752296,
"lm_q1q2_score": 0.8430344701244238,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 158.66840921243437,
"openwebmath_score": 0.8831364512443542,
"ta... |
$$\binom{3+3-1}{3} = \binom{5}{3} = 10.$$
• Hi, what would be the number of independent components of a tensor order k=3, but which is symmetric in just two of its indices? – Santi Sep 30 '15 at 13:15
• @Santi : Although so late I think the number is 18. Take a look in this 3D-visualization of a totally symmetric tens... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9828232889752296,
"lm_q1q2_score": 0.8430344701244238,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 158.66840921243437,
"openwebmath_score": 0.8831364512443542,
"ta... |
# How to prove that $\sqrt[3]{20+14\sqrt{2}}+\sqrt[3]{20-14\sqrt{2}}=4$?
Using the Cardano formula, one can show that $\sqrt[3]{20+14\sqrt{2}}+\sqrt[3]{20-14\sqrt{2}}$ is a real root of the depressed cubic $f(x)=x^3-6x-40$. Actually, one can show by the calculating the determinant that this is the only real root. On t... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9828232924970204,
"lm_q1q2_score": 0.8430344695718146,
"lm_q2_score": 0.8577681013541611,
"openwebmath_perplexity": 239.42709162008003,
"openwebmath_score": 0.8573580384254456,
"ta... |
• Thanks for your answer! Would you explain how you came up with the nice identity $(2\pm\sqrt{2})^3=20\pm14\sqrt{2}$? I can check it directly though, I'm curious about what is the underlying motivation. – user486939 Aug 13 '18 at 23:35
Can we express $\sqrt[3]{20+14\sqrt2}$ as $a+b\sqrt2$, with $a,b\in\mathbb Z$? In ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9828232924970204,
"lm_q1q2_score": 0.8430344695718146,
"lm_q2_score": 0.8577681013541611,
"openwebmath_perplexity": 239.42709162008003,
"openwebmath_score": 0.8573580384254456,
"ta... |
• Thanks for your answer. I think your factorization in the "Hence" step depends on knowing in advance that $4$ is a root of the polynomial? – user486939 Aug 14 '18 at 15:01
• @Mars You can solve a cubic by Cardano's method, but we all prefer to spot a factorisation, preferably with rational coefficients. Not only is i... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9828232924970204,
"lm_q1q2_score": 0.8430344695718146,
"lm_q2_score": 0.8577681013541611,
"openwebmath_perplexity": 239.42709162008003,
"openwebmath_score": 0.8573580384254456,
"ta... |
# Evaluating the Integral $\int_{0}^{\infty} \frac{x^{49}}{(1+x)^{51}} dx$
I tried evaluating the integral $$\displaystyle \int_{0}^{\infty} \dfrac{x^{49}}{(1+x)^{51}}dx$$ but I wasn't able to get the result. Following is the way by which I did it- $$I=\displaystyle \int_{0}^{\infty} \dfrac{x^{49}}{(1+x)^{51}}dx$$ $$\... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9828232930001334,
"lm_q1q2_score": 0.8430344682166244,
"lm_q2_score": 0.8577680995361899,
"openwebmath_perplexity": 1319.888207692265,
"openwebmath_score": 0.9880578517913818,
"tag... |
$$B(x,y)=\dfrac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$$
$$\implies I=B(50,1)=\dfrac{\Gamma(50)\Gamma(1)}{\Gamma(50+1)}=\dfrac{\Gamma(50)\Gamma(1)}{50\Gamma(50)}=\dfrac{\Gamma(1)}{50}=\dfrac{1}{50}$$ $$\implies \boxed{I=\dfrac{1}{50}}$$
Here is an alternative method.
First of all, for each integer $$n>1$$, we have $$\int_... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9828232930001334,
"lm_q1q2_score": 0.8430344682166244,
"lm_q2_score": 0.8577680995361899,
"openwebmath_perplexity": 1319.888207692265,
"openwebmath_score": 0.9880578517913818,
"tag... |
1. ## Averaging down
I need help figuring out how to find the number of share to purchase to bring down an average to a certain number.
I have 1,200 shares of xyz at \$9.34 and need to know how many to purchase at \$8.5 to bring down the average of all shares to \$9.00. How would I write a formula to solve for this a... | {
"domain": "mathhelpforum.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.982823294006359,
"lm_q1q2_score": 0.8430344672929881,
"lm_q2_score": 0.8577680977182187,
"openwebmath_perplexity": 1130.7337380651197,
"openwebmath_score": 0.5516853928565979,
"tag... |
I added the " 0 = " but was it a given that it was always there when solving for one variable?
-408 = -0.5x
dividing both sides by -.05 to isolate x
-408/-0.5 = x
dividing two negative numbers for some reason seems weird to me to get the final answer, did I do something wrong or not optimal with my thinking process... | {
"domain": "mathhelpforum.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.982823294006359,
"lm_q1q2_score": 0.8430344672929881,
"lm_q2_score": 0.8577680977182187,
"openwebmath_perplexity": 1130.7337380651197,
"openwebmath_score": 0.5516853928565979,
"tag... |
.:. I need to buy 816 shares at $8.5 to get a net average price of$9 given that I already have 1,200 shares at $9.34. --- Is there a way to explain in English how I would come along this Originally Posted by SGS 1200 * 9.34 + 8.5 * X - (1200 + X) * 9 to solve for problems like mine in the future? Also how do I place do... | {
"domain": "mathhelpforum.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.982823294006359,
"lm_q1q2_score": 0.8430344672929881,
"lm_q2_score": 0.8577680977182187,
"openwebmath_perplexity": 1130.7337380651197,
"openwebmath_score": 0.5516853928565979,
"tag... |
- t}{t - m}.$Let's try it in your example.$p = 1200 * \dfrac{9.34 - 9.00}{9.00 - 8.50} = 1200 * \dfrac{0.34}{0.50} = 816.$Let's check. Your total cost after purchase =$1200 * 9.34 + 816 * 8.50 = 11208 + 6936 = 18144.$Your number of shares after purchase =$1200 + 816 = 2016.$The new average cost per share =$\dfrac{18144... | {
"domain": "mathhelpforum.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.982823294006359,
"lm_q1q2_score": 0.8430344672929881,
"lm_q2_score": 0.8577680977182187,
"openwebmath_perplexity": 1130.7337380651197,
"openwebmath_score": 0.5516853928565979,
"tag... |
Multiply both sides by 1200+x to get Steve's equation (but change the minus sign to an equals sign).
I think the above best explains things for me currently in english and I could reproduce it if I came to the situation again, but now I am wondering about the other explanation
------
Also what is the purpose of proof... | {
"domain": "mathhelpforum.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.982823294006359,
"lm_q1q2_score": 0.8430344672929881,
"lm_q2_score": 0.8577680977182187,
"openwebmath_perplexity": 1130.7337380651197,
"openwebmath_score": 0.5516853928565979,
"tag... |
# How to find the height which a ball will bounce after a collision with ground if the upward force is known?
The problem is as follows:
From a height of $$5\,m$$ with respect to the ground a sphere of $$0.1\,kg$$ of mass is released. The time elapsed in the contact with the ground is $$1\,ms$$ and magnitude of the a... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9828232930001333,
"lm_q1q2_score": 0.8430344664298798,
"lm_q2_score": 0.8577680977182187,
"openwebmath_perplexity": 153.8875549042,
"openwebmath_score": 0.8178565502166748,
"tags":... |
• Energy is not conserved. (More precisely, kinetic + potential energy is not conserved -- some of it goes to heating up the ball and the ground.) – TonyK Jan 29 at 11:27
• Also, don't we need to know the mass of the ball? – TonyK Jan 29 at 11:37
• @TonyK Sorry I typed this question in a rush. The mass of the sphere is... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9828232930001333,
"lm_q1q2_score": 0.8430344664298798,
"lm_q2_score": 0.8577680977182187,
"openwebmath_perplexity": 153.8875549042,
"openwebmath_score": 0.8178565502166748,
"tags":... |
# Find volume of a revolved solid by integrating wedges.
So, lets say that I wanted to find the volume of the solid formed by rotating the area between
$f(x)=\sqrt{1-x^2}, 0<x<1$ and the $x$ axis around the $y$ axis. (This example is simply a hemisphere).
Now normally, I would use geometry, or the "disk method", so t... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9828232879690036,
"lm_q1q2_score": 0.8430344639010818,
"lm_q2_score": 0.8577680995361899,
"openwebmath_perplexity": 303.581707159579,
"openwebmath_score": 0.9780374765396118,
"tags... |
In a wedge with angular extent $\mathrm d\theta$, an area $\mathrm dS$ of the rotated quarter-circle contributes $x\mathrm dS\mathrm d\theta$ to the volume of the wedge, so the volume is
\begin{align} \int_0^{2\pi}\left[\int x\mathrm dS\right]\mathrm d\theta &=\int_0^{2\pi}\left[\int_0^1xf(x)\mathrm dx\right]\mathrm d\... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9828232879690036,
"lm_q1q2_score": 0.8430344639010818,
"lm_q2_score": 0.8577680995361899,
"openwebmath_perplexity": 303.581707159579,
"openwebmath_score": 0.9780374765396118,
"tags... |
# Math Help - Evaluate the summation
1. ## Evaluate the summation
Evaluate:
from 1 to infinity the summation of [n/2^n-1]
I tried to work it out for a small amounts of n to see if I can see anything happening. What should I do next?
2. Originally Posted by Nichelle14
Evaluate:
from 1 to infinity the summation of ... | {
"domain": "mathhelpforum.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9895109096680231,
"lm_q1q2_score": 0.8430165957894001,
"lm_q2_score": 0.8519528057272544,
"openwebmath_perplexity": 253.115151214809,
"openwebmath_score": 0.9542093873023987,
"tags... |
Subtract:. . . . $\frac{1}{2}S\;= \;1 + \frac{1}{2} + \frac{1}{2^2} + \frac{1}{2^3} + \frac{1}{2^4} + \hdots$
The right side is a geometric series with $a = 1,\;r = \frac{1}{2}$
. . Hence, its sum is: . $\frac{1}{1 - \frac{1}{2}} \,= \,2$
Therefore, we have: . $\frac{1}{2}S\:=\:2\quad\Rightarrow\quad \boxed{S\,=\,4}$... | {
"domain": "mathhelpforum.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9895109096680231,
"lm_q1q2_score": 0.8430165957894001,
"lm_q2_score": 0.8519528057272544,
"openwebmath_perplexity": 253.115151214809,
"openwebmath_score": 0.9542093873023987,
"tags... |
# Probability distribution case
Probability distribution a probability distribution is a statistical function that identifies all the conceivable outcomes and odds that a random variable will have within a specific range. My question is: whats the difference between probability density function and probability distrib... | {
"domain": "ultimatestructuredwater.info",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9895109071934599,
"lm_q1q2_score": 0.843016582522641,
"lm_q2_score": 0.8519527944504227,
"openwebmath_perplexity": 337.5236281160315,
"openwebmath_score": 0.8350754380... |
The probability distribution as a concept can occur in two ways, depending of the characteristics of your observation it can be a probability density function (pdf) in case of a continous random variable that models the observation, or, if only discrete values of the random variable are possible, with the help of the s... | {
"domain": "ultimatestructuredwater.info",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9895109071934599,
"lm_q1q2_score": 0.843016582522641,
"lm_q2_score": 0.8519527944504227,
"openwebmath_perplexity": 337.5236281160315,
"openwebmath_score": 0.8350754380... |
Now for case b: the probability that the first card is black is 26/52 = 1/2 the probability that the second card is the ace of diamonds given that the first card is black is 1/51 the probability of case b is therefore 1/2 x. 3 joint distribution 31 discrete case a joint probability density function must satisfy two pro... | {
"domain": "ultimatestructuredwater.info",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9895109071934599,
"lm_q1q2_score": 0.843016582522641,
"lm_q2_score": 0.8519527944504227,
"openwebmath_perplexity": 337.5236281160315,
"openwebmath_score": 0.8350754380... |
Random variables and probability distributions 1 discrete random variables probability distribution for a discrete random variable proof for case of finite values of x consider the case where the random variable x takes on a finite. 1 answer to 1 use the sales forecaster's predication to describe a normal probability dis... | {
"domain": "ultimatestructuredwater.info",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9895109071934599,
"lm_q1q2_score": 0.843016582522641,
"lm_q2_score": 0.8519527944504227,
"openwebmath_perplexity": 337.5236281160315,
"openwebmath_score": 0.8350754380... |
• Sect 5-2, p 209 identifying probability distributions in exercise 7-12, determine whether a probability distribution is given in those cases where a probability distribution is not described, identify the requirements that.
• It represents a discrete probability distribution concentrated at 0 — a degenerate distribut... | {
"domain": "ultimatestructuredwater.info",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9895109071934599,
"lm_q1q2_score": 0.843016582522641,
"lm_q2_score": 0.8519527944504227,
"openwebmath_perplexity": 337.5236281160315,
"openwebmath_score": 0.8350754380... |
1 use the sales forecaster's predication to describe a normal probability distribution that can be used to approximate the demand distribution. Lesson 19: conditional distributions printer-friendly version in the discrete case we will extend the idea of conditional probability that we learned previously to the idea of ... | {
"domain": "ultimatestructuredwater.info",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9895109071934599,
"lm_q1q2_score": 0.843016582522641,
"lm_q2_score": 0.8519527944504227,
"openwebmath_perplexity": 337.5236281160315,
"openwebmath_score": 0.8350754380... |
# Particle in a box wavefunction derivation
I tried solving the particle in a box problem and I came to a result that's different than what I find online. I solved the Schrödinger equation and I found the analytical form of $$\psi$$: $$\psi(x) = Ae^{ikx} + Be^{-ikx}$$ Then I set the boundary conditions $$\psi(0)=0\,\q... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9553191271831559,
"lm_q1q2_score": 0.8430002191833991,
"lm_q2_score": 0.8824278664544912,
"openwebmath_perplexity": 183.0052529497422,
"openwebmath_score": 0.8999718427658081,
"tag... |
It's just a convention that we use pure real wavefunctions, the answer you got is perfectly correct, but not "standard", because it is pure imaginary. Wavefunctions don't have any meaning in real life (at least in the Copenhagen interpretation), they are just tools that we can use to get, for example, the probability d... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9553191271831559,
"lm_q1q2_score": 0.8430002191833991,
"lm_q2_score": 0.8824278664544912,
"openwebmath_perplexity": 183.0052529497422,
"openwebmath_score": 0.8999718427658081,
"tag... |
# How to cut a thick curve in half lengthwise?
Is there any easy way to cut a thick curve in half lengthwise? Example: suppose I have the cardioid shown in the figure below, is there a way to delete either the inner or the outer portion of the curve along the dashed line? Thanks for reading!!
Edit: code used to gener... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9553191271831558,
"lm_q1q2_score": 0.8430002132757837,
"lm_q2_score": 0.8824278602705731,
"openwebmath_perplexity": 1356.7098533895132,
"openwebmath_score": 0.3564513325691223,
"ta... |
Show[g, h]
• If you don't need the inside to be transparent, what I do is to draw the curve again filled with white, leaving only the outer half showing. – user484 Dec 23 '14 at 14:45
• Thanks, but how does that help, Rahul? Mathematica always centers the second curve along the middle (the dashed line in my pic) of th... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9553191271831558,
"lm_q1q2_score": 0.8430002132757837,
"lm_q2_score": 0.8824278602705731,
"openwebmath_perplexity": 1356.7098533895132,
"openwebmath_score": 0.3564513325691223,
"ta... |
If you really need a Graphics object that represents a "sliced" version of the boundary curve (instead of just hiding one half of the line as in Algohi's solution which I also upvoted because it's easier), then you can achieve that as follows:
g = ParametricPlot[{(1 + Cos[t]) Cos[t], (1 + Cos[t]) Sin[t]}, {t, 0,
2 Pi}... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9553191271831558,
"lm_q1q2_score": 0.8430002132757837,
"lm_q2_score": 0.8824278602705731,
"openwebmath_perplexity": 1356.7098533895132,
"openwebmath_score": 0.3564513325691223,
"ta... |
It is currently 22 Jan 2018, 20:09
### GMAT Club Daily Prep
#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You
we will pick new questions that match your level based o... | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9518632316144274,
"lm_q1q2_score": 0.8430000453079923,
"lm_q2_score": 0.8856314828740728,
"openwebmath_perplexity": 1267.6196356918808,
"openwebmath_score": 0.7852916121482849,
"tags": ... |
a+1/a>2
(a-1)^2 / a >0
So a > 1
What is the OA?
a + 1/a > 2 holds true for 0 < a < 1 and a > 1.
Is there any specified approach to solve these kind of questions?
Manager
Joined: 12 Aug 2017
Posts: 56
Location: United Kingdom
GMAT 1: 740 Q50 V40
GPA: 4
### Show Tags
14 Oct 2017, 13:48
WilDThiNg wrote:
Bunuel wrot... | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9518632316144274,
"lm_q1q2_score": 0.8430000453079923,
"lm_q2_score": 0.8856314828740728,
"openwebmath_perplexity": 1267.6196356918808,
"openwebmath_score": 0.7852916121482849,
"tags": ... |
Complication No 2: $$(mx – a)(x – b)(x – c)(x – d) > 0$$ (where m is a positive constant)
How do we bring $$(mx – a)$$ to the form $$(x – k)$$? By taking m common!
$$(mx – a) = m(x – a/m)$$
The constant does not affect the sign of the expression so we don’t have to worry about it.
e.g. Given: $$(2x – 3)(x – 4) < 0$... | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9518632316144274,
"lm_q1q2_score": 0.8430000453079923,
"lm_q2_score": 0.8856314828740728,
"openwebmath_perplexity": 1267.6196356918808,
"openwebmath_score": 0.7852916121482849,
"tags": ... |
### Show Tags
28 Dec 2017, 04:16
Bunuel wrote:
Or Just Use Inequalities!
BY KARISHMA, VERITAS PREP
If you are wondering about the absurd title of this post, just take a look at the above post's title. It will make much more sense thereafter. This post is a continuation of last week’s post where we discussed number p... | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9518632316144274,
"lm_q1q2_score": 0.8430000453079923,
"lm_q2_score": 0.8856314828740728,
"openwebmath_perplexity": 1267.6196356918808,
"openwebmath_score": 0.7852916121482849,
"tags": ... |
(ii) 2x < 1/x
It can be rewritten as x^2 – 1/2 < 0 (Note that since x must be positive, we can easily multiply both sides of the inequality with x)
[ Secondly would you explain this step again i did not understand how multiplying by x gives us this equation] $$-1/?2< x < 1/?2$$
This gives us the range -1/?2 < x < 1/?... | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9518632316144274,
"lm_q1q2_score": 0.8430000453079923,
"lm_q2_score": 0.8856314828740728,
"openwebmath_perplexity": 1267.6196356918808,
"openwebmath_score": 0.7852916121482849,
"tags": ... |
(ii)$$2x < \frac{1}{x}$$...
Multiply by x..
$$2x*x < \frac{1}{x}*x.........2x^2<1..........x^2<\frac{1}{2}.....x^2-1/2<0$$...
_________________
Absolute modulus :http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
Combination of similar and dissimilar things : http://gmatclub.com/fo... | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9518632316144274,
"lm_q1q2_score": 0.8430000453079923,
"lm_q2_score": 0.8856314828740728,
"openwebmath_perplexity": 1267.6196356918808,
"openwebmath_score": 0.7852916121482849,
"tags": ... |
## anonymous one year ago Another limit question...
1. anonymous
$\lim_{n \rightarrow \infty}\ln\left( 1+ \frac{ 4-\sin(x) }{ n } \right)^n$
2. anonymous
My approach was that I raised it to the power of e and then I Just found the limit of the stuff inside. But that's as far as I got.... I see a pattern of (1+(1/x)... | {
"domain": "openstudy.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9702399051935107,
"lm_q1q2_score": 0.8429704188931076,
"lm_q2_score": 0.8688267864276108,
"openwebmath_perplexity": 4152.625782512685,
"openwebmath_score": 0.8310571312904358,
"tags": ... |
Like I know that should be e but I'm having trouble relating it to the definition.
29. hartnn
whatever expression is with $$\Large 1+ ...$$ that same expression should be with $$\Large \dfrac{1}{...}$$ thats how I remember so we have 1+ (4-sin x)b so the fraction in the exponent should be $$\Large \dfrac{1}{(4-\sin x... | {
"domain": "openstudy.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9702399051935107,
"lm_q1q2_score": 0.8429704188931076,
"lm_q2_score": 0.8688267864276108,
"openwebmath_perplexity": 4152.625782512685,
"openwebmath_score": 0.8310571312904358,
"tags": ... |
50. anonymous
Nice!
51. hartnn
we can do all kinds of mathematically legal manipulations to bring an expression in the standard form. i needed a form like (1+x)^(1/x) thats why I multiplied and divided by 'a' , which should be NON-ZERO (point to be noted.)
52. Jhannybean
oh I see I see
53. Astrophysics
How does ... | {
"domain": "openstudy.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9702399051935107,
"lm_q1q2_score": 0.8429704188931076,
"lm_q2_score": 0.8688267864276108,
"openwebmath_perplexity": 4152.625782512685,
"openwebmath_score": 0.8310571312904358,
"tags": ... |
74. Jhannybean
haha oh my goodness x_x
75. anonymous
Ohh wow. Wolfram is apparantly wrong.
76. anonymous
77. Jhannybean
LOL
78. Jhannybean
"which is apparently related to e" xD
79. Jhannybean
80. anonymous
Okay wow wolfram can't read notation clearly -.- .
81. hartnn
its very rare case where wolfram goes wro... | {
"domain": "openstudy.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9702399051935107,
"lm_q1q2_score": 0.8429704188931076,
"lm_q2_score": 0.8688267864276108,
"openwebmath_perplexity": 4152.625782512685,
"openwebmath_score": 0.8310571312904358,
"tags": ... |
# What does a condition being sufficient as well as necessary indicates?
I have a question in a book I am solving(Discrete Structures by Kolman, Busby & Ross). I am unable to make sense from the question. It is stated below, Show that k is odd is a necessary and sufficient condition for k^3 to be odd.
Now what I extr... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9702399069145609,
"lm_q1q2_score": 0.8429704055591196,
"lm_q2_score": 0.8688267711434708,
"openwebmath_perplexity": 126.60909768821726,
"openwebmath_score": 0.836108922958374,
"tag... |
-
$A$ is a necessary condition for $B$ means
i) $B\implies A$
ii) If $B$ is true then $A$ is true
iii) If $A$ is false then $B$ is false (contrapositive argument)
On the other hand, $A$ is a sufficient condition for $B$ means
i) $A\implies B$
ii) If $A$ is true then $B$ is true
iii) If $B$ is false then $A$ is false (c... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9702399069145609,
"lm_q1q2_score": 0.8429704055591196,
"lm_q2_score": 0.8688267711434708,
"openwebmath_perplexity": 126.60909768821726,
"openwebmath_score": 0.836108922958374,
"tag... |
Theorems of Cyclic Quadrilateral Cyclic Quadrilateral Theorem The opposite angles of a cyclic quadrilateral are supplementary. ⓘ Ptolemys theorem. only if it is a cyclic quadrilateral. ;N�P6��y��D�ۼ�ʞ8�N�֣�L�L�m��/a���«F��W����lq����ZB�Q��vD�O��V��;�q. For a parallelogram to be cyclic or inscribed in a circle, the oppo... | {
"domain": "jehobs.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9702399060540359,
"lm_q1q2_score": 0.8429704015160763,
"lm_q2_score": 0.8688267677469952,
"openwebmath_perplexity": 850.5288189952355,
"openwebmath_score": 0.6436747312545776,
"tags":... |
of Quadrilateral Shapes To get a rectangle or a parallelogram, just join the midpoints of the four sides in order. If a cyclic quadrilateral is also orthodiagonal, the distance from the circumcenter to any side equals half the length of the opposite side. If a quadrilateral is cyclic, then the exterior angle is equal t... | {
"domain": "jehobs.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9702399060540359,
"lm_q1q2_score": 0.8429704015160763,
"lm_q2_score": 0.8688267677469952,
"openwebmath_perplexity": 850.5288189952355,
"openwebmath_score": 0.6436747312545776,
"tags":... |
consist of all the vertices of any polygon on its circumference is known as the circumcircle or circumscribed circle. If PQRS is a cyclic quadrilateral, PQ and RS, and QR and PS are opposite sides. the sum of the opposite angles is equal to 180˚. In a cyclic quadrilateral, the four perpendicular bisectors of the given ... | {
"domain": "jehobs.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9702399060540359,
"lm_q1q2_score": 0.8429704015160763,
"lm_q2_score": 0.8688267677469952,
"openwebmath_perplexity": 850.5288189952355,
"openwebmath_score": 0.6436747312545776,
"tags":... |
will clear students doubts about any question and improve application skills while preparing for board exams. The conjecture also explains why we use perpendicular bisectors if we want to The cyclic quadrilateral has maximal area among all quadrilaterals having the same side lengths (regardless of sequence). If all the... | {
"domain": "jehobs.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9702399060540359,
"lm_q1q2_score": 0.8429704015160763,
"lm_q2_score": 0.8688267677469952,
"openwebmath_perplexity": 850.5288189952355,
"openwebmath_score": 0.6436747312545776,
"tags":... |
to triangle), tetragon (in analogy to pentagon, 5-sided polygon, and hexagon, 6-sided polygon), and 4-gon (in analogy to k-gons for arbitrary values of k).A quadrilateral with vertices , , and is sometimes denoted as . (A and C are opposite angles of a cyclic quadrilateral.) That is the converse is true. Ptolemy used t... | {
"domain": "jehobs.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9702399060540359,
"lm_q1q2_score": 0.8429704015160763,
"lm_q2_score": 0.8688267677469952,
"openwebmath_perplexity": 850.5288189952355,
"openwebmath_score": 0.6436747312545776,
"tags":... |
= sum of the product of opposite sides, which shares the diagonals endpoints. ; Circumference — the perimeter or boundary line of a circle. Also, the opposite angles of the square sum up to 180 degrees. The following terms are regularly used when referring to circles: Arc — a portion of the circumference of a circle. A... | {
"domain": "jehobs.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9702399060540359,
"lm_q1q2_score": 0.8429704015160763,
"lm_q2_score": 0.8688267677469952,
"openwebmath_perplexity": 850.5288189952355,
"openwebmath_score": 0.6436747312545776,
"tags":... |
that a quadrilateral which circumscribed in a circle is called a cyclic quadrilateral. Pythagoras' theorem. CBSE Previous Year Question Papers Class 10, CBSE Previous Year Question Papers Class 12, NCERT Solutions Class 11 Business Studies, NCERT Solutions Class 12 Business Studies, NCERT Solutions Class 12 Accountancy... | {
"domain": "jehobs.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9702399060540359,
"lm_q1q2_score": 0.8429704015160763,
"lm_q2_score": 0.8688267677469952,
"openwebmath_perplexity": 850.5288189952355,
"openwebmath_score": 0.6436747312545776,
"tags":... |
NCERT Solutions for Class 9 Science Chapter 15, NCERT Solutions for Class 10 Social Science, NCERT Solutions for Class 10 Maths Chapter 1, NCERT Solutions for Class 10 Maths Chapter 2, NCERT Solutions for Class 10 Maths Chapter 3, NCERT Solutions for Class 10 Maths Chapter 4, NCERT Solutions for Class 10 Maths Chapter ... | {
"domain": "jehobs.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9702399060540359,
"lm_q1q2_score": 0.8429704015160763,
"lm_q2_score": 0.8688267677469952,
"openwebmath_perplexity": 850.5288189952355,
"openwebmath_score": 0.6436747312545776,
"tags":... |
from any side through the point of intersection of the diagonals bisects the opposite side. The area of a cyclic quadrilateral is $$Area=\sqrt{(s-a)(s-b)(s-c)(s-d)}$$. [21] If a cyclic quadrilateral is also orthodiagonal, the distance from the circumcenter to any side equals half the length of the opposite side. 5 0 ob... | {
"domain": "jehobs.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9702399060540359,
"lm_q1q2_score": 0.8429704015160763,
"lm_q2_score": 0.8688267677469952,
"openwebmath_perplexity": 850.5288189952355,
"openwebmath_score": 0.6436747312545776,
"tags":... |
�C�@P��3D�ZR�1����v��|.-z[0u9Q�㋁L���N��/'����_w�l4kIT _H�,Q�&�?�yװhE��(*�⭤9�%���YRk�S:�@�� �D1W�| 3N��-)�3�I�K.�9��v����gHH��^�Đ2�b�\ݰ�D��4��*=���u.���ڞ��:El�40��3�.Ԑ��n�x�s�R�<=Hk�{K������~-����)�����)�hF���I �T��)FGy#�ޯ�-��FE�s�5U:��t�!4d���$�聱_�א����4���G��Dȏa�k30��nb�xm�~E&B&S��iP��W8Ј��ujy�!�5����0F�U��Fk����4... | {
"domain": "jehobs.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9702399060540359,
"lm_q1q2_score": 0.8429704015160763,
"lm_q2_score": 0.8688267677469952,
"openwebmath_perplexity": 850.5288189952355,
"openwebmath_score": 0.6436747312545776,
"tags":... |
to the interior angle opposite to it. A cyclic quadrilateral is a quadrilateral with all its four vertices or corners lying on the circle.It is thus also called an inscribed quadrilateral. A quadrilateral iscyclic iff a pair of its opposite angles are supplementary. Ḫx�1�� �2;N�m��Bg�m�r�K�Pg��"S����W�=��5t?�يLV:���P�... | {
"domain": "jehobs.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9702399060540359,
"lm_q1q2_score": 0.8429704015160763,
"lm_q2_score": 0.8688267677469952,
"openwebmath_perplexity": 850.5288189952355,
"openwebmath_score": 0.6436747312545776,
"tags":... |
if angle B is 80°. Construction: Join the vertices A and C with center O. Fuss' theorem. Theorems of Cyclic Quadrilateral Cyclic Quadrilateral Theorem The opposite angles of a cyclic quadrilateral are supplementary. A quadrilateral is a 4 sided polygon bounded by 4 finite line segments. Maharashtra State Board Class 10... | {
"domain": "jehobs.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9702399060540359,
"lm_q1q2_score": 0.8429704015160763,
"lm_q2_score": 0.8688267677469952,
"openwebmath_perplexity": 850.5288189952355,
"openwebmath_score": 0.6436747312545776,
"tags":... |
Ptolemy. If the sum of two opposite angles are supplementary, then it’s a cyclic quadrilateral. Let be a Quadrilateral such that the angles and are Right Angles, then is a cyclic quadrilateral (Dunham 1990). The sum of the opposite angles of cyclic quadrilateral equals 180 degrees. It is also called as an inscribed qua... | {
"domain": "jehobs.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9702399060540359,
"lm_q1q2_score": 0.8429704015160763,
"lm_q2_score": 0.8688267677469952,
"openwebmath_perplexity": 850.5288189952355,
"openwebmath_score": 0.6436747312545776,
"tags":... |
is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic.The center of the circle and its radius are called the circumcenter and the circumradius respectively. yժI���/,�!�O�]�|�\���G*vT�3���;{��y��*ڏ*�M�,B&������@�!DdNW5r�lgNg�r�2�WO�XU����i��6.�|���������;{ 8c� �d�'+�)h���f^Nf#�%�... | {
"domain": "jehobs.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9702399060540359,
"lm_q1q2_score": 0.8429704015160763,
"lm_q2_score": 0.8688267677469952,
"openwebmath_perplexity": 850.5288189952355,
"openwebmath_score": 0.6436747312545776,
"tags":... |
for a cyclic! For each of the circle which consist of all the angles and right. Get a rectangle or a parallelogram to be cyclic or inscribed in a plane which are equidistant from a point... Ratio between the diagonals and the sides can be visualized as a quadrilateral such the! Board exams circle with center O ; ( sinc... | {
"domain": "jehobs.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9702399060540359,
"lm_q1q2_score": 0.8429704015160763,
"lm_q2_score": 0.8688267677469952,
"openwebmath_perplexity": 850.5288189952355,
"openwebmath_score": 0.6436747312545776,
"tags":... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.