# Tagged Questions

Homework questions are welcome as long as they are asked honestly, explain the problem, and show sufficient effort. Please do not use this as the only tag for a question. For the answers on homework questions, helpful hints or instructions are preferred to a complete solution. Please do not add ...

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### Given $n$ points on the plane, find a circle which contains only three

Given $n$ points on $\mathbb{R}^{2}$, s.t no three are on the same straight line, and not all the points are on the same circle, prove that there exists a circle which contains only three of those ...
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### Show $\lim\limits_{n\to\infty}\int_{0}^{\infty}e^{-x}\sin(\frac{n}{x})~\text{d}x=0$

I'm having trouble showing $$\lim_{n\to\infty}\int_{0}^{\infty}e^{-x}\sin\left(\frac{n}{x}\right)~\text{d}x=0$$ The integrand doesn't converge for any $x$ so I don't know how to use the standard ...
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### Applying analysis to solve a line-of-sight problem

This was an optional h.w. problem: You are at the origin in $\mathbb{Z}\times\mathbb{Z}$. There are trees of a fixed finite radius at each point in $\mathbb{Z}\times\mathbb{Z}$ (other than the ...
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### Measurability of the composition of a measurable map with a surjective map satisfying an expansion condition

I am trying to figure out the following problem in measure theory and am stuck. It seems like it should be very easy, so I must be missing something. Let $g: \mathbb{R} \to \mathbb{R}$ be a mapping ...
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### 6 point lying on a common circle

$Z$ is an interior point of segment $XY$. Three semicircles are drawn over segments $XY$, $XZ$ and $ZY$ on the same side. The midpoints of the arcs are $M1$, $M2$ and $M3$ respectively. A circle ...
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### Galois Theory and Galois Groups

Show that $\mathbb{Q}[x]/\langle x^{3}-2\rangle = [{a + b\alpha + c\alpha^{2}: a, b, c \in \mathbb{Q}, \alpha^{3} = 2}]$ is not a Galois extension of $\mathbb{Q}$. In particular, show that every ...
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### Solving ODEs: The Frobenius Method, worked examples

I find the Frobenius Method quite beautiful, and I would like to be able to apply it. In particular there are three questions in my text book that I have attempted. In each question my limited ...
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### How can I calculate $\int \frac{\sec x\tan x}{3x+5}\,\mathrm dx$

How can I calculate $\displaystyle \int \frac{\sec x\tan x}{3x+5}\,\mathrm dx$ My Try:: $\displaystyle \int \frac{1}{3x+5}\left(\sec x\tan x \right)\,\mathrm dx$ Now Using Integration by Parts:: We ...
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### Is it true that $\lvert \sin z \rvert \leq 1$ for all $z\in \mathbb{C}$?

Is it true that $\left\lvert \sin z \right \rvert \leq 1$ for all $z \in \mathbb{C}$ ? I think that is not true, can anyone help me?
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### 7th class question my daughter asked and need answer if possible

My daughter has asked me to solve this question but I am unable to than I thought to post it here may be someone help. Q : The watchman works 4 days a week and has a rest on the fifth day. He had ...
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### If $R$ and $S$ are fields, either prove or disprove that $R\times S$ is a field

That's the question from my homework. I am thinking $R\times S$ is not a field, but I'm not sure. I understand the definition of a field, but I am not sure how to proceed.
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### Evaluation of the limit $\lim\limits_{n \to \infty } \frac1{\sqrt n}\left(1 + \frac1{\sqrt 2 }+\frac1{\sqrt 3 }+\cdots+\frac1{\sqrt n } \right)$

Evaluate the limit : $$\lim_{n \to \infty } {1 \over {\sqrt n }}\left( {1 + {1 \over {\sqrt 2 }} + {1 \over {\sqrt 3 }} + \cdots + {1 \over {\sqrt n }}} \right)$$ I can use the sandwich principle, ...
### If $x^3 =x$ then $6x=0$ in a ring
Let $R$ be a ring with unity where $$x^3=x,\;\;\; \forall x \in R$$ How do I prove that $$x+x+x+x+x+x=0$$