# All Questions

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### Constructible Solutions

We know that if a cubic equation with a rational coefficients has a constructible root, then the equation has a rational root. Now let; $$x^3-6x+2\sqrt{2}=0$$ A hint tells us to make the correct ...
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### Find distance of opposing peaks in hypercube

First of all, I am sorry for my english, it is very bad and I realize that. We have a hypercube of k order ( k=3 in this case ...
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### General Solution of differential equation.

Find the general solution of the differential equation: $$\frac{d^2y}{dx^2}+2\frac{dy}{dx}+5y=4\cos x.$$ Hence determine the solution which satisfies, $y=0$, $\frac{dy}{dx}=0$ when $x=0$.
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### Showing that similar matrices have the same minimal polynomial

I am in the process of proving the title. The hint says, for any polynomial $f$, we have $$f(P^{-1}AP) = P^{-1}f(A)P.$$ A is an $n \times n$ matrix over $F$ while $P$ is an invertible matrix such ...
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### Computing a limit of root difference

How do you compute the following limit by only using known limits, basic limit properties and theorems, such as the squeeze theorem. $\lim\limits_{n\to\infty} \sqrt[]{n}·(\sqrt[n]{3}-\sqrt[n]{2})$ ...
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### Intuition: Why is continuous decay expressed as the inverse of the equivalent continuous growth rate?

I understand $e$ as $\lim_{n \to \infty} \big(1+\frac{1}{n}\big)^n$. I also (finally) understand the idea that continuous growth is "a rate that is applied constantly to the amount present at any ...
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### Property of divisibility of numbers

How to prove that if $d=GCD(ac,b)$ and $GCD(a,b)=1$ then $d|c$.
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### Learning Differential Geometry

What levels of mathematics does one need to nail down before successfully studying Differential Geometry?
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### Canonical map $R/(I\cap J)\rightarrow R/I\times _{R/(I+J)} R/J$ is an isomorphism

From this MSE question I understand the canonical map $R/(I\cap J)\rightarrow R/I\times _{R/(I+J)} R/J$ is an isomorphism for $R$ a commutative ring and $I,J$ ideals. I tried proving this directly ...
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### The elliptic integral $\frac{K'}{K}=\sqrt{2}-1$ is known in closed form?

Has anybody computed in closed form the elliptic integral of the first kind $K(k)$ when $\frac{K'}{K}=\sqrt{2}-1$? I tried to search the literature, but nothing has turned up. This page ...
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### Integrate and find domain $\int\arccos(\sqrt{\frac{x-4}{x+6}})dx$

I need integrate and find domain of integral: $$\int\arccos(\sqrt{\frac{x-4}{x+6}})dx$$ How can i solve it? is it good way substitute argument of arccos? $$t=\sqrt{\frac{x-4}{x+6}}$$
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### $\iiint_W\sqrt{x^2+z^2}\,\mathrm{d}V$ Where $W$ is the solid delimited by $y=4$ and $y=\sqrt{x^2+z^2}$

$\iiint_W\sqrt{x^2+z^2}\,\mathrm{d}V$, $W$ is limited by the plane $y=4$ and the paraboloid $y=\sqrt{x^2+z^2}$. I'm trying to solve with spherical coordinates, however I got stuck in the following ...
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### Sample size for a desired accuracy

I recently revised for my statistics paper, and in a sample paper given by my lecturer, I am puzzled by how he derives with the answer. So. here is the question A study based on a sample size of 36 ...
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### Linear operator between $l^\infty$ and $l^2$

Let $T:\mathcal{l}^{\infty}(\mathbb{R})\to\mathcal{l}^2 (\mathbb{R})$ be given by $$T\left((x_n)_{n\in \mathbb{N}}\right) \colon= \left(\dfrac{1}{2^n} x_{2^n}\right)_{n\in \mathbb{N}}.$$ Find ...
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### Using basic knowledge about the roots of a quintic polynomial to determine if the polynomial is solvable by radicals

Assume that $f(x)$ is a quintic polynomial with integer coefficients and is irreducible over $\mathbb{Q}$. If $f(x)$ has three distinct real roots and two non-real complex roots, then $f(x)$ is not ...
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### Show both are equivalent.

$\exists xP(x)\implies P(y)$ is equivalent to $\forall x(P(x)\implies P(y))$ What can I show in this question to prove that they are equivalent. Any help is appreciated. Thanks.
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### Does this condition imply that $f'(z) \neq 0$?

Suppose $f:A \to B$ is bijective and analytic with analytic inverse. Then $f'(z) \neq 0$ for all $z \in A$. We know $f^{-1} \circ f (z) = z$ and so $(f^{-1})'( f^{-1} \circ f)(z) = 1$. why from ...
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### conditional probability and expectation of random variables

It is given that $E[Y/A]$ where $A = {X \in C}, C \in B(R)$; Want to indicate how to compute $E[Y/A]$. Here $X$ and $Y$ are random variables with joint density $f(x,y)$. Any ideas will be very ...
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### Show that $K_1K_2/k$ is Galoisienne if $K_1$ and $K_2$ are Galoisienne.

Let $k$ a field and $K_1/K$ and $K_2/k$ two Galois extension. 1) Show that $K_1K_2/k$ is a Galois extension. 2) Let $\varphi: Gal(K_1K_2/k)\longrightarrow Gal(K_1/k)\times Gal(K_2/k)$ defined by ...
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### What are the roots of $x^{2} = 2^{x}$?

What are the roots of $x^{2} = 2^{x}$? I drew the graphs and found $x = 2$ and $x = 4$, and there is one other root in $[-1,0]$. Can anyone describe an algebraic method to obtain all roots?
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### How do you compute the determinant of the square matrix with $1$'s on the diagonal and the entry's next to it?

How do you compute the determinant of the square matrix with $1$'s on the diagonal and the entry's next to it (in other words, the entry in the $i$-th row and the $j$-th column equals $1$ if ...
### Solve the differential equation for obtaining $x$ as a relation of $t$: $\frac{d^2x}{dt^2}=\alpha\sqrt{x}$
Question: Solve the differential equation for obtaining $x$ as a relation of $t$: $$\frac{d^2x}{dt^2}=\alpha\sqrt{x}$$ My attempt: $$\frac{d^2x}{dt^2}=\alpha\sqrt{x}$$ \Rightarrow ...
### Continuity of the limit of ${f_n}(x) = \cos(2^n\pi x)$.
I am trying to prove that the metric space of continuous functions from [0,1] to $\mathbb{R}$ is not compact by examining $\lim_{n \to \infty} {f_n}(x) = \cos(2^n\pi x)$. I cannot think of a direct ...