$$\begin{bmatrix}3 & -2\\-1 & -3\end{bmatrix}X + \begin{bmatrix}1 & 5\\0 & 7\end{bmatrix} = \begin{bmatrix}-5 & 8\\-1 & 3\end{bmat...

I am trying to solve the following nonlinear ODE: $$\frac{dy}{dx} = \frac{1}{x(ayx-b)},$$ where $a, b$ are constants and $a>0$. Moreover, you may assum...

Today I was thinking about proving this statement, but I really could not come up with an idea at all. I want to prove that $\sin10^\circ$ is irrational. ...

Suppose that $K$ is a non-trivial, alternating knot. Is it possible that $\det K = 1$ where $\det K=\Delta_K(-1)$? Using knotinfo, I checked that all non-...

I am a bit unsure where I have gone wrong in working this out. Sech$(x)=2/(e^x+e^{-x}).$ Maclaurin expansions: $e^x = 1+ x + x^2/2+ x^3/6 + x^4/24;\; e^{-...

I was going through this question given to my friend at university, and I can do it with a little calculation. Is there a set algorithm that will work or ...

Can someone explain how the equation in the below image is possible? As far as I remember $Var[aX] = a^2.Var[X]$ You can access the image

We can multiply $a$ and $n$ by adding $a$ a total of $n$ times. $$ n \times a = a + a + a + \cdots +a$$ Can we define division similarly using only additi...

Claim: $\color{red}{2^\sqrt2<e}$ Note: $2^\sqrt2=e^{\sqrt2\ln2}$ Different approach: We show $e^{x-1}>x^\sqrt x$ for $x>2$. Let $f(x) = x -1 - \s...

An open box is to be constructed so that the length of the base is $3$ times larger than the width of the base. If the cost to construct the base is $5$ d...