Articles of algebra precalculus

Solution of a Lambert W function

The question was : (find x) $6x=e^{2x}$ I knew Lambert W function and hence: $\Rightarrow 1=\dfrac{6x}{e^{2x}}$ $\Rightarrow \dfrac{1}{6}=xe^{-2x}$ $\Rightarrow \dfrac{-1}{3}=-2xe^{-2x}$ $\Rightarrow -2x=W(\dfrac{-1}{3})$ $\therefore x=\dfrac{-1}{2} W(\dfrac{-1}{3})$ But when i went to WolframAlpha, it showed the same result but in the graph: WolframAlpha Graph The curves intersect at a point… And hence there is a second […]

Probability that no two consecutive heads occur?

A fair coin is tossed $10$ times. What is the probability that no two consecutive tosses are heads? Possibilities are (dont mind the number of terms): $H TTTTTTH$, $HTHTHTHTHTHTHT$. But except for those, let $y(n)$ be the number of sequences that start with $T$ $T _$, there are two options, $T$ and $H$ so, $y(n) […]

$p$ prime, $1 \le k \le p-2$ there exists $x \in \mathbb{Z} \ : \ x^k \neq 0,1 $ (mod p)

I found this problem in my algebra book, but unfortunately, there is no solution included. Here it is: Let $p$ be a prime, $1 \le k \le p-2$. Show that there exists $x \in \mathbb{Z} \ $ such that $\ x^k \neq 0,1$ (mod $p$). It looks like a very nice problem, but I have […]

Conditions for a unique root of a fifth degree polynomial

Fifth degree polynomials cannot generally be solved analytically, but at least one solution always exists. Given the normal form $$ax^5+bx^4+cx^3+dx^2+ex+f=0,$$ is it possible to find sufficient conditions on $a,b,c,d,e,f$ that guarantee a unique solution? One sufficient condition is obviously $a,b,c,d,e>0$, so that the LHS of the above equation is strictly increasing in $x$. In my […]

How to solve $\mathrm{diag}(x) \; A \; x = \mathbf{1}$ for $x\in\mathbb{R}^n$ with $A\in\mathbb{R}^{n \times n}$?

I would like to solve the following equation for $x\in\mathbb{R}^{n}$ $$\mathrm{diag}(x) \; A \; x = \mathbf{1}, \quad \text{with $A\in\mathbb{R}^{n\times n}$},$$ where $\mathrm{diag}(x)$ is a diagonal matrix whose diagonal elements are the elements of $x$ and $\mathbf{1}$ is a vector whose elements are equal to 1. I will already be very happy to find a […]

Find area bounded by two unequal chords and an arc in a disc

Math people: This question is a generalization of the one I posed at . Below is an image of a unit circle with center $O$. $\theta_1, \theta_2 \in (0, \pi)$ and $\gamma \in (0, \min(\theta_1,\theta_2))$. $\theta_1 = \angle ROS$, $\theta_2 = \angle POQ$, and $\gamma = \angle ROQ$. I want to find the area […]

$a,b,c$ are positive reals and distinct with $a^2+b^2 -ab=c^2$. Prove $(a-c)(b-c)<0$

This question already has an answer here: If $a,b,c$ are positive integers, with $a^2+b^2-ab=c^2$ prove that $(a-b)(b-c)\le0$. 6 answers

$2d^2=n^2$ implies that $n$ is multiple of 2

I’m reading a proof of the irrationality of $\sqrt 2$. In a step it states that $2d^2=n^2$ implies that $n$ is multiple of 2. How?

Solving this equation $10\sin^2θ−4\sinθ−5=0$ for $0 ≤ θ<360°$

The first part of the question asks me to square both sides of the equation: $$3 \cos θ=2 − \sin θ$$ So that I can obtain and solve the quadratic: $$10\sin^2θ−4\sinθ−5=0 \;\;\text{for}\;\; 0 ≤ θ<360°$$ solutions obtained in the interval are: $69.2°, 110.8°, 212.3°, 327.7°$ However the second part of this question stumps me, it […]

(Dis)prove that this system has only integral solutions: $\sqrt x+y=7$and $\sqrt y+x=11$

This is the system of equations: $$\sqrt { x } +y=7$$ $$\sqrt { y } +x=11$$ Its pretty visible that the solution is $(x,y)=(9,4)$ For this, I put $x={ p }^{ 2 }$ and $y={ q }^{ 2 }$. Then I subtracted one equation from the another such that I got $4$ on RHS and […]