# Tagged Questions

Linear, exponential, logarithmic, polynomial, rational, and trigonometric functions, conic sections, binomial, surds, graphs and transformations of graphs, equation- and system-solving, and other symbolic-manipulation topics.

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### How to prove the inequalities between $20^{70^2},30^{60^2},40^{50^2}$

Let $$M=\{ 20^{70^2}, 30^{60^2},40^{50^2}\}$$. What number is the greatest and which is the smallest? I thought about beginning by assuming certain inequalities and trying to prove them, for example: ...
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### $2k-1$ is an odd integer if $k$ is an integer

I am working on this advanced power rule problem: This is the image of the problem I understand everything up until step 4 in the problem hint. I am getting stuck with the statement that says: "...
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### Prove That If $(a + b)^2 + (b + c)^2 + (c + d)^2 = 4(ab + bc + cd)$ Then $a=b=c=d$

If the following equation holds $$(a + b)^2 + (b + c)^2 + (c + d)^2 = 4(ab + bc + cd)$$ Prove that $a$,$b$,$c$,$d$ are all the same. What I did is I let $a$,$b$,$c$,$d$ all equal one number. Then I ...
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### How to find area of a polygon built on the roots of a given polynomial?

How to find the area of a (maximum area convex) polygon, built on the roots of a given polynomial in the complex plane? For example, consider the equation: $$2x^5+3x^3-x+1=0$$ It has one real and ...
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### Time and work issue

Question: 45 men can complete a work in 16 days. Six days after they started working, 30 more men joined them. How many days will they now take to complete the remaining work ? Answer of this ...
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### Find the values of $b$ for which the equation $2\log_{\frac{1}{25}}(bx+28)=-\log_5(12-4x-x^2)$ has only one solution

Find the values of 'b' for which the equation $$2\log_{\frac{1}{25}}(bx+28)=-\log_5(12-4x-x^2)$$ has only one solution. =$$-2/2\log_{5}(bx+28)=-\log_5(12-4x-x^2)$$ My try: After removing the ...
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### Find the value of $P(1)$

Let $P (x) = x^2 + bx + c$, where $b$ and $c$ are integer. If $P(x)$ is a factor of both $x^4 + 6x^2 + 25$ and $3x^4 + 4x^2 + 28x + 5$, find the value of $P(1)$. I am not being able to solve ...
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### What's the relation between earth coordinates and angles?

I've been looking for an answer for a specific question, a part of my question maybe related to this: Calculate the angle of a vector in compass (360) direction However, my question is more specific, ...
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### An inequality involving two complex numbers

Let $z_1, z_2 \in \mathbb C$ and $a,b \in \mathbb{R} \setminus \{0\}$. Prove that $$|z_1|^2+|z_2|^2-|z_1^2+z_2^2|\le 2\dfrac{|az_1+bz_2|^2}{a^2+b^2}\le |z_1|^2+|z_2|^2+|z_1^2+z_2^2|$$ ...
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### Quadratic Equation Based Problem:Prove either $a = 2l$ & $b = m$ or $b + m = al$

If by eleminating $x$ between the equation $x² + ax + b = 0$ & $xy + l (x + y) + m = 0$, a quadratic in $y$ is formed whose roots are the same as those of the original quadratic in $x$. Then ...
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### Find pressure in a sinusoidal function

Tiffany is a model rocket enthusiast. She has been working on a pressurized rocket filled with laughing gas. According to her design, if the atmospheric pressure exerted on the rocket is less than 10 ...
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### Is $0.999999999\ldots = 1$?

I'm told by smart people that $0.999999999\ldots = 1$, and I believe them, but is there a proof that explains why this is?
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### Finding the sum of $\cos45°$ + $i\cos135°$ + … + $i^{n}\cos(45+90n)°$ + … + $i^{40}\cos3645°$

My question is as follows: If $i^{2}$ = -1, find the value of $$\cos45° + i\cos135° + \ ...\ + i^{n}\cos(45+90n)° + \ ...\ + i^{40}\cos3645°$$ without the aid of a calculator. In terms of my attempts ...
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### Find a point on $y=\frac{1}{x^2}$ such that $y'=16$

I'm very new in this forum and I hope I don't ask something silly, which is asked many times before. I have to answer this question: Find the coordinates of the point(s) at which the curve has ...
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### Progressions modulo $n$

I don't understand how to do these 2 tasks: 1) Prove that any arithmetic progression modulo $n$ has a period that divides $n$. 2) Prove that any geometric progression modulo a prime number $p$ has a ...
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In an election, $70\%$ of males were registered voters and $40\%$ females were registered voters, all registered males casted their votes. But only $65\%$ registered females casted their votes. If ...
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### Why do non-linear functions have to be in V or U shape, why can't it just be one line? [closed]

The title says it all really, I just don't get why you can't just use one line.
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$f(x) = \begin{cases} 2x+3, & \text{if x$\lt$3} \\[2ex] x^2, & \text{if$x \ge 3$} \end{cases}$ $,\qquad$ $g(x) = \begin{cases} 7-2x, & \text{if x$\le$2 } \\[2ex] x+1, & \... 0answers 13 views ### Is$\mu = \frac{\beta sin(\frac{\pi}{\beta})cosh\zeta R - 1}{\langle\kappa\rangle N}$given the equation? I had sent this question before but with more unnecessary details that made it look like a complex problem so here I just want to clarify one thing. Given the equation:$\langle k \rangle = \frac{2\...
Now, I know this has been asked here but my question is something else so please bear with me. Question:- If the vertices of a triangle are represented by $z_1, z_2, z_3$ respectively then show ...
### Taylor expansion $f(x)=f(0)$
The following taylor expansion of the function $f(x)$, requires $f(x)$ to have a derivative up to what order? $$f(x)=f(0)+f'(0)x+f''(0)x^2/2+\mathcal{O}(x^3)$$ My solution: Based on the Taylor'...