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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2
votes
2answers
114 views

Using the basic laws of exponent

I have some problems with this question. Please help me. Thanks Simplify given expression$$ a^2 (abc)^{-2} a^3 b^7 $$ What are exponents of $a$, $b$, and $c$? I get $3,5,-2$ as exponents of $a,...
0
votes
1answer
29 views

Find parameter m knowing that the values of the function are in a interval of length 4

Please give me a hint on how to find the parameter $m$ knowing that the function values are in an interval of length $4$: $f(x)=\frac{x^2 + mx + 1 }{x^2-x+1}$.
12
votes
4answers
1k views

Why can a quartic polynomial never have three real and one complex root?

It seems that a quartic polynomial (degree $4$) either can have $0$ real, $1$ real, $2$ real, or $4$ real roots, and the rest is complex roots. Why can't it have $3$ real roots and $1$ complex?
2
votes
6answers
293 views

Prove that the derivative of $x^w$ is $w x^{w-1}$ for real $w$

Can anyone give a rigorous proof of the derivative of this type of function? Specifically showing, $\frac{d(x^w)}{dx} = wx^{w-1}$ for a real $w$? I tried to use the Taylor series expansion for $(x+...
3
votes
8answers
88 views

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: ...
0
votes
1answer
23 views

$2k-1$ 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: "...
2
votes
2answers
40 views

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 ...
5
votes
0answers
72 views

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 ...
1
vote
3answers
1k views

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 ...
1
vote
1answer
69 views

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 ...
1
vote
2answers
50 views

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 ...
0
votes
1answer
27 views

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, ...
4
votes
1answer
95 views

An inequality involving two complex numbers

Let $z_1, z_2 \in \mathbb C$ and $a,b \in \mathbb{R} \setminus \{0\}$. Prove that $$|z_1|^2+|z_2|^2-|z_1^2+z_2^2|\le 2\dfrac{|az_1+bz_2|^2}{a^2+b^2}\le |z_1|^2+|z_2|^2+|z_1^2+z_2^2|$$ ...
1
vote
1answer
34 views

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 ...
0
votes
1answer
480 views

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 ...
153
votes
22answers
28k views

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?
0
votes
2answers
50 views

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 ...
2
votes
1answer
66 views

Inequality on a sequence of $n$ reals whose sum is $0$

Consider $n\geq3$ real numbers $a_1,a_2,\dots ,a_n$ satisfying $a_1+a_2+\cdots+a_n=0$ and $$2a_k \leq a_{k-1}+a_{k+1}$$ for all $2\leq k\leq n-1$. Prove that $$|a_k|\leq\frac{n+1}{n-1}\,\max\big\{|a_{...
1
vote
0answers
42 views

Proving $a^ma^n=a^{m+n}$ by induction when $n$ or $m$ is negative (or both)

Suppose we have already proved this exponent law for when $m,n\in\mathbb{Z^+}$ as in here. Also suppose $x^{-n}=\frac{1}{x^n}$ is given as a definition. Let $m=-\lambda$ and $n=-\gamma$, where $\...
2
votes
3answers
126 views

Evaluate $\cos 36^\circ - \cos 72^\circ$ without the aid of a calculator [duplicate]

I have a quick question about a difficult trigonometric functions problem that I have been assigned. The problem is as follows: Evaluate $$\cos36° - \cos72°$$ without the aid of a calculator. In terms ...
1
vote
2answers
47 views

Prove that this is one-one, but not onto $\Bbb R$.

$\Bbb R$ stands for real numbers. $ f(x) = \begin{cases} 2-x, & \text{if $x \le 1 \qquad \text{is one to one but not onto } \Bbb R $ } \\ \frac{1}{x} , & \text{if $x >1$ } \end{cases}...
0
votes
1answer
34 views

Sides of triangle are in A.P., find its perimeter

The sides of a triangle are in Arithmetic Progression $(A.P.).$ If the smallest angle of the triangle is $\alpha$ and largest angle of the triangle exceeds smallest angle by $\beta$ , then what is the ...
2
votes
2answers
26 views

How to Solve $2.3856 + \log r = \log(364r - 363)$

I am solving geometric sequence and series problem, but got stuck on the logarithm part. we haven't tackled logarithm yet so this is supposed to be a challenge problem. can anyone advise on how to ...
3
votes
1answer
55 views

Simplifying $\sqrt[5]{1+g+g^3}=\frac {\sqrt{1+g^2}}{\sqrt[10]{5}}$ and similar ones

I saw that Ramanujan simplified many radicals such as: For $g^5=2$ $$\sqrt[5]{1+g+g^3}=\frac {\sqrt{1+g^2}}{\sqrt[10]{5}}\tag{1}$$ For $g^4=5$ $$\frac {\sqrt[5]{3+2g}-\sqrt[5]{4-4g}}{\sqrt[5]{3+2g}+\...
2
votes
2answers
49 views

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 ...
1
vote
3answers
44 views

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 ...
0
votes
0answers
22 views

Percentage of votes received in an election

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 ...
2
votes
1answer
70 views

Why must $|z|\gt 1$ be the necessary condition

Question:- If $\left|z+\dfrac{1}{z} \right|=a$ where $z$ is a complex number and $a\gt 0$, find the greatest value of $|z|$. My solution:- From triangle inequality we have $$|z|-\left|\dfrac{1}{...
1
vote
2answers
32 views

How to solve this inequality problem?

Given that $a^2 + b^2 = 1$, $c^2 + d^2 = 1$, $p^2 + q^2 = 1$, where $a$, $b$, $c$, $d$, $p$, $q$ are all real numbers, prove that $ab + cd + pq\le \frac{3}{2}$.
-1
votes
1answer
68 views

Is there a basis which spans the real numbers?

Is there a finite set of real numbers $S=\{a_1, a_2, ..., a_n \}$ such that every real number can be written as a linear combination (with integer coefficients) of the elements of $S$? If no, is there ...
2
votes
1answer
299 views

Reducing an indicator function summation into a simpler form.

Context I am attempting to reduce the space I need to store in an array in a program. I have made it so that the indices are always sorted. There are no indices where they are equal, and no indices ...
2
votes
1answer
58 views

Sum of series $1−ω^2+ω^4−ω^6+ω^8−ω^{10}+ω^{12}+⋯+ω^{600}−ω^{602}+ω^{604}$

I need to find sum of the series involving cube roots of unity $1−ω^2+ω^4−ω^6+ω^8−ω^{10}+ω^{12}+⋯+ω^{600}−ω^{602}+ω^{604}$. Found it in an old test paper. I applied Geometric Progression Sum Formula....
1
vote
4answers
152 views

Find $LK_1^2 + LK_2^2 + \dots + LK_{11}^2$.

$K_1 K_2 \dotsb K_{11}$ is a regular $11$-gon inscribed in a circle, which has a radius of $2$. Let $L$ be a point, where the distance from $L$ to the circle's center is $3$. Find $LK_1^2 + LK_2^2 + \...
3
votes
4answers
421 views

Geometry with complex numbers.

Let $a$, $b$, $c$, and $d$ be four complex numbers on the unit circle, such that the line joining $a$ and $b$ is perpendicular to the line joining $c$ and $d$. Find a simple expression for $d$ in ...
3
votes
1answer
125 views

Let $A_1 A_2 \dotsb A_{11}$ be a regular 11-gon inscribed in a circle of radius 2. [duplicate]

Let $A_1 A_2 \dotsb A_{11}$ be a regular 11-gon inscribed in a circle of radius 2. Let $P$ be a point, such that the distance from $P$ to the center of the circle is 3. Find $PA_1^2 + PA_2^2 + \dots +...
-1
votes
1answer
23 views

what will be following equation for A, B, C interms of m, n and X, Y, Z [closed]

$$X = A + B+C$$ $$Y = mA + nB$$ $$Z = nA + mB$$ then $A = $? in terms of $m$, $n$ and $X$, $Y$, $Z$ and $B = $? in terms of $m$, $n$ and $X$, $Y$, $Z$ and $C = $? in terms of $m$, $n$ and $...
1
vote
1answer
30 views

The greatest common divisor of $(O_n, T_n+2)$ where $O_n$ and $T_n$ are the oblong and triangular numbers respectively.

Suppose that $T_n$ is odd. Can we find infinitely many $n$ such that $(O_n, T_n+2)=1$? Is it trivial and obvious? My hunch based on some hand calculations is to look at $n$ congruent to $0$ or $2$ ...
3
votes
3answers
228 views

Non-induction proof of $2\sqrt{n+1}-2<\sum_{k=1}^{n}{\frac{1}{\sqrt{k}}}<2\sqrt{n}-1$

Prove that $$2\sqrt{n+1}-2<\sum_{k=1}^{n}{\frac{1}{\sqrt{k}}}<2\sqrt{n}-1.$$ After playing around with the sum, I couldn't get anywhere so I proved inequalities by induction. I'm however ...
1
vote
3answers
48 views

Find the following one-sided limits?

$$\text{a)} \ \ \lim_{x\to-2^+}(x+3)\frac{|x+2|}{x+2}$$ $$\text{b)} \ \ \lim_{x\to-2^-}(x+3)\frac{|x+2|}{x+2}$$ The answers are: $$\text{a)} \ \ 1$$ $$\text{b)} -1$$ How do you find them? It is ...
5
votes
1answer
35 views

Given a finite sequence, can we always find a relation that generates that sequence?

This is just something I've been wondering about, but I have no idea what the answer is. I suspect it's yes. Given an arbitrary finite sequence, can we always find a relation that generates that ...
0
votes
2answers
125 views

How does $x^4+y^4=z^2 \implies x^4+y^4=z^4$?

Why is the statement "the following cannot be satisfied" for $x^4+y^4=z^2$ more strong than for $x^4+y^4=z^4?$ More specifically, how does $x^4+y^4=z^2 \implies x^4+y^4=z^4?$ This statement was ...
4
votes
2answers
63 views

How many distinct ways are there to $2$-color the $8$ vertices of a cube?

How many distinct ways are there to $2$-color the $8$ vertices of a cube, with colorings only considered distinct up to rotation?
0
votes
3answers
53 views

N is a four digit number. If the leftmost digit is removed, the resulting three digit number is 1/9th of N. How many such N are possible? [closed]

N is a four digit number. If the leftmost digit is removed, the resulting three digit number is 1/9th of N. How many such N are possible with solution?
2
votes
0answers
36 views

Minimize a huge two-variable logarithmic-trigonometric-radical expression (MSU entrance early July 2016)

Minimize \begin{align}R(a,x)&=\sqrt{13+\log_a\left(\cos\left(\frac xa\right)\right)^2+\log_a\left(\cos\left(\frac xa\right)^4\right)}+\sqrt{97+\log_a\left(\sin\left(\frac xa\right)\right)^2-\...
3
votes
3answers
273 views

The number of distinct real roots of a polynomial

I have trying to solve this problem for a long time now. After having read related concepts, I am still stuck. The problem is as follows- Find the number of distinct real roots of the equation $$x^4-...
2
votes
1answer
76 views

Trouble Finding $f \circ g \; \text{ and } \; g\circ f$ for this function?

$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, & \...
0
votes
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\...
1
vote
1answer
50 views

Finding the orthocentre of a trinagle.

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 ...
1
vote
1answer
61 views

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'...