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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0
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1answer
17 views

Angle of view based on height and distance to a determined object

I'm trying to determine what angle of view is needed for a photo shoot so that I can determine which super telephoto lens to rent. I'm photographing an object thats 2,600 meters across from an ...
4
votes
2answers
89 views

Let $a,b,c>0$ so that $a+b+c=1$…

Let $a,b$ and $c$ be positive real numbers such that $a+b+c=1$. Prove that $$\frac{a}{b}+\frac{b}{a}+\frac{b}{c}+\frac{c}{b}+\frac{c}{a}+\frac{a}{c}+6\geq 2\sqrt{2}\left ( ...
5
votes
4answers
145 views

Why are there only 2 solutions for $x^n=1$?

(where $n>0$) I have been taught that an equation with the highest power $n$ will always have $n$ solutions. This does not appear to be the case with: $$x^n=1 \implies x=\pm1$$ Where $n$ is even, ...
1
vote
1answer
34 views

Finding values of $a$ with which a simple system has exactly 2 solutions

The problem is: Find such values of $a$ with which the system will have exactly two solutions I understand the solution provided at the Resuhege.ru website (problem no. 484630): First ...
0
votes
1answer
15 views

Converting word problems with speed into algebra

'A rower travels upstream at $6$ km per hour and back to the starting place at $10$ km per hour. The total journey takes $48$ minutes. How far upstream did the rower go?' I'm struggling turning the ...
2
votes
4answers
72 views

Why does basic algebra provide one value for $x$ when there should be two?

I have the equation $x^2=x$. If I divide $x$ from both sides I get $x=1$. Yet clearly $x$ can also equal $0$. What step in this process is wrong? It seems to me that there's only one step. And ...
3
votes
5answers
53 views

Quadratics question

To solve $-3x^2 +2x +1=0$, I'd normally break the middle term and then factorise. But I was wondering if there was a way to skip the factorising step? The factors I'd use in place of the middle term ...
1
vote
1answer
20 views

Create a set of system of linear equations to answer the following.

A factory is currently running at $85\%$ of its original capacity, and management is considering upgrading the equipment. The upgrade will take $6$ months, during which time the factory will not ...
5
votes
3answers
50 views

Line for set of three-dimensional vectors

If there is a set for 3D vectors $v$ where $ v \times \begin{pmatrix} -1 \\ 1 \\ 4 \end{pmatrix} = \begin{pmatrix} 5 \\ -27 \\ 8 \end{pmatrix}$ is a line, what is this line's equation? I'm not sure ...
2
votes
1answer
47 views

Let $a^n = a^{n - 1} + a^{n -2}$. Show that for any $A, B$, $F(n) = Aa^n + Bb^n$ satisfies Fibonacci recurrence relation.

$$\begin{align*} F(n) &= Aa^n + Bb^n\\ &= A(a^{n-1}+a^{n-2}) + B(b^{n-1}+b^{n-2}) \\ &= Aa^{n -1} + Aa^{n-2} + Bb^{n -1} + Bb^{n-2}\\ &= a^{n -1} (A + A^{a-1}) + b^{n - 2} (B + bB) ...
0
votes
1answer
34 views

Find the number of sets satisfying the conditions

Let $ N$ be the number of ordered pairs of nonempty sets $ \mathcal{A}$ and $ \mathcal{B}$ that have the following properties: • $ \mathcal{A} \cup \mathcal{B} = ...
1
vote
1answer
28 views

Finding a Recurrence Relation.

This is from AMC 2015 . For each positive integer n, let S(n) be the number of sequences of length n consisting solely of the letters A and B, with no more than three As in a row and no more than ...
2
votes
1answer
38 views

How to find solutions for this nonlinear equation?

I want to find an analytical solution $x$ as a function of parameters $(e,u,r,t)\in\mathbb{R}^4$ that satisfies the following condition: ...
1
vote
1answer
36 views

Find circumradius of $\Delta DEC$

$A(0,0),B(4,0)$ and $C(5,-2\sqrt 6)$ are the vertices of $\Delta ABC$. Incircle of the triangle touches side $AC$ and $BC$ at $D$ and $E$ respectively. Find the circumradius of the triangle $DEC$. Is ...
3
votes
3answers
48 views

Number of Non - Decreasing functions?

Let A={1,2,3.....10} & B={1,2,3....20}. We have to find the number of non decreasing functions from A-->B. What I tried :No. Of non decreasing functions = (Total functions) - (Number of ...
1
vote
1answer
47 views

Solve the equation: $(9x^2+6x-8)\sqrt{3x+2}+6x+23=27x^2+3\sqrt{10+3x}$

Solve the equation: $(9x^2+6x-8)\sqrt{3x+2}+6x+23=27x^2+3\sqrt{10+3x}$ I used wolframalpha.com and got only solution $x=-\dfrac{1}{3}$. And this is my try: Condition: $x\ge-\dfrac{2}{3}$. ...
3
votes
1answer
37 views

Prove that $\left (\sum_{k=1}^{n}\frac{1+x^{2k}}{1+x^{4k}} \right )\left ( \sum_{k=1}^{n}\frac{1+y^{2k}}{1+y^{4k}} \right )< \frac{1}{(1-x)(1-y)}.$

Let $n$ be a positive integer, and let $x$ and $y$ be positive real numbers such that $x^{n}+y^{n}=1$ Prove that $$\left (\sum_{k=1}^{n}\frac{1+x^{2k}}{1+x^{4k}} \right )\left ( ...
-1
votes
1answer
38 views

Finding values of $a, b$ such that $0\le x^4 +x^3 +ax+b\le (x^2-1)^2$

Given real values of $a, b$ such that for all $x\ge0$, $$0\le x^5+x^3+ax+b\le (x^2-1)^2\ ,$$ find the value of $ab$. What I've done is let $x=1$, thus $$0\le2+a+b\le0$$ this forces $a+b=-2$. let ...
0
votes
2answers
78 views

find the value of $x^{(x^{2})}+x^{(x^{8})}$

If $$x^{(x^{4})}=4$$ Then find the value of $$x^{(x^{2})}+x^{(x^{8})}$$ I did solve this, I want to see more solutions, thanks
1
vote
2answers
35 views

quadratic reduction problem

A train is travelling between two stations that are $100$ km apart at a speed of $v$ km/h. Express the time taken for the journey in terms of $v$. Here I got $\ t=\dfrac{100}{v}$. On the return ...
3
votes
2answers
78 views

Why am I getting two answers for 8th root of continued fraction

Find value of $x$: $x=\sqrt[8]{2207-\frac{1}{2207-\frac{1}{2207-....and\,so\, on}}}$ On solving ,we have $x^8=2207-\frac{1}{x^8}$ $x^8+\frac{1}{x^8}=2207$ $x^4+\frac{1}{x^4}=47$ ...
0
votes
2answers
50 views

How do I isolate/solve for $\theta$ in $\sin (2\theta) = 4 \cos (2\theta)$

Isolate the variable/solve for $\theta$: $$\sin (2\theta) = 4 \cos (2\theta)$$ Like which $\cos$ double angle formula would I use? Because there are three of them. Thanks in advance.
0
votes
3answers
27 views

If the coefficients of the y-terms are equal and the coefficients of the x-terms are equal, the graphs of the two lines will be parallel.

So a student has a claim that for any pair of linear relations, if the coefficients of the y-terms are equal and the coefficients of the x-terms are equal, the graphs of the two lines will be ...
3
votes
1answer
56 views

Maximize the Cyclic sum

Let $x_1,x_2,\dots ,x_6$ be nonnegative real numbers such that $x_1+x_2+x_3+x_4+x_5+x_6=1$, and $x_1x_3x_5+x_2x_4x_6 \geq \frac{1}{540}$. Let $p$ and $q$ be positive relatively prime integers such ...
2
votes
2answers
33 views

find the limit of $ \lim_{(x,y) \rightarrow (0,0)}\frac{2xy^3+x^2y^3}{x^4+2y^4}$

find the limit of $$ \lim_{(x,y) \rightarrow (0,0)}\frac{2xy^3+x^2y^3}{x^4+2y^4}$$ I have absolutely no idea how to proceed with that. I would prefer a solution that would involve use of squeeze ...
0
votes
1answer
37 views

finding the value of x from a complex form of absolute value

How do I find the value of x for this one, $$ |x-3|^{\frac{x+1}{4}} = |x-3|^{\frac{x-2}{3}}$$ I tried equating the exponents when I found out that the base of both sides are equal but I don't think ...
-1
votes
2answers
64 views

Problem involving geometric progression [closed]

Question: The bacteria in a certain culture double every $7.3$ hours. The culture has $7,500$ bacteria at the start. How many bacteria will the culture contain after $3$ hours? Possible Answers: a. ...
1
vote
1answer
53 views

a very basic question on finding the discriminant for $x^2+2(a-3)x-3a-7=0$

Sorry for asking such a basic question. In the following quadratic equation $$x^2+2(a-3)x-3a-7=0$$ by my calculations, $$D=\left(\frac{b}{2}\right)^2-ac=(a-3)^2-1(-3a-7)=a^2-6a+9+3a+7=a^2-3a+16$$ ...
0
votes
3answers
45 views

limit of $\frac{x^3y+xy^2}{x^2+y^2}$

find the limit of $$ \lim_{(x,y) \rightarrow (0,0)}\frac{x^3y+xy^2}{x^2+y^2}$$ no idea how to deal with that. I've tried to use squeeze theorem however this attempt was unsuccessful.
5
votes
2answers
119 views

Calculate $S=3\sqrt{\sqrt[3]{5}-\sqrt[3]{4}}-\sqrt[3]{2}-\sqrt[3]{20}+\sqrt[3]{25}$

Calculate $$S=3\sqrt{\sqrt[3]{5}-\sqrt[3]{4}}-\sqrt[3]{2}-\sqrt[3]{20}+\sqrt[3]{25}$$ $\color{red}{\text{without using calculator}.}$ Please help me, I can't find any solution to sovle it.
2
votes
2answers
22 views

Then the value of $ [f(2)] $ where [.] represents the greatest integer function is?

A differentiable function f is satisfying the relation $$f(x+y) = f(x) + f(y) + 2xy(x+y) - \dfrac{1}{3} $$ $ \forall $ $ x , y $ belongs to $\Re$ and $$lim_{h \to 0} \dfrac{3f(h)-1}{6h} = ...
0
votes
2answers
32 views

If $f(x) = \max\left|2\sin y-x\right|,$ Then $\min.$ value of $f(x)$

If $f(x) = \max\left|2\sin y-x\right|\;,$ Where $y\in \mathbb{R}\;,$ Then $\min.$ value of $f(x)$ $\bf{My\; try}$ We know that $-2 \leq 2\sin y\leq 2$. Now I did not Understand How Can I open ...
3
votes
2answers
32 views

Area of shaded region circle help

Find the area of the shaded region Area of the sector is $240^\circ$ or $\frac{4\pi}{3}$ Next find $\frac{b\cdot h}{2}$ which is $\frac{2\cdot2}{2}$ which is $2$. Then subtract the former ...
2
votes
4answers
56 views

Prove the existance of the following limit: $\frac{xy+yz+zx}{x^2+y^2+z^2}$

Find the following limit: $$\lim_{(x,y,z) \rightarrow (0,0,0)}\frac{xy+yz+zx}{x^2+y^2+z^2}$$ I have completely no idea what to do with this. Use iterated limits and this will be enough? How to solve ...
0
votes
2answers
34 views

Uniqueness of sum of exponentials

I would like to know if there is an example of two non-trivial sets of real numbers (for the definition of "non-trivial" see below) $X=\{ x_1, \ldots x_n \}$ and $Y = \{ y_1, \ldots y_m \}$, with $m$ ...
1
vote
1answer
22 views

How to re-arrange this formula to make “Y” the subject

$$p =\left(\frac{Y+K}{(Y/p_1)+(K/p_2)}\right)$$ Transpose this formula to make Y the subject The answer is $$Y = \left(\frac{Kp_1(p_2-p)}{p_2(p-p_2)}\right)$$ I have been unable to reach this, my ...
1
vote
2answers
82 views

Elementary Algebra - Distribute value amongst a group of people

I appreciate your help with this question. I'm ready to respond to any question you might have to help solve this problem. If you have an alternate solution it will be accepted as an answer (if ...
11
votes
3answers
163 views

Determine all functions satisfying $f\left ( f(x)^{2}y \right )=x^{3}f(xy)$

Denote by $\mathbb{Q}^{+} $ the set of all positive rational numbers. Determine all functions $f: \mathbb{Q}^{+} \rightarrow \mathbb{Q}^{+}$ which satisfy the following equation for all $x,y \in ...
1
vote
0answers
37 views

Olympiad question [duplicate]

Let $a$ and $b$ be positive integers such that $ab + 1$ divides $a^{2} + b^{2}$. Show that $$ \frac {a^{2} + b^{2}}{ab + 1} $$ is the square of an integer. It's Olympiad question, any help?
1
vote
1answer
21 views

Find the minimum value of $\frac{(\sum_{i=1}^n x_i)^2}{\sum_{i=1}^n \sqrt{x^2_i - i^2}}$

We have $x_i >i$ for all $1 \le i \le n$. Find the minimum value of $$\frac{(\sum_{i=1}^n x_i)^2}{\sum_{i=1}^n \sqrt{x^2_i - i^2}}$$ any help guys please?
3
votes
2answers
65 views

If $x,y,z>0$ and $xyz=32,$ Then the minimum of $x^2+4xy+4y^2+4z^2$ is

If $x,y,z$ are positive real no. and $xyz= 32\;,$ Then Minimum value of $$x^2+4xy+4y^2+4z^2$$ is $\bf{My\; Try::}$ Here I have Used $\bf{A.M\geq G.M}$ Inequality So $$\displaystyle ...
1
vote
3answers
51 views

Solve the system of equations $\begin{cases} xy-2y-3 &=\sqrt{y-x-1}+\sqrt{y-3x+5} \\ (1-y)\sqrt{2x-y}+2(x-1) &=(2x-y-1)\sqrt{y}. \end{cases}$

Solve the following system of equations ($x,y \in \Bbb R$): $$\begin{cases} xy-2y-3 &=\sqrt{y-x-1}+\sqrt{y-3x+5} \\ (1-y)\sqrt{2x-y}+2(x-1) &=(2x-y-1)\sqrt{y}. \end{cases}$$ I think this ...
-4
votes
3answers
43 views

Difference of powers of two [closed]

Is there a simple way (involving minimal calculations) to calculate $2^{987}-2^{986}=?$ Answer: $2^{986}$
-1
votes
2answers
22 views

System of equations help

How do you get $a=2$ and $d=5$ from the two equations (see where I marked it)? Thank you!
0
votes
2answers
47 views

Equation that handles diminishing returns

I've tried desperately to figure this out, but to no avail. I need an equation that will effectively reduce a number by $0.2 \%$ each time it is added to itself (The original number). In other ...
1
vote
3answers
31 views

Solve the following systems of equations by elimination. Verify the solutions.

The first question I came across was $r + 2s + 1 = 0$ and $r + 5s + 28 = 0$ and I had no trouble solving this and I verified the solution of s = -9 and r = 15. The next system is $4m - 3n = 27$ and ...
1
vote
2answers
29 views

Intersection between two three-dimensional planes

The intersection of the planes defined by $x \bullet \begin{pmatrix} 8 \\ 1 \\ -12 \end{pmatrix} = 35$ and $x \bullet \begin{pmatrix} 6 \\ 7 \\ -9 \end{pmatrix} = 70$ is a line. Find an equation of ...
0
votes
1answer
25 views

The second and the fifth terms of a geometric sequence are 16 and 1024…

The second and the fifth terms of a geometric sequence are $16$ and $1024$, respectively. Which term of this sequence is $4,096$? Can someone show me how to do this? I did: $$ar^4 = 1024$$ ...
1
vote
2answers
30 views

Find all vertical and horizontal asymptotes.

Find all vertical and horizontal asymptotes: $$f(x)= \frac{4x}{x^2+16}$$ My Work: (Vertical) 1) $x^2 + 16 = 0$ 2) $x^2=-16$ 3) not possible so there is no vertical asymptote My Work: (Horizontal) ...
0
votes
1answer
75 views

Hookes Law, Calulus and Bungee Jumping

A rope is specially designed and its modulus of elasticity is known from specifications. For the purpose of this problem, assume that the rope is stretched to twice its natural length by a person of ...