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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Math question related to Train

Two trains X and Y, leaves from point A and B towards B and A at the same time, after meeting each other they takes 4 hr 48 min and 3 hr 20 min to reach the point B and A. If the speed of train X is ...
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1answer
18 views

Graphing inequalities on a number line

What software or websites for graphing inequalities on a real number line?
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3answers
62 views

Is it possible to find [on hold]

If $$\frac {(a-b)(c-a)}{(b-c)(d-c)}=\frac {2012}{2013}$$ then find the value of $\dfrac {(a-c)(b-d)}{(a-b)(c-d)}$ in terms of numbers Note: $a,b,c,d$ are real numbers
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1answer
37 views

An easy question regarding Algebra [on hold]

Three schools $A, B, C$ have a total of $480$ students. Ten per cent of the students of school $A$ are going camping and the percentages for school $B$ and school $C$ are 8.5% and 15% respectively. ...
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2answers
30 views

how to prove $a+b-ab \le 1$ if $a,b \in [0,1]$?

Given: $0 \le a \le 1$ $0 \le b \le 1$ Prove: $a + b - ab \le 1$
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3answers
46 views

Factorising quadratics - coefficient of $x^2$ is greater than $1$

In factoring quadratics where the coefficient of $x^2$ is greater than $1$, I use the grouping method where we multiply the coefficient and constant together and then factor. My question is can ...
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1answer
65 views

Questions about $f(n)=3+\frac{12}n$

Experimental Psychology: To study the rate at which animals learn, a psychology student performed an experiment in which a rat was sent repeatedly through a laboratory maze. Suppose the time in ...
5
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2answers
207 views

Quadratics with roots as integers; possible values of a

Suppose $a$, $b$ are real numbers such that $a+b=12$ and both roots of the equation $x^2+ax+b=0$ are integers. Determine all possible values of $a$. I don't know how to go about doing this without ...
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1answer
20 views

Remainders and polynomial division

Completley impromptu, one of my extended middle school students asked a question about her additional maths she was studying outside of school. For a certain polynomial, f(x), the remainder on ...
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1answer
29 views

Solving for the roots of a polynomial

Suppose we have a polynomial of the form: $$-x^3+3x^2+9x-27=0$$ Is there an easy way to find the solutions of $x$? I know that they will be factors of $27$, so I begin by factoring $27$ into ...
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2answers
22 views

Solving equations with exponentials and trig algebraically

Is it possible to algebraically solve an equation of the following form? $A\sin(x)+Be^x=C$ If so, how?
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2answers
43 views

Faster way to for $z^3 = -2 (1+i \sqrt 3) \bar z$ than complex algebra

What is the fastest way to solve for $z^3 = -2 (1+i \sqrt 3) \bar z$? I know how to do this using complex algebra. but that takes a long time. Can someone show me a faster way?
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2answers
39 views

Simplifying $\frac{\sqrt{3}}{2\sqrt{3}+1} + \frac{\sqrt{3}}{11}$

I realize that is a basic math problem, but I am still having problems with it. The expression $$\frac{\sqrt{3}}{2\sqrt{3}+1} + \frac{\sqrt{3}}{11}$$ equals one of the following: $2\sqrt{3}-1$ ...
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0answers
23 views

Let $X = \mathbb{R}$ and $Y = \left \{ x \in \mathbb{R}\mid x ≥ 1 \right \}$. Define $G : X → Y$ by $G(x) = e^{x^2}$. Prove that $G$ is onto. [duplicate]

Let $X = \mathbb{R}$ and $Y = \left \{ x \in \mathbb{R}\mid x ≥ 1 \right \}$. Define $G : X → Y$ by $G(x) = e^{x^2}$. Prove that $G$ is onto.
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1answer
53 views

no. of real solution of the equation $1+8^x+27^x = 2^x+12^x+9^x.$

The no. of real solution of the equation $1+8^x+27^x = 2^x+12^x+9^x.$ $\bf{My\; Try::}$ Let $2^x=a>0$ and $3^x=b>0\;,$ where $x\in \mathbb{R}$ So equation convert into $1+a^3+b^3 = a+a^2b+b^2$ ...
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2answers
30 views

Rearrange $y = xa-zc$ so that $a-c$ is on one side of the equation.

Is it possible to rearrange the following equation so that $a - c$ is on one side of the equation? $$ y = xa-zc $$ Thanks!
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1answer
7 views

Is every zonal homogeneous polynomial a polynomial on the unit sphere?

Let $$P_k(x_1\ldots x_n)=\sum_{\lvert \alpha\rvert=k} c_\alpha x_1^{\alpha_1}\ldots x_n^{\alpha_n}, \qquad (x_1\ldots x_n)\in \mathbb{R}^n$$ be a homogeneous polynomial of degree $k$. Assume that ...
2
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4answers
69 views

Why $(x-5)^2-4$ can be factorised as $(x-5-2)(x-5+2)$

I would like to understand why $(x-5)^2-4$ can be factorised as $(x-5-2)(x-5+2)$ I am particularly concerned with the term, $-4$.
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2answers
23 views

Solving Inequalities with the use of their properties and cases [on hold]

Solve following inequality $$\dfrac4x + 3 \gt \dfrac2x + 1$$ and then graph the solution set on real number line.
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4answers
88 views

Prove that $13\vert(3^{n+1} +3^{n} +3^{n-1})$

Prove that $3^{n+1} +3^{n} +3^{n-1}$ is divisible by $13$ for all positive integral values of $n$
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2answers
22 views

Using the basic laws of exponent [on hold]

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 ...
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2answers
18 views

Express the given expression as a single logarithm

Express $$2 \ln (2 - x) + 3 \ln (x^2 - 5)$$ as a single logarithm. Can anyone help me with this question? Thanks
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3answers
45 views

Show that $2(a^3+b^3+c^3)>a^2(b+c)+b^2(c+a)+c^2(a+b)>6abc$

If $a,b,c$ are positive real numbers, not all equal, then prove that $$2(a^3+b^3+c^3)>a^2(b+c)+b^2(c+a)+c^2(a+b)>6abc$$ How can I show this?
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3answers
65 views

valid proof of series $\sum \limits_{v=1}^n v$

$$\sum \limits_{v=1}^n v=\frac{n^2+n}{2}$$ please don't downvote if this proof is stupid, it is my first proof, and i am only in grade 5, so i haven't a teacher for any of this 'big sums' proof: if ...
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1answer
50 views

Find exact value of $\sin\left(\dfrac x2\right) $

I have tried this problem over and over but can not get it. Can anyone provide a solution? Given $\sin(x) = -\dfrac67$ and $\tan(x)\gt0$ , find the exact value of $\sin\left(\dfrac x2\right) $.
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2answers
31 views

Solve and put in interval notation $4x^3 - 81x< 0$

The question is: Solve $4x^3 - 81x < 0$ and express the solution set in interval notation. I got $(-9/2,0)\cup(9/2,\infty)$ but I don't think its right. I factored it out to $x(2x+9)(2x+9)$
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3answers
64 views

Minimum value of $ f(x) = \frac{2+\sin x}{2+\cos x}$.

Minimum value of $\displaystyle f(x) = \frac{2+\sin x}{2+\cos x}$. My try: let $$\displaystyle y = \frac{2+\sin x}{2+\cos x}\Rightarrow 2y+y\cdot \cos x = 2+\sin x$$ So $$y\cdot \cos x-\sin x= ...
0
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2answers
47 views

If $\lim_{x\rightarrow \infty}\left[\left(x^5+7x^4+2\right)^c-x\right]$ is a finite, Then limit is

For a certain value of $'c',\lim_{x\rightarrow \infty}\left[\left(x^5+7x^4+2\right)^c-x\right]$ is a finite and non-zero, Then value of limit is $\bf{My\; Try::}$ Let $\displaystyle ...
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2answers
59 views

What is the angle that an Archimedean conical spiral makes with the floor?

I have a spiral in the form $$r = r_0(1-{\theta\over2\pi k }) \{r \ge 0\}$$ where $r_0$ is an initial radius, and $k$ is the number of turns. (It is a spiral that decays from $r_0$ to $0$ as $\theta$ ...
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2answers
882 views

Interesting Question on Ants

A horizontal stick is one metre long. Fifty ants are placed in random positions on the stick, pointing in random directions. The ants crawl head first along the stick, moving at one metre per minute. ...
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1answer
15 views

Unclear Application of Cauchy's Inequality

I was looking for a solution to a problem (both found here), where I came across the following ($a, b, c > 0$): Applying Cauchy's inequality, we get $(\frac{c}{a+2b} + \frac{a}{b+2c} + ...
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1answer
25 views

How can I solve this expression for x?

I would like to solve for $x$ given that \begin{equation} e^{-x}-\gamma-\eta e^{-\lambda(z-x)} = 0 \end{equation} where $\gamma, \eta, \lambda$ are positive constants and $z$ is a real number.
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1answer
25 views

questions related to progression [on hold]

Along a road lie an odd number of stones and distance between consecutive stones is 10m. A person can carry only one stone at a time and his job is to assemble all the stones around the middle stone. ...
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1answer
24 views

Giving a geometric representation of Cartesian products

What is being asked of me? Question 4 of Zorich(page 11) is exactly the following Give geometric representations of the following Cartesian products a) The Product of two line segments (a ...
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0answers
14 views

Is there a way to arrive at a funtion or a formula based on the outcome

The following table shows the input and the output. I'm trying to create a function that would relate the input and the output. SNU C020 C100 C300 C600 0 0 0 0 0 ...
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3answers
69 views

Solve: $\sin x - y\cos x = z$ for $x$.

I am working on programming a series of algorithms into a project, however I have run into trouble trying to solve this equation for $x$: $$ \sin x - y\cos x = z $$ It should be noted that $y$ and ...
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3answers
44 views

Best argument to prove $|x|\le a \iff -a\le x \le a$

$$|x|\le a \iff -a\le x \le a$$ I can only verify the integrity of this by talking about distances on the number line. But is there a algebraic argument that proves this?
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0answers
10 views

Locus of intersection between $y= 8\lambda/(\lambda ^2 + 4)$ and $y =2 \lambda x/(4-\lambda^2)$

I have the equations $$y=\frac{4\lambda}{\frac{1}{2}\lambda^2+2}\quad \text{and}\quad y=\frac{\lambda x}{-\frac{1}{2}\lambda ^2 + 2}$$ each representing a line. I'm asked to find the locus of the ...
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3answers
47 views

simplifying $-\pi i/8 (e^{i\pi/8} + e^{i3\pi/8} + e^{i5\pi/8} + e^{i7\pi/8})$

simplifying $-\pi i/8 (e^{i\pi/8} + e^{i3\pi/8} + e^{i5\pi/8} + e^{i7\pi/8})$ in my lecture notes somehow my lecture got from$-\pi i/8 (e^{i\pi/8} + e^{i3\pi/8} + e^{i5\pi/8} + e^{i7\pi/8})$ to ...
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0answers
40 views

Neutron-Density cross-plot interpretation

I have a question about solving a particular graphical problem. This is a picture of a Neutron-Density cross-plot: It's a little bit confusing as plots go, so allow me to try to explain the salient ...
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0answers
22 views

Problem about a focal chord

Given parabola $y^2=4ax$ with length of the focal chord equal to $l$ and the length of the perpendicular from vertex to the chord is $p$. Which one of these statements is true? 1) $l⋅p$ is constant ...
4
votes
2answers
59 views

Solving $\sin(2v) = \sin(v)$

$$\sin(2v) = \sin(v)$$ Why can't this equation be solved by setting: $$2v = v + 2\pi n \quad \leftrightarrow \quad v = 2\pi n\\2v = \pi - v + 2\pi n \quad\leftrightarrow \quad 3v = \pi + 2\pi n ...
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2answers
39 views

Inequality using only algebraic ''moves''

How can I verify the following inequality using only algebraic passages? $$ 5^\frac{1}{3} + 6^\frac{1}{2} > or < 4 $$
1
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1answer
47 views

Beautiful sines equation

If $θ_1,θ_2,θ_3,θ_4$ are four real numbers, then any root of the equation $\sinθ_1z^3+\sinθ_2z^2+\sinθ_3z+\sinθ_4$=3, lying inside the unit circle $\vert z\vert$=1, satisfies which inequality? ...
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0answers
9 views

no. of solution of the equation $\arccos(1-x)+m\cdot \arccos(x) = \frac{n\pi}{2}\;$

(1) The no. of solution of the equation $\displaystyle \arccos\left(\frac{1-2x-x^2}{(x+1)^2}\right) = \pi\left(1-\{x\}\right)\;,$ Where $x\in \left[\;0,76\;\right]$ Where $\{x\}$ denote fractional ...
2
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1answer
61 views

Beautifully looking little geometry/trigonometry problem

Given triangle ABC, a,b,c as its sides, p is a half perimeter, such that $\dfrac{p-a}{11}=\dfrac{p-b}{12}=\dfrac{p-c}{13}$. We need to find $(\tan\dfrac{A}{2})^2$ (A)$\dfrac{143}{432}$ ...
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5answers
49 views

solve this equation for $x$ : $y=x-6\sqrt{x}$

solve for $x$ this equation : $$y=x-6\sqrt{x}$$ I've tried raising everything to the power of two but it doesn't work $x$ shouldn't have two values.
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1answer
26 views

How to find $f^{−1}([9,0])$ and $f([1,4])$ for $f(x)=x-6\sqrt{x}$?

$f$ is a the function defined by $$\eqalign{ f\colon& \Bbb R &\rightarrow \Bbb R_+\\ & x&\mapsto x-6\sqrt{x} }$$ Find $f^{−1}([-9,0])$ and $f([1,4])$.
0
votes
1answer
27 views

simple problem of calculus.

A company wishes to manufacture a box with a volume of $36ft^3$ that is open on top and twice as long as it is wide.Find the dimensions of the box produced from the minimum amount of material. My ...
3
votes
4answers
56 views

Finding inverse of a function $h(x) = \frac{1-\sqrt{x}}{1+\sqrt{x}}$

I have a function: $$h(x) = \frac{1-\sqrt{x}}{1+\sqrt{x}}$$ With just pen and paper, how can I determine if there exists an inverse function? Am I supposed to sketch it on paper to see if it can ...