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
votes
2answers
28 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!
0
votes
1answer
6 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
votes
4answers
63 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, $-4$.
-1
votes
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.
5
votes
4answers
87 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 $n$
0
votes
2answers
19 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 ...
0
votes
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
1
vote
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?
5
votes
3answers
54 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 ...
0
votes
1answer
46 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) $.
0
votes
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 $x(2x+9)(2x+9)$
0
votes
3answers
60 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
votes
2answers
45 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 ...
1
vote
2answers
48 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 $\theta$ ...
13
votes
2answers
865 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. ...
0
votes
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} + ...
0
votes
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.
0
votes
1answer
24 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. ...
0
votes
1answer
22 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 ...
0
votes
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 ...
3
votes
3answers
66 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 ...
0
votes
3answers
42 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?
0
votes
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 ...
0
votes
3answers
43 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 ...
0
votes
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 ...
-1
votes
0answers
20 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 ...
1
vote
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
vote
1answer
44 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? ...
0
votes
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
votes
1answer
59 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)$\dfrac{143}{432}$ ...
1
vote
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.
0
votes
1answer
25 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
26 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 ...
4
votes
3answers
189 views

Why do non-real solutions of a polynom occur pairwise complex-conjugated?

So if I have a polynom with real coefficients and the solution $x+iy$, why is $x-iy$ always a solution too? Let $z$ and $w$ be complex numbers, with $w^{\ast}$ = complex-conjugated of $w$, then ...
3
votes
1answer
25 views

Specific piecewise-function SAT2 question

Taken from Barron's SAT Math Level 2 prep book: If f(x) = i, where i is an integer such that i ≤ x < i + 1, the range of f(x) is ...
0
votes
1answer
15 views

Solving for joint angles in 2-segment robot leg

I am trying to program a robot leg with 2 segments and two joints, such that for a given location of the foot, I can calculate the angles of both joints. From here on out, the positive Y direction is ...
3
votes
1answer
34 views

$\phi(v)/\Phi(v)$ is decreasing for $\phi$ and $\Phi$ being the PDF and CDF of $N(0,1)$

Let $\phi(v)$ and $\Phi(v)$ denote, respectively, the PDF and CDF of the standard normal distribution. How would one show that $$ \frac{\phi(v)}{\Phi(v)} $$ is decreasing? I tried the quotient rule ...
-4
votes
3answers
60 views

Multiplying a fraction [on hold]

Example: $\frac{x-1}{2}\cdot \frac{1}{3} =$ I'm really starting to hate math even though I've aced couple of times still can't calculate simple math. Please show work thanks! Okay basically what I ...
-2
votes
1answer
36 views

Can somebody simplify this radical? [on hold]

how do you simplify the radical shown above, please show as many steps as you can. I am very confused!!!
0
votes
2answers
22 views

Formal power series question

$$(1-t)^d \sum_{k = 0}^{\infty} \binom{d+k-1}{d-1} t^k = 1$$ How can this be proven? Thanks in advance.
0
votes
0answers
35 views

What is the proper term to describe algebraic techniques of equation manipulation?

Is there a term to describe the category of algebraic "tricks" that include: polynomial division completing the square quadratic formula partial fraction expansion etc. These are related since ...
0
votes
4answers
49 views

Solve system of equations

$$\sin(x+y)+1.6x=0$$ $$x^2+y^2=-1$$ Can this system be solved? Please help me with it. I managed to make graphs of it but can't get it solved without graph. Graph:
2
votes
4answers
50 views

Determining whether $\sum_{k=1}^\infty \frac{x^k}k$ converges [on hold]

$$\sum_{k=1}^\infty \frac{x^k}k$$ Does this series converge, if yes, then for what values of $x$?
0
votes
0answers
10 views

Calculate ratio of volumes in mixture with given ratio of masses

I have two components of a mixture: $a$ and $b$. I know that using $3.5 \text{ kg}$ of component $a$ and $0.5 \text{ kg}$ of component $b$ will give a proper mixture, and its density will equal $1.45 ...
1
vote
2answers
51 views

Find the domain of $\frac{x}{\sqrt{6x^2+3x+3/4}+x}$.

My attempt: Let's assume that $\sqrt{6x^2+3x+\frac{3}{4}}+x$ $>$ 0$$ \rightarrow\sqrt{6x^2+3x+\frac{3}{4}} > -x \rightarrow {6x^2+3x+\frac{3}{4}} > x^2\\ \rightarrow ...
0
votes
0answers
18 views

On Equivalence Relations and monoids

Let $R$ be an equivalence relation on a monoid $G$ such that $$ a_1 R a_2 \; \; and \; \; b_1 R b_2 \implies a_1b_1 R a_2b_2 $$ for all $a_i,b_i \in G$. Then the set $G / R = \{ [g] : g \in G \}$ ...
1
vote
3answers
56 views

Using mathematical induction to show that for any $n\ge$ 2 then $\prod_{i=2}^n\left(1-\frac{1}{i^2}\right)=\binom{n+1}{2 \cdot n}$

I'm trying to work through some practice problems but I've been stuck on this for god knows how long now and I've no idea where to even start. Just wondering if it would be possible for someone to ...
2
votes
2answers
42 views

If $(1-i)^n = 2^n$ , then find $n$.

If $$(1-i)^n = 2^n$$ then find $n$. If anything raised to $0$ is $1$, but according to my book $ n \ne 0$. Is the print wrong?