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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2answers
16 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: $a + b - ab \le 1$
2
votes
1answer
34 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 ...
0
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1answer
50 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 ...
3
votes
2answers
67 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 ...
1
vote
1answer
19 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 ...
0
votes
1answer
27 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 ...
1
vote
2answers
21 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?
0
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2answers
40 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?
2
votes
2answers
38 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: $2\sqrt{3}-1$ ...
-1
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0answers
22 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.
5
votes
1answer
52 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 $1+a^3+b^3 = a+a^2b+b^2$ ...
0
votes
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!
0
votes
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
votes
4answers
68 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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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
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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votes
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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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
59 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
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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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
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
votes
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 ...
1
vote
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 $\theta$ ...
13
votes
2answers
878 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.
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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
23 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
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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 ...
3
votes
3answers
68 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
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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 ...
0
votes
3answers
46 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
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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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votes
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 ...
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
45 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
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)$\dfrac{143}{432}$ ...
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
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 ...
4
votes
3answers
190 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 ...