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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0answers
10 views

Difficult nonlinear system based on max value

Let $ (a,b,c)$ be the real solution of the system of equations $ x^3 - xyz = 2$, $ y^3 - xyz = 6$, $ z^3 - xyz = 20$. The greatest possible value of $ a^3 + b^3 + c^3$ can be written in the form $ ...
4
votes
6answers
57 views

find x in $\sqrt[3]{6+\sqrt x} + \sqrt[3]{6-\sqrt x} = \sqrt[3] {3}$

Which one satisfies the equation $\sqrt[3]{6+\sqrt x} + \sqrt[3]{6-\sqrt x} = \sqrt[3] {3}$ (A)$27$ (B)$32$ (C)$45$ (D)$52$ (E)$63$ let $a = 6+\sqrt x , b=6-\sqrt x$ cube each side ...
2
votes
3answers
46 views

When I was teaching absolute function properties ,I made this question suddenly …

I was teaching absolute function properties in k-12 class. I made this question in my mind . Suppose $f(x)$ is a one-to-one function ,and it's definition is $f(x)=max\left \{ x,3x\right \}=ax+b|x|+c$ ...
-1
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0answers
21 views

Can someone confirm if the solutions are correct?

For the first one, I did 163 / 1200.. For the second one, I got 24% I think both of mine are correct, but solutions say otherwise.
2
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1answer
13 views

Commutative Monoid - matrix set

Let $M$={$\begin{bmatrix} a & b & c \\ c & a & b \\ b & c & a \end{bmatrix}|a,b,c\in \mathbb{R}, a+b+c=0$}. The matrices in $M$ are a special kind of Toeplitz matrices ...
1
vote
5answers
43 views

Is it possible to solve for $m$ in a linear equation without knowing $b$?

Suppose you know certain points on a line say $(5,2)$ up to $(8,10)$ but you don't know exactly where the $y$ intercept would be being somewhere down there at like $-25$ area. How would you solve for ...
0
votes
1answer
37 views

General Question about number of functions

I am wondering if there is any sort of algorithm , or if not, at least some general approach to the following; Lets say we have two finite sets $$A=\{a_1,a_2,…a_n\}$$ and $$B=\{b_1,b_2,…,b_m\}$$ ...
0
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2answers
29 views

In the doubling time formula, what does the a stand for?

$A(t) = P(2)^{t/a}$ what does the lower case a stand for?
2
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2answers
24 views

Differentials where the variable undergoes a percentage increase. Where am I wrong?

Let $R = \frac{k}{r^4}$, where $k$ is some constant. Find the change in $R$ as $r$ is increased by 10%. $R$ is the resistance of blood flow, $r$ is the radius of a vein. This problem seems easy ...
0
votes
1answer
24 views

Probability the range is disjoint

Let $A=\{1,2,3,4\}$, and $f$ and $g$ be randomly chosen (not necessarily distinct) functions from $A$ to $A$. The probability that the range of $f$ and the range of $g$ are disjoint is ...
3
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5answers
53 views

Simplify Square Root Expression $\sqrt{125} - \sqrt{5}$

$\sqrt{125}-\sqrt5$ simplify it. I thought it would be $\sqrt {5\cdot5\cdot5}-\sqrt 5$ which would be the square root of 25 which is 5 but it is not. Can you show how to simplify this?
5
votes
4answers
87 views

coefficient of $x^{17}$ in the expansion of $(1+x^5+x^7)^{20}$

I found this questions from past year maths competition in my country, I've tried any possible way to find it, but it is just way too hard. find the coefficient of $x^{17}$ in the expansion of ...
2
votes
4answers
129 views

find $\left( \frac{x}{x+y} \right)^{2007} + \left( \frac{y}{x+y} \right)^{2007}$

I found this questions from past year maths competition in my country, I've tried any possible way to find it, but it is just way too hard. if $x, y$ are non-zero numbers satisfying $x^2 + xy + ...
3
votes
3answers
33 views

How to solve this equality? [3]

$$4x^2 - 6x^4 + \frac{8x^6 - 2x^2 - \frac{1}{x^2}}{16} = 0$$ The equation has a strange look, and as such is probably as it should not be solved. Maybe the roots of trigonometric functions are ...
3
votes
6answers
119 views

evaluate $\frac 1{1+\sqrt2+\sqrt3} + \frac 1{1-\sqrt2+\sqrt3} + \frac 1{1+\sqrt2-\sqrt3} + \frac 1{1-\sqrt2-\sqrt3}$ [on hold]

Evaluate $\frac 1{1+\sqrt2+\sqrt3} + \frac 1{1-\sqrt2+\sqrt3} + \frac 1{1+\sqrt2-\sqrt3} + \frac 1{1-\sqrt2-\sqrt3}$ How to evalute this equation without using calculator?
2
votes
3answers
88 views

High computation in probability

Six men and some number of women stand in a line in random order. Let $p$ be the probability that a group of at least four men stand together in the line, given that every man stands next to at ...
0
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1answer
18 views

Counting with potency and simplifing

So I have the question: Simplify $(6^{n+4}) / 2^{n+5} \cdot 3^{n+2}$ I tried to write the expresion as $6^{n+4-(2n+7)}/6$, but that is wrong. So I guess I should factor it out. Perhaps $2^{2} + ...
1
vote
1answer
38 views

How to show that a curve passes through the origin?

It is given that the tangent to curve at points $x=1$ and $x=-1$ are perpendicular. I've managed to find the equation of the curve: y=$\frac{4}{3}x- \frac{5}{6}x^2$ but how do I show that the curve ...
1
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2answers
55 views

Square root equation

I have the equation $\sqrt{(7-x)} - \sqrt {(x+13)} = 2 $ The square root should be expanded so it is square root of $7-x$ - square root of $x+13 = 2$. When i square both sides i get: $7-x - x-13 = 4 ...
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0answers
34 views

Explanation for how C was solved in shown question.

I have been given the following question to solve: This was my attempt. Seems I was way off track: I cannot work out how c was found in the following image. I follow the working up until the ...
2
votes
1answer
19 views

Determine the domain and range of the following relations using set builder notation.

I have been given the following relations to find the domain and range of using builder notation. I am just beginning to learn the whole concept of set builder notation, and I am running into a ...
2
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2answers
82 views

Algebraic proof that $\sum\limits_{i=0}^n \binom{i}{k} = \binom{n + 1}{k + 1}$

I'm looking for an algebraic proof of this identity for $n, k \in \mathbb{N}$: $$\sum\limits_{i=0}^n \binom{i}{k} = \binom{n + 1}{k + 1}$$ So far, I've turned the left hand side of the equality into ...
2
votes
1answer
39 views

Checking logarithm inequality.

Which one of the following is true. $(a.)\ \log_{17} 298=\log_{19} 375 \quad \quad \quad \quad (b.)\ \log_{17} 298<\log_{19} 375\\ (c.)\ \log_{17} 298>\log_{19} 375 \quad \quad ...
0
votes
1answer
24 views

How to calculate mileage and diesel consumption? [on hold]

What do i need to calculate mileage and diesel consumption? For example if the distance is 20km and the speed is 35 mph, what ...
-4
votes
1answer
40 views

A system with modular arithmetic [on hold]

How do I solve this system? Note: (y mod 10) = (x mod 10). $$\begin{cases} 2y - x + (x \bmod 10) = 42\\[1ex] y + (x \bmod 10)= 32 \end{cases}$$ for x and y?
0
votes
2answers
31 views

Arter there any 'Horizontal Asymptote' rule exceptions?

An equation I have is $$F(x) = \frac{9x(x-9)}{3x^2-11x-4}.$$ Upon calculating using the rules taught in class, There is an H.A. at $y = 3$ and a V.A. at $x = -\frac13$ and at $4.$ After graphing, ...
1
vote
1answer
24 views

Finding in which division a point is in a wheel?

I've got a wheel with $38$ divisions. If I place a random point on the wheel and get the angle where it's located, is there a formula that I could use to figure out in which division is the point ...
0
votes
2answers
30 views

Fish population growth question: Can someone check my work and answers?

I'm reviewing for a math test this Tuesday and just want to make sure I'm doing things right. If someone could check my work that would be great. Here's the question (work below): Here's my work: ...
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votes
0answers
44 views

Would you please give me your opinion about solving this equation? [on hold]

Would you please give me your opinion about solving this equation? [![enter image description here][1]][1] $\sum_{i=0}^{1}\left (-X \right )^{i}*\sum_{J=0}^{7}\sum_{i=0}^{j}\, \gamma _{j}*X^{i}=-T$ ...
-3
votes
3answers
25 views

E or natural log problem, solve the equation [duplicate]

If someone could explain to me the first step or two so I could solve this that would be great. All the e's are confusing me Solve the equation. (Round your answer to four decimal places.) $$e^x − ...
-3
votes
0answers
42 views

Solving circle advanced problem [on hold]

I have the following: $X\bmod \sqrt3 = 0$ $Y\bmod 3 = 0$ $X^2 + Y^2 = R^2$ So someone suggested: $m,n$ in $\mathbb{Z}$ So the equation is: $3m^2+9n^2=R^2$ how do i solve this? I need to find ...
1
vote
2answers
23 views

Linear Approximation of a function at point a with change of x

So I know this problem is supposed to be very basic, but I cannot for the life of me get the answer my teacher and book gets. I would very much appreciate if a solution could be posted on how this ...
1
vote
3answers
87 views

proving that $g(x)=0$ has one real root

Given $g(x)=1+x+\dfrac{x^2}{2!}+\dfrac{x^3}{3!}+\cdots+\dfrac{x^{2n+1}}{(2n+1)!}$, Need to prove that $g(x)=0$ has one real root. I thought to use the fact that $e^x<T_{2n}(x)$ for all $x<0$, ...
2
votes
2answers
87 views

Sum of remainders of $2^n$

Hints Only Let $R$ be the set of all possible remainders when a number of the form $2^n$, $n$ a nonnegative integer, is divided by $1000$. Let $S$ be the sum of all elements in $R$. Find the ...
3
votes
4answers
105 views

Proving that $1\cdot3+3\cdot5+5\cdot7+\cdots+(2n-1)(2n+1)={n(4n^2+6n-1) \over 3}$ by induction for $n\geq 1$

Prove using mathematical induction that $$1\cdot3+3\cdot5+5\cdot7+\cdots+(2n-1)(2n+1)= {n(4n^2+6n-1) \over 3}.$$ Step 1: If we assume that the equation is true for a natural number, $n=k$, ...
4
votes
3answers
295 views

Finding roots of cubic equation

If $\alpha,\beta,\gamma $ are the roots of the equation $2x^3-3x^2-12x+1=0$.Then find the value of [$\alpha$]+[$\beta$]+[$\gamma$],where [.] denotes greatest integer function. My attempt: I first ...
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votes
0answers
66 views

Finding points on a circle.

I have a circle: $X^2 + Y^2 = R^2$. $R$ is known and can be expressed in the solution. $X/\sqrt{3} \in\mathbb{Z}$, $Y /3 \in\mathbb{Z}$. ($X$ divided by square root of 3 is a round number and $Y$ ...
4
votes
2answers
40 views

Number of divisors of the form $(4n+1)$

Find the number of divisors of $$2^2\cdot3^3\cdot5^3\cdot7^5$$ which are of the form $(4n+1)$ I know how to find the total number of divisors. But, to find the number of divisors of the form ...
4
votes
5answers
86 views

$f(x) =ax^6 +bx^5+cx^4+dx^3+ex^2+gx+h $ find f(7)

Problem : $f(x) =ax^6 +bx^5+cx^4+dx^3+ex^2+fx+3$ Given that : $f(1)= 1, f(2) =2 , f(3) = 3, f(4) =4, f(5)=5, f(6) =6$ find $f(7) =?$ My approach : We can put the values of f(1) = 1 in the ...
1
vote
3answers
55 views

find total integer solutions for $(x-2)(x-10)=3^y$

I found this questions from past year maths competition in my country, I've tried any possible way to find it, but it is just way too hard. How many integer solutions ($x$, $y$) are there of the ...
8
votes
1answer
100 views

Evaluate $a^2+b^2+c^2$

I found this questions from past year maths competition in my country, I've tried any possible way to find it, but it is just way too hard. If $a, b, c$ are distinct numbers such that $a^2 - bc = ...
3
votes
6answers
66 views

Solve $\sin A +\sin 2A +\sin 3A + \sin 4A = 0$, for $0 \leq A \leq 180$

I've tried using factor formula but still did not manage to get the answer, not sure if factor formula is the right method. I rearrange to $\sin 4A + \sin 2A + \sin 3A + \sin A = 0$, and after ...
0
votes
2answers
54 views

Prove that if $a<1/a<b<1/b$ then $a<-1$

The following is Exercise 3.2.8 from Velleman: Suppose that $a$ and $b$ are nonzero real numbers. Prove that if $a<1/a<b<1/b$ then $a<-1$. I solved it using the hint in the back of ...
3
votes
4answers
71 views

Given $\tan A + \tan B = 3x$ and $\tan A \tan B = 2x^2$, find $\tan A - \tan B$ [on hold]

Given $$\tan A + \tan B = 3x$$ and $$\tan A \tan B = 2x^{2}$$ How to find $\tan A - \tan B$ ? I've tried substitution but still couldn't find. EDIT: Can you solve this problem using the formulas for ...
3
votes
4answers
96 views

Solve $10x^4-7x^2(x^2+x+1)+(x^2+x+1)^2=0$

How to solve this equition? $$10x^4-7x^2(x^2+x+1)+(x^2+x+1)^2=0$$ My attempt: $$ 10x^4 - (7x^2+1)(x^2+x+1)=0$$ Thats all i can Update Tried to open brakets and simplify: $$(7x^2+1)(x^2+x+1) = ...
1
vote
3answers
30 views

Graphing of $y= \csc(x)+ \cot(x)$

What's the graph or table of values of $y=\csc(x) + \cot(x)$? I have already solved and graphed the values of $\csc(x)$ and $\cot(x)$.
2
votes
0answers
53 views

Upperbound for $\sum_{i=1}^n\frac{1}{x_i^2}$?

Suppose that $x_i>0$, $i=1,\ldots,n$. I'm looking for an upperbound (doesn't have to be particularly tight) of $\sum_{i=1}^n\frac{1}{x_i^2}$ in terms of some symmetric function of ...
0
votes
0answers
54 views

System of equations to solve this nested radical.

The nested radical $$1.75793\approx\sqrt{1+\sqrt{2+\sqrt{3+\cdots}}}$$ has yet to be given a closed form. However, nested radicals of the form, $$\sqrt{A+B\sqrt{A+B\sqrt{A+\cdots}}}$$ have the ...
3
votes
1answer
30 views

What is the closed form of the following expansion

I need some help figuring out the closed form of the following expansion. T[n]=T[n-1]+T[1]*T[n-2]+T[2]*T[n-3]+T[3]*T[n-4]+...+T[n-1] I haven't done this type of ...
2
votes
1answer
36 views

Why does this sequence of operations give $x^{\frac{1}{x-1}}$?

I found (purely from experimentation) that if you start with a random number and successively: Exponentiate, Raise to the power of $x$, Take the log with the same base as step one, Take the $x$-th ...