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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-2
votes
1answer
23 views

How would you divide a polynomial by another polynomial whose power is greater than its nominator?

I have a polynomial which is: $$\frac{(x^3-4x)}{(4x^2-4x+1)} = -10$$ Is there a way to do this? I have thought about doing long division which was not helpful...
0
votes
2answers
23 views

Finding the correct slope.

To determine the slope of the graph of this relation do I take the two points as (4,20), (0,0) and then proceed to take 20-0=20 and 4-0=4, to divide 20 by 4 to get the slope of 5m? For the ...
0
votes
1answer
46 views

Harder-Than-Seems Inverse of $f(x)=x^3-x-12$?

This may seem simple but I have had long days of frustration with finding the inverse of this: $$f(x)=x^3-x-12.$$ I got this on some homework and it did not ask for the inverse. However i wanted to ...
1
vote
3answers
32 views

How do I interpret this question: Do I multiply, divide, subtract first?

Which of the following expresses $6p+2py-4p$ in its simplest form? (A) $2p+2py$ (B) $4py$ (C) $4p^3y$ (D) $10p+2py$ Im not really sure how to go about it...
0
votes
4answers
38 views

Lawn mowing problem solving

Kate can mow the lawn in 45 minutes. Kate's sister takes twice as long to mow the same lawn. If they both have a mower and mow the lawn together, how many minutes will it take them? I know the answer ...
-3
votes
1answer
24 views

Express as a single logarithm [on hold]

Hi I need to express the following and have no clue how to do so. $$\ln(x+3)-3\ln(x-7)-\ln(x+8)$$ Can someone please help
-4
votes
2answers
54 views

How to simplify $(x+1) / (x^3-x)$ [on hold]

For all $x$ in the domain of the function $\frac{x+1}{x^3-x}$, this function is equivalent to which of the following? (A) $\dfrac{1}{x^2}-\dfrac{1}{x^3}$ (B) ...
1
vote
6answers
79 views

Sum of cosines of complementary/suplementary angles

Why are $(\cos(2^{\circ})+\cos(178^{\circ})), (\cos(4^{\circ})+\cos(176^{\circ})),.., (\cos(44^{\circ})+\cos(46^{\circ}))$ all equal zero? Could you prove it by some identity?
9
votes
7answers
119 views

Evaluating the indefinite integral $\int\sqrt{16-9x^2}\,dx$

I need to solve the integral below, but I just can't figure how. $$\int \sqrt{16-9x^2}\,dx$$ I have tried to replace $9x^2$ with $16\sin^2\theta$. I get to a point where I have the function ...
-1
votes
0answers
15 views

Find the solution to the following LPP by solving its dual. [on hold]

Minimize : $ Z = 300X_1 + 110X_2$ Subject to : \begin{align*} 30X_1 + 5X_2 &\geq 6 \\ 20X_1 + 10X_2 &\geq 8 \\ X_1, X_2 &\geq 0 \end{align*}
0
votes
1answer
25 views

matrix multiplication manipulation

a,b $\in \mathbb{R^n}$ and C $\in \mathbb{R^{nxn}}$. I have $ab^TCab^TC$. I try to manipulate this multiplication into: $b^TCaab^TC$. I need help.
0
votes
5answers
100 views

Why is reminder of $8^{30} / 7$ same as that of $1^{30} / 7$

I am not able to figure out why the reminder of $8^{30} / 7$ is same as that of $1^{30} / 7$. I know Euclid division $a=bq+r$ but I don't know modular arithmetic, so please explain without referring ...
1
vote
0answers
16 views

How to solve problems on alligation and mixture when three types are given?

Suppose there are three qualities of rice, A(1 dollar per Kg), b(2 dollar per Kg) and C(3 dollar per Kg). The salesmen want to mix these in a certain ratio a:b:c so as to make the price 2.5 dollar per ...
4
votes
6answers
258 views

Sine/cosine series

$$\frac{\sin²(1°) + \sin²(2°) + \sin²(3°) + .. + \sin²(90°)}{\cos²(1°) + \cos²(2°) + \cos²(3°) + .. + \cos²(90°)} = ?$$ I tried to use multiple identities but I couldn't simplify the expression. ...
2
votes
2answers
53 views

Proper way to solve function notations?

I'm just starting to use function notation and I'm wondering if I'm solving correctly. If $f(x) = 4x - 11$, determine a. $f (1/4)$ $f(x) = 4x - 11$ $f(1/4) = 4 (1/4) - 11$ $f(1/4) = 1 - 11 $ ...
0
votes
4answers
38 views

How to find perpendicular vectors in 3D

Find all values of a such that the vector $q = \langle 2, a, –2\rangle$ is perpendicular to the vector $p = \langle –3, a, 5 \rangle$.
4
votes
2answers
63 views

Find the maximum value of the fraction

Let $a$ and $b$ be positive integers satisfying $\frac{ab+1}{a+b}<\frac{3}{2}$. The maximum possible value of $\frac{a^3b^3+1}{a^3+b^3}$ is $\frac{p}{q}$, where $p$ and $q$ are relatively prime ...
1
vote
1answer
14 views

Property of an almost additive sequence of functions

We say that a sequene of functions $\Phi=(\phi_n)_n$ is almost additive if there exists a constant $C > 0$ such that for every $n,m \in \mathbb{N}$ and $x\in \Lambda$ we have \begin{equation*} -C + ...
0
votes
2answers
49 views

A question about Idempotent functions [on hold]

some functions are such that $f\circ f(x)=f(x)$ like these 1) $$f(x)=x \implies f\circ f(x)=x=f(x)\\$$ 2)$$f(x)=\lvert x\rvert \implies f\circ f(x)=\lVert x\rVert=\lvert x\rvert=f(x)\\$$ 3) ...
0
votes
2answers
13 views

Finding function inverse

Hi, This question is actually from KhanAcademy - Algebra 2. I managed to solve it using the rote method by swapping the x and y. But I would like to find out the reason for swapping x and y for ...
2
votes
2answers
41 views

Find a recursion (combinatorial)

Consider sequences that consist entirely of $ A$'s and $ B$'s and that have the property that every run of consecutive $ A$'s has even length, and every run of consecutive $ B$'s has odd length. ...
0
votes
1answer
30 views

Help to manipulate and rearrange this inequality

I am working through a proof and I am trying to understand all of the steps. It uses one inequality to show another: Let $a_1, \ldots, a_k$ be given real numbers and $p_1, \ldots, p_k$ where $p_i \geq ...
0
votes
3answers
36 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
75 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 ...
12
votes
5answers
1k views

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

I was teaching absolute function properties in a K-12 class. I made this question in my mind. Suppose $f(x)$ is a one-to-one function, and its definition is $f(x)=max\left \{ x,3x\right ...
-2
votes
1answer
36 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
votes
1answer
14 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
38 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
votes
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
votes
2answers
25 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
votes
5answers
67 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
89 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
130 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
124 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
votes
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
vote
2answers
57 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 ...
-2
votes
0answers
35 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
votes
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
43 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
33 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: ...