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Questions tagged [algebra-precalculus]

For questions about algebra and precalculus topics, which include linear, exponential, logarithmic, polynomial, rational and trigonometric functions, conic sections, binomial, surds, graphs and transformations of graphs, solving equations and systems of equations and other symbolic-manipulation topics.

4
votes
1answer
42 views

$\cos(n\vartheta)=\frac{a_n}{3^n}$

I want to show, if i know that $\cos(\vartheta)=\frac{1}{3}$ than $\cos(n\vartheta)=\frac{a_n}{3^n}$ for $n\in \mathbb{N}$, where $a_n \in \mathbb{Z}$,$3 \nmid a_n $ My approach was to do it by ...
0
votes
1answer
54 views

$m-m\sqrt{m}=80$ $\Rightarrow m- 5\sqrt{m}=?$

$m-m\sqrt{m}=80$ $\Rightarrow m- 5\sqrt{m}=?$ I tried different approaches such as, substituting $\sqrt{m}=t$, squaring the given equations, substituting $\frac{m-m\sqrt{m}}{16}$ as $5$. Nothing ...
0
votes
2answers
17 views

Finding the ratio of 5th of two different arithmetic sequences

It is given that the ratio of the sum to the nth term of two different arithmetic sequences is $7n+2:n+3$. Find the ratio of the 5th term of the sequences. I have no idea where to start this pls help!
-3
votes
1answer
35 views

Floor function linear problem [on hold]

Solve $\lfloor\frac12x\rfloor +\lfloor\frac23x\rfloor=x$, where $x$ is a real number. Is there any method besides trial and error???
3
votes
3answers
57 views

find the maximum value of $xy + yz +zx$ [duplicate]

find the maximum value of $$xy + yz +zx$$given that $x+2y+z=4$ my attempt : $(x+y+z)^2=x^2+y^2+z^2+2(xy+yz+zx) $ or $2S=2(xy+zx+zy)=(x+y+z)^2 -x^2-y^2-z^2=(4-y)^2-x^2-y^2-z^2$ $2S=-x^2-z^2-8y+16=-...
0
votes
1answer
36 views

What constant $c$ will make $ \sum_{k=2}^{N}c^{\frac{1}{k\log k}}=N ?$

What constant $c$ will make this equality valid for any $N$ chosen? $$ \sum_{k=2}^{N}c^{\frac{1}{k\log k}}=N. $$ I tried getting a rough idea of what $c$ should be and got about $1.46$ when $N=1000$ ...
0
votes
3answers
38 views

Check if true : Atleast one of the integers, a, b, c must be even

Suppose a, b, c are integers such that the equation $ ax^2 + bx + c =. 0 $ has a rational root. Check if true : Atleast one of the integers, a, b, c must be even. I know for rational roots $ b^2 - ...
0
votes
1answer
39 views

For how many real numbers 'b', does $x^2 + bx + 6b = 0$ have one integral root?

The question states. For how many real numbers 'b', does $ f(x) =x^2 + bx + 6b = 0$ have one integral root ? My line of thinking : Let $\alpha , \beta$ be the roots of of $f(x)$. $\alpha + \beta = -b....
1
vote
0answers
63 views

How to find this area of this figure without using calculus?

Is it the same as the sector of Angle $\theta$ ? Can you find the shaded area for $\theta=90$ degree ?
0
votes
0answers
20 views
-3
votes
2answers
57 views

Number of integral solutions for the equation $x + 2y = 2xy$ is? [on hold]

How many integral solutions does the following equation have? $$x + 2y = 2xy$$ I have tried hit and trail method and I got only one solution, namley, $x=y=0$.. But Is there any other way to solve ...
0
votes
1answer
46 views

Is $\tan^{-1}(-1) = 3\pi /4$ or $=7\pi /4$? I understand they're both valid solutions, but what about places where the value is added/subtracted?

For example: calculate $\int^4_2 \tan^{-1} x \, dx$ If $\tan^{-1}(-1) = 3\pi /4$, then the final answer is $\frac{-\pi}{2}$. But if $\tan^{-1}(-1)= 7\pi /4$, then the final answer would be $\frac{3}{-...
0
votes
1answer
35 views

Distance of the point $(a,b,c)$ to the plane $z=0$

I'm trying to solve a calculus problem, I need to find the mass of a cylinder, I'm close to the answer, I got $8\pi$ but it should be $16\pi$. I think my mistake lies in the density function since it ...
0
votes
2answers
19 views

Find unknown matrix element using elementary row operation?

$A=\left(\begin{matrix}x & 5 & x \\ 1 & 3 & -2\\ -2 &-2 &2 \end{matrix}\right)$ $B=\left(\begin{matrix}0 & 0 & 21 \\ 1 & -1 & -14\\ 0 &\frac{4}{3} &...
2
votes
3answers
52 views

Find all the $z$ such that $(1+\frac{1}{z})^{4}=1$

Question if $S$ be the set of solution of $$\bigg(1+\frac{1}{z}\bigg)^{4}=1$$ then prove that the points are co-linear. Attempt $\bigg(1+\frac{1}{z}\bigg)^{4}=1$ $\implies z^4+4z^3+6z^2+4z+1=z^4$ $...
0
votes
1answer
57 views

Asymptotic growth of $\frac{k}{1}+\frac{k^2}{1\cdot 2}+\frac{k^3}{1\cdot 2\cdot 4}+\dots$

Let $k$ be a positive integer, and let $$n=\frac{k}{1}+\frac{k^2}{1\cdot 2}+\frac{k^3}{1\cdot 2\cdot 4}+\frac{k^4}{1\cdot 2\cdot 4\cdot 8}+\dots,$$ where the sum goes on until the next term in the sum ...
0
votes
1answer
19 views

Fractional Exponents Microeconomics

I need to find $X_a$ using this equation. I am having trouble working out how they got this answer. The question is $$ Y=X_a^{1/3} \left(\frac{w_a}{w_2} X_a \right)^{1/3} $$ The answer works out ...
0
votes
3answers
38 views

Factorization of $(m-n)p^3-(m-p)n^3+(n-p)m^3$ [on hold]

Factorize $(m-n)p^3-(m-p)n^3+(n-p)m^3$
1
vote
2answers
42 views

An inequality with positive integers

Let $s$ be a positive integer. Let $a_1, a_2,\ldots, a_s$ be distinct positive integers such that $a_i\geq2, \ \forall \ i\in\lbrace 1,2,\ldots, s\rbrace$. Show that $(a_1a_2\ldots a_s)^2-a_1a_2\ldots ...
0
votes
0answers
29 views

I have worked an equation as low as i can. I need help to solve for a variable.

I have been working on a problem in which I have been working for hours to no avail. This is something i'll be putting into python, but can't seem to figure the end result here. Original variables ...
4
votes
2answers
88 views

Number of real solutions to $x^7 + 2x^5 + 3x^3 + 4x = 2018$

Find the number of real solutions of $x^7 + 2x^5 + 3x^3 + 4x = 2018$? What is the general approach to solving this kind of questions? I am interested in the thought process. Few of my thoughts ...
0
votes
3answers
35 views

Transpose $x^2 + y^2 + 2x + 2y + 1 = 0$ to find $y$ in terms of $x$

I'm self-studying from Stroud & Booth's phenomenal "Engineering Mathematics", and am stuck on a problem from the end of the fourth chapter, "Graphs". Namely, I'm to transpose the following ...
1
vote
3answers
104 views

Algebra: Is $f(x) = \frac x x$ the same as $f(x) = 1$?

My question is basically: can we simplify $f(x) = \frac x x$ to $f(x) = 1$ without considering $f(0)$? Second, if they are not the same thing how can we simplify equations like $x + 2xy = 5x$ to $1 + ...
0
votes
0answers
18 views

Is a function obtained by superposing other continuous functions continuous? [on hold]

If $A,B,C,...$ continuous functions in the interval $[a,b]$, then $A+B+C+...$ is necessarily continuous in that interval?
1
vote
1answer
36 views

proving that x+y+z=0 and that $x^2-yz=y^2-zx=z^2-xy$

For the real numbers x, y, z which are different between themselves and the real numbers k, the following equalities are true: $x^2+y^2+kxy=y^2+z^2+kyz=z^2+x^2+kzx$, prove that $x+y+z=0$ $x^2-yz=y^...
2
votes
1answer
42 views

Example of mathematical antilogies involving the equality symbol ( always false statements, for all permissible values of the variables).

Logicians ( in propositional calculus) classify statements/formulas into 3 categories : tautologies ( always true) , contingent statements ( sometimes true, sometimes false) , antilogies ( always ...
1
vote
2answers
70 views

“For all X, X =A iff X=B, therefore the A = B”. Is this logically correct?

Suppose I want to prove ( in elementary arithmetics, or, maybe, in an abstract additive group) that : the additive inverse of (a+b) = -b + -a May I proceed as follows? Suppose X = - ( a+b). Now,...
3
votes
0answers
48 views

Problem about linear combinations of real numbers over $\Bbb Q$

I have the following problem that might be silly but I am not able to find a solution. Let $a_1,\dots,a_n$ be real numbers such that $a_1,\dots,a_n,\pi$ are linearly independent over $\Bbb Q$. Let $...
0
votes
1answer
34 views

Radical Inequalities

Find the number of positive integers $n$ so that $4 < \sqrt{n} < 10.$ The answer I have is 5 since 25,36,49,64,81 work but that's incorrect. Did I misread the question or did I miss something?
-1
votes
3answers
43 views

Two brothers fishing [on hold]

Two brothers A and B are fishing. Person A says he got 10 fish. Person B says he got 12 fish. However, person A says: You need to multiply any number B says by $\frac{3}{4}$ to get the real answer! ...
1
vote
1answer
12 views

Proportionally distribute a lump sum between a maximum and a minimum

I have to distribute a number, x, to a certain number of people, n. I know that I want to distribute the maximum, a, to the top person, and the minimum, b, to the lowest person. My question is, what ...
3
votes
4answers
60 views

$ax+by=x^2+y^2\implies a=x$ and $b=y$

I currently edit curriculum for high school geometry and I came across a mistake in one of their diagrams. After doing some work, I boiled down their mistake to an assumption that if $ax+by=x^2+y^2$ ...
0
votes
2answers
39 views

Is it possible to explain why $\sum e^{2n}/6^{n-1}$ is divergent without using any tests?

I'm pretty sure $\sum e^{2n}/6^{n-1}$ can be expressed as a geometric series with some algebra. Let's say a geometric series is defined as one that looks like: $\sum ar^{n-1}$, where $a$ is unchanging ...
0
votes
1answer
12 views

What is the value of x for given function

How we can initiate the problem.. by trial and error method no one is satisfying..
0
votes
1answer
43 views

How to show $\forall n \in \mathbb{N^*}, (\frac{2n}{3}+\frac{1}{3})\sqrt{n} \leq \sum_{k=1}^{n}\sqrt{k}$?

How to show by induction $\forall n \in \mathbb{N^*}, (\frac{2n}{3}+\frac{1}{3})\sqrt{n} \leq \sum_{k=1}^{n}\sqrt{k}$ Thanks for heping me :)
1
vote
2answers
26 views

Slide Rule Marking Proportion

The basic slide rule scale starts at 1, runs through all the positive numerals, then ends at 1. How are the distances between these whole numbers on a slide rule proportioned?
1
vote
0answers
47 views

What are the basic needed mathematical knowledge for a student finishing high school? [on hold]

I’m a high school student who wants to study mathematics in university. I’ve worked hard and have also acquired some knowledge of university level courses. That being said, I’d like to be perfect in ...
0
votes
1answer
49 views

Inequalities involved in the proof of transcendence of $[0,10,10^{2!}, \dots]$

In proof of transcendence of the simple continued fraction $[0,a_1,a_2, \dots]$ in which $a_k=10^{k!}$ (Hardy, et al's Theory of Numbers) it uses the following two inequalities: i. $(1+\frac{1}{10})(...
-6
votes
0answers
32 views

What is the greatest possible value? [on hold]

What is the greatest possible value? Susan wants to buy mangoes for $ 150$ cents each and $(3x+2)$ peaches at $80$ cents each. Form an inequality and find the greatest possible value of $x$ if she ...
0
votes
0answers
54 views

Differentiation inside integration

How to deal with the differentiation inside integration like this? $$ I = \int d\theta_1 d\theta_2 \frac{df(\theta_1 , \theta_2)}{d\theta_1 d\theta_2} cos\theta_1$$ I want to evaluate this ...
0
votes
0answers
42 views

Multiplying with plus or minus sign in front

I am having a hard time following what happens in between these steps: $\frac{L^2}{(1{\mp}e)}=L^2(1{\pm}e)$ How did the minus-plus sign move to the numerator after a change to a plus-minus sign?
0
votes
5answers
54 views

If $\frac{a}{\sin{A}}=\frac{b}{\cos{A}}$, show that $\sin{A}\cos{A}=\frac{ab}{a^2+b^2}$

I don't know how to go about solving this, I think I need to use $\sin^2\theta+\cos^2\theta=1$, but I'm not sure how to go about this. The closest I managed to get was: $$\frac{a}{\sin{A}}=\frac{b}{\...
4
votes
2answers
64 views

Is there a way to solve $x\left(\frac{e^x+1}{e^x-1}\right)=4$ for x besides just plugging numbers in?

This comes into play in the equation for the shift in Cosmic Microwave Background (CMB) photon frequency due to inverse Compton scattering: $\frac{\Delta T}{T_{CMB}} = y \left( x\left(\frac{e^x+1}{e^...
-2
votes
5answers
60 views

Square root without a calculator algorithm [duplicate]

Out of curiosity I'm trying to find an effective algorithm to find the value of a square root of a number(a) without a calculator. I'm trying to find a solution without searching it up. What I have ...
1
vote
2answers
28 views

How do I solve for $k = f(k)$ when it is raised to the power $v^2$?

To solve for $k$ in the following function: $$k^{v^2}=f(k)$$ Which of the algebraic manipulations is correct? (1) $$k= \pm f(k)^{1/v^2}$$ or (2) $$k^v= \pm f(k)^{1/2} \rightarrow k = \pm f(k)^{{1/...
1
vote
1answer
34 views

Express $x_1$ from the equation

Could you kindly help me express $x_1$ from the equation below? $$y = \frac{ x_1^3 - x_2^3 }{ x_1^2 + x_2^2 + 1}$$ Thank you.
1
vote
1answer
20 views

Let $S=${$1$,$2$,..,$n$}.In how many ways can we choose two subsets $A$ and $B$ of $S$ so that $B \neq \emptyset$ and $B \subseteq A \subseteq S$?

Let $S=${$1$,$2$,..,$n$}.In how many ways can we choose two subsets $A$ and $B$ of $S$ so that $B \neq \emptyset$ and $B \subseteq A \subseteq S$? My first approach involves finding the sum $\sum_{k=...
0
votes
3answers
41 views

Calculate the double integral $I=\iint_Dxy\,{\rm d}x\,{\rm d}y$

The problem is as follows: Calculate the double integral $$I=\iint_Dxy\,{\rm d}x\,{\rm d}y$$ when the region $D$ is defined by $\{(x,y):0\le x\le1, 0\le y\le1, xy\le\frac{1}{2} \}$. The region $D$ ...
1
vote
1answer
40 views

When to apply PEDMAS in reverse?

I was told that when undoing operations in an equation you should start by following the PEDMAS rules, but in reverse. So, for example: 8x + 16/x = 4x + 16 According to the advice, I should start ...
0
votes
1answer
44 views

When do we need to take absolute values and why? [closed]

When do we have to take the absolute value when dealing with radicals and why? Surely $\sqrt[2]{x^2}$ is $|x|$ but I would like to know the mathematical reason behind it (not just words), which might ...