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.

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11 views

Help with Fourier series problem

So, I know that it is even and half-wave symmetric and i have to integrate from $0$ to $3/2$. Is this the right function for the graph? $f(t)=\left\{\begin{array}{ll}4, & 0<t<1 \\ 2, &...
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3answers
44 views

Sum of complex roots' fractions

According to this: If $\omega^7 =1$ and $\omega \neq 1$ then find value of $\displaystyle\frac{1}{(\omega+1)^2} + \frac{1}{(\omega^2+1)^2} + \frac{1}{(\omega^3+1)^2} + ... + \frac{1}{(\omega^6+1)^2}=...
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1answer
20 views

A line $L_0: 2x+5y=11$ rotates about a point $P(\alpha, \beta)$ on the line $L_0$ such that $\alpha$ and $\beta$ are integers [CONT..]

A line $L_0: 2x+5y=11$ rotates about a point $P(\alpha, \beta)$ on the line $L_0$ such that $\alpha$ and $\beta$ are integers and $|\alpha+\beta|$ is least, through an angle $(-1)^n.n^{\circ}$ in $n^{...
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3answers
38 views

Proving $f(x,n)=\lfloor x \lfloor x \lfloor x \lfloor x …(\text{n times})\rfloor\rfloor \rfloor \rfloor $ is increasing for $x>0$

$$f(x,n)=\lfloor \underbrace{x \lfloor x \lfloor x \lfloor x \dots\rfloor\rfloor \rfloor \rfloor}_{\text{$n$ times}}$$ with $ n \in \mathbb{N}$. Can we prove this function is increasing for $x>0$? ...
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2answers
44 views

Find the minimun of $MN+\frac{3}{5}MP$, $MN$ and $MP$ is two sides of a quadrilateral.

In a quadrilateral $OPMN$ ,$\angle NOP=90^\circ$,$ON=1$,$OP=3$, and $M$ satisfy $\vec{MO}\cdot\vec{MP}=4$, find the minimum of $MP+\frac{3}{5}MN$ I choose the vertex $O$ of $OPMN$ as the origin of ...
3
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1answer
64 views

Find all x such that : $x^{x^{x^3({x^{x^{3}}+1)}+3}}=3^{81}$

This was the question : Find all x such that : $x^{x^{x^3({x^{x^{3}}+1)}+3}}=3^{81}$ By observation (error and trial) I was able to find that $x=\sqrt[3]3$ is a solution , also , I was able to ...
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1answer
50 views

Proving $|a+b|\le |a|+|b|$ from $-|a|\le a \le |a|$

In Spivak Calculus chapter 1, question no. 14, it is asked to prove the aforementioned inequality. However, the way I proved it is unnecessarily wrong. Can someone critique it for me, and mention an ...
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2answers
76 views

If positive integers $a$, $b$, $c$ satisfy $\frac1{a^2}+\frac1{b^2}=\frac1{c^2}$, then the sum of all values of $a\leq 100$ is …

I'm struggling to solve the following problem. I would like hint (just a hint, not a full solution please) on how to solve it: The positive integers $a$, $b$, and $c$ satisfy $$\dfrac1{a^2}+\...
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1answer
37 views

Intersection of Lines and Planes (Highschool math)

Question: a) Solve the system: b) Give a geometric interpretation of the solution(s). I tried doing part a) but I instantly got confused as I don't know what the question is asking me. For example:...
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2answers
33 views

What to do in this case when applying the law of sinus? I get an error 2 message.

I'm trying to applying the law of sinus in this triangle to get the angle of A. However when I do that I find myself having to do the $sin^{-1} 1.59$ to get A. This is obtained by having $\dfrac{22}{...
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1answer
107 views

Inequality involving $-x\log(x)$

Suppose that $$\exp(-a + b - \log(c)) \leq x \leq \exp(a + b -\log(c)).$$ Moreover, suppose that $0 \leq x \leq \frac{1}{e}$. I would like to conclude that $$-x\log(x) \leq \frac{-\exp(-a +b)}{c}(a +...
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0answers
56 views

find all pairs of natural numbers $(x, y)$ so that $121$ divides $x^2+y^2$

I have to find all pairs of natural numbers $(x, y)$ so that $121$ divides $$x^2+y^2$$ I thought of writing the amount as equal to $11z$, $z$ natural number. Then I noticed that the numbers must have ...
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1answer
35 views

How can $\frac{\tan \theta -1 }{\tan \theta + 1 } = \frac{1 - \cot \theta}{1 + \cot \theta}$ be proven using conjugates?

I can prove the this trig identity by working on both sides. But I am not sure how to prove this using one side and by conjugate. $$ \frac{\tan \theta -1 }{\tan \theta + 1 } = \frac{1 - \cot \theta}{...
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28 views

Matrice Algebra

I am reading a paper and stumbled upon this piece: $$(2u-1)^TQ(2u-1) = 4u^TQu - 4(1^TQ)u + 1^TQ1$$ I have two conflicting results but both results are different $4u^TQu-2u1^TQ-2u^TQ1+1^TQ1$ another ...
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1answer
41 views

Contradiction problem from $P(x)=a_nx^n+a_{n-1}x^{n-1}+ \dots+ a_0$

Prove that there is no polynomial $$P(x)=a_nx^n+a_{n-1}x^{n-1}+ \dots+ a_0$$ with integer coefficients and of degree at least $1$ with the property that $P(0), P(1), P(2), \dots$ are all prime numbers....
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3answers
58 views

Solve in $\mathbb Z$ the equation $(x+y)(x^2+y^2)=4xy+3$

So I have $$(x+y)(x^2+y^2)=4xy+3$$ I tried to develop it and I got $$x^3+y^3+xy(x+y-4)=3$$ what can I do next?
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6answers
120 views

How to find roots of quartic equation $x^4-x+1=0$

How to solve $x^4-x+1$? My attempt: $x^4-x+1=0$ $\implies x^4-x^3-x+1+x^3=0$ $\implies x^3(x-1)-(x-1)+x^3=0$ $\implies (x^3-1)(x-1)+x^3=0$ But I couldn't find a way to combine $x^3$ into that ...
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2answers
25 views

Quadratic with missing Linear Coefficient

Let $x^2-mx+24$ be a quadratic with roots $x_1$ and $x_2$. If $x_1$ and $x_2$ are integers, how many different values of $m$ are possible? I'm assuming we can use Vieta's Formula. We can say $x_1+...
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2answers
38 views

prove that $(2^k5^{k+1}+1,2^{n+1}5^k+1) \ne 1$

prove that for any $k,n \in N$ we have $$(2^k5^{k+1}+1,2^{n+1}5^k+1) \ne 1$$
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1answer
29 views

How to prove $\sum_{i=1}^n i.i! = (n+1)!-1$ with mathematical induction? [duplicate]

I'm trying to prove $$ \sum_{i=1}^n i.i! = (n+1)!-1 $$ with mathematical induction. The first step I did after prove it for 1 was: $$ \sum_{i=1}^{n+1} i.i! = (n+2)!-1= (n+2).(n+1)!-1 $$ but I can't ...
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1answer
30 views

How to solve this algebra problem related to Pythagorean triplets? [duplicate]

If $a^2+b^2=c^2 $ and $a+b+c=1000$ where a,b,c are positive integers, find the product $abc$. First I tried this: $$(a+b)^2=(1000-c)^2$$ $$a^2+b^2+2ab=c^2+1000^2-2000c$$ $$2ab=1000^2-2000c$$ But I ...
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0answers
45 views

How to prove $-|a|\le a\le |a|$ based on the fact that $|a|\le b ⇔ -b\le a\le b$

I tried doing this: If $a\ge 0$, then it's obvious that $-a\le |a|$ If $a\le 0$, then it's obvious that $a\le |a|$ Combining two cases we can see, $-|a|\le a\le |a|$ However, I am definitely sure that ...
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2answers
47 views

Continuity and the definition of a function. [duplicate]

I am reading through one of my maths textbooks at the moment and the following example is given. Determine whether the following functions are continuous at $x=2$ $f(x) = \frac{x^2 - 4}{x - 2}$...
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1answer
26 views

Spivak Calculus chapter 1 question 14

Question: Prove that $|a|=|-a|$ (The trick is not to become confused by too many cases. First prove the statement for $a\ge 0$. Whys is it then obvious for $a\le0$?) My proof for $a\ge0$: $|a|=a\\|-a|...
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2answers
65 views

How do I prove that $\max(x,\max(y,z)) = \max(\max(x,y),z))$ using an algebraic formula?

The maximum of two numbers can be expressed by $$\max(x,y) = \frac12\left(x+y+|x-y|\right)$$ Consequently, we can write $$\max(x,\max(y,z))=\frac{1}{4}\left(2x+y+z+|y-z|+|2x-y-z-|y-z||\right)$$ $$\...
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1answer
55 views

How may one solve problems over expressions like $(2+px)^6$ without the binomial theorem?

A friend of mine posed a problem on a mathematics discord server. The coefficient of the $x^2$ term in the expansion of $(2+px)^6$ is $60$. Find the value of the positive constant $p$. I ...
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0answers
28 views

Product of Factors Formula?

The product of the proper positive integer factors of $n$ can be written as $n^{(ax+b)/c}$, where $x$ is the number of positive divisors $n$ has, $c$ is a positive integer, and the greatest common ...
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31 views

At which values of $x_0$ does $\lim\limits_{x \rightarrow x_0} g(x)$ exist?

My answer is $(-\infty , -4) \cup (-4, \infty)$ but I am not sure if my answer is right.
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2answers
31 views

Sine Parametric function exercise

Find the biggest negative value of $a$ , for which the maximum of $f(x) =sin(24x+\frac{πa}{100})$ is at $x_0=π$ The answer is $a=-150$, but I don't understand the solving way. I would appreciate if ...
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0answers
62 views

How can $\sqrt{x} < 9 = [0,9)$?

I have tried solving this but seem to be missing something. I can't seem to figure out how to solve this problem. I get $x<81$ as my answer but I am obviously missing something. The answers are ...
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1answer
20 views

Let $s_k(n)$ denote number of digits in $(k+2)^n$ in base $k$ , evaluate $\lim_{n→∞}\frac{s_6(n)s_4(n)}{n^2}$.

Let $s_k(n)$ denote number of digits in $(k+2)^n$ in base $k$ , evaluate $\lim_{n→∞}\frac{s_6(n)s_4(n)}{n^2}$. How to find out the number of digits in a particular base? Any hint for the problem is ...
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2answers
49 views

How to solve $(x \times e^{x^2})^2$? [closed]

We found two solutions: $1$. $ x ^ 2 \times e ^ {2x ^ 2} $ $2$. $ x ^ 2 \times e ^ {x ^ 4} $. Which one is correct?
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0answers
30 views

What's the value of x in this equation?

What is the value of $x$: $$\frac{3^{3x^2}-4\cdot 3^{6x^2}+2x-20}{3^{(27-3x)(5x-7)}}=\frac{3^{-8x}\cdot 3^6}{3^{5+15x}}$$ enter image description here Thank you
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1answer
33 views

Picewised defined function with Inverse.

Let $$f(x) = \begin{cases} k(x) &\text{if }x>3, \\ x^2-6x+12&\text{if }x\leq3. \end{cases} $$ Find the function $k(x)$ such that $f$ is its own inverse. I really need a start to this. ...
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5answers
101 views

If $a, b, c\in\mathbb R^+, $ then prove that $a^3b+b^3c+c^3a\ge abc(a+b+c) .$

While trying to prove it, I proved the following two inequalities: $a^4+b^4+c^4\ge abc(a+b+c)$ and $(a^2b+b^2c+c^2a)(ab+bc+ca)\ge abc(a+b+c)^2.$ The later one, on some simplification gives $a^3b+b^...
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0answers
13 views

necessary and sufficient conditions?

What are the conditions under which a real-valued function of real variables $f(x,(y_1,...y_n))$can be written as $\sum_{i=1}^{N} g_i(x,y_i)$
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1answer
51 views

Why is considering only quadratic in one of the variables of a two variable quadratic sufficient for calculating roots

Find the $positive$ integral solutions to $7x^2-2xy+3y^2-27=0$ My solution: Assuming the quadratic in $x$ , if we assume one root to be integral , the other has to be rational (as y must be an ...
3
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2answers
55 views

All integer solutions of $x^3-y^3=2020$.

Find all integer pairs $(x,y)$ satisfying $x^3-y^3=2020$. First, $x^3-y^3=(x-y)(x^2+xy+y^2)=2020$ and $2020=2^2\cdot 5 \cdot 101$. But what next? Can it be worked out by using modulo? Or how? Any ...
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2answers
41 views

Prove that $\exists !c \in \mathbb{R} \exists ! x \in \mathbb{R} (x^2 + 3x + c = 0)$

This is an exercise from Velleman's "How To Prove It". I am struggling with how to finish the final part of the uniqueness proof, so any hints would be appreciated! a. Prove that there is a ...
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2answers
23 views

Finding Maclaurin series f(x)

Can anyone please help me with finding Maclaurin series for this $$f(x) = x^3 \tan^{-1}(2x); \quad |x|<\frac12$$ https://i.stack.imgur.com/bUhxk.jpg
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1answer
38 views

Can a real 2 by 2 matrix have one eigenvalue with geometric multiplicity 2?

Given the a real matrix $A=\begin{bmatrix} a & b \\ c& d\end{bmatrix}$, we assume that it has only one real eigenvalue $\lambda$. I am wondering if it is possible that the eigenvalue $\lambda$ ...
3
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3answers
106 views

Spivak Calculus chapter 1 problem 13 proof critique

Question paraphrased: Prove that the maximum between two numbers $x$ and $y$ is given by: $$\max(x,y)=\frac{x+y+|y-x|}{2}$$ Proof: Let $x$ and $y$ be two arbitrary numbers. Then, one and only one ...
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0answers
31 views

Can I use this rule in Spivak Calculus chapter 1?

In proof question no. 12 (iv), I used $|\frac{x}{y}| = -\frac{x}{y}$ if $x>0$ and $y<0$. However I am not sure if Spivak has defined that $\frac{1}{-y} = -\frac{1}{y}$. Should I take this as a ...
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1answer
27 views

How does the exponent behave when changing numerator and denominator?

I have seen this equation: $$\left( \frac{ n }{ n-1 } \right)^{ n-1 } = \left( \frac{ n-1 }{ n } \right)^{ 1-n }$$ As you can see the numerator switched with the denominator and I wonder how. I know ...
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0answers
45 views

does p = np in the hard set of problems [closed]

just got asked this in my year 6 homework questions. Apparently it is really easy according to my teacher. Any help would be appreciated. thnx
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2answers
52 views

find function f(x) such that the following holds [closed]

find function f(x) such that the following holds \begin{align} e^{\sum_{i=1}^{n} f(x_i)} = \sum_{i=1}^{n} x_i \end{align} where $0<x_i<1 \quad \forall I$ and n is an integer n>1. or, if the ...
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1answer
28 views

How to solve this implicit equation? (use trial and error)

I want to solve this implicit equation and find $f$ When $Re$ is constant: $$\frac{1}{\sqrt{f}}=2\log({Re.\sqrt{f})}- 0.8$$ I tried to make the equation simple By using: $\sqrt{f}=t>0,Re=a$: $$\...
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0answers
13 views

Limit of maximum of double indexed seqeunce

Suppose we have a uniformly bounded non-negative double indexed sequence $\{x_{m,n}\}_{m,n}$ such that $\forall m\in\mathbb N$ $\lim_{n\rightarrow\infty}x_{m,n}=1$. Then what can be said about the ...
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0answers
8 views

Create a table of values and an associated graph that are increasing, with a point of inflection.

X Y 3 .. 4 9 5 14 6 17 7 19 8 .. The question is how I can find the missing value. As I did the problem, I got 1 and 21 but they are the wrong answers in this case.
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0answers
20 views

linear equation and inequality in $n$ variables

Let $ c_1, \cdots, c_n$ be given non-zero real numbers, and not all of the same sign. Does there exists a solution to the equation $$ c_1 x_1 + \cdots + c_n x_n =0, $$ with $ x_i \in (0,1)$ for all ...

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