# Tagged Questions

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### Prove that $\sqrt{n}$ is irrational [on hold]

Question: Using fundamental theorem of integers and the fact that every natural number that is not prime, prove that $\sqrt{n}$ is irrational unless $n=m^2$ for some $m\in\mathbb N$. Here is how I ...
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### Rearrange equation with integrating factor

I'm trying to do the following in the middle of a huge question involving a differential equation - I need to rearrange this equation for t, but have no idea where to start. First image is the ...
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### Derivates and Limits in the Same Problem are an Issue.

I am working on the following problem:- Evaluate lim x→1 [( x^1/4 - 1 ) / ( x^1/3 - 1 )] by relating it to the derivatives of functions. Now this is quite a ...
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### Solve the algebra equation- unsure about order of operations, how to go about solving, solve for x

The question states: solve the equation. State the solution set and check your answer. I've spent a good 45 minutes on this, to know avail. If someone could sort of walk me through this I would be ...
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### Prove that this Newton sum value is unique

\begin{align}a+b+c+d&=1\\ a^2+b^2+c^2+d^2&=2\\ a^3+b^3+c^3+d^3&=3\\ a^4+b^4+c^4+d^4&=4\\ a^5+b^5+c^5+d^5&- ?\end{align} The usual method I see for solving this kind of ...
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### Rational solutions to a system of equations

I have a system of equations \begin{align} xy + 3zw = 0; \\ xz + 2yw = 0; \\ xw + yz = 0. \\ \end{align} Plugging it into a CAS, I see that all the rational solutions to this system have ...
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### Concerning the point $(7,a)$ on the line containing $(0,0)$ and $(4,2)$

I have recently been studying to take the GRE's and while working through the math section I find a lot of problems similar to this: Now I know it is supposed to be assumed that point $O$ is marked ...
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### How to solve a system of two linear equations with two unknowns?

How do I solve this system of equations? $$\begin{cases} 7(a+b)=b-a \\4(3a+2b)=b-8\end{cases}$$ Progress I tried both substitution and elimination, but when I set $a$ or $b$ free on one side, I ...
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### At what time and distance from Delhi will the mall train completely cross the goods train?

A goods train $158$ metres long, and traveling at the average speed of $32$ km/hr leaves Delhi at $6:00$ A.M. Another mall train $130$ metres long and traveling at the average speed of $80$ km/hr ...
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### Hey guys. Given the graph below, find the equation of the transformed parent function. [closed]

It would be great if there is a detailed explanation. Also, is there a standard method I can use to answer all kinds of graphs including exponents and logs? Thanks
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### Sum involving integer part and cosine function

How to find the close form of sum and eliminate $k$? $$\sum_{k=1}^{n} \frac{n \left[ \cos \left( \frac{n}{k}- \left[\frac{n}{k} \right]\right) \right]}{k}$$
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### Show that $ax^2+2hxy+by^2$ is positive definite when $h^2<ab$

The question asks to "show that the condition for $P(x,y)=ax^2+2hxy+by^2$ ($a$,$b$ and $h$ not all zero) to be positive definite is that $h^2<ab$, and that $P(x,y)$ has the same sign as $a$." Now ...
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### Transformation matrix from a translated-rotated coordinate system to the general coordinate system

In Figure 1, suppose $XYZ$ (in black) as my general coordinate system and $X'Y'Z'$ (orange) as another system with parallel axes respect to $XYZ$. Consider $xyz$ (green) is my 3rd coordinate system ...
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### Any idea how to linearize this equation? $X^2-Y^2=aZ+bZ^2$

The intention is to linearize this equation $X^2-Y^2=aZ+bZ^2$ into something which looks like $Z=mX+nY+c$ so that a graph of $Z$ against $X$ or $Y$ can be plotted. X,Y,Z are variables while a,b,c are ...
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### What is the non-trivial, general solution of these equal ratios? [closed]

Provide non-trivial solution of the following: $$\frac{a}{b+c}=\frac{b}{c+a}=\frac{c}{a+b}$$ $a=?, b=?, c=?$ The solution should be general.
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### Find the following formulas for the images of the line $ax+by+C=0$ under translation and rotation.

We have lamar lines in the form of $ax+by+c=0$, where $a,$ $b,$ and $c$ are fixed reals satisfying $a^2+b^2\neq0$. We need to find the following formulas for the images of the line $ax+by+C=0$: ...
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### The meeting of Cars

Three cars, A, B and C move towards north in a particular straight track (consider the length of the tract infinite). Another car D comes from a certain distance towards south. The car A meets B at 8 ...
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### Showing that planes intersect

let there be two planes $$2x-y-5z+11=0$$ and$$2x+2y+z-1=0$$ show that they intersect attempt at a solution: If planes do not intersect they are parralel hence there is a $t\in R$ such that ...
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### Given a satisfactory real number = [any integer]/(2b) where a and b are integers, how would one find the minimum value of b?

For instance, 0.625 = 5/(2*4). Given 0.625, how would one find 4? 0.75 = 1/(2*2). Given 0.75, how would one find 2? I should ...
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### finding the symmetric point

let there be $4$ points. $A(-1,1,1), B(2,0,-1), C(1,3,-2), D(-2,-1,0)$. the $4$ points are not on the same line. the plane which goes through the points $A$ and $B$, and which is also paralel to the ...
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### Solve system of equations for the ratios of the vectors

(Sorry for the bad title, didn't think of a better way to describe the problem). I have a system $\mathbf{A}\in\mathbb{C}$ that forms the problem $\mathbf{Ax}=\mathbf{b}$, for which I want to find an ...
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### Graph exponential function

I am having problems understanding why $xe^x + 10e^x$ has two $(x,y)$ intercepts. I understand why there is one $(0,10)$, but am unclear on how to return $(-10,0)$. Any help would be much ...
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### Finite sum equaling Kronecker Delta

could anyone help understand how $$\sum_{j=0}^{n-r}\binom{n-r}{j}*(-1)^{j} = [1 + (-1)]^{n-r}$$ I see that if $j=0$, i get $1=1^{n-r}$, and if $j=n-r$, i get $(-1)^{n-r},$ but what about the rest of ...
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### How to solve this graphing question?

$\frac{|x-2|} {(x^2-4)}+\frac{(x-2)} {|x-2|} = b$ determine for which values of $b$ the equation has one and only solution. I tried sketching the graph, but was unable to do so accuratly...also, ...
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### For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at each x -intercept.

For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at each x -intercept. f(x) = (1/5)x^4(x^2 - 3) the choice 1- 0, ...
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### Using Algebra with Trig Functions

Using Algebra with Trig Functions I'm trying to find the correct 1 second audio signal I would need to apply to a 1 second known noise signal to have the output signal be a sin wave. The basic ...
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### How can I factor $x^2 + 2\sqrt{3}\,x + 3$? [closed]

$$x^2 + 2\sqrt{3}\,x + 3$$ Anyone could tell me how may I factor this? Thanks a lot
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### algebraic representation of a line in 3d

Is an algebraic representation of a line in 3d possible, or there can be only a parametric one?
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### the volume of pyramid value

when calculating the volume of pyramid using a determinnat, is it ok to take the determinanat in absloute value so that every negative result would be converted to positive volume number?
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### Vector calculation question

the points a b c d are concordantly ( 1,2,-3) , (-1,2,1) , ( 0,1,-2) , ( 2,-1,1) find formula of the plane going thorugh d and which is pararlel to plane abc calculate the volume of pyramid abcd. ...
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### explain this confusing algebraic identity?

Can anyone show, step-by-step, how the expression on the LHS can be turned into the expression on the RHS? $x^ay^b=a^ab^b(a+b)^{-(a+b)}(x+y)^{a+b}$
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### If $f\in V$ of degree $n$ then for every $g \in P_n(\Bbb R)$ there exist scalars $c_0,c_1,..,c_n$ such that $g = c_0f + c_1f'+ … + c_nf^{(n)}$

Let $V=P(\Bbb R)$ and $1 ≤ i$ be the vector space of the polynomials with real coefficients, on the field of real numbers $\Bbb R$. Let $T_i(f)=f^{(i)}$ the $i$th derivate of $f$. a) I have to show ...
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### $\{ v_1,v_2,…,v_n\}$ is basis of $V$ if and only if $\{ v_1,v_1 + v_2,…,v_1 + v_2+…+v_n,\}$ is a basis of $V$

Let $V$ a vector space over a field $K$. Is it true $\{ v_1,v_2,...,v_n\}$ is basis of $V$ if and only if $\{ v_1,v_1 + v_2,...,v_1 + v_2+...+v_n,\}$ is a basis of $V$ ? I made some examples and ...
183 views

### Linear algebra calculus trick.

I have a matrix and a vector: $$A=\begin{bmatrix} a &b\\ c&d \end{bmatrix},$$ $$\vec v=\begin{bmatrix} a+b\\ c+d \end{bmatrix}$$ Is there an algebraic operation that produce the ...
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### Interpreting & Analysing a Transitional Matrix

How do you interpret such a problem Are we expect to add the rows, and that would be the one with larger number of goats in the long term. Therefore A(row 1) and b(row 2)... therefore the answer is ...
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### Explanation on characterstic polynomial

$A_2 = \begin{pmatrix} 1 & 1 \\ a & 1 \end{pmatrix}$ So the characteristic polynomial of $A_2$ is $P_a(t) = (t-1)^2 - a$ Then, $P_a(t) = t^2 -2t +1 -a$ ...
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### What am I doing wrong in searching for the intersection point between 2 linear equations?

y=4x+5 y=3x-7 First I take equation 1 and set y to 0: 0 = 4x + 5 -5 = 4x -5/4 = x I get that the information above only ...
1k views

### Finding an equation of circle which passes through three points

How to find the equation of a circle which passes through these points $(5,10), (-5,0),(9,-6)$ using the formula $(x-q)^2 + (y-p)^2 = r^2$. I know i need to use that formula but have no idea how to ...
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### Subset Sum represented as a perfect number

Can we form a set of $29$ distinct integer elements such that every subset of elements possible has a sum which is a perfect power? A perfect power is a positive integer that can be represented a p^q ...
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### How should I prove $(a+b)^3= a^3+3ab(a+b)+b^3$ — Model or figure?

In what way can I prove/verify $(a+b)^3= a^3+3ab(a+b)+b^3$ ? Should I make a 3D model, or create 2D figure? In the case of 3D model, I have made $a^3$ and $b^3$; i.e cube'a' and cube'b'. I don't know ...