-4
votes
0answers
65 views

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 ...
0
votes
1answer
20 views

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 ...
0
votes
1answer
17 views

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 ...
2
votes
2answers
21 views

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 ...
4
votes
1answer
36 views

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 ...
1
vote
1answer
28 views

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 ...
0
votes
3answers
29 views

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 ...
0
votes
6answers
301 views

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 ...
-1
votes
2answers
28 views

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 ...
-2
votes
2answers
40 views

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
2
votes
1answer
44 views

Sum involving integer part and cosine function

How to find the close form of sum and eliminate $k$? $$ \sum_{k=1}^{n} \frac{n \left[ \cos \left( \frac{n}{k}- \left[\frac{n}{k} \right]\right) \right]}{k} $$
2
votes
4answers
66 views

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 ...
1
vote
1answer
38 views

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 ...
2
votes
2answers
93 views

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 ...
4
votes
4answers
510 views

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.
0
votes
0answers
24 views

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 $ax+by+C=0$: ...
2
votes
3answers
71 views

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 ...
1
vote
1answer
145 views

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 ...
0
votes
4answers
45 views

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 ...
1
vote
0answers
22 views

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 ...
0
votes
0answers
54 views

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 ...
0
votes
1answer
22 views

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 ...
0
votes
1answer
29 views

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 ...
1
vote
1answer
72 views

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, ...
0
votes
1answer
50 views

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, ...
0
votes
0answers
35 views

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 ...
0
votes
4answers
64 views

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
0
votes
4answers
53 views

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?
0
votes
1answer
21 views

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?
2
votes
1answer
46 views

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. ...
0
votes
2answers
62 views

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? $x^ay^b=a^ab^b(a+b)^{-(a+b)}(x+y)^{a+b}$
2
votes
1answer
57 views

Isomorphism implies direct sum of Kernel and Image

If $f: U \rightarrow V$ and $g: V \rightarrow W$ are linear transformations between vector spaces over a field $K$ such that $ g \circ f$ is an isomorphism, then $V = \operatorname{Im}f \oplus ...
1
vote
0answers
40 views

$T (x_1,x_2,x_3,…,x_n) = (-x_3,x_3,x_4,x_5,…) $ then $ W \ne ker T$

Let $V$ the vector space of all sequences of real numbers and $W$ the subspace given by $W = \{(a,a,0,0,...) | a \in R\}$ , and $T : V \rightarrow V$ given by $T (x_1,x_2,x_3,...,x_n) = ...
1
vote
2answers
29 views

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 ...
1
vote
4answers
87 views

$\{ v_1,v_2,…,v_n\}$ 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 ...
3
votes
2answers
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 ...
0
votes
0answers
14 views

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 ...
0
votes
3answers
41 views

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$ ...
0
votes
3answers
42 views

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 ...
0
votes
4answers
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 ...
0
votes
0answers
32 views

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 ...
3
votes
3answers
233 views

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 ...
0
votes
1answer
23 views

Expressing units of time

How would you express 8/3 seconds as time after 3pm ? 8/3 = 2.66666 0.66*60 =40 miliseconds = 0.04 seconds so 2.04 seconds after 3 3:00:02:04 pm ? Is this correct?
1
vote
3answers
37 views

Is this triangle question missing information?

In the $\Delta KLP$, find $a+b$: My question is that: isn't some information missing from the question? Because all I can see is is that $ \usepackage{ gensymb } \angle SKP = \angle LTS = ...
0
votes
2answers
35 views

Re-writing a a differential function

I don't understand the concept of this... how do I derive a an equation written in terms of a function? How do I differentiate f(function inside) ...?
0
votes
0answers
33 views

Finding the constant of a function in terms of the gradient of a tangent.

Let $f : \Bbb R \to \Bbb R, f (x) = e^x+ k$, where $k$ is a real number. The tangent to the graph of $f$ at the point where $x = a$ passes through the point $(0, 0)$. Find the value of $k$ in terms of ...
-1
votes
2answers
36 views

How to solve $\frac12 \sec^2 \frac x2 = 1$ under restricted domain?

solve: $$\frac12 \sec^2 \left(\frac x2\right) = 1$$ and domain $x: (-\pi,\pi) \cup (\pi,3\pi)$. sec^2 (x/2) = 2 sec^2 (x/2) can be re-written as tan(x/2)^2 + 1, therefore tan^2(x/2) + 1 = 2 ...
0
votes
1answer
55 views

An algebra/linear algebra question

Suppose 8 real numbers $a,b,c,d$ and $x,y,z,w$ satisfy \begin{equation*} a^2+b^2+c^2+d^2=x^2+y^2+z^2+w^2=1,\quad ax+by+cz+dw=0. \end{equation*} Is it true that \begin{equation*} ...
0
votes
1answer
36 views

General Solution for Cosine (negative angles)

cos2(x+pi/3)=1/2 2(x+pi/3)=pi/3 x+pi/3=pi/6 x+2pi/6=pi/6 x=-pi/6 x=5pi/6 (is this step correct) ... ?? x = +/- pi/6 +kpi , k is a subset of Z x = +/- 5pi/6 +kpi , k is a subset of Z can someone ...
0
votes
1answer
41 views

Finding equations when given new center of a circle

$y = −x + \sqrt{2}$, $y = −x − \sqrt{2}$, $y = x + \sqrt{2}$, and $y = x − \sqrt{2}$. These equations determine lines, which in turn bound a diamond shaped region in the plane. Construct a diamond ...