Various problems - help,suggestion?

[R A V E N]

Problem 1:Prove that the following system of linear equations has unique solutions(sorry,I don`t know precise english term for this kind of system) and if it has,solve it:

$$x_1+x_2+x_3+x_4=4$$

$$x_1+2x_2+3x_3+x_4=7$$

$$x_2+x_3+2x_4=4$$

$$x_1-x_2+x_3-2x_4=-1$$

$$2x_1-2x_2+x_3-x_4=0$$

So, four variables,five equations.I don`t see how any method could be applied here:Gaussian elimination,Cramer`s rule,matrix method.Any hint just to get stared would be enough.

 

[dirk_mec1]

Look for dependent equations and see if there are four independent equations then calculate the determinant.

 

[R A V E N]

What is dependent equation?If I am right,it is one equation from the system multiplied by some constant to produce additional equation.But I do not see any.

R A V E N: stared

*started

 

[HallsofIvy]

Problem 1:Prove that the following system of linear equations has unique solutions(sorry,I don`t know precise english term for this kind of system) and if it has,solve it:

$$x_1+x_2+x_3+x_4=4$$

$$x_1+2x_2+3x_3+x_4=7$$

$$x_2+x_3+2x_4=4$$

$$x_1-x_2+x_3-2x_4=-1$$

$$2x_1-2x_2+x_3-x_4=0$$

So, four variables,five equations.I don`t see how any method could be applied here:Gaussian elimination,Cramer`s rule,matrix method.Any hint just to get stared would be enough.

Well, try to solve it! Since you have 4 variables, you should be able to solve any 4 of the equations for those variables. Once you have done that, do they satisfy the fifth equation?
A slightly more formal way, though not any better, would be to set up the 5 by 4 matrix of coefficients and row-reduce it. If the last row turns out to be all 0's then there is a solution, otherwise, there is not.

 

[R A V E N]

Yeah,choosing random four equations and solving system by using Cramer`s rule works.Thanks,HallsofIvy.

 

[R A V E N]

Problem 2:Solve this equation($$z$$ - complex number):

$$(z^2-2)^6+z^6=0$$

Any hints?

 

[Dick]

z is not a root. So it's safe to divide both sides by z^6. That gives you ((z^2-2)/z)^6=(z-2/z)^6=(-1). So z-2/z is equal to one of the six sixth roots of (-1). Solve z-2/z=k and set k to be each of the six sixth roots. You'll get a total of twelve roots (as you should, it's a twelfth degree equation). That's pretty tedious. But you can do it.

 

[R A V E N]

Thanks,Dick.

Problem 3:
Calculate:

$$\lim_{x\to\infty}\left[\tan\left(\frac{\pi}{4}+x\right)\right]^{\cot{2x}}$$

Even hint how to start would be appreciated.

 

[R A V E N]

Actually,correct form is:

$$\lim_{x\to 0}\left[\tan\left(\frac{\pi}{4}+x\right)\right]^{\cot{2x}}$$

By using trigonometric identities,I transformed above into:

$$\lim_{x\to 0}\left(\tan\frac{1}{\cos{2x}}\right)^{\frac{\cos{2x}}{\sin{2x}}}$$

 

[Dick]

I think you slipped up on the trig identities, but you don't really need them anyway. You have a 1^(infinity) type limit. Take the log to change it to an infinity*0 limit and rearrange it to apply l'Hopital's rule.

 

[R A V E N]

Yeah,instead of attempting anything and ramble in various faulty courses,it`s better to wait advice for correct approach.

On what basis did you concluded that it is $$1^\infty$$ type of a limit?

 

[Dick]

No, it's better for you to ramble in faulty courses, otherwise I won't help at all. :) x-> 0, x+pi/4 -> pi/4. tan(pi/4)=1. 2x -> 0, cos(2x) -> 1, sin(2x) -> 0, cos(2x)/sin(2x)=cot(2x)->infinity. Hence 1^infinity.

 

[R A V E N]

And it was that simple?(By the way,everything is simple if you have enough time and energy to invest into studying,thinking and looking for advice on various places).

I just wonder why those 800+ pages mathematical books which have zillion things in it don`t have any example which would for the moment escape rigorous mathematical world and demonstrate things like that and other in more human-friendly fashion?

For the sake of more successful and contented both teachers and students.

 

[Dick]

And it was that simple?(By the way,everything is simple if you have enough time and energy to invest into studying,thinking and looking for advice on various places).

I just wonder why those 800+ pages mathematical books which have zillion things in it don`t have any example which would for the moment escape rigorous mathematical world and demonstrate things like that and other in more human-friendly fashion?

For more successful and contented both teachers and students.

What I gave just shows you what KIND of limit it is. It doesn't solve the limit. Did you figure out the value?

 

[R A V E N]

I will try. I know that solution is $$e$$.