Math Is Fun Forum

FP1Student

Guest

Several Questions.

Hey - I'm a little stuck on some of these questions. I would really appreciate it if you led me in the right direction - thank you. ^_____^

Firstly.

I have the cubic equation x³+ax+b=0, where and a b are constants, and has roots α, β and γ.

I know that a=αβ+βγ+αγ and that b=-αβγ

It tells me that αβ=γ and to express a and b in terms of γ only.

I can get b=-γ²

But I can't find a. I keep of getting more and more α's and β's.

I can get a down to: a=γ+γ(α+β) But then I get stuck. >_<

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Secondly.

I have 3 simultaneous equations:

-x+2y+z=a
x+y+z=b
4x+y+2z=c

And I know that the determinant is 0 and so it has no unique solution. But what I am asked to find is the relation between a,b and c for which the equations are consistent. And I have no idea how to do that. o_O

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Thirdly.

I have a point P with column matrix

and the point Q is the image of P under the transformation of:

.

I found that the column matrix of Q would be:

.

And then I need to show that the line joining P to Q makes and angle θ with the x-axis. I can do it numerically, but not algebraically. What would be the algebraic method?

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And that's it. ^_^ Thank you very very much for your help in advance. It will be greatly appreciated.

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Several Questions.

You can find another relationship between α, β and γ by equating coefficients of the x² term of the equation. That will allow you to express α+β in terms of γ and so you'll be done.

Your three simultaneous equations have a determinant of 0 because the first two have LHS's that can be combined in some way to form the LHS of the third one.
ie. m(-x+2y+z) + n(x+y+z) = 4x+y+2z, for some value of m and n.
Find those and the relationship between a, b and c will follow.

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Why did the vector cross the road?
It wanted to be normal.

FP1Student

Guest

Re: Several Questions.

Oh yes. >_< Gah, how did I forget that? Haha, okay, I've got the first question down.

Is it really that easy? I thought that it would have been like that because I remember reading that they had to be multiples of each other. It is worth 4 marks. But okay, I will use that method. Thank you. ^_^