• Coming soon...BoS Trial exams
    Watch this space!

Complex Question and another question (1 Viewer)

DraconisV

Christopher Fife
Joined
Mar 11, 2005
Messages
186
Gender
Male
HSC
2006
Ok i have this complex number question that im confused with.

Ok.

1. Solve the equation Z^5 + 81Z = 0, expressing the answers in the form a + ib, where a and b are real.

Ok im confused can someone show me how this is done.

Also

2. The graph y = (ax^2 + bx + c)/(x^2 + qx + r) has line x=1, x=3 and y=2 as the asymptotes and (0,1) as turning point.

Use this information to show that y = (2x^2 - 4x +3)/(x^2 - 4x + 3)

Im totally lost with this one, ive tried doing all sorts of things but nothing has made me progress at all.

Can someone please show me, very closely how the hell this dam bloody question is done.

Thank you guys.
 
P

pLuvia

Guest
First one, factorise it, then use DeMoivre's theorem to get the four roots of unity. Then you should end up with four equations then just change them back to a+bi form

Second one, you should know that the denominator when equalled to zero are yields the horizontal asymptotes i.e. y=2 and that the numerator yields the vertical asymptotes i.e. x=1, x=3. So you can sub those numbers into those equations, and then using the first derivative and subbing in the (0,1) you have a second equation, then solve simultaneously for the pronumerals

Hope that helped
 

Trev

stix
Joined
Jun 28, 2004
Messages
2,035
Location
Pine Palace, St. Lucia, Brisbane.
Gender
Male
HSC
2005
Since it's 'x' assymptotes are x=1 and x=3 that means the denominator (as it can't equal zero) will be (x-1)(x-3)=x<sup>2</sup>-4x+3.

You are given (0,1) as a point, so sub this into the equation you have so far:
1=c/r, and you figured out r=3, so c=3.
You now have y=(ax<sup>2</sup> - bx +3)/(x^2 - 4x + 3)
Differentiate this to get:
dy/dx=[(x²-4x+3)(2ax+b)-(ax²+bx+3)(2x-4)]/[(x²-4x+3)²]
Since (0,1) is a turning point, then when dy/dx=0, x=0. So:
0=3b+12, b=-4.

(I can't be bothered typing the rest) but, make 'x' the subject and do the same as for the 'x' assymptoes, the denominator should equal zero when you sub in y=2, so you will then find out the value for a.
 

Users Who Are Viewing This Thread (Users: 0, Guests: 1)

Top