man0005's specialist question thread :)

[man0005]

what about this one? ><
z^3 + az^2 + bz + 10 - i
find the constants a and b given they are real, if z = (1+i) is a zero of the polynomial

[m@tty]

You have (z-1-i) as a factor.

Long divide, then use the quadratic formula on the result.

[man0005]

hm if a and b are real, wouldnt another factor be (z-1+i)?
cant i do that?

[man0005]

If someone could also help me with this one, that would be great
Find the set of real values for k, k doesnt equal -1, for which the roots of the equation x^2 +4x -1 + k ( x^2 + 2x + 1 ) = 0
are:
a) real and distinct
b) real and equal
c) complex with positive part

[m@tty]

hm if a and b are real, wouldnt another factor be (z-1+i)?
cant i do that?

No you can't there is an imaginary unit in the expression... right at the end. complicates it a bit.

For the conjugate root theorem to apply, ALL coefficients must be real (no i's present when simplified).

And, IF you could apply the conjugate theorem, it would be (z+(1+i)).

[man0005]

ohhh, om gosh how did i miss that ._.
thanks !

do you have any ideas for the next one?

[m@tty]

Use discriminant



Solve for:
(real and equal)

(complex)

(real and distinct)

[xZero]

If someone could also help me with this one, that would be great
Find the set of real values for k, k doesnt equal -1, for which the roots of the equation x^2 +4x -1 + k ( x^2 + 2x + 1 ) = 0
are:
a) real and distinct
b) real and equal
c) complex with positive part

expand k,



now use and if it =0 then its real and equal, if its >0 then real and distinct and <0 will be complex

Edit: I Got Ninja'ed oO

[man0005]

ahh thanks
got another one
Let z be a complex number with | z | = 6
Let A be the point representing z. Let B be the point representing (1+i)z
Prove that OAB is an isosceles right-angled triangle

(sorry if im annoying you with all the qs D:  )

[VCEMan94]

Can someone help me with this one as well? D:
Let S = { z : | z - (2root2 + i (2root2) | < (or equal to) 2
If z belongs to S, find the maximum and minimum values of |z|
If z belongs to S, find the maximum and minimum values of Arg (z)

[xZero]

change 1+i and z into polar form so and



well since the new line moved by 45 degrees, then the adjacent side would be which will get you 6, so we have 2 sides with magnitude of 6 and a 45 degree angle between the hypotenuse and the opposite, hence it is a isosceles right-angled triangle

[xZero]

Can someone help me with this one as well? D:
Let S = { z : | z - (2root2 + i (2root2) | < (or equal to) 2
If z belongs to S, find the maximum and minimum values of |z|
If z belongs to S, find the maximum and minimum values of Arg (z)

since my brain is dead ill just tell you what i would do,

convert S into a Cartesian equation and graph it by letting z = x + yi.

The value of |z| is asking you what is the maximum and the minimum distance from the graph to the origin.
The Arg(z) is the angle from origin to the graph, maximum and minimum occurs when the line from the origin to the graph is a tangent, so just work out where on the graph the line will be a tangent which goes through the origin

[VCEMan94]

oh okay, but how would you find the tangent? :/

and would you know how to do this one by any chance?
Let w = cis theta   and z = w + (1/w)
Show that z lies on ellipse with equation (x^2/25) + (y^2/9) = 1/4

[m@tty]

change 1+i and z into polar form so and



well since the new line moved by 45 degrees, then the adjacent side would be which will get you 6, so we have 2 sides with magnitude of 6 and a 45 degree angle between the hypotenuse and the opposite, hence it is a isosceles right-angled triangle

Did you assume a right angle in your explanation?

[xZero]

change 1+i and z into polar form so and



well since the new line moved by 45 degrees, then the adjacent side would be which will get you 6, so we have 2 sides with magnitude of 6 and a 45 degree angle between the hypotenuse and the opposite, hence it is a isosceles right-angled triangle

Did you assume a right angle in your explanation?

Nope, the angle between the new line and z is 45 degrees (from the pi/4 + theta part) and work out the magnitude of the line connecting the new line and z, which is 6. Then use these facts to proof it