p294 q16 b c

question:

it is given that the equation

$\displaystyle z^2-2z+k =0 $(k is real ) --------- (*)

has no real roots.

a) find the range of k

b) find the quadratic equation whose roots are the cubes of the roots of (*) and show that the discriminant of this equation is $\displaystyle 4(1-k)(4-k)^2$

if this equation has real roots , deduce the value of k.

c) find, in terms of k , the squares of the roots of (*), expressing the answers in the form x+iy where x and y are real.

as k varies, find the equation of the locus of the points in the Argand plane representing the squares of the roots of (*).

i have problems in b) and c)

b) $\displaystyle 4(1-k)(4-k)^2 \ge 0$

$\displaystyle (k-1)(k-4)^2 \le 0$

$\displaystyle k \le 1$ or$\displaystyle k = 4 $

But the answer says it is k = 4. i don't understand.

c)i get the two points $\displaystyle 2 - k + 2\sqrt{1-k} $and $\displaystyle 2 - k - 2\sqrt{1-k}$ ie $\displaystyle -2 + 2 \sqrt 3 i $and $\displaystyle -2 - 2 \sqrt 3 i $

but i don't know how to represent the locus of them as k varies. it seems that i have problems in the idea of locus.

thanks!