Number Theory T...

Let $$\alpha,\ \beta$$ be real and $$\mathrm{z}$$ be a complex number. If $$\mathrm{z}^{2}+\alpha \mathrm{z}+\beta=0$$ has two distinct roots on the line $$Re(z) =1$$, then it is necessary that: 

Let z be a complex number such that $$\left|\dfrac{z-i}{z+2i}\right|=1$$ and $$|z|=\dfrac{5}{2}$$. Then the value of $$|z+3i|$$ is?

If $$ z$$ is a complex number such that $$ |z|\geq 2$$, then the minimum value of $$ |z+\displaystyle \frac{1}{2}|$$

If $$z = x + iy$$ and $$\omega = \dfrac{(1 -iz)}{(z-i)}$$, then $$\left|\omega\right| = 1$$ implies that in the complex plane

Let z be a complex number such that the imaginary part of z is nonzero and a = $$z^2 + z + 1$$ is real. Then a cannot take the value

$$\displaystyle \left|\dfrac{\sqrt{3}+i}{(1+i)(1+\sqrt{3}i)}\right|=$$

The modulus of $$\sqrt{2}i-\sqrt{-2}i$$ is:

If $$z =3+5i$$, then $$z^3+z+198=$$

If $$z=2-3i$$ then $$z^2-4z+13=$$

The complex number $$\displaystyle \frac{1+2i}{1-i}$$ lies in the quadrant :