Complex Numbers...

The equation $$\sqrt {3x^2+x+5}=x-3$$, where $$x$$ is real, has 

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 $$z = x + iy$$ 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

Express $$\dfrac{1}{(1 -  cos  \theta  +  2  i  sin  \theta)}$$ in the form $$x + iy$$

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

If $$a, b$$ and $$c$$ are real numbers then the roots of the equation $$(x - a)(x - b) + (x - b)(x - c) + (x - c)(x - a) = 0$$ are always

$$ax^2+bx+c=0$$, where a, b, c are real, has real roots if.

If $$z =3+5i$$, then $$z^3+z+198=$$