Complex Numbers...

Discuss the nature of the roots of the equations  $$\displaystyle (a+c-b)x^{2}+2cx+(b+c-a)=0$$

If $$-2$$ is a root of the quadratic equation $$x^2 + px + 2 = 0$$ and the quadratic equation $$2x^2 + px+ k = 0$$ has equal roots, find the value of $$k$$.

If $$z = re^{i\theta}$$, then the value of $$|e^{iz}|$$ is equal to

Find the value of $$p$$ for which the quadratic equation $$x^2 + p(4x + p - 1) + 2 = 0$$ has equal roots ?

Find the modulus and amplitude of $$-2i$$

Find the modulus and amplitude of $$-2 + 2 \sqrt 3i$$

Find the modulus and amplitude of $$-\sqrt 3-i$$

The amplitude of

$$\displaystyle sin \frac{\pi}{5} + i \left ( 1 - cos \frac{\pi}{5} \right )$$

If $$z_1, z_2, \varepsilon C$$ are such that $$|z_1 + z_2|^2 = |z_1|^2 + |z_2|^2$$ then $$\displaystyle \frac{z_1}{z_2}$$ is

Find the value of $$x^3 + 7x^2 -x + 16$$, where $$x = 1 + 2i$$