Complex Numbers...

If $$x + i  y = \dfrac{3}{2 + cos  \theta + i  sin  \theta}$$, then $$x^2 + y^2$$ is equal to

Let $$z = x + iy$$, where $$x$$ and $$y$$ are real. The points $$(x, y)$$ in the $$X-Y$$ plane for which $$\dfrac {z + i}{z - i}$$ is purely imaginary lie on

The roots of the equation $${ x }^{ 2 }-8x+16=0$$

The expression $$\dfrac {(1 + i)^{n}}{(1 - i)^{n - 2}}$$ equals.

The value of $$a$$ for which the quadratic equation $$3{x}^{2} + 2{a}^{2}x + 2x + {a}^{2} - 3a + 2 = 0$$ possesses roots of opposite signs lies in

If $$z_1$$ and $$z_2$$ are complex numbers with $$|z_1|=|z_2|$$, then which of the following is/are correct?
1. $$z_1=z_2$$
2. Real part of $$z_1 =$$ Real part of $$z_2$$
3. Imaginary part of $$z_1 =$$ Imaginary part of $$z_2$$
Select the correct answer using the statements given below :

Let a, b be non-zero real numbers. Which of the following statements about the quadratic equation $$ax^2 + (a+b) x + b = 0$$ is necessarily true?
(I) It has at least one negative root.
(II) It has at least one positive root.
(III) Both its roots are real.

The quadratic equation $${ x }^{ 2 }+bx+4=0$$ will have real roots if

The roots of the equation $$2a^2x^2 - 2abx +b^2=0$$ when a < 0 and b > 0 are : 

If ................. then the roots of the quadratic equation are equal.