Complex Numbers...

Find the root of equation x² + kx + 2 = 0 where, k is the arithmetic mean of root of equation x² + 6x + 8 = 0. 

If the roots of the equation \(3 a x^{2}+2 b x+c=0\) are in the ratio \(3: 2\), then which one of the following is correct?

For any positive integer \(n,(-1 \sqrt{-1})^{4 n+5}\) is:

Find the nature of roots of x²+ 7x - 10 = 0.

Find the value of \(\sqrt{(5+12 i )}\), where \(i =\sqrt{-1}\).

What is the modulus is \(\frac{\sqrt{2}+ i }{\sqrt{2}- i }\) where \(i =\sqrt{-1}\)?

Find the roots of \(a^{2} x^{2}-3 a b x+2 b^{2}=0\).

Find the value of \(\theta\) for which \(z =\frac{3-2 i \sin \theta}{2- i \sin \theta}\) is purely real.

If \(\alpha\) and \(\beta\) are the roots of the equation \(x^{2}-6 x+3=0\), what is the value of \(\frac{\alpha}{\beta}+\frac{\beta}{\alpha} ?\)

The equation which has the sum of its roots as 3 is: