Complex Numbers...

sinx + i cos 2x and cos x – i sin 2x are conjugate to each other for:

The real value of α for which the expression \(\frac{1 - i sin \alpha}{1 + 2 i sin \alpha}\) is purely real is :

If z is a complex number, then

If z = x + iy lies in the third quadrant, then \(\frac{\bar z}{z}\) also lies in the third quadrant if

The value of (z + 3) (\(\bar z\) + 3) is equivalent to

If \((\frac{1 + i}{1 - i})^x = 1\), then

A real value of x satisfies the equation \((\frac{3 - 4ix}{3 + 4ix})\) \( = \alpha - i\beta (\alpha, \beta \in R)\) if \(\alpha^2 + \beta^2 = \)

Which of the following is correct for any two complex numbers \( z _1\) and \(z _2\) ?

The point represented by the complex number (2, – i) is rotated about origin through an angle \(\frac{\pi}{2}\) in the clockwise direction, the new position of point is:

Let x, y ∈ R, then x + iy is a non real complex number if:

If a + ib = c + id, then

The complex number z which satisfies the condition \(|\frac{i + z}{i - z}| = 1\) lies on

\(|z_1 + z_2|=|z_1|+|z_2|\) is possible if

The real value of θ for which the expression \(\frac{1 + i cos \theta}{1 - 2i cos \theta}\) is a real number is:

The value of arg (x) when x < 0 is: