The value of \(\sum_{n=1}^{13}\left(i^{n}+i^{n-1}\right)\) is

If |z - 2 + i | ≤ 2, then the greatest value of |z| is

If |z₁| = 1, |z₂| = 2, |z₃| = 3 and |9z₁z₂ + 4z₁z₃ + z₂z₃| = 12, then the value of |z₁+z₂+z₃| is 

If z = cos\(\frac { \pi }{ 4 } \) + i sin\(\frac { \pi }{ 6 } \), then ______

If z = \(\frac { 1 }{ 1-cos\theta -isin\theta } \), the Re(z) = ___________

Find the square roots of 4+3i

Obtain the Cartesian form of the locus of z = x + iy in each of the following cases:
\(\left[ Re\left( iz \right) \right] ^{ 2 }=3\)

If (cosθ + i sinθ)² = x + iy, then show that x²+y² =1

Find the argument of -2

Find the values of the real number x and y if 3x + (2x - 3y) i = 6 + 3i⁹.

If z₁ = 3, z₂ = -7i, and z₃ = 5 + 4i, show that z₁(z₂ + z₃) = z₁ z₂ + z₁ z₃

If z₁= 2 + 5i, z₂ = -3 - 4i, and z₃ = 1 + i, find the additive and multiplicate inverse of z₁, z₂ and z₃

Find the principal value of -2i.

Find the locus of z if |3z - 5| = 3 |z + 1| where z = x + iy.

If \(\frac { (a+i)^{ 2 } }{ 2a-i } \) = p + iq, show that p²+q² = \(\frac { ({ a }^{ 2 }+i)^{ 2 } }{ 4a^{ 2 }+1 } \).

If z = x + iy and arg \(\left( \frac { z-i }{ z+2 } \right) =\frac { \pi }{ 4 } \), then show that x² + y² + 3x - 3y + 2 = 0

Show that \(\left( \frac { \sqrt { 3 } }{ 2 } +\frac { i }{ 2 } \right) ^{ 5 }+\left( \frac { \sqrt { 3 } }{ 2 } -\frac { i }{ 2 } \right) ^{ 5 }=-\sqrt { 3 } \)

Verify that arg(1+i) + arg(1-i) = arg[(1+i) (1-i)]

Find all the roots \((2-2i)^{ \frac { 1 }{ 3 } }\) and also find the product of its roots.