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

If z is a non zero complex number, such that 2iz² = \(\bar { z } \) then |z| is

z₁, z₂ and z₃ are complex number such that z₁ + z₂ + z₃ = 0 and |z₁| = |z₂| = |z₃| = 1 then z₁² + z₂² + z₃³ is

If z₁, z₂, z₃ are the vertices of a parallelogram, then the fourth vertex z₄ opposite to z₂ is _____

If x_r = \(cos\left( \frac { \pi }{ 2^{ r } } \right) +isin\left( \frac { \pi }{ 2^{ r } } \right) \) then x₁, x_2, x₃ ... x_∞ is _________

Find the values of the real numbers x and y, if the complex numbers (3−i)x−(2−i)y+2i +5 and 2x+(−1+2i)y+3+ 2i are equal.

If z₁ = 1 - 3i, z₂ = - 4i, and z₃ = 5 , show that (z₁ + z₂) + z₃ = z₁+ (z₂ + z₃)

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

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

If z₁ and z₂ are 1-i, -2+4i then find Im\(\left( \frac { { z }_{ 1 }{ z }_{ 2 } }{ \bar { { z }_{ 1 } } } \right) \).

If \(\frac { z+3 }{ z-5i } =\frac { 1+4i }{ 2 } \), find the complex number z in the rectangular form

If z₁= 3 - 2i and z₂ = 6 + 4i, find \(\frac { { z }_{ 1 } }{ z_{ 2 } } \) in the rectangular form.

Find z⁻¹, if z = (2 + 3i) (1− i).

Explain the falacy:

Find the circle roots of -27.

Find the value of the real numbers x and y, if the complex number (2+i)x+(1−i)y+2i −3 and x+(−1+2i)y+1+i are equal

Find the following \(\left| \overline { (1+i) } (2+3i)(4i-3) \right| \)

If 1, ω, ω² are the cube roots of unity then show that (1+5ω²+ω⁴) (1+5ω+ω²) (5+ω+ω⁵) = 64