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

If z₁, z₂ and z₃ are complex numbers such that |z₁| = |z₂| = |z₃| = |z₁+z₂+z₃| = 1 find the value of \(\left| \frac { 1 }{ { z }_{ 1 } } +\frac { 1 }{ z_{ 2 } } +\frac { 1 }{ { z }_{ 3 } } \right| \)

If |z| = 2 show that \(3\le \left| z+3+4i \right| \le 7\)

Find the rectangular form of the complex numbers
\(\left( cos\frac { \pi }{ 6 } +isin\frac { \pi }{ 6 } \right) \left( cos\frac { \pi }{ 12 } +isin\frac { \pi }{ 12 } \right) \)

If (x₁ + iy₁)(x₂ + iy₂)(x3 + iy₃)...(x_n+ iy_n) = a + ib, show that
(x₁² + y₁²)(x₂² + y₂²)(x₃² + y₃²)...(x_n² + y_n²) = a² + b²​

Obtain the Cartesian form of the locus of z in each of the following cases.
|z| = |z - i|

Find the product \(\frac { 3 }{ 2 } \left( cos\frac { \pi }{ 3 } +isin\frac { \pi }{ 3 } \right) .6\left( cos\frac { 5\pi }{ 6 } +isin\frac { 5\pi }{ 6 } \right) \)in rectangular from

If z = 2−2i, find the rotation of z by θ radians in the counter clockwise direction about the origin when \(\theta =\frac { 2\pi }{ 3 } \).

Simplify the following:
i ¹⁷²⁹

Given the complex number z = 3 + 2i, represent the complex numbers z, iz, and z + iz in one Argand diagram. Show that these complex numbers form the vertices of an isosceles right triangle.