If \(z=\cfrac { \left( \sqrt { 3 } +i \right) ^{ 3 }\left( 3i+4 \right) ^{ 2 } }{ \left( 8+6i \right) ^{ 2 } } \) , then |z| is equal to 

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

If |z| = 1, then the value of \(\frac { 1+z }{ 1+\overline { z } }\) is

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

The principal argument of \(\cfrac { 3 }{ -1+i } \) is

Evaluate the following if z = 5−2i and w = −1+3i
z + w

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

If \(\omega \neq 1\) is a cube root of unity, then the show that \(\cfrac { a+b\omega +c{ \omega }^{ 2 } }{ b+c\omega +{ a\omega }^{ 2 } } +\cfrac { a+b\omega +{ c\omega }^{ 2 } }{ c+a\omega +b{ \omega }^{ 2 } } =-1\)

The complex numbers u, v, and w are related by \(\frac { 1 }{ u } =\frac { 1 }{ v } +\frac { 1 }{ w } \) If v = 3−4i and w = 4+3i, find u in rectangular form.

If \(\left| z-\frac { 2 }{ z } \right| =2\) show that the greatest and least value of |z| are \(\sqrt { 3 } +1\) and \(\sqrt { 3 } -1\) respectively.

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 \(2cos\alpha=x+\frac { 1 }{ x } \) and \(2cos\ \beta =y+\frac { 1 }{ y } \), show that ​\({ x }^{ m }{ y }^{ n }+\frac { 1 }{ { x }^{ m }{ y }^{ n } } =2cos(m\alpha +n\beta )\)​

Simplify: \(\left( -\sqrt { 3 } +3i \right) ^{ 31 }\)