Prove that the values of \(\sqrt [ 4 ]{ -1 } arr\ \pm \frac { 1 }{ \sqrt { 2 } } \left( 1\pm i \right) \). Let z = (-1)

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

Show that \(\left( \frac { i+\sqrt { 3 } }{ -i+\sqrt { 3 } } \right) ^{ 2\omega }+\left( \frac { i-\sqrt { 3 } }{ i+\sqrt { 3 } } \right) ^{ 2\omega }\) = -1

Verify that 2 arg(-1) ≠ arg(-1)²

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.

Find the radius and centre of the circle \(z\bar { z } \)-(2+3i)z-(2-3i)\(\bar { z } \)+9 = 0 where z is a complex number.

Find the radius and centre of the circle, \(z \bar{z}-(2+3 i) z-(2-3 i) \bar{z}+9=0\)

If the imaginary part of \(\frac{2 z+1}{i z+1}\) is -2 then prove that the locus of the point representing z in the complex plane is a straight line.

Let z₁ and z₂ be two complex numbers such that \(\bar z_{1}+i \bar z_{2}=0\) and arg \(\left(z_{1} z_{2}\right)=\pi\), then find (z₁).

Show that the complex number 'z' satisfying \(\arg \left(\frac{z-1}{z+1}\right)=\frac{\pi}{4}\) lies on a circle.

If a = \(\cos \alpha+i \sin \alpha, b=\cos \beta+i \sin \beta\) and c = \(\cos \gamma+i \sin \gamma \text { and } \frac{b}{c}+\frac{c}{a}+\frac{a}{b}=1\) then that \(\cos (\beta-\gamma)+\cos (\gamma-\alpha)+\cos (\alpha-\beta)=1\).

If \(\sin \alpha+\sin \beta+\sin \gamma=0=\cos \alpha+\cos \beta+\cos \gamma\), then show that \(\sin ^{2} \alpha+\sin ^{2} \beta+\sin ^{2} \gamma=\frac{3}{2}\)

lf (\(\omega\)) is a complex cube root of unity, then find the value of \(\frac{(-1+i \sqrt{3})^{15}}{(1-i)^{20}}+\frac{(-1-i \sqrt{3})^{15}}{(1+i)^{20}}\).

Solve \(z^{4}=1-\sqrt{3 i}\)