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

Show that \(\left( \frac { 19+9i }{ 5-3i } \right) ^{ 15 }-\left( \frac { 8+i }{ 1+2i } \right) ^{ 15 }\) is purely imaginary.

Show that \(\left( 2+i\sqrt { 3 } \right) ^{ 10 }-\left( 2-i\sqrt { 3 } \right) ^{ 10 }\) is purely imaginary

Let z₁, z₂ and z₃ be complex numbers such that \(\left| { z }_{ 1 } \right\| =\left| { z }_{ 2 } \right| =\left| { z }_{ 3 } \right| =r>0\) and z₁+ z₂+ z₃ \(\neq \) 0 prove that \(\left| \frac { { z }_{ 1 }{ z }_{ 2 }+{ z }_{ 2 }{ z }_{ 3 }+{ z }_{ 3 }{ z }_{ 1 } }{ { z }_{ 1 }+{ z }_{ 2 }+{ z }_{ 3 } } \right| \) = r

If z₁, z_2, and z₃ are three complex numbers such that |z₁| = 1, |z₂| = 2|z₃| = 3 and |z₁ + z₂ + z₃| = 1, show that |9z₁z₂ + 4z₁z₂ + z₂z₃| = 6

If the area of the triangle formed by the vertices z, iz and z + iz is 50 square units, find the value of |z|

 If z = x + iy is a complex number such that Im \(\left( \frac { 2z+1 }{ iz+1 } \right) =0\) show that the locus of z is 2x²+ 2y²+ x - 2y = 0

If \(\frac { 1+z }{ 1-z } =cos2\theta +isin2\theta \), show that z = i tan\(\theta\)

If \(cos\alpha +cos\beta +cos\gamma =sin\alpha +sin\beta +sin\gamma =0\) then show that 
(i) \(cos3\alpha +cos3\beta +cos3\gamma =3cos(\alpha +\beta +\gamma )\)
(ii) \(sin3\alpha +sin3\beta +sin3\gamma +sin3\gamma =3sin\left( \alpha +\beta +\gamma \right) \)

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

Find the value of \(\left( \cfrac { 1+sin\frac { \pi }{ 10 } +icos\frac { \pi }{ 10 } }{ 1+sin\frac { \pi }{ 10 } -icos\frac { \pi }{ 10 } } \right) ^{ 10 }\)

If \(\omega \neq 1\) is a cube root of unity, show that the roots of the equation (z −1)³ + 8 = 0 are \(-1,1-2\omega ,1-2{ \omega }^{ 2 }\).

If \(\omega \neq 1\) is a cube root of unity, show that \(\left( 1-\omega +{ \omega }^{ 2 } \right) ^{ 6 }+\left( 1+\omega -{ \omega }^{ 2 } \right) ^{ 6 }=128\)

Find the quotient \(\frac { 2\left( cos\frac { 9\pi }{ 4 } +isin\frac { 9\pi }{ 4 } \right) }{ 4\left( cos\left( \frac { -3\pi }{ 2 } + \right) isin\left( \frac { -3\pi }{ 2 } \right) \right) } \) in rectangular form

Suppose z₁, z₂ and z₃ are the vertices of an equilateral triangle inscribed in the circle |z| = 2. If z₁ = 1 + i\(\sqrt { 3 } \) then find z₂ and z₃.

Solve the equation z³+ 8i = 0, where \(z \in \mathbb{C}\)

Find the fourth roots of unity.

Find the cube roots of unity.

Simplify: (1+i)¹⁸

Show that \(\left( \frac { 19-7i }{ 9+i } \right) ^{ 12 }+\left( \frac { 20-5i }{ 7-6i } \right) ^{ 12 }\) is real

If \(\omega \neq 1\) is a cube root of unity, show that \(\left( 1+\omega \right) \left( 1+{ \omega }^{ 2 } \right) \left( 1+{ \omega }^{ 4 } \right) \left( 1+{ \omega }^{ 8 } \right) ...\left( 1+{ \omega }^{ { 2 }^{ 11 } } \right) =1\).

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

Show that \(\left( 2+i\sqrt { 3 } \right) ^{ 10 }+\left( 2-i\sqrt { 3 } \right) ^{ 10 }\) is real