iⁿ+iⁿ⁺¹+iⁿ⁺²+iⁿ⁺³ is

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

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

The solution of the equation |z| - z = 1 + 2i is

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

If z is a complex number such that \(z \in \mathbb{C} \backslash \mathbb{R}\) and \(z+\frac { 1 }{ z } \epsilon R\), then |z| is

If z = x + iy is a complex number such that |z+2| = |z−2|, then the locus of z is

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

The principal argument of the complex number \(\frac { \left( 1+i\sqrt { 3 } \right) ^{ 2 } }{ 4i\left( 1-i\sqrt { 3 } \right) } \) is

If \(\alpha \) and \(\beta \) are the roots of x²+x+1 = 0, then \({ \alpha }^{ 2020 }+{ \beta }^{ 2020 }\) is

The product of all four values of \(\left( cos\cfrac { \pi }{ 3 } +isin\cfrac { \pi }{ 3 } \right) ^{ \frac { 3 }{ 4 } }\) is

If \(\omega \neq 1\) is a cubic root of unity and \(\left| \begin{matrix} 1 & 1 & 1 \\ 1 & { -\omega }^{ 2 }-1 & { \omega }^{ 2 } \\ 1 & { \omega }^{ 2 } & { \omega }^{ 7 } \end{matrix} \right| \) = 3k, then k is equal to 

If z = 1-cos θ + i sin θ, then |z| = _____________

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

Write the following in the rectangular form:
\(\overline { \left( 5+9i \right) +\left( 2-4i \right) } \)

If z = x + iy, find the following in rectangular form.
\(Re\left( \frac { 1 }{ z } \right) \)

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

If z₁ = 1-3i, z₂ = - 4i, and z₃ = 5, show that (z₁z₂)z₃ = z₁(z₂z₃)

Write the following in the rectangular form:
 \(\overline { 3i } +\frac { 1 }{ 2-i } \).

Find the modulus of the following complex number \(\frac { 2-i }{ 1+i } +\frac { 1-2i }{ 1-i } \)

Show that the following equations represent a circle, and, find its centre and radius
\(\left| 2z+2-4i \right| =2\)

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

Find the values of the real number x and y if 3x + (2x - 3y) i = 6 + 3i⁹.

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

Obtain the Cartesian form of the locus of z = x + iy in each of the following cases:
 Im[(1−i)z+1] = 0

Obtain the Cartesian form of the locus of z = x + iy in each of the following cases:
|z + i| = |z - 1|

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

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

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.

Explain the falacy:

Show that the complex numbers 3 + 2i, 5i, -3 + 2i and -i form a square.

Find the locus of z if |3z - 5| = 3 |z + 1| where z = x + iy.

If \(\frac { (a+i)^{ 2 } }{ 2a-i } \) = p + iq, show that p²+q² = \(\frac { ({ a }^{ 2 }+i)^{ 2 } }{ 4a^{ 2 }+1 } \).

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 z = x + iy and arg \(\left( \frac { z-i }{ z+2 } \right) =\frac { \pi }{ 4 } \), then show that x² + y² + 3x - 3y + 2 = 0

Show that \(\left( \frac { \sqrt { 3 } }{ 2 } +\frac { i }{ 2 } \right) ^{ 5 }+\left( \frac { \sqrt { 3 } }{ 2 } -\frac { i }{ 2 } \right) ^{ 5 }=-\sqrt { 3 } \)

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

If z = x + iy and arg\(\left( \frac { z-1 }{ z+1 } \right) =\frac { \pi }{ 2 } \), then show that x² + y² = 1.

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