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

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

If \(\frac { z-1 }{ z+1 } \) is purely imaginary, then |z| is

If (1+i)(1+2i)(1+3i)...(1+ni) = x + iy, then \(2\cdot 5\cdot 10...\left( 1+{ n }^{ 2 } \right) \) is

If \(\omega =cis\cfrac { 2\pi }{ 3 } \), then the number of distinct roots of \(\left| \begin{matrix} z+1 & \omega & { \omega }^{ 2 } \\ \omega & z+{ \omega }^{ 2 } & 1 \\ { \omega }^{ 2 } & 1 & z+\omega \end{matrix} \right| \)=0

If, i² = -1, then i¹ + i² + i³ + ....+ up to 1000 terms is equal to ________

If a = 3 + i and z = 2 - 3i, then the points on the Argand diagram representing az, 3az and - az are ___________

When z = x + iy, then iz is
(1) x-iy
(2) i(x+iy)
(3) -y+ix
(4) Rotation of z by 90° in the counter clockwise direction

(1+3i) (1-3i)
(i) (1)² - (3i)²
(2) 1 + 9
(3) 10
(4) -8

Simplify the following
i¹⁹⁴⁷+ i¹⁹⁵⁰

Simplify \(\left( \frac { 1+i }{ 1-i } \right) ^{ 3 }-\left( \frac { 1-i }{ 1+i } \right) ^{ 3 }\) into rectangular form

Find z⁻¹, if z = (2 + 3i) (1− i).

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

Obtain the Cartesian form of the locus of z = x + iy in each of the following cases:
\(\left[ Re\left( iz \right) \right] ^{ 2 }=3\)

Write in polar form of the following complex numbers
\(2+i2\sqrt { 3 } \)

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

Show that the points 1, \(\frac { -1 }{ 2 } +i\frac { \sqrt { 3 } }{ 2 } ,\) and \(\frac { -1 }{ 2 } -i\frac { \sqrt { 3 } }{ 2 } \)  are the vertices of an equilateral triangle.

If |z| = 3, show that \(7\le \left| z+6-8i \right| \le 13\).

If z = x + iy is a complex number such that \(\left| \frac { z-4i }{ z+4i } \right| =1\) show that the locus of z is real axis.

 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

Solve the equation z³+ 27 = 0

Find all cube roots of \(\sqrt { 3 } +i\)

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