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 \(\alpha \) 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

The least positive integer n such that \(\left( \frac { 2i }{ 1+i } \right) ^{ n }\) is a positive integer is ____________

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

If x + iy = \(\frac { 3+5i }{ 7-6i } \), they y = ___________

The value of (1+i)⁴ + (1-i)⁴ is __________

The complex number z which satisfies the condition \(\left| \frac { 1+z }{ 1-z } \right| \)  = 1 lies on _________

If ω is the cube root of unity, then the value of (1-ω) (1-ω²) (1-ω⁴) (1-ω⁸) is _________

\(\frac { (cos\theta +isin\theta )^{ 6 } }{ (cos\theta -isin\theta )^{ 5 } } \) = ________

If a = cos α + i sin α, b = -cos β + i sin β then \(\left( ab-\frac { 1 }{ ab } \right) \) is _________

If x = cos θ + i sin θ, then xⁿ + \(\frac { 1 }{ { x }^{ n } } \) is ______

If z₁, z₂, z₃ are the vertices of a parallelogram, then the fourth vertex z₄ opposite to z₂ is _____

If x_r = \(cos\left( \frac { \pi }{ 2^{ r } } \right) +isin\left( \frac { \pi }{ 2^{ r } } \right) \) then x₁, x_2, x₃ ... x_∞ is _________

Find z⁻¹, if z = (2 + 3i) (1− i).

Obtain the Cartesian equation for the locus of z = x + iy in each of the following cases:
|z - 4| = 16

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

Show that the following equations represent a circle, and, find its centre and radius
|3z-6+12i| = 8

Simplify the following:
 \(\sum _{ n=1 }^{ 102 }{ { i }^{ n } } \)

Find the following \(\left| \overline { (1+i) } (2+3i)(4i-3) \right| \)

Find the modulus and principal argument of the following complex numbers.
\(-\sqrt { 3 } +i\)

If z₁ and z₂ are two complex numbers, such that |z₁| = Iz₂|, then is it necessary that z₁ = z₂?

If 1, ω, ω² are the cube roots of unity show that (1+ω²)³ - (1+ω)³ = 0

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

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 z₁ = 3 + 4i, z₂ = 5 -12i, and z₃ = 6 + 8i, find |z₁|, |z₂|, |z₃|, |z₁+z₂|, |z₂-z₃| and |z₁+z₃|

Which one of the points 10 − 8i, 11+ 6i is closest to 1 + i.

Show that the equation \({ z }^{ 3 }+2\bar { z } =0\) has five solutions

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.

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

Find the circle roots of -27.

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

Find the locus of z if Re\(\\ \left( \frac { \bar { z } +1 }{ \bar { z } -i } \right) \) = 0.

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

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

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

Verify that arg(1+i) + arg(1-i) = arg[(1+i) (1-i)]