Find the value of the real numbers x and y, if the complex number (2+i)x+(1−i)y+2i −3 and x+(−1+2i)y+1+i are equal

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

If z₁ = 3, z₂ = -7i, and z₃ = 5 + 4i, show that z₁(z₂ + z₃) = z₁ z₂ + z₁ z₃

Write \(\frac { 3+4i }{ 5-12i } \) in the x + iy form, hence find its real and imaginary parts.

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

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

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

Find z⁻¹, if z = (2 + 3i) (1− i).

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|

If |z| = 2 show that \(3\le \left| z+3+4i \right| \le 7\)

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.

For any two complex number z₁ and z₂ such that |z₁| = |z₂| = 1 and z₁z₂ \(\neq \) -1, then show that \(\frac { { z }_{ 1 }+{ z }_{ 2 } }{ 1+{ z }_{ 1 }{ z }_{ 2 } } \) is real number.

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

If |z| = 1, show that \(2\le \left| { z }^{ 2 }-3 \right| \le 4\)

If \(\left| z-\frac { 2 }{ z } \right| =2\) show that the greatest and least value of |z| are \(\sqrt { 3 } +1\) and \(\sqrt { 3 } -1\) respectively.

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

Show that the following equations represent a circle, and find its centre and radius \(\left| z-2-i \right| =3\)

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

Find the rectangular form of the complex numbers
\(\left( cos\frac { \pi }{ 6 } +isin\frac { \pi }{ 6 } \right) \left( cos\frac { \pi }{ 12 } +isin\frac { \pi }{ 12 } \right) \)

If (x₁ + iy₁)(x₂ + iy₂)(x3 + iy₃)...(x_n+ iy_n) = a + ib, show that
(x₁² + y₁²)(x₂² + y₂²)(x₃² + y₃²)...(x_n² + y_n²) = a² + b²​

If \(\omega \neq 1\) is a cube root of unity, then the show that \(\cfrac { a+b\omega +c{ \omega }^{ 2 } }{ b+c\omega +{ a\omega }^{ 2 } } +\cfrac { a+b\omega +{ c\omega }^{ 2 } }{ c+a\omega +b{ \omega }^{ 2 } } =-1\)

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

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

Solve the equation z³+ 27 = 0

Find the value of \(\sum _{ k=1 }^{ 8 }{ \left( cos\frac { 2k\pi }{ 9 } +isin\frac { 2k\pi }{ 9 } \right) } \).

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

Show that |3z−5+i| = 4 represents a circle, and, find its centre and radius.

Show that |z+2−i|<2 represents interior points of a circle. Find its centre and radius.

Represent the complex number −1−i

Find the principal argument Arg z, when z = \(\frac { -2 }{ 1+i\sqrt { 3 } } \)

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

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

Simplify \(\left( \frac { 1+cos2\theta +isin2\theta }{ 1+cos2\theta -isin2\theta } \right) ^{ 30 }\)

If \(z=\left( cos\ \theta +isin\ \theta \right) \), show that \({ z }^{ n }+\frac { 1 }{ { z }^{ n } } =2cos\ n\theta \) and \({ z }^{ n }-\frac { 1 }{ { z }^{ n } } =2i\ sin\ n\theta \)

Simplify \(\left( sin\frac { \pi }{ 6 } +icos\frac { \pi }{ 6 } \right) ^{ 18}\)

Given the complex number z = 2 + 3i, represent the complex numbers in Argand diagram z, −iz , and z−iz

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

If z₁ = 3, z₂ = -7i, and z₃ = 5 + 4i, show that (z₁ + z₂)z₃ = z₁z₃ + z₂ + z₃

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

Obtain the Cartesian equation for the locus of z = x + iy in each of the following cases:
|z - 4|²- |z -1 |² = 16

Find the rectangular form of the complex numbers
\(\frac { cos\frac { \pi }{ 6 } -isin\frac { \pi }{ 6 } }{ 2\left( cos\frac { \pi }{ 3 } +isin\frac { \pi }{ 3 } \right) } \)

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

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

Represent the complex numbe \(1+i\sqrt { 3 } \) in polar form.

If | z | = 2 show that \(8 \leq|z+6+8 i| \leq 12\)

If (x₁ + iy₁)(x₂ + iy₂)(x₃ + iy₃)...(x_n+ iy_n) = a + ib, show that 
\(\sum _{ r=1 }^{ n }{ tan^{ -1 } } \left( \frac { { y }_{ r } }{ { x }_{ r } } \right) ={ tan }^{ -1 }\left( \frac { b }{ a } \right) +2k\pi ,k\epsilon Z\)

If \(2\ cos\ \alpha=x+\frac { 1 }{ x } \) and \(2\ cos\ \beta =y+\frac { 1 }{ y } \), show that \(\frac { x }{ y } +\frac { y }{ x } =2cos\left( \alpha -\beta \right) \)

If \(2cos\ \alpha=x+\frac { 1 }{ x } \) and \(2cos\ \beta =y+\frac { 1 }{ y } \), show that ​\(xy-\frac { 1 }{ xy } =2isin\left( \alpha +\beta \right) \)​

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