If |z1| = |z2| = ….. = |zn| = 1, thenShow that |z1 + z2 + z3 + …. + zn|

Given |z1| = |z2| = … = |zn| = 1⇒ |z1|2 = |z2|2 = … = |zn|2 = 1⇒ z1 z̅ 1= z2 z̅ 2= z3 z̅ 3= … = zn z̅ n= 1Now,Hence proved.

Q19 of 66 Page 91

If |z₁| = |z₂| = ….. = |z_n| = 1, then
Show that |z₁ + z₂ + z₃ + …. + z_n|

Given |z₁| = |z₂| = … = |z_n| = 1

⇒ |z₁|² = |z₂|² = … = |z_n|² = 1

⇒ z₁ z̅ ₁= z₂ z̅ ₂= z₃ z̅ ₃= … = z_n z̅ _n= 1

Now,

Hence proved.

If |z₁| = 1 (z₁ ≠ –1) and then show that the real part of z₂ is zero.

If z₁, z₂ and z₃, z₄ are two pairs of conjugate complex numbers, then find

If for complex numbers z₁ and z₂, arg (z₁) – arg (z₂) = 0, then show that |z₁ – z₂| = |z₁| – |z₂|.

Solve the system of equations Re (z₂) = 0, |z| = 2.