Prove that following inequalities:
(i) `|(z)/(|z|) -1| le |arg z|` (ii) `|z-1|le|z| |arg z | + |z|-1|`

Prove that following inequalities: |z/(|z|)-1|lt=|a rgz| (ii) |z-1|lt=|z|+||z|-1|

If |z-(4 + 3i) |=1, then find the complex number z for each of the following cases: (i) |z| is least (ii) |z| is greatest (iii) arg(z) is least (iv) arg(z) is greatest

Consider the region R in the Argand plane described by the complex number. Z satisfying the inequalities |Z-2| le |Z-4| , |Z-3| le |Z+3| , |Z-i| le |Z-3i| , |Z+i| le |Z+3i| Answer the followin questions : Minimum of |Z_(1)-Z_(2)| given that Z_(1) , Z_(2) are any two complex numbers lying in the region R is

Find the value of z, if |z |= 4 and arg (z) = (5pi)/(6) .

Solve the inequation : 3z - 5 le z + 3 le 5z - 9, z in R .

The value of lambda for which the inequality |z_(1)/|z_(1)|+z_(2)/|z_(2)|| le lambda is true for all z_(1),z_(2) in C , is

State true or false for the following. Let z_(1) " and " z_(2) be two complex numbers such that |z_(2) + z_(2)| = |z_(1) | + |z_(2)| , then arg (z_(1) - z_(2)) = 0

Let z_(1),z_(2) and z_(3) be three complex number such that |z_(1)-1|= |z_(2) - 1| = |z_(3) -1| and arg ((z_(3) - z_(1))/(z_(2) -z_(1))) = (pi)/(6) then prove that z_(2)^(3) + z_(3)^(3) + 1 = z_(2) + z_(3) + z_(2)z_(3) .

Statement-1: for any non-zero complex number |z-1| le ||z|-1|+|z| arg (z) Statement-2 : For any non-zero complex number z |z/(|z|)-1| le "arg"(z)

z_(1) "the"z_(2) "are two complex numbers such that" |z_(1)| = |z_(2)| . "and" arg (z_(1)) + arg (z_(2) = pi," then show that "z_(1) = - barz_(2).