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

If z is a complex number such that -pi//2lt=a r g zlt=pi//2, then which of the following inequality is true? (a) |z- z |lt=|z|(a r g z-a r g z ) b. |z- z |geq|z|(a r g z-a r g z ) c. |z- z |<(a r g z-a r g z ) d. none of these

If |z_(1)+z_(2)|=|z_1|+|z_2| then

Lt a be a complex number such that |a|<1a n dz_1, z_2z_, be the vertices of a polygon such that z_k=1+a+a^2+...+a^(k-1) for all k=1,2,3, T h e nz_1, z_2 lie within the circle (a) |z-1/(1-a)|=1/(|a-1|) (b) |z+1/(a+1)|=1/(|a+1|) (c) |z-1/(1-a)|=|a-1| (d) |z+1/(a+1)|=|a+1|

If z_1, z_2, z_3, z_4 are the affixes of four point in the Argand plane, z is the affix of a point such that |z-z_1|=|z-z_2|=|z-z_3|=|z-z_4| , then z_1, z_2, z_3, z_4 are

Find the Area bounded by complex numbers arg|z|lepi/4 and |z-1|lt|z-3|

|z_(1)|=|z_(2)|" and "arg z_(1)+argz_(2)=0 then

If z_1 is a root of the equation a_0z^n+a_1z^(n-1)+........+(a_(n-1)z+a_n=3, where |a_1| lt 2 for i=0,1,.....n, then (a). |z|>1/3 (b). |z| lt 1/4 (c). |z| gt 1/4 (d). |z| lt 1/3

If z_1, z_2 and z_3 , are the vertices of an equilateral triangle ABC such that |z_1 -i = |z_2 -i| = |z_3 -i| .then |z_1 +z_2+ z_3| equals:

The maximum area of the triangle formed by the complex coordinates z, z_1,z_2 which satisfy the relations |z-z_1|=|z-z_2| and |z-(z_1+z_2 )/2| |z_1-z_2| is

If n is a positive integer, prove that |I m(z^n)|lt=n|Im(z)||z|^(n-1.)