If z is any complex number, prove that : `|z|^2= |z^2|`.

If z_1 and z_2 (ne 0) are two complex numbers, prove that : |z_1 z_2|= |z_1||z_2| .

If z_1 and z_2 (ne 0) are two complex numbers, prove that : |z_1/z_2|= (|z_1|)/ (|z_2|), z_2 ne 0 .

If z is any non-zero complex number, prove that the multiplicative inverse of z is bar z/(|z|^2) . Hence express the following numbers in the form x + iy, where x , y in R : (-3+2i)/(4-5i) .

If z is any non-zero complex number, prove that the multiplicative inverse of z is bar z/(|z|^2) . Hence express the following numbers in the form x + iy, where x , y in R : (4- sqrt(-9))^(-1) .

For any complex number z, prove that : -|z| leR_e (z) le |z| .

For any complex number z, prove that : -|z| leI_m (z) le |z| .

For complex numbers z and w, prove that |z|^2 w -|w|^2 z = z - w if and only if z=w or z bar w= 1 .

If z^2+z+1=0 where z is a complex number, then the value of (z+1/z)^2+(z^2+1/z^2)^2+....+(z^6+1/z^6)^2 is

If Z is a complex number, then |Z-8| lt= |Z-2|implies:

If two complex numbers z_1,z_2 are such that |z_1|= |z_2| , is it then necessary that z_1 = z_2 ?