[" 9) For all "z in C" ,prove that "],[[" (1) "zbar(z)=|z|^(2)," (i) "(z-bar(z))" is "0" or imaginary "]]

For all z in C, prove that (i) (1)/(2)(z+bar(z))=Re(z), (ii) (1)/(2i)(z-bar(z))=Im(z), (iii) z bar(z)=|z|^(2), (iv) z+bar(z))"is real", (v) (z-bar(z))"is 0 or imaginary".

Prove that |z|=|-z|

Prove that |z|=|-z|

Prove that : z= bar(z) iff z is real.

If |z + i| = |z - i|, prove that z is real.

For all two complex numbers z1 and z2,prove that Re(z1z2)=Re z1 Re z2-Im z1 Im z2

If a^x=b ,\ b^y=c\ a n d\ c^z=a , prove that x y z=1

The set of all alpha in R for which w = (1+(1-8 alpha)z)/(1-z) is a purely imaginary number, for all z in C , satisfying |z| = 1 and Re(z) ne 1 , is

Let R be the relation of the set Z of all integers defined by : R = {(a,b) : a, b in Z and (a-b) " is divisible by " n in N} Prove that : (i) (a,a) in R for all a in Z (ii) (a,b) in R rArr (b,a) in R for all a, b in Z (iii) (a,b) in R and (b,c) in R rArr (a,c) in R " for all" a,b,c in Z