Properties of a complex no. If `z;z_1;z_2` are complex no.; then (vii)`bar(z_1+z_2)=barz_2+barz_1` (viii)`bar(z_1-z_2)=barz_1-barz_2` (ix)`bar(z_1z_2)=barz_1barz_2` (x) `(barz_1)/z_2=barz_1/barz_2` where `z_2!=0`

Which of the following is (are) correct? (A) bar(z_1-z_2)-a(barz_1-barz_2)=0 (B) bar(z_1-z_2)+a(barz_1-barz_2)=0 (C) bar(z_1-z_2)+a(barz_1-barz_2)=-b (D) bar(z_1-z_2)+a(barz_1-barz_2)=-b

If z, z_1 and z_2 are complex numbers, prove that (i) arg (barz) = - argz (ii) arg (z_1 z_2) = arg (z_1) + arg (z_2)

If z is a nonzero complex number then (bar(z^-1))=(barz)^-1 .

If z_(1),z_(2) are tow complex numberes (z_(1) ne z_(2)) satisfying |z_(1)^(2)- z_(2)^(2)|=|barz_(1)^(2)+barz_(2)^(2) - 2barz_(1)barz_(2)| , then

Which of the following is (are) correct? (A) barz_1+abarz_2=2b (B) barz_1+abarz_2=b (C) barz_1+abarz_2=-b (D) barz_1+abarz_2=-2b

If z_(1) = 3+ 2i and z_(2) = 2-i then verify that (i) bar(z_(1) + z_(2)) = barz_(1) + barz_(2)

Conjugate of a complex no and its properties. If z, z_1, z_2 are complex no.; then :- (i) bar(barz)=z (ii)z+barz=2Re(z)(iii)z-barz=2i Im(z) (iv)z=barz hArr z is purely real (v) z+barz=0implies z is purely imaginary (vi)zbarz=[Re(z)]^2+[Im(z)]^2

Show that the equation of the circle in the complex plane with z_1 & z_2 as its diameter can be expressed as , 2z barz-(barz_1+barz_2)z-(barz_1+barz_2)barz+z_1barz_2+z_2barz_1=0

Find the complex number z if zbarz = 2 and z + barz=2

Modulus of a Complex Number & its properties If z;z_1;z_2inCC then (i)|z|=0hArrz=0 i.e. Re(z)=Im(z)=0 (ii)|z|=|barz|=|-z| (iii) -|z|leRe(z)le|z|;-|z|leIm(z)le|z| (iv) zbarz=|z|^2 (v)|z_1z_2|=|z_1||z_2| (vi)|(z_1)/(z_2)|=|z_1|/|z_2|; z_2!=0