If `z_(1), z_(2)` are `1-i, -2 + 4i` respectively, find `I_(m)((z_(1)z_(2))/(bar(z_(1))))`.

Let z_1 = 2 -i, z_2 = -2 + i . Find Im(1/(z_1barz_1))

Let z_1 = 2 -i, z_2 = -2 + i . Find Re((z_1z_2)/barz_1)

If z_(1)=2+5i,z_(2)=3-i,where" "i=sqrt(-1), find (i) Z_(1)*Z_(2) (ii) Z_(1)xxZ_(2) (iii) Z_(2)*Z_(1) (iv) Z_(2)xxZ_(1) (v) acute angle between Z_(1)andZ_(2) . (vi) projection of Z_(1)onZ_(2) .

Let z_(1),z_(2) and z_(3) be three non-zero complex numbers and z_(1) ne z_(2) . If |{:(abs(z_(1)),abs(z_(2)),abs(z_(3))),(abs(z_(2)),abs(z_(3)),abs(z_(1))),(abs(z_(3)),abs(z_(1)),abs(z_(2))):}|=0 , prove that (i) z_(1),z_(2),z_(3) lie on a circle with the centre at origin. (ii) arg(z_(3)/z_(2))=arg((z_(3)-z_(1))/(z_(2)-z_(1)))^(2) .

Show that the triangle whose vertices are z_(1)z_(2)z_(3)andz_(1)'z_(2)'z_(3)' are directly similar , if |{:(z_(1),z'_(1),1),(z_(2),z'_(2),1),(z_(3),z'_(3),1):}|=0

If z_(1) , z_(2) , z_(3) are any three complex numbers on Argand plane, then z_(1)(Im(barz_(2)z_(3)))+z_(2)(Imbarz_(3)z_(1)))+z_(3)(Imbarz_(1)z_(2))) is equal to

If z_1 = 2 -i and z_2= -2+i , find Re((z_1 z_2)/z_1) .

bb"statement-1" " Let " z_(1),z_(2) " and " z_(3) be htree complex numbers, such that abs(3z_(1)+1)=abs(3z_(2)+1)=abs(3z_(3)+1) " and " 1+z_(1)+z_(2)+z_(3)=0, " then " z_(1),z_(2),z_(3) will represent vertices of an equilateral triangle on the complex plane. bb"statement-2" z_(1),z_(2),z_(3) represent vertices of an triangle, if z_(1)^(2)+z_(2)^(2)+z_(3)^(2)+z_(1)z_(2)+z_(2)z_(3)+z_(3)z_(1)=0