Fsc Part 1 Mathematics (Complete Solution)
Q6
Show that ∀ z ∈ C
(i) z2+Z2is a real number.
SOLUTION:
Let z=x+yι
z2=(x+yι)2
z2=(x)2+(yι)2+2(x)(yι)
z2=x2+y2ι2+2xyι
z2=x2+y2(-1)+2xyι
z2=x2-y2+2xyι --------------->Eq(1)
and let Z=x-yι
Z2=(x-yι)2
Z2=(x)2+(yι)2-2(x)(yι)
Z2=x2+y2ι2-2xyι
Z2=x2+y2(-1)-2xyι
Z2=x2-y2-2xyι ------------------->Eq(2)
Eq(2) + Eq(1)
z2+Z2=x2-y2+2xyι +x2-y2-2xyι
=2x2-2y2
this is shows that z2+Z2 is a real number.
(ii) (z-Z)2 is a real number.
SOLUTION:let z=x+yι and Z=x-yι
(z-Z)2=[x+yι -(x-yι)]2
=(x+yι -x+yι)2
=(2yι)2
=4y2ι2
=4y2(-1)
=-4y2
thus (z-Z)2 is a real number.