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.

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