If `z=|-5 3+4i5-7i3-4i6 8+7i5+7i8-7i9|,t h e nz` is purely real purely imaginary `a+i b ,w h e r ea!=0,b!=0` d. `a+i b ,w h e r eb=4`

Prove that a^4+b^4+c^4> a b c(a+b+c),w h e r ea ,b ,c > 0.

If z=x+i ya n dw=(1-i z)//(z-i) , then show that |w|=1 z is purely real.

Show that ((19 + 9i)/(5 - 3i))^(15) - ((8 + i)/(1 + 2i))^(15) is purely imaginary

Consider two complex numbers alphaa n dbeta as alpha=[(a+b i)//(a-b i)]^2+[(a-b i)//(a+b i)]^2 , where a ,b , in R and beta=(z-1)//(z+1), w here |z|=1, then find the correct statement: (a)both alphaa n dbeta are purely real (b)both alphaa n dbeta are purely imaginary (c) alpha is purely real and beta is purely imaginary (d) beta is purely real and alpha is purely imaginary

Consider the equation 10 z^2-3i z-k=0,w h e r ez is a following complex variable and i^2=-1. Which of the following statements ils true? (a) For real complex numbers k , both roots are purely imaginary. (b) For all complex numbers k , neither both roots is real. (c) For all purely imaginary numbers k , both roots are real and irrational. (d) For real negative numbers k , both roots are purely imaginary.

If z=(3+7i)(a+ib) , where a , b in Z-{0} , is purely imaginery, then minimum value of |z|^(2) is

Reduce (4+3i)/(5-4i) in x+iy form, find its real and imaginary parts.

Show that ((19+9i)/(5-3i))^(15)-((8+i)/(1+2i))^(15) is purely imaginary.

Prove that : (1 + i)^(4n) and (1 + i)^(4n + 2) are real and purely imaginary respectively.

If z=(7-i)/(3-4i)" then "z^(14)=