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

The quadratic equation p(x)=0 with real coefficients has purely imaginary roots. Then the equation p(p(x))=0 has A. only purely imaginary roots B. all real roots C. two real and purely imaginary roots D. neither real nor purely imaginary roots

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

Given z is a complex number with modulus 1. Then the equation [(1+i a)/(1-i a)]^4=z has (a) all roots real and distinct (b)two real and two imaginary (c) three roots two imaginary (d)one root real and three imaginary

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

If a is a complex number such that |a|=1 , then find thevalue of a, so that equation az^2 + z +1=0 has one purely imaginary root.

If a , b are complex numbers and one of the roots of the equation x^(2)+ax+b=0 is purely real whereas the other is purely imaginery, and a^(2)-bara^(2)=kb , then k is

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

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.

Prove that z=i^i, where i=sqrt-1, is purely real.

Let z_1a n dz_2 be complex numbers such that z_1!=z_2 and |z_1|=|z_2|dot If z_1 has positive real part and z_2 has negative imaginary part, then (z_1+z_2)/(z_1-z_2) may be (a)zero (b) real and positive (c) real and negative (d) purely imaginary