Show that the equation `a z^3+b z^2+ barb z+ bara =0` has a root `alpha` such that `|alpha|=1,a ,b ,za n dalpha` belong to the set of complex numbers.

Prove that the equation Z^3+i Z-1=0 has no real roots.

Show that the equation Z^4+2Z^3+3Z^2+4Z+5=0 has no root which is either purely real or purely imaginary.

If alpha is a complex constant such that alpha z^2+z+ baralpha=0 has a real root, then (a) alpha+ bar alpha=1 (b) alpha+ bar alpha=0 (c) alpha+ bar alpha=-1 (d)the absolute value of the real root is 1

If the roots of equation x^(3) + ax^(2) + b = 0 are alpha _(1), alpha_(2), and alpha_(3) (a , b ne 0) . Then find the equation whose roots are (alpha_(1)alpha_(2)+alpha_(2)alpha_(3))/(alpha_(1)alpha_(2)alpha_(3)), (alpha_(2)alpha_(3)+alpha_(3)alpha_(1))/(alpha_(1)alpha_(2)alpha_(3)), (alpha_(1)alpha_(3)+alpha_(1)alpha_(2))/(alpha_(1)alpha_(2)alpha_(3)) .

Find the centre and radius of the circle formed by all thepoints represented by z = x + iy satisfying the relation |(z-alpha)/(z-beta)|= k (k !=1) , where alpha and beta are the constant complex numbers given by alpha = alpha_1 + ialpha_2, beta = beta_1 + ibeta_2 .

The roots of the cubic equation (z + alpha beta )^3 = alpha ^3 , alpha is not equal to 0, represent the vertices of a triangle of sides of length

Show that the solution of the equation [(x, y),(z, t)]^(2)=O is [(x,y),(z,t)]=[(pm sqrt(alpha beta),-beta),(alpha,pm sqrt(alpha beta))] where alpha, beta are arbitrary.

If n >1 , show that the roots of the equation z^n=(z+1)^n are collinear.

If the roots of equation x^3+a x^2+b=0a r ealpha_1,alpha_2 and alpha_3(a ,b!=0) , then find the equation whose roots are (alpha_1alpha_2+alpha_2alpha_3)/(alpha_1alpha_2alpha_3),(alpha_2alpha_3+alpha_3alpha_1)/(alpha_1alpha_2alpha_3),(alpha_1alpha_3+alpha_1alpha_2)/(alpha_1alpha_2alpha_3)

Let z=(11-3i)/(1+i) . If alpha is a real number such z-ialpha is real, then the value of alpha is