If one root of the equation `z^2-a z+a-1=0i s(1+i),w h e r ea` is a complex number then find the root.

Let x_1, x_2 are the roots of the quadratic equation x^2+a x+b=0,w h e r ea ,b are complex numbers and y_1, y_2 are the roots of the quadratic equation y^2+|a|y_|b|=0. If |x_1|=|x_2|=1, then prove that |y_1|=|y_2|=1

Let z_(1)" and "z_(2) be two roots of the equation z^(2)+az+b=0 , z being complex number further, assume that the origin, z_(1)" and "z_(2) form an equilateral triangle, then

If the roots of the equation x^3+P x^2+Q x-19=0 are each one more that the roots of the equation x^3-A x^2+B x-C=0,w h e r eA ,B ,C ,P ,a n dQ are constants, then find the value of A+B+Cdot

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.

Consider the equation z^(2)-2z+4=0 . (i) Find two complex numbers z_(1)" and "z_(2) satisfying the above equation. (ii) Simplify (z_1)/(z_2)+(z_2)/(z_1) .

Match the following for the equation x^2+a|x1=0,w h e r ea is a parameter. Column I, Column II No real roots, p. a<-2 Two real roots, q. varphi Three real roots, r. a=-2 Four distinct real roots, s. ageq0

Let z be a complex number satisfying the equation z^2-(3+i)z+m+2i=0,w h e r em in Rdot Suppose the equation has a real root. Then root non-real root.

If the equation |z-a|+|z-b|=3 represents an ellipse and a ,b in C ,w h e r ea is fixed, then find the locus of bdot

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

If (1+i) is a root of the equation x^(2)-x+(1-i)=0 , then the other root is