If `alpha` is a complex constant such that `alphaz^2+z+baralpha=0` has areal root then

If alpha is a complex number such that alpha^(2)+alpha+1=0 , then the value of alpha^(61) is

If |2z-1|=|z-2|a n dz_1, z_2, z_3 are complex numbers such that |z_1-alpha|lt alpha , |z_2-beta|ltbeta, then= |(z_1+z_2)/ (alpha+beta)| a) lt|z| b.

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

Let f(x)=a_(5)x^(5)+a_(4)x^(4)+a_(3)x^(3)+a_(2)x^(2)+a_(1)x, "where" a_(i),s are real and f (x) = 0 has a positive root alpha_(0). Then f^(prime)(x)=0 has a positive root alpha_1 such that 0 < alpha_1 < alpha_0 f^(prime)(x)=0 has at least two real roots f^('')(x)=0 has at least one real root none of these

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

Let f(x)=ax^3+bx^2+cx+1 has exterma at x=alpha,beta such that alpha beta (a)three equal real roots (b)one negative root if f(alpha) > 0 and f(beta) (c)one positive root if f(alpha) 0 (d) none of these

If x+y alphaz, when y is constant, (x+z) alpha y when z is constant, then show that (x+y+z) alpha yz when y and z both are constant.

If alpha and beta are the complex cube roots of 1 show that, alpha^(4)+beta^(4)+alpha^(-1).beta^(-1)=0.

If alpha is a root of the equation 4x^2+2x-1=0, then prove that 4alpha^3-3alpha is the other root.

Let alpha,beta be real and z be a complex number. If z^2+alphaz+beta=0 has two distinct roots on the line Re z = 1, then it is necessary that