Find the range of real number `alpha` for which the equation `z+alpha|z-1|+2i=0` has a solution.

Find the range of K for which the equation |z+i| - |z-i | = K represents a hyperbola.

Find the range of lambda for which equation x^3+x^2-x-1-lambda=0 has 3 real solution.

IF agt=1 , find all complex numbers z satisfying the equation z+a|z+1|+i=0

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

Find the number of complex numbers which satisfies both the equations |z-1-i|=sqrt(2)a n d|z+1+i|=2.

The number of values of alpha in [-10pi, 10pi] for which the equations (sin alpha)x-(cos alpha)y+3z=0, (cos alpha)x+(sin alpha)y-2z=0 and 2x+3y+(cos alpha)z=0 have nontrivial solution is

Find a complex number z satisfying the equation z+sqrt(2)|z+1|+i=0.

Find a complex number z satisfying the equation z+sqrt(2)|z+1|+i=0.

Find a complex number z satisfying the equation z+sqrt(2)|z+1|+i=0.

Let lambda and alpha be real. Then the numbers of intergral values lambda for which the system of linear equations lambdax +(sin alpha) y+ (cos alpha) z=0 x + (cos alpha) y+ (sin alpha) z=0 -x+(sin alpha) y -(cos alpha) z=0 has non-trivial solutions is