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

The range of real number alpha for which the equation z+alpha|z-1|+2i=0 has a solution is :

If the least and the largest real values of alpha , for which the equation z + alpha |z – 1| + 2i = 0 (z in C and 1 = sqrt(-1) ) has a solution, are p and q respectively, then 4(p^2 + q^2) is equal to ____

The number of real values of alpha for which the system of equations x + 3y + 5z = alpha x 5 x + y + 3z = alpha y 3 x + 5y + z = alpha z has infinite number of solutions is

Find the complex number z for which |z| = z + 1 + 2i.

The solution of the equation |z|-z=1+2i

Let lambda and alpha be real. Find the set of all values of lambda for which the system of linear equations lambdax + ("sin"alpha)y + ("cos" alpha)z =0 , x + ("cos"alpha)y + ("sin" alpha) z =0 and -x + ("sin" alpha)y -("cos" alpha)z =0 has a non-trivial solution. For lambda =1 , find all values of alpha .

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

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