If the expression `(1 + ir)^3` is of the form of `s(1+ i)` for some real 's' where 'r' is also real and `i = sqrt(-1)` then the value of r can be

Express ((1)/(3)+3i)^(3) in the form a+ib .

Express (1)/(1-i) in the form a+ib .

If the imaginary part of (2+i)/(ai-1) is zero, where a is a real number, then the value of a is equal to

If r is a real number |r| lt 1 and if a= 5(1-r) , then

If a continous function of defined on the real line R, assumes positive and negative values in R, then the equation f(x)=0 has a root in R. For example, if it is known that a continuous function f on R is positive at some point and its minimum values is negative, then the equation f(x)=0 has a root in R. Considetr f(x)=ke^(x)-x for all real x where k is real constant. The positive value of k for which ke^(x)-x=0 has only root is

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

Let f(x) = x^4 + ax^3 + bx^2 + cx + d be a polynomial with real coefficients and real roots. If |f(i)|=1where i=sqrt(-1) , then the value of a +b+c+d is

If z=i^(i^(i)) where i=sqrt-1 then |z| is equal to

Prove that z=i^i, where i=sqrt-1, is purely real.