Outer electronic configuration of Gd (Atomic no : 64) is -

The outer electron configuration of $$Gd$$ (Atomic No. $$64$$) is :

If $$A = {\sin ^2}x + {\cos ^4}x,$$ then for all real $$x$$:

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