A random variable which takes only one value is referred to as a constant random variable or degenerate random variable. You can read this wiki article.
So, the reasoning for part vi is okay.
In part viii, we are trying to prove Var(-X) = -Var(X). Note variance of any RV is always non-negative. So this relation can never be true unless the RV is itself a constant. So, I agree that the solution can be better structured
Last edited: May 16, 2013