Free variables of (λx.xy)x and bound variables of λxy.x
Your answer is correct, $y$ is surely free. The model one is wrong. Maybe there was a typo and the answer was meant for $(\lambda y.xy)x$
It depends on the precise definition of $BV$. Often only $FV$ is defined formally, because it's more important. (Bound variables can be freely renamed in a subterm, but free variables can't.) So if $BV$ is defined as
\begin{eqnarray*}
      BV(x) &amp;=&amp; \{\} \\
      BV(M N) &amp;=&amp; BV(M)\cup BV(N) \\
      BV(\lambda x.M) &amp;=&amp; \{x\}\cup BV(M) \\
    \end{eqnarray*}
then the model answer is correct. Note that this definition makes sense: If you have a term $M$ and substitute for a free variable $x$ another term $N$ with no bound variables ($BV(N)=\{\}$), you expect that $BV(M)=BV(M[x:=N])$. If you'd consider only bound variables that also appear under the $\lambda$, this property wouldn't hold.