Solution

For $S_1$:

$\forall xP(x)\vee \forall xQ(x)$ means either $P(x)$ is true for all $x$, or $Q(x)$ is true for all $x$.

But $\forall x(P(x)\vee Q(x))$ means for every $x$, at least one of $P(x)$ or $Q(x)$ is true.

These two statements are not logically equivalent.

So, $S_1$ is true.

For $S_2$:

$\exists xP(x)\wedge \exists xQ(x)$ means there exists some $x$ for $P(x)$ and there exists some $x$ for $Q(x)$. These may be different values.

But $\exists x(P(x)\wedge Q(x))$ means there exists the same $x$ for which both $P(x)$ and $Q(x)$ are true.

These two statements are also not logically equivalent.

So, $S_2$ is true.