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Solution

Statement I:

For the Vertex Cover problem, there is a standard polynomial time approximation algorithm.

It repeatedly chooses an edge and includes both endpoints in the vertex cover.

This gives a solution whose size is at most $2$ times the optimal solution.

So, Vertex-cover has a polynomial time $2$-approximation algorithm.

Therefore, Statement I is true.

Statement II:

For Travelling Salesman Problem with triangle inequality, the standard MST based TSP-tour approximation algorithm gives a $2$-approximation.

It does not give a $3$-approximation as the standard result.

So, the statement saying it is a polynomial time $3$-approximation algorithm is not correct in the standard form.

Therefore, Statement II is false.