Proving sets of regular expressions and context free grammars are decidable [duplicate]

This question already has an answer here:

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Consider below languages:

  1. $ L_1=\{<M>|M$ is a regular expression which generates at least one string containing an odd number of 1’s$ \}$
  2. $ L_2=\{<G>|G$ is context free grammar which generates at least one string of all 1’s$ \}$

Its given that both above languages are decidable, but no proof is given. I tried guessing. $ L_1$ is decidable, its a set of regular expressions containing

  • odd number of $ 1$ ‘s, or
  • even number of $ 1$ ‘s and $ 1^+$ or
  • $ 1^*$

So we just have to parse regular expression for these characteristics. Is this right way to prove $ L_1$ is decidable?

However, can we have some algorithm to check whether given input CFG accepts at least one string of all 1’s? I am not able to come up with and hence not able prove how $ L_2$ is decidable.