finite automata – NFA designing for strings starting with $01$

The question was asked

Construct an NFA with set of all strings that start with $10$.

The solution provided to me is
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But my question is what if the automaton receives an input $0$ at the starting? Also there is no option for $q_1$ to transit after receiving $1$. So I think the solution should be

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Please correct me if I am wrong.