Converting this NFA-e to DFA
Alright we've to remove the unreachable state that is $Q$, for which we have to follow these steps:


Initialize a set $R$ containing the initial state.
Create a set $M$ which'll contain the states which are reachable from states in $R$ in single transition.
Perform $R = R\cup M$.
Repeat from step 2 until $R$ stops getting new states.
Remove states from complement of $R$.


Create set $R = \{q_1\}$.

Calculate $M$ by finding states which can be reached from $\{q_1\} = \{P\}$.

Perform $R = R\cup M:\{q_1\}\cup\{P\}=\{q_1, P\}$.

Calculate $M$ again from $\{q_1,P\}=\{P\}$.

Perform $R=R\cup M:\{q_1,P\}\cup\{P\}=\{q_1,P\}$. $R$ has no new states to add. Therefore, we'll remove states from complement of $R: \{q_1, P, Q\} - \{q_1,P\}=\{Q\}$. The FSM will be: 

I learned this here http://www.cs.um.edu.mt/gordon.pace/Research/Software/Relic/Transformations/FSA/remove-unreachable.html