The other answers are truly excellent and have settled the
intended question. For a bit of fun, however, allow me to
mention the following paradoxical solution.

Namely, with a certain precise and reasonable understanding
of the rules of your game, which I shall presently give, I
claim that no additional questions are required for the
lie-telling game over the truth-telling game. In
particular, in your case 10 questions still suffice!

Specifically, to be a bit more definite about what it means
to give a wrong answer, I propose that the rules should
allow that the second player, at most once during the game,
decides that a given round will be a lie-telling round, for
which he will privately ponder the correct truthful answer,
but then give as his answer precisely the opposite of the
correct answer. So if a truthful answer would have been
*Yes*, then on this lie-telling round he says *No*, and
conversely. (In particular, in this version of the game,
the wrong answer is not a random answer in any sense,
although it could be that the choice of which round is to
be a lie-telling round is determined randomly.) On the
other rounds, he tells the truth. Secondly, I note that you didn't insist that the questions of player 1 must have a particular form.

With these rules for the liar game, I claim that no
additional questions over the fully truthful case are
required to determine the secret number.

The reason is a simple logic trick: if in the fully
truthful game, one would want on a round to ask a question
$Q$, then in this liar game, one should instead ask the
question $P$:

 - *If I were to have asked $Q$ on this round, would you have said Yes?*

Consider how the second player will react. First, if he is
on a truth-telling round, then he will give the same answer
to this question that he would have given to $Q$. If in
contrast he is on a lie-telling round, then he ponders
question $P$, and considers that if the first player had
asked $Q$ on this round and a truthful answer had been Yes,
then he would have said No, since this is a lie-telling
round, and so a truthful answer to $P$ is No, but since it
is a lie-telling round, he answers Yes to $P$. Similarly,
if a truthful answer to $Q$ would have been No, then a
lie-telling answer to $Q$ would be Yes, and so a truthful
answer to $P$ would be Yes, but since it is a lie-telling
round, he answers No. Thus, because of the double-negation
effect, the lie-telling answer to $P$ is the same as the
truth-telling answer to $Q$.

Therefore, the first player can in effect gain exactly the
same information from the second player in the liar game
that he can in the fully truth-telling game.

The same argument shows that, in fact, it doesn't matter
how often the second player decides to lie, as long as he
lies by stating each time the exact opposite of a truthful
answer. Indeed, the second player could randomly decide for
each round whether he will lie or tell the truth on that
round, but the double-negation trick of question $P$ allows
the first player nevertheless to gain exactly the same
information, and so no additional questions beyond the
truth-telling case are required, even if the second player decides randomly at the beginning of every round whether to lie or tell the truth on that round.

Ha!