UPDATE (regarding duplicate) :
This question is not a duplicate of another question. Sure, the situation in both the questions is same and, yes, both questions ultimately provide a methodology to solve the problem and finding the correct value of friction, but the moderator should realize that here I was NOT asking for a method to solve for friction. The problem was that I did not even realize that I will have to solve for friction.
I had solved a lot of such problems way back in school and I had got into a habit of assuming static friction to be the equal and opposite to whatever force is applied against friction (upto a maximum limit of friction). This was a blunder I made, and this is what made it appear like a "paradox". Moreover, as it turns out, I asked the same problem to a few friends of mine and many of them made the same mistake.
So, essentially the "another" problem is just asking for a general methodology to solve such problems, while this problem is like a puzzle which presents the user a methodology of solving the problem by considering different systems and the contradictions that arise due to them. I believe, a user who knows the general methodology presented in the other question is susceptible to the confusion/paradox that this problem presents.
ORIGINAL QUESTION:
This is a familiar problem with the setting as given below:
$\mu$ is the coefficient of friction.
Now, with the given applied force of $F = 10N$, and taking $g = 10m/s^2$, we know that the maximum friction force between $m_1$ and $m_2$ can be $30N$. Since, $F = 10N$ is less than the maximum friction force $30N$, friction force will be $f = 10N$ and the 2 blocks will, therefore, move together.
The acceleration of the combined system will therefore be:
$$ a = F/(m_1+m_2) $$ $$ a = 10/(5+3) = \frac{10}{8} m/s^2$$
Now, if we only consider the $m_1$ block and create its free-body diagram, we see that in the horizontal direction, there is only one force : force applied by $m_2$ block due to friction. And this friction is $f = 10N$. Now, if we calculate the acceleration of $m_1$ block, we find that:
$$a = f/m_1 = \frac{10}{5} = 2 m/s^2$$
This is paradoxical. How come the value of acceleration comes out to be different when we only consider the second block in our problem?