Is there a way to determine the exact value of static friction in this situation? 
Consider the following situation. Two blocks A and B connected by an inextensible string rest on a rough horizontal surface. The mass of A is 2kg and that of B is 3kg. The coefficient of static friction between block A and the surface is 0.1 and the coefficient of static friction between block B and the surface is 0.2
Two external forces of magnitudes 1N and 8N are now applied on A and B. The situation is shown in the figure below:



My goal is to find the tension in the string connecting the blocks and the friction acting on each block after the forces are applied.

First, I calculated the values of limiting friction corresponding to A and B, using (friction coefficient)*(normal contact force). For A it turns out to be 2N and for B 6N.
Now, considering the two blocks plus the string as a single system, the total external force is 8-1=7N rightwards and the maximum possible static friction is 6+2=8N leftwards. From this I concluded that the system cannot accelerate under the application of the given forces. Moreover, the sum of the static friction forces acting on A and B must be exactly 7N in order to balance out the external force.
To find the tension in the string, we need to find the individual magnitudes of the two friction forces. But I can't think of any reasonable way to obtain this using Newton's laws. I would like to know the right way to approach this problem.
The correct answer:

*

*Friction on block B is 6N leftwards.

*Friction on block A is 1N leftwards.

*Tension in the string is 2N, the system is in equilibrium.

From the solution, it seems that the friction on block B has been taken as the limiting(or maximum) value, 6N.
How did they arrive at this conclusion? Is there a way to explain this just using Newton's laws and motion constraints, or is some additional reasoning and extra conceptual insight required? Are these the ONLY possible values of friction forces? It seems like the forces could be anything as long as they add up to 7N and do not exceed their respective limiting values(although my gut feeling tells me that only one solution is possible for a practical situation such as this). The tension would then be decided accordingly, depending on the values of the friction forces (assuming that the breaking stress of the string is large enough, of course). In other words, Newton's laws predict an INFINITE number of solutions. What would be the most accurate and way of arriving at the answer to the question?
Final question: Is my way of concluding that the system does not accelerate correct? Or is there a more solid argument?
 A: Such problems of indeterminate forces (eg Block on a block problem, with friction and Paradox in the two block problem) can be dealt with in 2 stages :
stage 1 : look at the system as a whole to determine whether it accelerates, and if so by how much
stage 2 : apply $F=ma$ to each part of the system individually.
Regarding the 2 blocks and string together as the system, the net applied force on it is 7N to the right. The maximum friction force is 8N (left or right). So the system does not accelerate. The net force on each block must be zero.
Applying the equilibrium conditions for each block we have
block A : $T-1 \le 2$ and $1-T \le 2$ giving $-1 \le T \le 3$
block B : $8-T \le 6$ and $T-8 \le 6$ giving $2 \le T \le 14$.
There is also a constraint for the string : $T \ge 0$.
Combining the inequalities, the only feasible solutions are in the interval
$2 \le T \le 3$.  
I agree with Pirx : a unique solution does not exist. The answer given is not the only solution. If you think the IITJEE exam question proves otherwise, please upload the question and official solution.
A: The answer will indeed depend on the initial conditions. Here is how I think about this problem to arrive at the solution suggested to you: Very roughly, if you consider a situation with the rope slack at the beginning, then block B will start moving until the rope is tight. While the block is moving, the friction force on it will be equal to its weight times the kinetic friction coefficient for the moving surface (which in general will be lower than the limiting value for static friction). At some point the rope will tighten, and exert a force on Block B, retarding its motion and ultimately stopping it. The assumption seems to be that, at the point when block B stops, the friction force returns to the limiting value of static friction. Sounds reasonable to me, but reality might be more complex than this.
