In the question (the second part), why doesn't tension change? In the following question (the second part), why doesn't tension change? I've solved the question. In the first part, when particle A is about to move, tension is $2 N$. In the second part, $X = 2.8$, but tension is still $2 N$ (and is used to calculate X).
I don't understand why tension hasn't changed, since a new horizontal force is acting on Particle B, in the direction opposite to B. Shouldn't that change how much force the string has to apply on B? I have understanding gaps about the tension concept in general, I think.

 A: In the second part of the question they are again hinting to the fact that body A is in equilibrium . That's why the tension is again $2  N $.
The other hint that the body A is about to move upward points that friction acting on the body  B is leftward.
A: Why should the tension force change ?
Both the questions are talking about equilibrium condition and in both the question , $A$  is at rest and is about to move either downward or upward. So , tension in both the case is counter balancing weight of $A$(thus equals $2N$)
For B :
When the new Force is introduced , it has a perpendicular as well as a horizontal component . The component perpendicular to the surface leads to a decrease in the normal force acting on $B$ due to the rough surface. And hence the magnitude of the friction force on $B$ is lesser in the second question than in the first one ( you can calculate this).
So the horizontal component of your external force couldn't change the tension force because the friction force changed accordingly to your external force . If there was no component of the externally applied force , the tensions would have changed and system will not be in equilibrium.
Hope it helps.
