BTW the first question in PEB, how did you solve it?

Did you make a straight line x1+x2=100 and then used hit and trial to observe that if Aditya has x between 20 and 70 ( or Gaurav has x between 30 and 80) then he cannot be made better off hence the first allocation is pareto efficient?

20th
first take a two people economy and see the optimum, since it is unregulated we know the optimal for each person is v'(x)=0 since the function v is same for all they will maximize at the same x. ( given it is inverted U) so now even if the economy has more population say 3 people the answer will be the same regardless that all of the three operate at that point where v(xi) is maximized.

now the question is asking km driven per person
in first case it was 2x/2= x
and second case 3x/3=x

so it's same.

Others I did not attempt for the exam. I'll try and let you know.

No, I wrote the utility of both as a function of Aditya's work hours, and graphed both functions (its basic graphing idk if eco students are familiar with it). Identified both locations, and from there it was easy to see if I could keep one's utility constant and increase the other's. You'll get it. In case you don't, I'll graph it out for you and put up a picture.

Not parabolas, each graph is a pair of straight lines with a kink at the point where it touches the x axis. So like a parabola but both "arms" of the parabola are straight lines.

For 5th you could just observe that for a=0 he will never undertake any effort, so it has to be a root of the equation. Similarly the other factor should vanish there.

If we fix the utility of the second person such that the end points of concave fn lie on the upper and right edge then max utility of the first person will be at the corner points of the first?

alibaba .. i think you got this equation in 5th after doing bit of shifting from LHS to RHS ...
i have done it myself ... i got the ans. thanx for hint . :)