Observe the following recurrences, use master theorem to solve those that can be solved(Please indicate which case). For those that cannot be solved directly, try to devise some guess of the solution, and then use substitution method to prove your guess.

a) T(n)=4T(n/3)+nlg⁡n.

b) T(n)=4T(n/2)+n^2 √n.

c) T(n)=T(n/2)+T(n/4)+T(n/8)+n.

Two flowing streams of argon gas are adiabatically mixed to form a single flow/stream. one stream is 1.5 kg/s at 400 kpa and 200 c while the second stream is 2kg/s at 500 kpa and 100 ? . it is stated that the exit state of the mixed single flow of argon gas is 150 c and 300 kpa. assuming there is no work output or input during the mixing process, does this process violate either the first or the second law or both? explain and state all your assumptions.

Adouble-strand no. 60 roller chain is used to transmit power between a 13-tooth driving sprocket rotating at 300 rev/min and a 52-tooth driven sprocket. a) what is the allowable horsepower of this drive? b) estimate the center-to-center distance if the chain length is 82 pitches. c) estimate the torque and bending force on the driving shaft by the chain if the actual horsepower transmitted is 30 percent less than the corrected (allowable) power.

Agas mixture consists of 8 kmol of h2 and 2 kmol of n2. determine the mass of each gas and the apparent gas constant of the mixture.