Here, mass of monkey m = 40 kg
Maximum tension the rope can stand, T = 600 N. In each case, actual tension in the rope will be equal to the apparent weight of the monkey (R).
The rope will break when R exceeds T.
(a) When monkey climbs up with a = 6m s^-2

R = m(g + a) = 40(10 + 6) = 640 N (which is greater than T) Hence the rope will break.

(b) When monkey climbs down with a = 4 m s^-2 

R = m(g-a) = 40(10 – 4) = 240 N (which is less than T) :. The rope will not break.

(c) When monkey climbs up with a uniform speed v = m s^-1, its acceleration a = 0

So,  R = mg = 40 x 10 = 400 N (which is less than T)
Hence, The rope will not break.

(d) When monkey falls down the rope nearly freely under gravity, a = g

R = m(g -a) = m(g -g) = zero
Hence the rope will not break.