6th and 7th one

6)
from conservation of mechanical energy, we can say at any point the algebraic sum of KE and PE will be equal to mgH where m is the mass of the object.

it is given that at some instant KE=2PE
let height bt h' at that instant,
KE + PE = mgH
​replacing KE with 2PE
3PE = mgH 

so the first part is correct

7) we know from the work energy theorem that total work done is equivalent to the net change in the KE of the body .

s = t² + 2t
v = d(s)/dt = 2t+2.

at t=2 
v= 2*2 + 2 = 6
at t=4
v=2*4+2 = 10

change in KE = 1/2 * 2 * (10²-6²) =  64J

​3mgh'=mgH
h'=H/3
and KE =2PE
​putting the values

1/2 m v² = mgH/3

v= (2gH/3)^1/2