The number of all possible values of , where 0 < < , for which the system of equations array r (y+z) 3 =(x y z) 3 x 3 = 2 3 y + 2 3 z array and (x y z) 3 =(y+2 z) 3 +y 3 have a solution (x₀, y₀, z₀ ) with y₀ z₀ 0 , is

Step-by-step solution

Given equations can be written as gathered x 3 - 3 y - 3 z =0 x 3 - 2 3 y - 2 3 z =0 and x 3 - 2 y 3 - 1 z ( 3 + 3 )=0 gathered Eqs. (ii) and (iii), implies aligned & 2 3 = 3 + 3 & 3 = 3 & 3 =1 & 3 = 4 , 5 4 , 9 4 or = 12 , 5 12 , 9 12 . aligned