If the system of equations $$11x + y + \lambda z = -5$$, $$2x + 3y + 5z = 3$$, $$8x - 19y - 39z = \mu$$ has infinitely many solutions, then $$\lambda^4 - \mu$$ is equal to :

For infinitely many solutions, $$D = 0$$ and consistency conditions. Here we set $$D = 0$$, where $$D = \begin{vmatrix} 11 & 1 & \lambda \\ 2 & 3 & 5 \\ 8 & -19 & -39 \end{vmatrix}.$$ Expanding, we get $$D = 11(-117 + 95) - 1(-78 - 40) + \lambda(-38 - 24)$$ $$= 11(-22) + 118 + \lambda(-62)$$ $$= -242 + 118 - 62\lambda = -124 - 62\lambda.$$ Setting $$D = 0$$ gives $$-124 - 62\lambda = 0 \Rightarrow \lambda = -2.$$

With $$\lambda = -2$$, the system becomes $$11x + y - 2z = -5,\quad 2x + 3y + 5z = 3,\quad 8x - 19y - 39z = \mu.$$ From the first two equations, multiplying the first by 3 gives $$33x + 3y - 6z = -15$$ and subtracting the second yields $$31x - 11z = -18.$$ Multiplying the first equation by 19 gives $$209x + 19y - 38z = -95$$ and adding the third equation leads to $$(209+8)x +(-38-39)z = -95 + \mu,$$ i.e., $$217x - 77z = -95 + \mu.$$ Meanwhile, multiplying $$31x - 11z = -18$$ by 7 also gives $$217x - 77z = -126.$$ Equating these expressions for $$217x - 77z$$, we have $$-95 + \mu = -126 \Rightarrow \mu = -31.$$

Therefore, $$\lambda^4 - \mu = (-2)^4 - (-31) = 16 + 31 = 47.$$

The correct answer is Option (3): 47.