Three numbers are chosen at random, one after another with replacement, from the set S = {1, 2, 3, ......, 100}. Let p₁ be the probability that the maximum of chosen numbers is at least 81 and p₂ be the probability that the minimum of chosen numbers is at most 40.

The value of $${{125} \over 4}{p_2}$$ is ___________.

Let $$\alpha$$, $$\beta$$ and $$\gamma$$ be real numbers such that the system of linear equations

x + 2y + 3z = $$\alpha$$

4x + 5y + 6z = $$\beta$$

7x + 8y + 9z = $$\gamma $$ $$-$$ 1

is consistent. Let | M | represent the determinant of the matrix

$$M = \left[ {\matrix{ \alpha & 2 & \gamma \cr \beta & 1 & 0 \cr { - 1} & 0 & 1 \cr } } \right]$$

Let P be the plane containing all those ($$\alpha$$, $$\beta$$, $$\gamma$$) for which the above system of linear equations is consistent, and D be the square of the distance of the point (0, 1, 0) from the plane P.

The value of | M | is _________.

Let $$\alpha$$, $$\beta$$ and $$\gamma$$ be real numbers such that the system of linear equations

x + 2y + 3z = $$\alpha$$

4x + 5y + 6z = $$\beta$$

7x + 8y + 9z = $$\gamma $$ $$-$$ 1

is consistent. Let | M | represent the determinant of the matrix

$$M = \left[ {\matrix{ \alpha & 2 & \gamma \cr \beta & 1 & 0 \cr { - 1} & 0 & 1 \cr } } \right]$$

Let P be the plane containing all those ($$\alpha$$, $$\beta$$, $$\gamma$$) for which the above system of linear equations is consistent, and D be the square of the distance of the point (0, 1, 0) from the plane P.

The value of D is _________.

Consider the lines L₁ and L₂ defined by

$${L_1}:x\sqrt 2 + y - 1 = 0$$ and $${L_2}:x\sqrt 2 - y + 1 = 0$$

For a fixed constant $$\lambda$$, let C be the locus of a point P such that the product of the distance of P from L₁ and the distance of P from L₂ is $$\lambda$$². The line y = 2x + 1 meets C at two points R and S, where the distance between R and S is $$\sqrt {270} $$. Let the perpendicular bisector of RS meet C at two distinct points R' and S'. Let D be the square of the distance between R' and S'.

The value of $$\lambda$$² is __________.