In the pyramid $P - ABCD$, $PD \perp$ base $ABCD$, $CD \parallel AB$, $AD = DC = CB = 1$, $AB = 2$.
(1) Prove that $BD \perp PA$;
(2) Find the sine of the angle between $PD$ and plane $PAB$.

In the tetrahedron $A B C D$ , $A D \perp C D , A D = C D$ , $\angle A D B = \angle B D C$ , and $E$ is the midpoint of $A C$.
(1) Prove: Plane $B E D \perp$ plane $A C D$ ;
(2) Given $A B = B D = 2 , \angle A C B = 60 ^ { \circ }$ , point $F$ is on $B D$ . When the area of $\triangle A F C$ is minimized, find the volume of the tetrahedron $F - A B C$ .

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 $$M = \begin{pmatrix} \alpha & 1 & 0 \\ \beta & 2 & 1 \\ \gamma & 1 & 0 \end{pmatrix}.$$ 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 ____.

Problem 4
In a three-dimensional Cartesian coordinate system $x y z$, consider the positional relationship among three planes defined by Equations (1)-(3), and the positional relationship among the three planes and a sphere defined by Equation (4).
$$\begin{aligned} & a _ { 11 } x + a _ { 12 } y + a _ { 13 } z = b _ { 1 } , \\ & a _ { 21 } x + a _ { 22 } y + a _ { 23 } z = b _ { 2 } , \\ & a _ { 31 } x + a _ { 32 } y + a _ { 33 } z = b _ { 3 } , \\ & x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 3 , \end{aligned}$$
where $a _ { i j }$ and $b _ { i } ( i , j = 1,2,3 )$ are constants.
For the three planes, let $\mathrm { A } = \left( \begin{array} { l l l } a _ { 11 } & a _ { 12 } & a _ { 13 } \\ a _ { 21 } & a _ { 22 } & a _ { 23 } \\ a _ { 31 } & a _ { 32 } & a _ { 33 } \end{array} \right)$ be the coefficient matrix and $\mathbf { B } = \left( \begin{array} { l l l l } a _ { 11 } & a _ { 12 } & a _ { 13 } & b _ { 1 } \\ a _ { 21 } & a _ { 22 } & a _ { 23 } & b _ { 2 } \\ a _ { 31 } & a _ { 32 } & a _ { 33 } & b _ { 3 } \end{array} \right)$ be the augmented coefficient matrix.
I. Let $\mathbf { A } = \left( \begin{array} { l l l } 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{array} \right)$ and $\mathbf { B } = \left( \begin{array} { c c c c } 1 & 1 & 1 & 3 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & - c \end{array} \right)$ where $c$ is a positive constant.

1. Find $\operatorname { rank } ( \mathrm { A } )$ and $\operatorname { rank } ( \mathrm { B } )$.
2. Among the three planes, the plane that is tangential to the sphere defined by Equation (4) at a point $\mathrm { P } ( 1,1,1 )$ is called Plane 1. Between the other two planes, the plane with the shorter distance to P is called Plane 2. Find the distance between P and Plane 2. Then, find the volume of the part of the sphere existing between Planes 1 and 2.

II. When the three planes intersect in a line, find $\operatorname { rank } ( \mathrm { A } )$ and $\operatorname { rank } ( \mathrm { B } )$.
III. Suppose that the three planes are tangential to the sphere at three different points. Illustrate all possible positional relationships among the three planes and the sphere. In addition, for each relationship, find $\operatorname { rank } ( \mathrm { A } )$ and $\operatorname { rank } ( \mathrm { B } )$.