16. If the minimum value of the function $f ( x ) = | x + 1 | + 2 | x - a |$ is 5, then the real number $a = $ $\_\_\_\_$ .\\
III. Solution Questions: This section contains 6 questions, for a total of 75 points. Show your work, proofs, or calculation steps.\\
(17) (This question is worth 13 points: part (I) is worth 5 points, part (II) is worth 8 points)\\
Eating zongzi on Dragon Boat Festival is a traditional custom in China. A plate contains 10 zongzi: 2 with red bean paste, 3 with meat, and 5 plain. The three types of zongzi look identical. Three zongzi are randomly selected.\\
(I) Find the probability that one zongzi of each type is selected.\\
(II) Let $X$ denote the number of red bean paste zongzi selected. Find the probability distribution and mathematical expectation of $X$.\\
(18) (This question is worth 13 points: part (I) is worth 7 points, part (II) is worth 6 points)\\
Given the function $f ( x ) = \sin \left( \frac { \pi } { 2 } - x \right) \sin x - \sqrt { 3 } \cos ^ { 2 } x$\\
(I) Find the minimum positive period and maximum value of $f ( x )$.\\
(II) Discuss the monotonicity of $f ( x )$ on $\left[ \frac { \pi } { 6 } , \frac { 2 \pi } { 3 } \right]$.\\
(19) (This question is worth 13 points: part (I) is worth 4 points, part (II) is worth 9 points)\\
As shown in question (19), in the triangular pyramid $P - A B C$, $P C \perp$ plane $A B C$, $P C = 3$, $\angle A C B = \frac { \pi } { 2 }$. Points $D$ and $E$ are on segments $A B$ and $B C$ respectively, with $C D = D E = \sqrt { 2 }$, $C E = 2 E B = 2$.\\
(I) Prove that $D E \perp$ plane $P C D$.\\
(II) Find the cosine of the dihedral angle $A - P D - C$.\\
(20) (This question is worth 12 points: part (I) is worth 7 points, part (II) is worth 5 points)\\
Let the function $f ( x ) = \frac { 3 x ^ { 2 } + a x } { e ^ { x } } ( a \in R )$.\\
(I) If $f ( x )$ has an extremum at $x = 0$, determine the value of $a$ and find the equation of the tangent line to the curve $y = f ( x )$ at the point $( 1 , f ( 1 ) )$.\\
(II) If $f ( x )$ is decreasing on $[ 3 , + \infty )$, find the range of values for $a$.\\
(21) (This question is worth 12 points: part (I) is worth 5 points, part (II) is worth 7 points)

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\caption{Question (19) Figure}
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As shown in question (21), the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ {