[1] Let $x$ be a positive real number satisfying $x ^ { 2 } + \frac { 4 } { x ^ { 2 } } = 9$. Then

$$\left( x + \frac { 2 } { x } \right) ^ { 2 } = \text { A B }$$

Therefore, $x + \frac { 2 } { x } = \sqrt { \text { A B } }$. Furthermore,

$$\begin{aligned}
x ^ { 3 } + \frac { 8 } { x ^ { 3 } } & = \left( x + \frac { 2 } { x } \right) \left( x ^ { 2 } + \frac { 4 } { x ^ { 2 } } - \right. \\
& = \text { D } \sqrt { \text { E F } }
\end{aligned}$$

□ C

Also,

$$x ^ { 4 } + \frac { 16 } { x ^ { 4 } } = \text { G H }$$

[2] Let $p$ and $q$ be two conditions regarding the real number $x$:

$$\begin{array} { l l } 
p : & x = 1 \\
q : & x ^ { 2 } = 1
\end{array}$$

Also, let $\bar { p }$ and $\bar { q }$ denote the negations of conditions $p$ and $q$, respectively.\\
(1) Choose one from the options (0)–(3) below for each of the blanks K, L, M, N. You may select the same option more than once.\\
$q$ is a \text{____} condition for $p$ (K)\\
$\bar { p }$ is a \text{____} condition for $q$ (L)\\
($p$ or $\bar { q }$) is a \text{____} condition for $q$ (M)\\
($\bar { p }$ and $q$) is a \text{____} condition for $q$ (N)\\
(0) necessary but not sufficient\\
(1) sufficient but not necessary\\
(2) necessary and sufficient\\
(3) neither necessary nor sufficient

(2) Let $r$ be a condition regarding the real number $x$:

$$r : x > 0$$

Choose one from options ⓪–⑦ below for the blank O.

Consider the three propositions:

A: ``($p$ and $q$) $\Longrightarrow r$''\\
B: ``$q \Longrightarrow r$''\\
C: ``$\bar { q } \Longrightarrow \bar { p }$''

The correct statement about the truth values of these propositions is □ O.\\
(0) A is true, B is true, C is true\\
(1) A is true, B is true, C is false\\
(2) A is true, B is false, C is true\\
(3) A is true, B is false, C is false\\
(4) A is false, B is true, C is true\\
(5) A is false, B is true, C is false\\
(6) A is false, B is false, C is true\\
(7) A is false, B is false, C is false

[3] Let $a$ be a constant, and let $g ( x ) = x ^ { 2 } - 2 \left( 3 a ^ { 2 } + 5 a \right) x + 18 a ^ { 4 } + 30 a ^ { 3 } + 49 a ^ { 2 } + 16$. The vertex of the parabola $y = g ( x )$ is

$$\text { ( P } a ^ { 2 } + \text { Q } a , \text { R } a ^ { 4 } + \text { S T } a ^ { 2 } + \text { U V } \text { ) }$$

When $a$ varies over all real numbers, the minimum value of the $x$-coordinate of the vertex is $- \frac { \text { W X } } { \text { Y Z } }$.

Next, let $t = a ^ { 2 }$. Then the $y$-coordinate of the vertex can be expressed as

$$\text { R } t ^ { 2 } + \text { S T } t + \text { U V }$$

Therefore, when $a$ varies over all real numbers, the minimum value of the $y$-coordinate of the vertex is AA.