For natural numbers $a , b$, let P and Q be the points where the curve $y = a ^ { x + 1 }$ and the curve $y = b ^ { x }$ meet the line $x = t ( t \geq 1 )$, respectively. Find the number of all ordered pairs $( a , b )$ of $a , b$ satisfying the following condition. For example, $a = 4 , b = 5$ satisfies the following condition. [4 points]
(A) $2 \leq a \leq 10, 2 \leq b \leq 10$
(B) For some real number $t \geq 1$, $\overline { \mathrm { PQ } } \leq 10$.

For natural numbers $a , b$, let P and Q be the points where the curve $y = a ^ { x + 1 }$ and the curve $y = b ^ { x }$ meet the line $x = t ( t \geq 1 )$ respectively. Find the number of all ordered pairs $( a , b )$ satisfying the following conditions. For example, $a = 4 , b = 5$ satisfies the following conditions. [4 points] (가) $2 \leq a \leq 10, 2 \leq b \leq 10$ (나) For some real number $t \geq 1$, $\overline { \mathrm { PQ } } \leq 10$.

Let A be the point where the graph of the exponential function $y = a ^ { x } ( a > 1 )$ meets the line $y = \sqrt { 3 }$. For the point $\mathrm { B } ( 4,0 )$, if the line OA and the line AB are perpendicular to each other, what is the product of all values of $a$? (Here, O is the origin.) [4 points]
(1) $3 ^ { \frac { 1 } { 3 } }$
(2) $3 ^ { \frac { 2 } { 3 } }$
(3) 3
(4) $3 ^ { \frac { 4 } { 3 } }$
(5) $3 ^ { \frac { 5 } { 3 } }$

For a constant $a$ ($a > 1$), let A be a point in the first quadrant on the curve $y = a ^ { x } - 2$. Let B be the point where the line passing through A and parallel to the $y$-axis meets the $x$-axis, and let C be the point where this line meets the asymptote of the curve $y = a ^ { x } - 2$. If $\overline { \mathrm { AB } } = \overline { \mathrm { BC } }$ and the area of triangle AOC is 8, what is the value of $a \times \overline { \mathrm { OB } }$? (Here, O is the origin.) [4 points]
(1) $2 ^ { \frac { 13 } { 6 } }$
(2) $2 ^ { \frac { 7 } { 3 } }$
(3) $2 ^ { \frac { 5 } { 2 } }$
(4) $2 ^ { \frac { 8 } { 3 } }$
(5) $2 ^ { \frac { 17 } { 6 } }$

16. A line $l$ with slope $k ( k < 0 )$ passes through point $F ( 0,1 )$ and intersects the curve $y = \frac { 1 } { 4 } x ^ { 2 } ( x \geq 0 )$ and the line $y = - 1$ at points $A$ and $B$ respectively. If $| F B | = 6 | F A |$, then $k = $ \_\_\_\_.
III. Solution Questions: This section contains 6 questions, totaling 70 points. Show your working, proofs, or calculation steps. Questions 17-21 are required questions that all candidates must answer. Questions 22 and 23 are optional questions; candidates should answer according to the requirements. (I) Required Questions: Total 60 points.

On the coordinate plane, $\Gamma$ is a square with side length 4, centered at the point $(1,1)$, with sides parallel to the coordinate axes. The graph of the function $y = a \times 2 ^ { x }$ intersects $\Gamma$, where $a$ is a real number. The maximum possible range of $a$ is (22)(23) $\leq a \leq$ (24).