For a certain financial product, the expected asset $W$ after $t$ years of investing an initial asset $W _ { 0 }$ is given as follows: $$W = \frac { W _ { 0 } } { 2 } 10 ^ { a t } \left( 1 + 10 ^ { a t } \right)$$ (where $W _ { 0 } > 0 , t \geq 0$, and $a$ is a constant.) When an initial asset $w _ { 0 }$ is invested in this financial product, the expected asset after 15 years is 3 times the initial asset. When an initial asset $w _ { 0 }$ is invested in this financial product, the expected asset after 30 years is $k$ times the initial asset. What is the value of the real number $k$? (where $w _ { 0 } > 0$) [4 points]
(1) 9
(2) 10
(3) 11
(4) 12
(5) 13

For a certain financial product, when an initial asset $W _ { 0 }$ is invested, the expected asset $W$ after $t$ years is given as follows. $$\begin{aligned} & W = \frac { W _ { 0 } } { 2 } 10 ^ { a t } \left( 1 + 10 ^ { a t } \right) \\ & \text { (where } W _ { 0 } > 0 , t \geq 0 \text {, and } a \text { is a constant.) } \end{aligned}$$ When an initial asset $w _ { 0 }$ is invested in this financial product, the expected asset after 15 years is 3 times the initial asset. When an initial asset $w _ { 0 }$ is invested in this financial product, the expected asset after 30 years is $k$ times the initial asset. What is the value of the real number $k$? (where $w _ { 0 } > 0$) [3 points]
(1) 9
(2) 10
(3) 11
(4) 12
(5) 13

Let $k$ be the $x$-coordinate of the intersection point of the curve $y = \left(\frac{1}{5}\right)^{x-3}$ and the line $y = x$. A function $f(x)$ defined on the set of all real numbers satisfies the following conditions. For all real numbers $x > k$, $f(x) = \left(\frac{1}{5}\right)^{x-3}$ and $f(f(x)) = 3x$. What is the value of $f\left(\frac{1}{k^{3} \times 5^{3k}}\right)$? [4 points]

If $$\lim _ { x \rightarrow 0 } \left[ 1 + x \ln \left( 1 + b ^ { 2 } \right) \right] ^ { \frac { 1 } { x } } = 2 b \sin ^ { 2 } \theta , b > 0 \text { and } \theta \in ( - \pi , \pi ]$$ then the value of $\theta$ is
(A) $\pm \frac { \pi } { 4 }$
(B) $\pm \frac { \pi } { 3 }$
(C) $\pm \frac { \pi } { 6 }$
(D) $\pm \frac { \pi } { 2 }$

Let $x _ { 0 }$、$y _ { 0 }$ be positive real numbers. If the point $\left( 10 x _ { 0 } , 100 y _ { 0 } \right)$ on the coordinate plane lies on the graph of the function $y = 10 ^ { x }$ , then the point $\left( x _ { 0 } , \log y _ { 0 } \right)$ will lie on the graph of the line $y = a x + b$ , where $a$、$b$ are real numbers. What is the value of $2 a - b$?
(1) 4
(2) 9
(3) 15
(4) 18
(5) 22

If $x, y$ are two positive real numbers satisfying $x^{-\frac{1}{3}} y^{2} = 1$ and $2\log y = 1$, then $\frac{x - y^{2}}{10} =$ (13--1) (13--2).

$$\frac { 3 ^ { x } } { 2 ^ { 2 x } } = \frac { 1 } { 5 }$$
Given this, what is the value of the expression $5 ^ { \frac { 1 } { x } }$?
A) $\frac { 3 } { 2 }$
B) $\frac { 4 } { 3 }$
C) $\frac { 9 } { 4 }$
D) $\frac { 9 } { 5 }$
E) $\frac { 5 } { 6 }$

$$\begin{aligned} & 2 ^ { x } = \frac { 1 } { 5 } \\ & 3 ^ { y } = \frac { 1 } { 4 } \end{aligned}$$
Given this, what is the value of the product $x \cdot y$?
A) $\frac { \ln 3 } { \ln 2 }$
B) $\frac { \ln 15 } { \ln 2 }$
C) $\frac { \ln 5 } { \ln 4 }$
D) $\frac { \ln 25 } { \ln 3 }$
E) $\frac { \ln 5 } { \ln 6 }$

$$4 ^ { x - 2 } = 6 ^ { 2 x - 2 }$$
Given that, what is the value of the expression $9 ^ { \mathbf { x } }$?
A) $\frac { 9 } { 2 }$
B) $\frac { 4 } { 3 }$
C) $\frac { 8 } { 3 }$
D) $\frac { 5 } { 4 }$
E) $\frac { 9 } { 4 }$