If $f ( x ) = \left( \frac { 3 } { 5 } \right) ^ { x } + \left( \frac { 4 } { 5 } \right) ^ { x } - 1 , x \in R$, then the equation $f ( x ) = 0$ has:
(1) No solution
(2) More than two solutions
(3) One solution
(4) Two solutions

Let $f ( x ) = a ^ { x }$ $( a > 0 )$ be written as $f ( x ) = f _ { 1 } ( x ) + f _ { 2 } ( x )$, where $f _ { 1 } ( x )$ is an even function and $f _ { 2 } ( x )$ is an odd function. Then $f _ { 1 } ( x + y ) + f _ { 1 } ( x - y )$ equals:
(1) $2 f _ { 1 } ( x ) f _ { 1 } ( y )$
(2) $2 f _ { 1 } ( x + y ) f _ { 1 } ( x - y )$
(3) $2 f _ { 1 } ( x ) f _ { 2 } ( y )$
(4) $2 f _ { 1 } ( x + y ) f _ { 2 } ( x - y )$

Let $\sum _ { k = 1 } ^ { 10 } f ( a + k ) = 16 \left( 2 ^ { 10 } - 1 \right)$, where the function $f$ satisfies $f ( x + y ) = f ( x ) f ( y )$ for all natural numbers $x , y$ and $f ( 1 ) = 2$. Then the natural number ' $a$ ' is:
(1) 3
(2) 16
(3) 4
(4) 2

$\lim _ { x \rightarrow 2 } \frac { 3 ^ { x } + 3 ^ { 3 - x } - 12 } { 3 ^ { - \frac { x } { 2 } } - 3 ^ { 1 - x } }$ is equal to

The minimum value of $f ( x ) = a ^ { a ^ { x } } + a ^ { 1 - a ^ { x } }$, where $a , x \in R$ and $a > 0$, is equal to:
(1) $a + 1$
(2) $2 a$
(3) $a + \frac { 1 } { a }$
(4) $2 \sqrt { a }$

The equation $e^{4x} + 8e^{3x} + 13e^{2x} - 8e^x + 1 = 0, x \in R$ has:
(1) four solutions two of which are negative
(2) two solutions and both are negative
(3) no solution
(4) two solutions and only one of them is negative

Let $S = \{x : x \in \mathbb{R}$ and $\left(\sqrt{3} + \sqrt{2}\right)^{x^2 - 4} + \left(\sqrt{3} - \sqrt{2}\right)^{x^2 - 4} = 10\}$. Then $n(S)$ is equal to
(1) 2
(2) 4
(3) 6
(4) 0

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a function defined by $f ( x ) = \log _ { \sqrt { m } } \{ \sqrt { 2 } ( \sin x - \cos x ) + m - 2 \}$, for some $m$, such that the range of $f$ is $[ 0,2 ]$. Then the value of $m$ is $\_\_\_\_$.
(1) 5
(2) 3
(3) 2
(4) 4

$\lim _ { x \rightarrow 0 } \frac { e - ( 1 + 2 x ) ^ { \frac { 1 } { 2 x } } } { x }$ is equal to
(1) 0
(2) $\frac { - 2 } { e }$
(3) e
(4) $e - e ^ { 2 }$

Let $f: R \rightarrow R$ be defined $f(x) = ae^{2x} + be^x + cx$. If $f(0) = -1$, $f'(\log_e 2) = 21$ and $\int_0^{\log 4} (f(x) - cx)\, dx = \frac{39}{2}$, then the value of $|a + b + c|$ equals:
(1) 16
(2) 10
(3) 12
(4) 8

If $\alpha = \lim _ { x \rightarrow 0 ^ { + } } \left( \frac { \mathrm { e } ^ { \sqrt { \tan x } } - \mathrm { e } ^ { \sqrt { x } } } { \sqrt { \tan x } - \sqrt { x } } \right)$ and $\beta = \lim _ { x \rightarrow 0 } ( 1 + \sin x ) ^ { \frac { 1 } { 2 } \cot x }$ are the roots of the quadratic equation $a x ^ { 2 } + b x - \sqrt { \mathrm { e } } = 0$, then $12 \log _ { \mathrm { e } } ( \mathrm { a } + \mathrm { b } )$ is equal to $\_\_\_\_$

If $f ( x ) = \frac { 2 ^ { x } } { 2 ^ { x } + \sqrt { 2 } } , \mathrm { x } \in \mathbb { R }$, then $\sum _ { \mathrm { k } = 1 } ^ { 81 } f \left( \frac { \mathrm { k } } { 82 } \right)$ is equal to
(1) $1.81 \sqrt { 2 }$
(2) 41
(3) 82
(4) $\frac { 81 } { 2 }$

If $\lim _ { x \rightarrow \infty } \left( \left( \frac { \mathrm { e } } { 1 - \mathrm { e } } \right) \left( \frac { 1 } { \mathrm { e } } - \frac { x } { 1 + x } \right) \right) ^ { x } = \alpha$, then the value of $\frac { \log _ { \mathrm { e } } \alpha } { 1 + \log _ { \mathrm { e } } \alpha }$ equals:
(1) $e ^ { - 1 }$
(2) $\mathrm { e } ^ { 2 }$
(3) $e ^ { - 2 }$
(4) e

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).

An organization introduced two different nutrients into culture dishes A and B at 12 o'clock. At this time, the bacterial counts in dishes A and B are $X$ and $Y$ respectively. The quantity in dish A doubles every 3 hours; for example, at 3 PM the quantity in A is $2X$. The quantity in dish B doubles every 2 hours; for example, at 2 PM the quantity in B is $2Y$, and at 4 PM the quantity in B is $4Y$. Part of the measurement results are recorded in the table below. At 6 PM, the organization measured that the quantities in dishes A and B are the same. To estimate the bacterial quantities in dishes A and B from 12 o'clock to 12 midnight using an exponential growth model, select the correct options.

Time (o'clock)	12	13	14	15	16	17	18	19	20	21	22	23	24
Quantity in A	$X$			$2X$
Quantity in B	$Y$		$2Y$		$4Y$

(1) $X > Y$ (2) At 1 PM, the quantity in A is $\frac{4}{3}X$ (3) At 3 PM, the quantity in B is $3Y$ (4) At 7 PM, the quantity in B is 1.5 times that of A (5) At 12 midnight, the quantity in B is 2 times that of A

Over the past five years, a country's total carbon emissions decreased from $X$ billion metric tons of CO2 equivalent (CO2e) in year 1 to $Y$ billion metric tons of CO2 equivalent (CO2e) in year 5, achieving an average annual carbon reduction of 5\%, that is, $Y = ( 1 - 0.05 ) ^ { 4 } X$. The five-year carbon emission totals and annual growth rates are recorded in the following table, where Year $n$ carbon emission growth rate $= \frac { \text {(Year } n \text { carbon emission total)} - \text {(Year } n - 1 \text { carbon emission total)} } { \text {Year } n - 1 \text { carbon emission total} }$, $n = 2,3,4,5$.

	Year 1	Year 2	Year 3	Year 4	Year 5
\begin{tabular}{ c } Carbon Emission Total
$($ billion metric tons $\mathrm { CO } 2 \mathrm { e } )$

& $X$ & $A$ & $B$ & $C$ & $Y$ \hline Annual Carbon Emission Growth Rate & & - 0.07 & $p$ & $q$ & $r$ \hline \end{tabular}
Select the correct options.
(1) $A = 0.93 X$
(2) $Y \leq 0.8 X$
(3) $\frac { - 0.07 + p + q + r } { 4 } = - 0.05$
(4) $\sqrt [ 4 ] { \frac { Y } { X } } - 1 = - 0.05$
(5) $0.93 ( 1 + p ) ( 1 + q ) ( 1 + r ) = ( 0.95 ) ^ { 4 }$

Let the exponential function $f(x) = 1.2^{x}$. Select the correct options.
(1) $f(0) > 0$
(2) $f(10) > 10$
(3) On the coordinate plane, the graph of $y = 1.2^{x}$ intersects the line $y = x$
(4) On the coordinate plane, the graphs of $y = 1.2^{x}$ and $y = \log(1.2^{x})$ are symmetric about the line $y = x$
(5) For any positive real number $b$, $\log_{1.2} b \neq 1.2^{b}$

Consider points $P ( x , y )$ on the coordinate plane satisfying the equation $\frac { 2 ^ { x ^ { 2 } } } { 8 } = \frac { 4 ^ { x } } { 2 ^ { y ^ { 2 } } }$. Select the correct options.
(1) When $x = 3$, there are 2 distinct solutions satisfying this equation
(2) If point $( a , b )$ satisfies this equation, then point $( - a , - b )$ also satisfies this equation
(3) All possible points $P ( x , y )$ form a circle
(4) Point $P ( x , y )$ may lie on the line $x + y = 4$
(5) For all possible points $P ( x , y )$, the maximum value of $x - y$ is $1 + 2 \sqrt { 2 }$

It is known that UVI values have an exponential relationship with altitude: for every 300-meter increase in altitude, the UVI value increases by 4\% of the value before the increase. At ground level, the ultraviolet radiation received from the sun is 400 joules per square meter. At a mountain 4500 meters above ground level, the UVI value of the ultraviolet radiation received is which of the following options? (Single choice question, 3 points)
(1) $4 ( 1 + 0.04 \times 15 )$
(2) $4 \left( 1 + 0.04 ^ { 15 } \right)$
(3) $4 ( 1 + 0.04 ) ^ { 15 }$
(4) $4 \times 100 ( 1 + 0.04 ) ^ { 15 }$
(5) $4 \times 100 \left( 1 + 0.04 ^ { 45 } \right)$

Let x be a real number such that
$$( \sqrt { 7 } + \sqrt { 3 } ) ^ { x } = 4$$
Given this, which of the following is the expression $( \sqrt { 7 } - \sqrt { 3 } ) ^ { x }$ equal to?
A) $2 ^ { - x }$
B) $2 ^ { - x + 1 }$
C) $4 ^ { x }$
D) $4 ^ { x - 1 }$
E) $4 ^ { x + 1 }$

The operation $\Delta$ is defined on the set of real numbers for all real numbers a and b as
$$a \Delta b = a ^ { 2 } + 2 ^ { b }$$
Given that $2 \Delta ( 1 \Delta x ) = 12$, what is x?
A) $\frac { 1 } { 2 }$
B) $\frac { 2 } { 3 }$
C) $\frac { 1 } { 4 }$
D) 1
E) 2

A function f is defined on the set of real numbers as
$$f ( x ) = 1 + e ^ { - x }$$
Accordingly, I. The range of function f is $( 1 , \infty )$. II. Function f is decreasing on its domain. III. The line $y = 0$ is a horizontal asymptote of function f. Which of the following statements are true?
A) Only II
B) Only III
C) I and II

For integers a and b
$$\frac { 6 ^ { a ^ { 2 } + b ^ { 2 } } } { 9 ^ { a b } } = 96$$
Given that, what is the product $\mathbf { a } \cdot \mathbf { b }$?
A) 1 B) 2 C) 3 D) 4 E) 6

$$\begin{aligned} & 4 ^ { x } + 4 ^ { y } = 10 \\ & 4 ^ { x } - 4 ^ { y } = 8 \end{aligned}$$
Accordingly, what is the value of the expression $\mathbf { 2 } ^ { \mathbf { x } + \mathbf { y } }$?
A) 2 B) 3 C) 4 D) 5 E) 6

Below; the graphs of linear functions $f$, $g$ and $h$ are shown in Figure 1 on a rectangular coordinate plane divided into unit squares, and the derivatives of these functions are shown in Figure 2.
Accordingly; what is the correct ordering of $f ( 0 ) , g ( 0 )$ and $h ( 0 )$?
A) $\mathrm { f } ( 0 ) < \mathrm { h } ( 0 ) < \mathrm { g } ( 0 )$
B) $g ( 0 ) < f ( 0 ) < h ( 0 )$
C) $g ( 0 ) < h ( 0 ) < f ( 0 )$
D) $h ( 0 ) < f ( 0 ) < g ( 0 )$
E) $h ( 0 ) < g ( 0 ) < f ( 0 )$