For every real $x$, the expression $2 + \frac { 3 \mathrm { e } ^ { - x } - 5 } { \mathrm { e } ^ { - x } + 1 }$ is equal to: a. $\frac { 5 - 3 \mathrm { e } ^ { x } } { 1 + \mathrm { e } ^ { x } }$; b. $\frac { 5 + 3 \mathrm { e } ^ { x } } { 1 - \mathrm { e } ^ { x } }$; c. $\frac { 5 + 3 \mathrm { e } ^ { x } } { 1 + \mathrm { e } ^ { x } }$; d. $\frac { 5 - 3 \mathrm { e } ^ { x } } { 1 - \mathrm { e } ^ { x } }$.

QUESTION 161
The value of $e^0 + \ln 1$ is
(A) 0
(B) 1
(C) 2
(D) $e$
(E) $e + 1$

Prove that, for every real number $t$, $\operatorname { ch } ( t ) \leqslant \exp \left( \frac { t ^ { 2 } } { 2 } \right)$.

We denote by $\mathcal{E}$ the set of functions $f : \mathbb{C} \rightarrow \mathbb{C}$ expandable as a power series with radius of convergence infinity. Justify that if $(f, g) \in \mathcal{E}^{2}$ and $(\lambda, \mu) \in \mathbb{C}^{2}$, then $\lambda f + \mu g \in \mathcal{E}$ and $fg \in \mathcal{E}$.

The sum of the series $\frac { 1 } { 2 ! } - \frac { 1 } { 3 ! } + \frac { 1 } { 4 ! } - \ldots$ upto infinity is
(1) $e ^ { - 2 }$
(2) $e ^ { - 1 }$
(3) $e ^ { - 1 / 2 }$
(4) $e ^ { 1 / 2 }$

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) $\frac { 81 } { 2 }$

[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.

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 $1 + 2 \sqrt { 2 }$