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) $e + 1$

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) $\frac { 81 } { 2 }$

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 $1 + 2 \sqrt { 2 }$