We consider the curve $\mathscr{C}$ with equation $y = \mathrm{e}^{x}$.
For every strictly positive real $m$, we denote by $\mathscr{D}_{m}$ the line with equation $y = mx$.

1. In this question, we choose $m = \mathrm{e}$.
   Prove that the line $\mathscr{D}_{\mathrm{e}}$, with equation $y = \mathrm{e}x$, is tangent to the curve $\mathscr{C}$ at its point with abscissa 1.
2. Conjecture, according to the values taken by the strictly positive real $m$, the number of intersection points of the curve $\mathscr{C}$ and the line $\mathscr{D}_{m}$.
3. Prove this conjecture.

Let $S$ be the set of all the natural numbers, for which the line $\frac { x } { a } + \frac { y } { b } = 2$ is a tangent to the curve $\left( \frac { x } { a } \right) ^ { n } + \left( \frac { y } { b } \right) ^ { n } = 2$ at the point $( a , b ) , ab \neq 0$. Then
(1) $S = \phi$
(2) $n ( S ) = 1$
(3) $S = \{ 2k : k \in N \}$
(4) $S = N$

Let $a , b$ be real numbers, and let $O$ be the origin of the coordinate plane. It is known that the graph of the quadratic function $f ( x ) = a x ^ { 2 }$ and the circle $\Omega : x ^ { 2 } + y ^ { 2 } - 3 y + b = 0$ both pass through point $P \left( 1 , \frac { 1 } { 2 } \right)$, and let point $C$ be the center of $\Omega$.
Prove that the graph of $y = f ( x )$ and $\Omega$ have a common tangent line at point $P$.

The line $y = 4 x - 2$ is tangent to the graph of the function $f ( x ) = x ^ { 4 } + 1$ at the point $\mathrm { P } ( \mathrm { a } , \mathrm { b } )$.
Accordingly, what is the sum $a + b$?
A) 3
B) 4
C) 5
D) 6
E) 7