One, Select (6 points per question, total 36 points)
1. The number of solutions to the equation $\left(\frac{3}{19}\right)^{x}+\left(\frac{5}{19}\right)^{x}+\left(\frac{11}{19}\right)^{x}=$ $2 \sqrt{x-1}$ is (
(A) 0
(B) 1
(C) 2
(D) Infinite
$-1 .(B)$
\| $x-1 \geqslant 0$, so $x \geq 1$.
$$
\because f(x)=\left(\frac{3}{19}\right)^{x}+\left(\frac{5}{19}\right)^{x}+\left(\frac{11}{19}\right)^{x} \text { in }[1,+\infty) \mathrm{E} .
$$

is a decreasing function,
$\therefore$ when $x=1 \|_{1}^{+}, f(x)_{\max }{ }^{2} 1$.
X $g(x)=2 \sqrt{x \cdots 1}$ in $[1,+\infty)$ is an increasing function,
$\therefore$ if $x=1$ then $\underline{g}(x)_{\min }=0$.