Q1. The number of proper subsets of the set $\left\{x \left\lvert\,-1 \leqslant \log _{\frac{1}{x}} 10<-\frac{1}{2}\right., x \in N\right\}$ is
1. The set $\left\{x \left\lvert\,-1 \leqslant \log _{\frac{1}{x}} 10<-\frac{1}{2}\right., x \in N\right\}=$
$$
\begin{array}{l}
\left\{x \left\lvert\,-1 \leqslant \frac{1}{\lg _{\frac{1}{x}}}<-\frac{1}{2}\right., x \in N\right\}=\quad\{x \mid 1 \leqslant \lg x<2, x \in N\}= \\
\{x \mid 10 \leqslant x<100, x \in N\} .
\end{array}
$$

Therefore, the number of its proper subsets is $C_{90}^{0}+C_{90}^{1}+\cdots+C_{90}^{89}=2^{90}-1$.