If $f(x) = \frac{x - |x|}{|x|}$,then $f(-1) = $

The domain of the function $f(x) = \frac{1}{\log_{10}(1 - x)} + \sqrt{x + 2}$ is

Let $P = \{ \theta : \sin \theta - \cos \theta = \sqrt{2} \cos \theta \}$ and $Q = \{ \theta : \sin \theta + \cos \theta = \sqrt{2} \sin \theta \}$ be two sets. Then

If $U = \{a, b, c, d, e, f\}$,$A = \{a, b, c\}$,$B = \{c, d, e, f\}$,and $C = \{c, d, e\}$,find $(A \cap B) \cup (A \cap C)$.

Each set $X_r$ contains $5$ elements and each set $Y_r$ contains $4$ elements and $\bigcup_{r = 1}^{24} X_r = S = \bigcup_{r = 1}^n Y_r$. If each element of set $S$ belongs to exactly $10$ of the $X_r$'s and to exactly $6$ of the $Y_r$'s,then $n$ is equal to:

Let $A = \{x : x \in N \text{ and } x \text{ is a multiple of } 2\}$,$B = \{x : x \in N \text{ and } x \text{ is a multiple of } 5\}$,and $C = \{x : x \in N \text{ and } x \text{ is a multiple of } 10\}$. Describe the set $A \cap (B \cup C)$.