Polynomial Inequalities
\( A\) and \( B \) are two subsets of \(\mathbb{R}\) defined by: \[\begin{align} A &= \{ x \mid x^2 - 6 x - 91 \leq 0 \}, \\ B &= \{ x \mid x^2 - 6 x - 91 = 0 \}. \end{align} \] How many integers does the set \( A \cap B^C \) contain?
Details and assumptions
\( B^C \) denotes the complement of set \( B \).
\(\mathbb{R}\) denotes the set of all real numbers.
\end{align} \] Then which of the following is the set that represents the shaded region, including the two axes, in the above graph?
The sets \( A = \{ x \mid x^2 - 5 x + 4 \leq 0 \} \) and \( B = \{ x \mid x^2 + ax + b < 0 \} \) satisfy the following conditions: \[ A \cap B = \emptyset \mbox{ and } A \cup B = \{ x \mid 1 \leq x < 15 \}. \] What is the value of \( a + b? \)
\( P,\ Q \) and \( R \) are three subsets of \(\mathbb{R}\) defined as follows: \[\begin{align} P &= \{ x \mid x^2 - 21x + 110 > 0 \}, \\ Q &= \{ x \mid x^2 - 15x + 44 < 0 \}, \\ R &= \{ x \mid x^2 - 11ax + 10a^2 < 0,\ a > 0 \}. \end{align} \] What is the sum of the minimum and maximum values of \(a\) such that \( (P \cap Q) \subset R \)?
Details and assumptions
\(\mathbb{R}\) denotes the set of all real numbers.
\( A\), \(B \) and \( C \) are subsets of \(\mathbb{R}\) defined as follows: \[\begin{align} A &= \{ x \mid x^2 - 10 x + 21 \geq 0 \}, \\ B &= \{ x \mid x^2 - 10 x + 21 > 0 \},\\ C &= \{ x \mid x^2 - 19 x + 48 = 0 \}. \end{align} \] What is the value of the sum of all the elements of the set \( (A \cap B^C) \cup C \)?
Details and assumptions
\( B^C \) denotes the complement of set \( B \).
\(\mathbb{R}\) denotes the set of all real numbers.