Algebra
# 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?

\( P,\ Q \) 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.

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