Question

Let \( S \) be the set of positive integral values of \( a \) for which \[ \frac{a x^2 + 2(a + 1)x + 9a + 4}{x^2 - 8x + 32} < 0, \quad \forall x \in \mathbb{R}. \] Then, the number of elements in \( S \) is:

• A. 1
• B. 0
• C. \(\infty\)
• D. 3

Answer 1

The Correct Option is B

Consider the inequality:

\[ ax^2 + 2(a + 1)x + 9a + 4 < 0 \quad \forall x \in \mathbb{R} \]

For the quadratic to be negative for all values of \( x \), the coefficient of \( x^2 \) must be negative:

\[ a < 0 \]

Since we are looking for positive integral values of \( a \), no such values exist.