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

Answer: (B)

0

Answer 2

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.