Quadratic Equations
Inequality Valid for All x — Positive Integer Values
nta_pyq_2024_jan
Grade 11
Question:
Let $S$ be the set of positive integral values of $a$ for which $\dfrac{ax^2+2(a+1)x+9a+4}{x^2-8x+32}<0$, $\forall x\in\mathbb{R}$. Then, the number of elements in $S$ is:
Step-by-Step Solution
Key Concept: Denominator $x^2-8x+32=(x-4)^2+16>0$ always. So need numerator $ax^2+2(a+1)x+9a+4<0$ for all $x$. Requires $a<0$ and discriminant $<0$.
$a<0$ required, but positive integers $a>0$. $S=\emptyset$, 0 elements.
Correct Answer: 2