Quadratic Equations
Quadratic Equations
Allen Star Batch
Grade 11
Question:
Suppose $A = \{x: 5x - a \leq 0\}$, $B = \{x: 6x - b > 0\}$, $a, b \in \mathbb{N}$ and $A \cap B \cap \mathbb{N} = \{2, 3, 4\}$. The number of such pairs $(a, b)$ is:
Step-by-Step Solution
Key Concept: Cross-multiplying a proportion of rational functions produces a quadratic equation whose solutions depend on the constraint.
From the inequalities $x \leq \frac{a}{5}$ and $x > \frac{b}{6}$, we have $4 b > c > d$ and the ordering constraint $r - a 0)$, yielding $3x^2 + 11x + 5 = 0$.
Correct Answer: 3