Sequences & Series
Sequence and Series
star_batch_jee_advanced_2025
Grade 11
Question:
If $x, y, z$ are three distinct positive real numbers and are in H.P., then $\frac{3x+2y}{2x-y}$ and $\frac{3z+2y}{2z-y}$ is greater then:
Step-by-Step Solution
Key Concept: Rationalize compound fractions by combining terms over common denominators, then express the result in terms of symmetric expressions like $\frac{z}{x} + \frac{x}{z}$.
The expression $\frac{3x + \frac{4xz}{x+z}}{2x - \frac{2xz}{x+z}} + \frac{3z + \frac{4xz}{x+z}}{2z - \frac{2xz}{x+z}}$ simplifies by finding common denominators in each fraction. After algebraic manipulation, this reduces to $3 + \frac{7}{2}\left(\frac{z}{x} + \frac{x}{z}\right)$, where $\frac{z}{x} + \frac{x}{z} \in [10, \infty)$ for positive $x$ and $z$.
Correct Answer: 1,2