If a, b, c are three distinct positive real n | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Given that $a, b, c$ are in $H.P.$,their reciprocals $\frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ are in $A.P.$Let $\frac{1}{a} = p - q$,$\frac{1}{b} =

Vedclass · Accepted Answer

Greater than $10$

Vedclass · Answer

(D) Given that $a, b, c$ are in $H.P.$,their reciprocals $\frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ are in $A.P.$Let $\frac{1}{a} = p - q$,$\frac{1}{b} = p$,and $\frac{1}{c} = p + q$,where $p, q > 0$ and $p > q$.Then $a = \frac{1}{p-q}$,$b = \frac{1}{p}$,and $c = \frac{1}{p+q}$.Substituting these into the expression:$\frac{3a + 2b}{2a - b} = \frac{\frac{3}{p-q} + \frac{2}{p}}{\frac{2}{p-q} - \frac{1}{p}} = \frac{3p + 2(p-q)}{2p - (p-q)} = \frac{5p - 2q}{p + q}$Similarly,$\frac{3c + 2b}{2c - b} = \frac{\frac{3}{p+q} + \frac{2}{p}}{\frac{2}{p+q} - \frac{1}{p}} = \frac{3p + 2(p+q)}{2p - (p+q)} = \frac{5p + 2q}{p - q}$Adding these: $\frac{5p - 2q}{p + q} + \frac{5p + 2q}{p - q} = \frac{(5p - 2q)(p - q) + (5p + 2q)(p + q)}{p^2 - q^2} = \frac{5p^2 - 7pq + 2q^2 + 5p^2 + 7pq + 2q^2}{p^2 - q^2} = \frac{10p^2 + 4q^2}{p^2 - q^2} = 10 + \frac{14q^2}{p^2 - q^2}$.Since $p > q > 0$,$p^2 - q^2 > 0$,so the expression is strictly greater than $10$.

If $a, b, c$ are three distinct positive real numbers which are in $H.P.$,then $\frac{3a + 2b}{2a - b} + \frac{3c + 2b}{2c - b}$ is

Similar Questions

If the harmonic mean between $a$ and $b$ is $H$,then the value of $\frac{1}{H - a} + \frac{1}{H - b}$ is

If ${A_1}, {A_2}$; ${G_1}, {G_2}$ and ${H_1}, {H_2}$ are $AMs$,$GMs$,and $HMs$ between two quantities,then the value of $\frac{{G_1 G_2}}{{H_1 H_2}}$ is

The sum to infinity of a geometric progression is $4/3$ and the first term is $3/4$. The common ratio is

$A$ series whose $n^{th}$ term is $\left( \frac{n}{x} \right) + y$,the sum of $r$ terms will be

The number $111...1$ ($91$ times) is a:

Vedclass Test Series

Exam Paper Generator

Online Exam Module