If a, b, c, d are in H.P.,then | vedclass

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

Question

(C) Given $a, b, c, d$ are in $H.P.$,their reciprocals $\frac{1}{a}, \frac{1}{b}, \frac{1}{c}, \frac{1}{d}$ are in $A.P.$For any three terms in $H.P.$

Vedclass · Accepted Answer

$ac + bd > b^2 + c^2$

Vedclass · Answer

(C) Given $a, b, c, d$ are in $H.P.$,their reciprocals $\frac{1}{a}, \frac{1}{b}, \frac{1}{c}, \frac{1}{d}$ are in $A.P.$For any three terms in $H.P.$,say $x, y, z$,we know that $y = \frac{2xz}{x+z}$. Since $A.M. > H.M.$,we have $\sqrt{xz} > y$,which implies $xz > y^2$.Applying this to the consecutive terms of the $H.P.$:$1$. For $a, b, c$ in $H.P.$,we have $ac > b^2$.$2$. For $b, c, d$ in $H.P.$,we have $bd > c^2$.Adding these two inequalities,we get:$ac + bd > b^2 + c^2$.

If $a, b, c, d$ are in $H.P.$,then

Similar Questions

If the roots of the equation $k x^3 - 18 x^2 - 36 x + 8 = 0$ are in harmonic progression,then $k =$

If $\tan B = \frac{2 \sin A \sin C}{\sin (A+C)}$,then $\tan A, \tan B$ and $\tan C$ are in

If $\cos (\theta-\alpha), \cos \theta$ and $\cos (\theta+\alpha)$ are in harmonic progression,then $2 \tan ^2 \theta=$

Given that the roots of $x^3+3px^2+3qx+r=0$ are in harmonic progression,then:

If $\sin (\theta-\alpha), \sin \theta$ and $\sin (\theta+\alpha)$ are in $H.P.$,then the value of $\cos ^2 \theta$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module