Search any question & find its solution
Question:
Answered & Verified by Expert
The harmonic mean H of two numbers is 4 and the arithmetic mean A and geometric mean G satisfy the equation $2 \mathrm{~A}+\mathrm{G}^{2}=27$. The two numbers are
Options:
Solution:
2274 Upvotes
Verified Answer
The correct answer is:
6,3
Let the two numbers be $\mathrm{a}, \mathrm{b}$.
Given, $H . M .=4 \Rightarrow \frac{2 a b}{a+b}=4 \Rightarrow a b=2(a+b)$ ...(i)
Also, given $2 \mathrm{~A}+\mathrm{G}^{2}=27$
$\Rightarrow 2\left(\frac{a+b}{2}\right)+a b=27$
$\Rightarrow \quad a+b+a b=27$
$\Rightarrow a+b+2(a+b)=27 \quad$ (from(i))
$\Rightarrow \quad 3 \mathrm{a}+3 \mathrm{~b}=27$
$\Rightarrow \quad a+b=9$ ...(ii)
From (i), $a b=2(9)=18$ ...(iii)
Solving (ii), (iii) we get
$$
\mathrm{a}=3, \mathrm{~b}=6 \text { or } \mathrm{a}=6, \mathrm{~b}=3
$$
Given, $H . M .=4 \Rightarrow \frac{2 a b}{a+b}=4 \Rightarrow a b=2(a+b)$ ...(i)
Also, given $2 \mathrm{~A}+\mathrm{G}^{2}=27$
$\Rightarrow 2\left(\frac{a+b}{2}\right)+a b=27$
$\Rightarrow \quad a+b+a b=27$
$\Rightarrow a+b+2(a+b)=27 \quad$ (from(i))
$\Rightarrow \quad 3 \mathrm{a}+3 \mathrm{~b}=27$
$\Rightarrow \quad a+b=9$ ...(ii)
From (i), $a b=2(9)=18$ ...(iii)
Solving (ii), (iii) we get
$$
\mathrm{a}=3, \mathrm{~b}=6 \text { or } \mathrm{a}=6, \mathrm{~b}=3
$$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.