JEE Main 2014 · 12 Apr · Shift 1 · Mathematics Q68

← JEE Main 2014 · 12 Apr · Shift 1

MathematicsSingle correct+4 / -1Question 68 of 90

Let G be the geometric mean of two positive numbers a and b, and M be the arithmetic mean of $\dfrac1a$ and $\dfrac1b$ . If $\dfrac1M:G$ is 4 : 5, then a : b can be :

1 : 4

1 : 2

2 : 3

3 : 4

Answer and worked solution

Official answer: A

$\frac1M=\frac{2ab}{a+b}$ (the HM). $\frac{2ab}{(a+b)\sqrt{ab}}=\frac45$ gives $\frac{2\sqrt{ab}}{a+b}=\frac45$ .

With $t=\sqrt{a/b}$ : $2t^2-5t+2=0$ , so $t=\frac12$ and $a:b=1:4$ .

Topics:AM-GM-HM inequalities

← Previous question Next question →

JEE Main 2014 · 12 Apr · Shift 1, Mathematics question 68