Statement 1 If a and b be two positive numbers, where `agtb` and `4xxGM=5xxHM` for the numbers. Then, `a=4b`.
Statement 2 `(AM)(HM)=(GM)^(2)` is true for positive numbers.

Options

A. Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1
B. Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1
C. Statement 1 is true, Statement 2 is false
D. Statement 1 is false, Statement 2 is true

Answer

C

Explanation

`:.A=(a+b)/(2),G=sqrt(ab) " and " H=(2ab)/(a+b)`
Given, `4G=5H" " "……(i)"`
and `G^(2)=AH`
`:. H=(G^(2)))/(A)" " "……(ii)"`
From Eqs. (i) and (ii), we get
`4G=(5G^(2))/(A) implies 4A=5G`
`implies 2(a+b)=5sqrt(ab)`
`implies4(a^(2)+b^(2)+2ab)=25ab`
`implies 4a^(2)-17ab+4b^(2)=0`
`implies (a-4b)(4a-b)=0`
`a=4b,4a-b ne 0" " [:.agtb]`
`:.` Statement 1 is true.
Statement 2 is true only for two numbers, if numbers more than two, then this formula`(AM)(HM)=(GM)^(2)` is true, if numbers are in GP.
Statement 2 is false for positive numbers.