NDA 2018 Maths · Mathematics · Sequences and Series · hard

For positive real numbers a and b with a≠ b: 1. (GM)^2=AMtimesHM. 2. The arithmetic mean of any two real numbers is at least their geometric mean.

Single correct option

Consider the following statements for positive real numbers

a

and

b

with

a\ne b

: 1.

(\text{GM})^2=\text{AM}\times\text{HM}

. 2. The arithmetic mean of any two real numbers is at least their geometric mean. Which of the statements given above is/are correct?

Show answer and solution

Answer: (A) 1 only

For

a,b

: AM

=\frac{a+b}2

, HM

=\frac{2ab}{a+b}

, so AM

\times

=ab=(\text{GM})^2

. Statement 1 is true. Statement 2 fails for negative numbers (and the GM may not even be real); e.g.

-1

and

-9

have AM

-5

but GM

3

. The AM-GM inequality needs non-negative numbers.

Original question written in the NDA 2018 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake