UPSC NDA Math Model Paper 9

Let, two numbers be a and b,
We know, AM=

(a+b)

2

and GM=√ab
Given that, GM : AM = n : m
⇒ AM : GM = m : n
⇒

a+b

2√ab

=

m

n

⇒

(a+b)2

4ab

=

m2

n2

……. . (i)
⇒

(a+b)2−4ab

4ab

=

m2−n2

n2

⇒

(a−b)2

4ab

=

m2−n2

n2

………. . (ii)
Since, on dividing eqn (i) and (ii), we get

(a+b)2

(a−b)2

=

m2

m2−n2

⇒

a+b

a−b

=

m

√m2−n2

⇒

(a+b)+(a−b)

(a+b)−(a−b)

=

m+√m2−n2

m−√m2−n2

⇒

2a

2b

=

a

b

=

m+√m2−n2

m−√m2−n2

Hence, option (2) is correct.