UPSC NDA Math Model Paper 5

Consider, the two numbers are x and y.
Given, the arithmetic mean and geometric mean of the x and y is A and G.
⇒A=

x+y

2

.......(1)
⇒G2=xy ......(2)
The harmonic mean of two number x and y is 4.
⇒

2xy

x+y

=4
⇒2xy=4(x+y)
⇒xy=2(x+y)
⇒G2=4A(∵x+y=2A)
⇒G2=4A .........(3)
Given, Their arithmetic mean A and the geometric mean G satisfy the relation 2A+G2=27.
⇒2A+G2=27
⇒6A=27
⇒A=

9

2

From equation (1), (2) and (3), we have
x + y = 9 and xy = 18
⇒ x = 6 and y = 3
Hence, the harmonic mean of two number is 4, Their arithmetic mean A and the geometric mean G satisfy the relation 2A+G2=27 , then the two numbers are 6 and 3.