KVPY Sample Paper KVPY Stream-SX Model Paper-30

Let n be number of ordered pairs (x, y) of integers such that \[0<x<y<100\] and A.M. of x and y exceeds their GM by 2. n is equal to

A) 5

B) 7

C) 9

D) 8

Correct Answer: B

Solution :

\[\frac{x+y}{2}=\sqrt{xy}+2\] \[\Rightarrow {{(\sqrt{y}-\sqrt{x})}^{2}}=4\]\[\Rightarrow \sqrt{y}-\sqrt{x}=2\]

Possible ordered pairs = \[({{1}^{2}},{{3}^{2}}),({{2}^{2}},{{4}^{2}}),\,....({{7}^{2}},{{9}^{2}})\]

Total ordered pairs = 7