GPREP #14

[hardikrs]

if x and y are positive, which of the following must be greater than 1/sqrt(x+y)

1. sqrt(x+y)/2x

2. (sqrt(x) + sqrt(y))/(x+y)

3. (sqrt(x) - sqrt(y))/(x+y)

1. None
2. 1 only
3. 2 only
4. 1 and 3 only
5. 2 and 3 only.

Please explain how to pick numbers or a easy way to solve this.
thanks,

[3Gs]

only equation 2 is greater...Hence answer is 3...please be noted that x and y are positive so we can cross multiply both sides without changing inequality sign (>)


1. sqrt(x+y)/2x > 1/sqrt(x+y)
x+y > 2x
y>x...doesnt satisfy (satisfies only when y is greater)


2. (sqrt(x) + sqrt(y))/(x+y) > 1/sqrt(x+y)
(sqrt(x) + sqrt(y)) = sqrt(x+y) sqrt(x+y)/sqrt(x+y)
(sqrt(x) + sqrt(y)) = sqrt(x+y)
squaring both sides, x + y + sqrt(2xy) > x+y ...True....Hence Satisfies


3. (sqrt(x) - sqrt(y))/(x+y) > 1/sqrt(x+y)
similar to 2...x + y - sqrt(2xy) > x+y ...Not true...doesnt satisfy