Data Sufficiency

Brent@GMATPrepNow wrote:

anandhimanshu85 wrote:IS X^ 2 + Y^2>100

1.2XY<100

2.(X + Y)^2>200

Target Question: Is x^2 + y^2 > 100?

Statement 1: 2xy < 100
The only way to show that this statement is not sufficient is by counter-example.
There are several pairs of values that satisfy the condition that 2xy < 100
case a: x=0 y=0, in which case x^2 + y^2 is less than 100
case b: x=0 y=100, in which case x^2 + y^2 is greater than 100
So, statement 1 is NOT SUFFICIENT

Statement 2: (x+y)^2 > 200
Expand to get: x^2 + 2xy + y^2 > 200
Rearrange to get: x^2 + y^2 + 2xy > 200

Aside: (this is the tricky part)
Notice that for any values of x and y, it must be true that (x-y)^2 > 0
Expand to get x^2 - 2xy + y^2 > 0
Rearrange to get x^2 + y^2 > 2xy
Now add x^2 + y^2 to both sides to get: x^2 + y^2 + x^2 + y^2 > x^2 + y^2 + 2xy
Simplify to get: 2(x^2 + y^2) > x^2 + y^2 + 2xy

Now notice that the right-hand side of this inequality matches the left-hand side of the inequality we derived from statement 2.
So, we can write: 2(x^2 + y^2) > x^2 + y^2 + 2xy > 200
Now ignore the middle part to get: 2(x^2 + y^2) > 200
Divide both sides by 2 to get x^2 + y^2 > 100
Since we can answer the target question with certainty, statement 2 is SUFFICIENT and the answer is B

Cheers,
Brent

GMATGuruNY wrote:I offered a solution (similar to Brent's) in my two posts here:

https://www.beatthegmat.com/x2-y2-t92678.html