rajudantuluri wrote:

Bunuel wrote:

arvindg wrote:

Problem source:

Veritas Practice Test

Is x^2 + y^2 > 100?

(1) 2xy < 100

(2) (x + y)^2 > 200

Is x^2 + y^2 > 100?(1) 2xy < 100 --> clearly insufficient: if \(x=y=0\) then the answer will be NO but if \(x=10\) and \(y=-10\) then the answer will be YES.

(2) (x + y)^2 > 200 --> \(x^2+2xy+y^2>200\). Now, as \((x-y)^2\geq{0}\) (square of any number is more than or equal to zero) then \(x^2+y^2\geq{2xy}\) so we can safely substitute \(2xy\) with \(x^2+y^2\) (as \(x^2+y^2\) is at least as big as \(2xy\) then the inequality will still hold true) --> \(x^2+(x^2+y^2)+y^2>200\) --> \(2(x^2+y^2)>200\) --> \(x^2+y^2>100\). Sufficient.

Answer: B.

Are you sure the OA is C?

Hi Bunuel,

I understand the algebraic solution now but I had a different reasoning when I did this problem.

Let's say (x + y)^2 = 200. In this case x+y would be 10\sqrt{2}. I considered the minimum possibility where x=y which makes each of them 5\sqrt{2}. So x^2+y^2=100.

Since it our question says (x + y)^2 > 200, based on above deduction, I assumed that their (x + y) should definitely be more than 10 \sqrt{2} implying that x^2+y^2 should also be greater than 100.

Is this a reasonable deduction? I honestly didn't even think of any other numbers to prove my case incorrect after this and I know that doesn't stand well with other questions maybe but I was wondering if it was a good method for this question.

rajudantuluri

hi

I have a similar question to Bunuel the great

Question stem:

Is "x^2 + y^2 > 100"

Now taking the minimum values for x and y

x^ + x^2 > 100

or, 2x^2 > 100

Now the question becomes:

Is 2x^2 > 100

statement 1 says:

twice the value, the product of x and y, is less than 100

If maximized, it becomes, 2x^2 < 100

but, we don't know whether the maximum value is less than 100

not sufficient

statement 2 says:

(x + y)^2 > 200

taking minimum values

(x + x)^2 > 200

or (2x)^ 2 > 200

or 4x^2 > 200

reducing by 1/2

2x^2 > 100

sufficient ...

B

But, rajudantuluri, here I have a question, why it has to be assumed that the minimum value of (x + y)^2 will exceed 200 ...?

thanks in advance