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anandhimanshu85: Junior | Next Rank: 30 Posts; Posts: 17; Joined: Mon Aug 23, 2010 8:52 am; Thanked: 1 times

Veritas Confusing Question

by anandhimanshu85 » Sat Feb 11, 2012 4:54 am

IS X^ 2 + Y^2>100

1.2XY<100

2.(X + Y)^2>200

IMO OA [spoiler]= B as in order to have second statement true a+b should be atleast 15 and any sum for x and y to 15 have their sum of the square greater than 100 but OA was C.Please suggest [/spoiler]

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Source: Beat The GMAT — Data Sufficiency | Sat Feb 11, 2012 4:54 am

GMAT/MBA Expert

Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pm; Location: Vancouver, BC; Thanked: 5254 times; Followed by:1268 members; GMAT Score:770

by Brent@GMATPrepNow » Sat Feb 11, 2012 7:32 am

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

Brent Hanneson - Creator of GMATPrepNow.com

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anandhimanshu85: Junior | Next Rank: 30 Posts; Posts: 17; Joined: Mon Aug 23, 2010 8:52 am; Thanked: 1 times

by anandhimanshu85 » Sat Feb 11, 2012 11:07 am

Thanks Brent, I also opted for a B but I was surprised to see a C as the Answer, so thought to verify here and you have confirmed it I was right...

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

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pm; Location: New York, NY; Thanked: 13060 times; Followed by:1906 members; GMAT Score:790

by GMATGuruNY » Sat Feb 11, 2012 1:42 pm

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

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

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Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.

As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.

For more information, please email me (Mitch Hunt) at [email protected].
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[email protected]: Legendary Member; Posts: 934; Joined: Tue Nov 09, 2010 5:16 am; Location: AAMCHI MUMBAI LOCAL; Thanked: 63 times; Followed by:14 members

by [email protected] » Wed Feb 15, 2012 11:39 pm

Hi Brent and Mitch!!! I solved the question in a different way... I hope its right...

IS X^ 2 + Y^2>100

1.2XY<100

2.(X + Y)^2>200

Statement 1: xy < 50

therefore when x = 5 and y = 6 yes it comes out to be 25 and 36.

But when when x = 9 and y = 5; xy = 45 but x^2 + y^2 = 81 + 25 which turns out to be greater than 100...

Hence statement 1 is insufficient...

STATEMENT 2:

(X + Y)^2 > 200

NOW lets us solve this equation first...

taking sq root on both sides,

(X + Y) > 10sqrt root of 2 or (X + Y) < -10 sqr root of 2

basically it comes out to be approximately + or - 15

Hence either (X + Y) > 15 or (X + Y) < -15

In both the cases take as many values as possible,

the answer comes out to be (X)^2 + Y^2 > 100

Hence statement 2 is sufficient...

Hope this really helped...... took a lot of time to figure out this question...

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GMAT/MBA Expert

Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pm; Location: Vancouver, BC; Thanked: 5254 times; Followed by:1268 members; GMAT Score:770

by Brent@GMATPrepNow » Thu Feb 16, 2012 6:47 am

[email protected] wrote:Hi Brent and Mitch!!! I solved the question in a different way... I hope its right...

IS X^ 2 + Y^2>100

1.2XY<100

2.(X + Y)^2>200

Statement 1: xy < 50

therefore when x = 5 and y = 6 yes it comes out to be 25 and 36.

But when when x = 9 and y = 5; xy = 45 but x^2 + y^2 = 81 + 25 which turns out to be greater than 100...

Hence statement 1 is insufficient...

STATEMENT 2:

(X + Y)^2 > 200

NOW lets us solve this equation first...

taking sq root on both sides,

(X + Y) > 10sqrt root of 2 or (X + Y) < -10 sqr root of 2

basically it comes out to be approximately + or - 15

Hence either (X + Y) > 15 or (X + Y) < -15

In both the cases take as many values as possible,

the answer comes out to be (X)^2 + Y^2 > 100

Hence statement 2 is sufficient...

Hope this really helped...... took a lot of time to figure out this question...

Looking for contradictory sets of values (as you did with statement 1) is a great way to show insufficiency. Unfortunately, you really can't do it to prove sufficiency.

There's a problem with the part in blue above.

To be certain that statement 2 is sufficient, it's not enough to say "In both the cases take as many values as possible, the answer comes out to be X^2 + Y^2 > 100"

Perhaps it's possible that out of the infinite possibilities out there, there's one pair of values for x and y such that X^2 + Y^2 is not greater than 100. We won't know for sure unless we try every pair of values.

That said, this method is useful for getting a general feeling about whether a statement is sufficient. For example, if you plug in several pairs of numbers and, each time, X^2 + Y^2 > 100, then you might feel confident that the statement is sufficient. However, you can't be absolutely certain of your answer.

I cover this in greater detail at the end of this video: https://www.beatthegmat.com/mba/2010/10/ ... ble-method

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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ArunangsuSahu: Master | Next Rank: 500 Posts; Posts: 382; Joined: Thu Mar 31, 2011 5:47 pm; Thanked: 15 times

by ArunangsuSahu » Fri Feb 17, 2012 12:18 pm

Statement 2:
Let's take x=y and
(x+y)^2=(10*2^1/2)^2

x=5*2^1/2=y that's the least

and x^2+y^2=100

So for all other cases for (x+y)^2>200
x^2+y^2>100

(B)..SUFFICIENT

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