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Is x^2 + y^2 > 3z

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Is x^2 + y^2 > 3z [#permalink] New post 01 Sep 2013, 10:16
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Is x^2 + y^2 > 3z

(1) (x + y)^2 = 9z and (x - y)^2 = z
(2) z = 0
[Reveal] Spoiler: OA
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Re: Is x^2 + y^2 > 3z [#permalink] New post 01 Sep 2013, 10:34
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Is x^2 + y^2 > 3z

(1) (x + y)^2 = 9z and (x - y)^2 = z --> x^2+2xy+y^2=9z and x^2-2xy+y^2=z. Add them up 2(x^2+y^2)=10z --> x^2+y^2=5z. If x=y=z=0, then x^2 + y^2=0=3z and the answer is NO but if x, y, and z are different from zero, then the answer is YES. Not sufficient.

(2) z = 0. If x=y=z=0, then x^2 + y^2=0=3z and the answer is NO but if x or y are different from zero, then the answer is YES. Not sufficient.

(1)+(2) From (1) we have that x^2+y^2=(non \ negative)+(non \ negative)=5z and since from (2) we have that z=0, then x=y=z=0. Thereofre the answer to the question is NO. Sufficient.

Answer: C.
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Re: Is x^2 + y^2 > 3z [#permalink] New post 26 Mar 2014, 02:40
pavan2185 wrote:
Is x^2 + y^2 > 3z

(1) (x + y)^2 = 9z and (x - y)^2 = z
(2) z = 0



1) given x^2+y^2+2xy = 9z --> x^2+y^2 = 9z- 2xy

given x^2+y^2-2xy = z --> x^2+y^2 = z+2xy

so 9z- 2xy = z+2xy --> 8z= 4xy --> z = (xy)/2

question now becomes is x^2 + y^2 > (3xy)/2

if x=y = 0 then answer is yes otherwise it is no

2)z= 0
x=y =z = 0 then yes other wise no

1+2

from 1 we know z= (xy)/2 and from 2 we know z = 0

so 0 = (xy)/2 --> xy = 0 so either x or y or both can be 0

if x=0 and y =0 then answer is no
if either one of x and y is not 0 then the answer is yes

I am getting E as the answer, can anyone tell me where is my error if any ?
Thank you
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Re: Is x^2 + y^2 > 3z [#permalink] New post 26 Mar 2014, 07:13
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qlx wrote:
pavan2185 wrote:
Is x^2 + y^2 > 3z

(1) (x + y)^2 = 9z and (x - y)^2 = z
(2) z = 0


1) given x^2+y^2+2xy = 9z --> x^2+y^2 = 9z- 2xy

given x^2+y^2-2xy = z --> x^2+y^2 = z+2xy

so 9z- 2xy = z+2xy --> 8z= 4xy --> z = (xy)/2

question now becomes is x^2 + y^2 > (3xy)/2

if x=y = 0 then answer is yes otherwise it is no

2)z= 0
x=y =z = 0 then yes other wise no

1+2

from 1 we know z= (xy)/2 and from 2 we know z = 0

so 0 = (xy)/2 --> xy = 0 so either x or y or both can be 0

if x=0 and y =0 then answer is no
if either one of x and y is not 0 then the answer is yes

I am getting E as the answer, can anyone tell me where is my error if any ?
Thank you


The scenario in red is not possible: From (1) we have that x^2+y^2=5z (is-x-2-y-2-3z-158984.html#p1262747) --> x^2+y^2=0. Both x and y must be zero in order that to hold true.

Hope it's clear.
_________________

NEW TO MATH FORUM? PLEASE READ THIS: ALL YOU NEED FOR QUANT!!!

PLEASE READ AND FOLLOW: 11 Rules for Posting!!!

RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis NEW!!!; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) NEW!!!; 12. Tricky questions from previous years. NEW!!!;

COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: Is x^2 + y^2 > 3z   [#permalink] 26 Mar 2014, 07:13
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