Author 
Message 
TAGS:

Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 43896

2
This post received KUDOS
Expert's post
2
This post was BOOKMARKED
Question Stats:
30% (01:09) correct 70% (01:29) wrong based on 27 sessions
HideShow timer Statistics



Math Expert
Joined: 02 Sep 2009
Posts: 43896

Re M3207 [#permalink]
Show Tags
17 Jul 2017, 03:30
2
This post received KUDOS
Expert's post
2
This post was BOOKMARKED
Official Solution:If \(x\) and \(y\) are integers and \(x \leq y \leq x\), does \(\sqrt{x^2  y^2} = x + y\)? This is a hard question. You should pay attention to every detail and read the solution very carefully First of all, \(x \leq y \leq x\) ensures two things: 1. \(x^2y^2\geq 0\), so the square root of this number will be defined. 2. \(x+y\geq 0\), so the square root won't be equal to negative number. Next, \(x \leq x\) implies that \(x \geq 0\). And finally, before moving to the statements, let's rephrase the question: Does \(\sqrt{x^2  y^2} = x + y\)? Square both sides: does \(x^2  y^2 = x^2+2xy + y^2\)? Does \(xy+y^2=0\)? Notice here that we cannot reduce this by \(y\), because we'll loose a possible root: \(y=0\). Does \(y(x+y)=0\)? Does \(y=0\) or \(x=y\)? (1) \(xy\) is NOT a square of an integer. If \(y = 0\) were true, then \(xy\) would be 0, which is a square of an integer. If \(x = y\) were true, then \(xy\) would be \(y^2\), which is a square of an integer (since we are told that \(y\) is an integer). Therefore, since we are told that \(xy\) is NOT a square of an integer, then neither \(y=0\) nor \(x=y\) is true. Sufficient. (2) Point \((x, y)\) is above xaxis If \(y = 0\) were true, then point \((x, y)\) would be ON the xaxis. If \(x = y\) were true, then then point \((x, y)\) would be \((x, x)\), so (nonnegative, nonpositive), which would mean that it's either on xaxis or below it. Therefore, since we are told that point \((x, y)\) is above xaxis, then neither \(y=0\) nor \(x=y\) is true. Sufficient. Answer: D
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
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.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



Manager
Joined: 02 Jun 2015
Posts: 192
Location: Ghana

Re: M3207 [#permalink]
Show Tags
17 Jul 2017, 05:23
Bunuel wrote: Official Solution:
If \(x\) and \(y\) are integers and \(x \leq y \leq x\), does \(\sqrt{x^2  y^2} = x + y\)?
This is a hard question. You should pay attention to every detail and read the solution very carefully
First of all, \(x \leq y \leq x\) ensures two things: 1. \(x^2y^2\geq 0\), so the square root of this number will be defined. 2. \(x+y\geq 0\), so the square root won't be equal to negative number.
Next, \(x \leq x\) implies that \(x \geq 0\).
And finally, before moving to the statements, let's rephrase the question: Does \(\sqrt{x^2  y^2} = x + y\)? Square both sides: does \(x^2  y^2 = x^2+2xy + y^2\)? Does \(xy+y^2=0\)? Notice here that we cannot reduce this by \(y\), because we'll loose a possible root: \(y=0\). Does \(y(x+y)=0\)? Does \(y=0\) or \(x=y\)?
(1) \(xy\) is NOT a square of an integer.
If \(y = 0\) were true, then \(xy\) would be 0, which is a square of an integer. If \(x = y\) were true, then \(xy\) would be \(y^2\), which is a square of an integer (since we are told that \(y\) is an integer).
Therefore, since we are told that \(xy\) is NOT a square of an integer, then neither \(y=0\) nor \(x=y\) is true. Sufficient.
(2) Point \((x, y)\) is above xaxis
If \(y = 0\) were true, then point \((x, y)\) would be ON the xaxis. If \(x = y\) were true, then then point \((x, y)\) would be \((x, x)\), so (nonnegative, nonpositive), which would mean that it's either on xaxis or below it.
Therefore, since we are told that point \((x, y)\) is above xaxis, then neither \(y=0\) nor \(x=y\) is true. Sufficient.
Answer: D Hi Bunuel, Can we say that x = y is the same as x = y in this case? And if that is the case, will that affect the fact that statement (2) is sufficient? Thanks
_________________
Kindly press kudos if you find my post helpful



Math Expert
Joined: 02 Sep 2009
Posts: 43896

Re: M3207 [#permalink]
Show Tags
17 Jul 2017, 05:35
duahsolo wrote: Bunuel wrote: Official Solution:
If \(x\) and \(y\) are integers and \(x \leq y \leq x\), does \(\sqrt{x^2  y^2} = x + y\)?
This is a hard question. You should pay attention to every detail and read the solution very carefully
First of all, \(x \leq y \leq x\) ensures two things: 1. \(x^2y^2\geq 0\), so the square root of this number will be defined. 2. \(x+y\geq 0\), so the square root won't be equal to negative number.
Next, \(x \leq x\) implies that \(x \geq 0\).
And finally, before moving to the statements, let's rephrase the question: Does \(\sqrt{x^2  y^2} = x + y\)? Square both sides: does \(x^2  y^2 = x^2+2xy + y^2\)? Does \(xy+y^2=0\)? Notice here that we cannot reduce this by \(y\), because we'll loose a possible root: \(y=0\). Does \(y(x+y)=0\)? Does \(y=0\) or \(x=y\)?
(1) \(xy\) is NOT a square of an integer.
If \(y = 0\) were true, then \(xy\) would be 0, which is a square of an integer. If \(x = y\) were true, then \(xy\) would be \(y^2\), which is a square of an integer (since we are told that \(y\) is an integer).
Therefore, since we are told that \(xy\) is NOT a square of an integer, then neither \(y=0\) nor \(x=y\) is true. Sufficient.
(2) Point \((x, y)\) is above xaxis
If \(y = 0\) were true, then point \((x, y)\) would be ON the xaxis. If \(x = y\) were true, then then point \((x, y)\) would be \((x, x)\), so (nonnegative, nonpositive), which would mean that it's either on xaxis or below it.
Therefore, since we are told that point \((x, y)\) is above xaxis, then neither \(y=0\) nor \(x=y\) is true. Sufficient.
Answer: D Hi Bunuel, Can we say that x = y is the same as x = y in this case? And if that is the case, will that affect the fact that statement (2) is sufficient? Thanks x = y is ALWAYS the same as x = y. But this won't affect the answer for (2) because we know from the stem that \(x \geq 0\). So, \((x, x)\), will be (nonnegative, nonpositive), which would mean that it's either on xaxis or below it.
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
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.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



Intern
Joined: 23 Feb 2017
Posts: 37

Re: M3207 [#permalink]
Show Tags
17 Jul 2017, 19:13
Hi, question :sqrt(x 2 −y 2) = x+y? squaring on both sides, we get x2y2 = (x+y)^2 (x+y)(xy) = (x+y)^2 => (xy) = (x+y) => y= y?
1.xy is not square of an integer. => I don't know how this information is useful to get to stem. 2. point(x,y) is above x  axis, means we have a value of y, so, answer to problem statement is No, = > S
Can someone help me out here.. atleast is my statement y= y? is correct?



Math Expert
Joined: 02 Sep 2009
Posts: 43896

Re: M3207 [#permalink]
Show Tags
17 Jul 2017, 20:39
sasidharrs wrote: Hi, question :sqrt(x 2 −y 2) = x+y? squaring on both sides, we get x2y2 = (x+y)^2 (x+y)(xy) = (x+y)^2 => (xy) = (x+y) => y= y?
1.xy is not square of an integer. => I don't know how this information is useful to get to stem. 2. point(x,y) is above x  axis, means we have a value of y, so, answer to problem statement is No, = > S
Can someone help me out here.. atleast is my statement y= y? is correct? Unfortunately nothing is correct. You cannot reduce (x + y)(x  y) = (x + y)^2 by x + y, because x + y can be 0, and we cannot divide by 0. By doing so you are loosing a root, namely x + y = 0, or which is the same x = y. Never reduce equation by variable (or expression with variable), if you are not certain that variable (or expression with variable) doesn't equal to zero. We cannot divide by zero. It seems that you did not read the solution above. Again, this is a hard question. You should pay attention to every detail and read the solution very carefully.
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
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.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



Manager
Joined: 02 Nov 2015
Posts: 174

Re: M3207 [#permalink]
Show Tags
18 Jul 2017, 07:49
buenel , can we expect questions of this difficulty in real GMAT ?? just wanted to know. btw very tough.



Math Expert
Joined: 02 Sep 2009
Posts: 43896

Re: M3207 [#permalink]
Show Tags
18 Jul 2017, 07:59



Intern
Joined: 06 Apr 2017
Posts: 27

Re: M3207 [#permalink]
Show Tags
21 Jul 2017, 16:35
For statement 2)
\((x,y)=(1,1)\), which satisfies the question stem, leads to \(1(11)=0\)
\((x,y)=(1,1)\), which satisfies the question stem, leads to \(1(1+1)=2\)
I've read the solution several times, but I still see this a proving insufficiency. Any clarification is appreciated!



Math Expert
Joined: 02 Sep 2009
Posts: 43896

Re: M3207 [#permalink]
Show Tags
22 Jul 2017, 01:10
spence11 wrote: If \(x\) and \(y\) are integers and \(x \leq y \leq x\), does \(\sqrt{x^2  y^2} = x + y\)?
This is a hard question. You should pay attention to every detail and read the solution very carefully
First of all, \(x \leq y \leq x\) ensures two things: 1. \(x^2y^2\geq 0\), so the square root of this number will be defined. 2. \(x+y\geq 0\), so the square root won't be equal to negative number.
Next, \(x \leq x\) implies that \(x \geq 0\).
And finally, before moving to the statements, let's rephrase the question: Does \(\sqrt{x^2  y^2} = x + y\)? Square both sides: does \(x^2  y^2 = x^2+2xy + y^2\)? Does \(xy+y^2=0\)? Notice here that we cannot reduce this by \(y\), because we'll loose a possible root: \(y=0\). Does \(y(x+y)=0\)? Does \(y=0\) or \(x=y\)?
(1) \(xy\) is NOT a square of an integer.
If \(y = 0\) were true, then \(xy\) would be 0, which is a square of an integer. If \(x = y\) were true, then \(xy\) would be \(y^2\), which is a square of an integer (since we are told that \(y\) is an integer).
Therefore, since we are told that \(xy\) is NOT a square of an integer, then neither \(y=0\) nor \(x=y\) is true. Sufficient.
(2) Point \((x, y)\) is above xaxis
If \(y = 0\) were true, then point \((x, y)\) would be ON the xaxis. If \(x = y\) were true, then then point \((x, y)\) would be \((x, x)\), so (nonnegative, nonpositive), which would mean that it's either on xaxis or below it.
Therefore, since we are told that point \((x, y)\) is above xaxis, then neither \(y=0\) nor \(x=y\) is true. Sufficient.
Answer: D
For statement 2)
(x,y)=(1,1), which satisfies the question stem, leads to \(1(11)=0\)
\((x,y)=(1,1)\), which satisfies the question stem, leads to \(1(1+1)=2\)
I've read the solution several times, but I still see this a proving insufficiency. Any clarification is appreciated! Please reread again. This time paying attention to the highlighted part.
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
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.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



Intern
Joined: 06 Apr 2017
Posts: 27

Re: M3207 [#permalink]
Show Tags
22 Jul 2017, 06:18
Bunuel,
Thanks  I see it now. I'm a little bit weak at simplifying compound inequalities I suppose. Back to the books!
SR
Posted from my mobile device



Intern
Joined: 04 Aug 2014
Posts: 33
GMAT 1: 620 Q44 V31 GMAT 2: 620 Q47 V28
GPA: 3.2

Re: M3207 [#permalink]
Show Tags
03 Aug 2017, 01:09
hi bb
can we re write the question stem as does x + y = x^2  y^2 ?










