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If a is non-negative, is x^2 + y^2 > 4a ? Updated on: 10 May 2013, 09:26
Originally posted by Marcab on 12 Dec 2012, 06:46.
Last edited by walker on 10 May 2013, 09:26, edited 3 times in total.
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E
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!

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95% (hard)

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33% (02:07) correct 67% (02:15) wrong based on 686 sessions

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If a is non-negative, is x^2 + y^2 > 4a ?

(1) (x + y)^2 = 9a
(2) (x - y)^2 = a
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E
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If a is non-negative, is x^2 + y^2 > 4a ? 12 Dec 2012, 06:55
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Marcab
If a is non-negative, is x^2 + y^2 > 4a ?

(1) (x + y)^2 = 9a
(2) (x - y)^2 = a

Source: Jamboree
I am not convinced with the OA.
Show more

If a is non-negative, is x^2 + y^2 > 4a ?

(1) (x + y)^2 = 9a --> x^2+2xy+y^2=9a. Clearly insufficient.

(2) (x – y)^2 = a --> x^2-2xy+y^2=a. Clearly insufficient.

(1)+(2) Add them up 2(x^2+y^2)=10a --> x^2+y^2=5a. Also insufficient as x, y, and a could be 0 and x^2 + y^2 > 4a won't be true, as LHS and RHS would be in that case equal to zero. Not sufficient.

Answer: E.
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If a is non-negative, is x^2 + y^2 > 4a ? 12 Dec 2012, 07:01
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If a is non-negative, is x^2 + y^2 > 4a ?
(1) (x + y)^2 = 9a
(2) (x - y)^2 = a

I am getting E
A is non negative, so a can be zero or positive.
x=0, y =0, a = 0...satisfies both 1) and 2) but not x^2 + y^2>4a
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If a is non-negative, is x^2 + y^2 > 4a ? 14 Dec 2012, 03:12
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Ans:


both the statements alone don’t give any solution . When we combine them and add we get x^2+y^2=5a which is greater than 4a , but a can be zero( a is non-negative), in that case the answer becomes may be. Therefore the answer is (E).
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If a is non-negative, is x^2 + y^2 > 4a ? 23 May 2013, 23:06
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Bunuel
Marcab
If a is non-negative, is x^2 + y^2 > 4a ?

(1) (x + y)^2 = 9a
(2) (x - y)^2 = a

Source: Jamboree
I am not convinced with the OA.

If a is non-negative, is x^2 + y^2 > 4a ?

(1) (x + y)^2 = 9a --> x^2+2xy+y^2=9a. Clearly insufficient.

(2) (x – y)^2 = a --> x^2-2xy+y^2=a. Clearly insufficient.

(1)+(2) Add them up 2(x^2+y^2)=10a --> x^2+y^2=5a. Also insufficient as x, y, and a could be 0 and x^2 + y^2 > 4a won't be true, as LHS and RHS would be in that case equal to zero. Not sufficient.

Answer: E.
Show more


Hi Bunel,

Can you please explain where am I going wrong:

(x^2+y^2) = x^2+2xy+y^2 = 9a..........(1)

x^2+y^2 >= 2xy ..........(2)

Substitute equation 2 in 1

Thus, (x^2+y^2+x^2+y^2) = 2(x^2+y^2) >= 9a

2(x^2+y^2) >= 9a

Finally, (x^2+y^2) >= 4.5a. Sufficient

(2) (x – y)^2 = a --> x^2-2xy+y^2=a. Clearly insufficient.

Answer: A
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If a is non-negative, is x^2 + y^2 > 4a ? 24 May 2013, 01:03
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Bunuel
Marcab
If a is non-negative, is x^2 + y^2 > 4a ?

(1) (x + y)^2 = 9a
(2) (x - y)^2 = a

Source: Jamboree
I am not convinced with the OA.

If a is non-negative, is x^2 + y^2 > 4a ?

(1) (x + y)^2 = 9a --> x^2+2xy+y^2=9a. Clearly insufficient.

(2) (x – y)^2 = a --> x^2-2xy+y^2=a. Clearly insufficient.

(1)+(2) Add them up 2(x^2+y^2)=10a --> x^2+y^2=5a. Also insufficient as x, y, and a could be 0 and x^2 + y^2 > 4a won't be true, as LHS and RHS would be in that case equal to zero. Not sufficient.

Answer: E.


Hi Bunel,

Can you please explain where am I going wrong:

(x^2+y^2) = x^2+2xy+y^2 = 9a..........(1)

x^2+y^2 >= 2xy ..........(2)

Substitute equation 2 in 1

Thus, (x^2+y^2+x^2+y^2) = 2(x^2+y^2) >= 9a

2(x^2+y^2) >= 9a

Finally, (x^2+y^2) >= 4.5a. Sufficient

(2) (x – y)^2 = a --> x^2-2xy+y^2=a. Clearly insufficient.

Answer: A
Show more

\(x^2+y^2\geq{4.5a}\) does NOT necessarily mean that \(x^2 + y^2 > 4a\). Consider x=y=a=0, in this case \(x^2 + y^2= 4a\).

Hope it's clear.
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If a is non-negative, is x^2 + y^2 > 4a ? 28 May 2013, 15:01
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Man i actually made a quick questimate and could tell within 15seconds that it would not work out. so answer I got was E. I have realised a lot of the gmat questions have the same concept and My question is though is this s dangerous way to appraoch the gmat exam
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If a is non-negative, is x^2 + y^2 > 4a ? 23 Oct 2014, 02:08
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Bunuel
Marcab
If a is non-negative, is x^2 + y^2 > 4a ?

(1) (x + y)^2 = 9a
(2) (x - y)^2 = a

Source: Jamboree
I am not convinced with the OA.

If a is non-negative, is x^2 + y^2 > 4a ?

(1) (x + y)^2 = 9a --> x^2+2xy+y^2=9a. Clearly insufficient.

(2) (x – y)^2 = a --> x^2-2xy+y^2=a. Clearly insufficient.

(1)+(2) Add them up 2(x^2+y^2)=10a --> x^2+y^2=5a. Also insufficient as x, y, and a could be 0 and x^2 + y^2 > 4a won't be true, as LHS and RHS would be in that case equal to zero. Not sufficient.

Answer: E.
Show more

Hi bunuel,
a clarification;
in inequalities that have multiple variables as is the case with this question and without any clear information about the variables i.e if they are negative, non-negative etc etc, the options are always insufficient, right?
the reason i am asking this question is because i approached this question the same way and got E.
nothing is mentioned about the variables and most importantly there is more than 1 variable. I mean the values for these variables could be anything. nothing is explicitly mentioned about them.
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If a is non-negative, is x^2 + y^2 > 4a ? 24 Oct 2014, 03:00
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Bunuel
Marcab
If a is non-negative, is x^2 + y^2 > 4a ?

(1) (x + y)^2 = 9a
(2) (x - y)^2 = a

Source: Jamboree
I am not convinced with the OA.

If a is non-negative, is x^2 + y^2 > 4a ?

(1) (x + y)^2 = 9a --> x^2+2xy+y^2=9a. Clearly insufficient.

(2) (x – y)^2 = a --> x^2-2xy+y^2=a. Clearly insufficient.

(1)+(2) Add them up 2(x^2+y^2)=10a --> x^2+y^2=5a. Also insufficient as x, y, and a could be 0 and x^2 + y^2 > 4a won't be true, as LHS and RHS would be in that case equal to zero. Not sufficient.

Answer: E.

Hi bunuel,
a clarification;
in inequalities that have multiple variables as is the case with this question and without any clear information about the variables i.e if they are negative, non-negative etc etc, the options are always insufficient, right?
the reason i am asking this question is because i approached this question the same way and got E.
nothing is mentioned about the variables and most importantly there is more than 1 variable. I mean the values for these variables could be anything. nothing is explicitly mentioned about them.
Show more

No, that's not correct. We do have some information about the variables: we know that a is non-negative, (x + y)^2 = 9a and (x - y)^2 = a. Yes, this info is not enough to answer whether x^2 + y^2 > 4a but if, for example, the question were whether \(x^2 + y^2 \geq{4a}\), then the answer would be C, not E.
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If a is non-negative, is x^2 + y^2 > 4a ? 27 Sep 2018, 01:14
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Hi Bunuel, thank you for the detailed explanation.
Another question,
If instead of a being non-negative, it was mentioned that 'a is positive' would the answer be A or C?
I may be wrong, but I suspected A similar to a deduction above i.e x^2+y^2 >= 4.5a
What are your thoughts?
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If a is non-negative, is x^2 + y^2 > 4a ? 02 Nov 2018, 15:04
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Bunuel
Marcab
If a is non-negative, is x^2 + y^2 > 4a ?

(1) (x + y)^2 = 9a
(2) (x - y)^2 = a

Source: Jamboree
I am not convinced with the OA.

If a is non-negative, is x^2 + y^2 > 4a ?

(1) (x + y)^2 = 9a --> x^2+2xy+y^2=9a. Clearly insufficient.

(2) (x – y)^2 = a --> x^2-2xy+y^2=a. Clearly insufficient.

(1)+(2) Add them up 2(x^2+y^2)=10a --> x^2+y^2=5a. Also insufficient as x, y, and a could be 0 and x^2 + y^2 > 4a won't be true, as LHS and RHS would be in that case equal to zero. Not sufficient.

Answer: E.
Show more


Would it be "C" , if the statement says a is a positive integer ?
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If a is non-negative, is x^2 + y^2 > 4a ? 03 Nov 2018, 01:38
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Bunuel
Marcab
If a is non-negative, is x^2 + y^2 > 4a ?

(1) (x + y)^2 = 9a
(2) (x - y)^2 = a

Source: Jamboree
I am not convinced with the OA.

If a is non-negative, is x^2 + y^2 > 4a ?

(1) (x + y)^2 = 9a --> x^2+2xy+y^2=9a. Clearly insufficient.

(2) (x – y)^2 = a --> x^2-2xy+y^2=a. Clearly insufficient.

(1)+(2) Add them up 2(x^2+y^2)=10a --> x^2+y^2=5a. Also insufficient as x, y, and a could be 0 and x^2 + y^2 > 4a won't be true, as LHS and RHS would be in that case equal to zero. Not sufficient.

Answer: E.


Would it be "C" , if the statement says a is a positive integer ?
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Yes, because in this case 5a will always be greater than 4a.
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If a is non-negative, is x^2 + y^2 > 4a ? 13 May 2020, 15:52
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If a is non-negative, is x^2 + y^2 > 4a ?

(1) (x + y)^2 = 9a
(2) (x - y)^2 = a

Statement 1: when we expand we get x^2 + y^2 +2xy=9a; x^2 + y^2=9a-2xy.
The question stem becomes is 9a-2xy>4a? Or is 5a>2xy.All we know is that a is
Non negative.x and y can take any number of values. Insufficient

Statement 2 : when we expand we get x^2 + y^2- 2xy=a; x^2 + y^2=a+2xy.
The question stem becomes is a+2xy>4a? Or is 3a<2xy.Again,All we know is that a is
Non negative.x and y can take any number of values. Insufficient


Together, when we add statements 1 and 2,we get 2x^2 + 2y^2 =10a; This equals x^2 + y^2 =5a;
The question stem becomes is 5a>4a or is a>0.However,we know a is non negative meaning it could be positive or 0.Insufficient

Answer is E

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If a is non-negative, is x^2 + y^2 > 4a ? 01 Apr 2025, 21:32
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Hi Bunuel,

I understand that options 1 and 2 are clearly not sufficient, but when we combine the two, we get:

(x+y)^2= 9(x-y)^2

taking their ratio: (x+y)^2/ (x-y)^2= 9/1

Upon taking square root on both sides, we get x+y=3 and x-y=1, which means x=2 and y=1. We can definitely say that x^2 + y^2 is greater than 4a. (since after this we know value of a=1).

Where am i going wrong here?

Bunuel
Marcab
If a is non-negative, is x^2 + y^2 > 4a ?

(1) (x + y)^2 = 9a
(2) (x - y)^2 = a

Source: Jamboree
I am not convinced with the OA.

If a is non-negative, is x^2 + y^2 > 4a ?

(1) (x + y)^2 = 9a --> x^2+2xy+y^2=9a. Clearly insufficient.

(2) (x – y)^2 = a --> x^2-2xy+y^2=a. Clearly insufficient.

(1)+(2) Add them up 2(x^2+y^2)=10a --> x^2+y^2=5a. Also insufficient as x, y, and a could be 0 and x^2 + y^2 > 4a won't be true, as LHS and RHS would be in that case equal to zero. Not sufficient.

Answer: E.
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If a is non-negative, is x^2 + y^2 > 4a ? 02 Apr 2025, 00:05
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Jeeya
Hi Bunuel,

I understand that options 1 and 2 are clearly not sufficient, but when we combine the two, we get:

(x+y)^2= 9(x-y)^2

taking their ratio: (x+y)^2/ (x-y)^2= 9/1

Upon taking square root on both sides, we get x+y=3 and x-y=1, which means x=2 and y=1. We can definitely say that x^2 + y^2 is greater than 4a. (since after this we know value of a=1).

Where am i going wrong here?

Bunuel
Marcab
If a is non-negative, is x^2 + y^2 > 4a ?

(1) (x + y)^2 = 9a
(2) (x - y)^2 = a

Source: Jamboree
I am not convinced with the OA.

If a is non-negative, is x^2 + y^2 > 4a ?

(1) (x + y)^2 = 9a --> x^2+2xy+y^2=9a. Clearly insufficient.

(2) (x – y)^2 = a --> x^2-2xy+y^2=a. Clearly insufficient.

(1)+(2) Add them up 2(x^2+y^2)=10a --> x^2+y^2=5a. Also insufficient as x, y, and a could be 0 and x^2 + y^2 > 4a won't be true, as LHS and RHS would be in that case equal to zero. Not sufficient.

Answer: E.
Show more

Read the solution carefully. x, y, and a could be 0.

First, you can’t divide both sides of (x + y)^2 = 9(x - y)^2 by (x - y)^2 unless you know for sure that (x - y) ≠ 0, which you don’t.

Second, when you take the square root of both sides, you get |x + y| = 3|x - y|, not (x + y) = 3(x - y).

Even if you assume (x + y) = 3(x - y), that doesn’t mean x + y = 3 and x - y = 1. It could just as well be x + y = 9 and x - y = 3. That equation only implies x = 2y, which still doesn't give specific values.

And finally, pure algebraic questions are no longer a part of the DS syllabus of the GMAT.

DS questions in GMAT Focus encompass various types of word problems, such as:

  • Word Problems
  • Work Problems
  • Distance Problems
  • Mixture Problems
  • Percent and Interest Problems
  • Overlapping Sets Problems
  • Statistics Problems
  • Combination and Probability Problems

While these questions may involve or necessitate knowledge of algebra, arithmetic, inequalities, etc., they will always be presented in the form of word problems. You won’t encounter pure "algebra" questions like, "Is x > y?" or "A positive integer n has two prime factors..."

Check GMAT Syllabus for Focus Edition

You can also visit the Data Sufficiency forum and filter questions by OG 2024-2025, GMAT Prep (Focus), and Data Insights Review 2024-2025 sources to see the types of questions currently tested on the GMAT.

So, you can ignore this question.

Hope it helps.­
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