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which of the following must be greater

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which of the following must be greater [#permalink] New post 23 Mar 2011, 03:55
00:00
A
B
C
D
E

Difficulty:

(N/A)

Question Stats:

50% (00:00) correct 50% (00:00) wrong based on 4 sessions
If x and y are positive, which of the following must be greater that 1/sqrt(x + y)?

I. sqrt(x + y)/2x
II. [sqrt(x) + sqrt(y)]/(x + y)
III. [sqrt(x) - sqrt(y)]/(x + y)

a) None
b) I only
c) II only
d) I and II
e) II and III

OA is provided.

Is there an easier to solve this than using numbers.
[Reveal] Spoiler: OA
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Re: which of the following must be greater [#permalink] New post 23 Mar 2011, 04:24
nikhilsrl wrote:
If x and y are positive, which of the following must be greater that 1/sqrt(x + y)?

I. sqrt(x + y)/2x
II. [sqrt(x) + sqrt(y)]/(x + y)
III. [sqrt(x) - sqrt(y)]/(x + y)

a) None
b) I only
c) II only
d) I and II
e) II and III

OA is provided.

Is there an easier to solve this than using numbers.


Looks like "C" to me. Not too sure though.

I. sqrt(x + y)/2x > 1/sqrt(x + y)
x+y > 2x

This is true only for y>x. What if y=x or y<x;
Say; x=2, y=1
3 < 4
Say x=2,y=2
4=4
Proved wrong.

II. [sqrt(x) + sqrt(y)]/(x + y)
[sqrt(x) + sqrt(y)]/(x + y) > 1/sqrt(x+y)
[sqrt(x) + sqrt(y)] > sqrt(x+y)

x=0.25; y=0.25

[sqrt(x) + sqrt(y)] = 0.5+0.5=1
sqrt(x+y) = sqrt(0.25+0.25) = some number between 0 and 1

x=9; y=9
[sqrt(x) + sqrt(y)] = 3+3=6
sqrt(x+y) = sqrt(9+9) = some number between 4 and 5

True.

III. [sqrt(x) - sqrt(y)]/(x + y)

This is simple. x=y; this become 0.

RHS will be some +ve number.

Not true.

Ans: "C"
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Re: which of the following must be greater [#permalink] New post 23 Mar 2011, 08:23
solving i ) Y > X ( which could be true or false)
sloving ii) 2 Sqrt(xy) > 0 ( which is true-given x and y are positive)
solving iii) -2 Squrt(xy) > 0 ( which is false)
Therefore C should be the answer.
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Re: which of the following must be greater [#permalink] New post 23 Mar 2011, 08:26
Each equation can be solved by using > sign against the 1/sqrt(x + y)
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Re: which of the following must be greater [#permalink] New post 23 Mar 2011, 22:13
let x=y=1
then only C is correct.
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Re: which of the following must be greater   [#permalink] 23 Mar 2011, 22:13
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