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Is |a| > a? (1) a^2 < a (2) (a/2) > (2/a)

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Bunuel
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Is |a| > a? (1) a^2 < a (2) (a/2) > (2/a) [#permalink] New post   09 Mar 2016, 13:25
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A
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E

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65% (hard)

Question Stats:

59% (02:09) correct 41% (02:03) wrong based on 146 sessions
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Is |a| > a?

(1) a^2 < a
(2) (a/2) > (2/a)
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A
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chetan2u
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Re: Is |a| > a? (1) a^2 < a (2) (a/2) > (2/a) [#permalink] New post   09 Mar 2016, 19:47
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Bunuel wrote:
Is |a| > a?

(1) a^2 < a
(2) (a/2) > (2/a)


HI,
A good Q testing our concepts on MOD and Inequalities..

lets see the INFO from Q


Is|a|>a?..
a can be -ive,0 or positive..
when a is positive, |a|=a, so |a| cannot be >a..
when a=0, both sides again will be 0 and equal..
when a is -ive, |a| is -a and will be positive and thus |a|>a..

we are asked about the THIRD scenario, so we have to find if a is -ive...

lets see the statements


(1) a^2 < a
a^2-a<0..
a(a-1)<0..
if a is negative, a(a-1) will become product of two -ive numbers and thus will be +ive..
so the answer is NO for "is |a|>a?"
Suff

(2) (a/2) > (2/a)
\((\frac{a}{2}) - (\frac{2}{a}) > 0\)..
\(\frac{(a^2-4)}{2a}>0\)..
this means both (a^2-4) and 2a are of same sign..
if a is -ive 2a will be negative, and a^2-4 will be negative with a between 0 and -2... SO -ive value possible
if a is +ive 2a will be +ive, and a^2-4 will be +ive with a >2.. SO +ive value possible
Different solutions possible
Insuff

ans A
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Re: Is |a| > a? (1) a^2 < a (2) (a/2) > (2/a) [#permalink] New post   11 Mar 2016, 10:24
Bunuel wrote:
Is |a| > a?

(1) a^2 < a
(2) (a/2) > (2/a)


question basically asking for whether a is negative??
becoz then only I-2I>-2(let a=-2)---------->2>-2

(1) a^2<a
a^2 >=0
then 0=a^2<a means a is non negative.
IaI=a ....
so we have a NO ans and is suff.

(2)a/2>2/a
(a/2)-(2/a)>0
(a+2)(a-2)/2a>0

sol. range will be -2<a<0 and a>2
so a could be -ve or +ve
insuff..

Ans A
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RastogiSarthak99
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Re: Is |a| > a? [#permalink] New post   23 May 2023, 14:52
is |a| > a?

so |a| will always be positive, but here magnitutde of a is important.

--> St 1 says a^2 < a

so a^2 is always positive, and a^2 < a meaning a is always +ve
use a number
a = 0.5, a^2= 0.25

Test out main equation: |0.5| < 0.5
This is a def no.

--> St 2
a/2>2/a

simplifying it we get a^2>2
use a number a^2 = 4. a = +- 2
|-2| < 2 Yes
|2| < 2 No
In suff

A
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Re: Is |a| > a? (1) a^2 < a (2) (a/2) > (2/a) [#permalink] New post   24 May 2023, 21:59
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Bunuel wrote:
Is |a| > a?
(1) a^2 < a
(2) (a/2) > (2/a)

Solution:
Pre Analysis:
  • We are asked if \(|a| > a\) or not
  • Which is true only when a is negative and not when it is positive or 0
  • Thus, we are asked if a is negative number or not

Statement 1: \(a^2 < a\)
  • \(a^2 < a\) is true for the 2nd region of the number line which is \(0<a<1\)
    • You can go through this article for more understanding on number line
  • So, we can be sure that a is not negative
  • Thus, statement 1 alone is sufficient and we can eliminate options B, C and E

Statement 2: \(\frac{a}{2} > \frac{2}{a}\)
    \(⇒\frac{a^2-4}{2a}>0\)
  • 2 cases possible:
    • Case 1: both \(a^2-4<0\) and \(2a<0\) which is true when \(-2<a<0\) which means a is negative
    • Case 2: both \(a^2-4>0\) and \(2a>0\) which is true when \(a>2\) which means a is positive
  • Thus, statement 2 alone is not sufficient

Hence the right answer is Option A
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Re: Is |a| > a? (1) a^2 < a (2) (a/2) > (2/a) [#permalink]
24 May 2023, 21:59
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Bunuel
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