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# If a < b , is a > 0 ?

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Director
Joined: 09 Mar 2018
Posts: 995
Location: India
If a < b , is a > 0 ?  [#permalink]

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01 Jan 2019, 21:13
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Difficulty:

65% (hard)

Question Stats:

54% (01:49) correct 46% (01:57) wrong based on 49 sessions

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If a < b , is a > 0 ?

(1) $$a^2 < b^2$$
(2) $$a^2 < ab < b^2$$

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Manager
Joined: 01 May 2017
Posts: 82
Location: India
If a < b , is a > 0 ?  [#permalink]

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02 Jan 2019, 00:55
1
If a < b , is a > 0 ?

(1) $$a^2<b^2$$

case(i)
Given a<b,
So, if a < 0 then $$a^2 > b^2$$
example a = -2 and b = -1, where a<b
$$a^2$$ =4
$$b^2$$ = 1
$$a^2 > b^2$$

But this is contradiction to (1) a^2<b^2 so a is non-negative

case (ii) if a = 0 or a lies between 0 to 1
then b>0 as b>a, which in turn implies $$b^2 > a^2$$

case (ii) a>0 (1) satisfies

for a = 0 it satisfies so I can not strictly say a>0

Insufficient

(2) $$a^2<ab<b^2$$
$$a^2 < ab$$
$$a^2-ab$$ < 0
a*(a-b)<0 ---- equation 1
given a < b => (a-b) < 0

since a-b < 0 => a >0 in equation 1 [as (-ve)*(+ve) < 0]
Sufficient

Option B is correct
GMAT Tutor
Joined: 07 Nov 2016
Posts: 56
GMAT 1: 760 Q51 V42
Re: If a < b , is a > 0 ?  [#permalink]

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02 Jan 2019, 04:51
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KanishkM wrote:
If a < b , is a > 0 ?

(1) $$a^2 < b^2$$
(2) $$a^2 < ab < b^2$$

St 1 :

a < b

$$a^2 < b^2$$

Case 1 : a = -2 , b = 3 No

Case 2 : a = 3 , b = 4 Yes

Not Sufficient

St 2 :

a < b .... Ineq1

$$a^2 <ab$$

i.e a * a < a * b

We have multiplied both sides of Ineq1 by a and the sign has not changed. So, a > 0

Sufficient

Choice B
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Posts: 7971
Re: If a < b , is a > 0 ?  [#permalink]

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02 Jan 2019, 23:00
If a < b , is a > 0 ?

(1) $$a^2 < b^2$$..
$$a^2<b^2$$ means |a|<|b|..
Let us see what |a|<|b| tells us
There can be two cases when |a|<|b| ..
a) b is positive.... then b>a
b) b is negative...then b<a..
These hold true irrespective of value of a

It is given b>a ... Therefore, we can just say b is positive and nothing about a..
example.. a=2 and b=4, OR a=-2 and b=4..
Insufficient

(2) $$a^2 < ab < b^2$$
$$a^2<ab......ab-a^2>0....a(b-a)>0$$
Now, we know b-a>0 as b>a, therefore a>0...
Since positive*positive>positive
Sufficient

B
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Re: If a < b , is a > 0 ?   [#permalink] 02 Jan 2019, 23:00
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