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# Is a < 0, if a and b are integers?

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Math Expert
Joined: 02 Aug 2009
Posts: 5537
Is a < 0, if a and b are integers? [#permalink]

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11 Dec 2017, 06:58
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55% (hard)

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56% (01:24) correct 44% (01:20) wrong based on 50 sessions

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Is a < 0, if a and b are integers?

(1) $$2a^2-8b^2>0$$
(2) $$a<2b$$
[Reveal] Spoiler: OA

_________________

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Re: Is a < 0, if a and b are integers? [#permalink]

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11 Dec 2017, 08:22
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Re: Is a < 0, if a and b are integers? [#permalink]

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11 Dec 2017, 08:41
chetan2u wrote:
Is a < 0, if a and b are integers?

(1) $$2a^2-8b^2>0$$
(2) $$a<2b$$

A) a^2-4b^2>0
(a+2b)(a-2b)>0
There are two conditions now
1.
a+2b>0 & a-2b>0 or a>0
or 2.
a+2b<0 & a-2b<0 or a<0. Hence insufficient.

B)
a<2b, a=3 and b=2 satisfies...a>0
a=-10 b=2 still satisfies a<0. Hence insufficient.

Combining A & B,
Since a<2b, condition 2 will satisfy.
a+2b<0 & a-2b<0 or a<0. Hence C.
Intern
Joined: 16 May 2017
Posts: 17
Location: India
Re: Is a < 0, if a and b are integers? [#permalink]

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12 Dec 2017, 14:50
chetan2u wrote:
Is a < 0, if a and b are integers?

(1) $$2a^2-8b^2>0$$
(2) $$a<2b$$

1 Stmt not sufficient
2(a^2-4b^2) > 0
2(a+2b^2)(a-2b^2)>0 Insuff

Sub a<0 will yield negative value

a>0 will yield positive value.

2 stmnt

a<2b with second stmt alone we cannot predict relation of a with 0.

combining 1 & 2 stmt

We will get to know the value of a in relation with b.
hence choose the negative root a=-2b.
hence IMO C
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Joined: 21 Aug 2013
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Re: Is a < 0, if a and b are integers? [#permalink]

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13 Dec 2017, 10:31
chetan2u wrote:
Is a < 0, if a and b are integers?

(1) $$2a^2-8b^2>0$$
(2) $$a<2b$$

(1) 2a^2 - 8b^2 > 0 . Dividing both sides by 2, we get a^2 - 4b^2 > 0 OR (a)^2 - (2b)^2 > 0 OR (a+2b)(a-2b) > 0
This doesnt tell us anything about whether a is positive or negative, so insufficient.

(2) a < 2b Or a-2b < 0. Insufficient on its own.

Combining the two statements, a-2b < 0 But (a+2b)(a-2b) > 0. This is possible only when a+2b is also < 0.
So we have both a-2b<0 and a+2b<0 OR we have a < 2b as well as a < -2b. Whether b is positive or negative, 2b and -2b will have opposite signs. a is less than both so a must be negative. And if b=0, both 2b=-2b=0, thus a<0. So in any case a will be negative only. Sufficient.

Re: Is a < 0, if a and b are integers?   [#permalink] 13 Dec 2017, 10:31
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# Is a < 0, if a and b are integers?

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