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If a × b is a positive number & b^2 > ab, then which of the

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archana21
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If a × b is a positive number & b^2 > ab, then which of the [#permalink] New post   16 Jul 2021, 09:38
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If a × b is a positive number and b^2 > ab, then which of the following can be true?

I. a^2 < b^2 < a < b

II. a < b < a^2 < b^2

III. b < a < a^2 < b^2

A) I only
B) II only
C) III only
D) I & III only
E) I, II & III
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E
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chetan2u
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Re: If a × b is a positive number & b^2 > ab, then which of the [#permalink] New post   16 Jul 2021, 11:14
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archana21 wrote:
If a × b is a positive number and b^2 > ab, then which of the following can be true?

I. a^2 < b^2 < a < b

II. a < b < a^2 < b^2

III. b < a < a^2 < b^2

A) I only
B) II only
C) III only
D) I & III only
E) I, II & III



a*b is positive => Both a and b have same sign.
\(b^2>ab……….b^2-ab>0……….b(b-a)>0\)
This means if b>0, then b-a>0……b>a>0…(i)
If b<0, then b-a<0 or b<a………b<a<0…(ii)

In both the above cases, we can say that |b|>|a|

I. \(a^2 < b^2 < a < b\)…… This will be the case (i), when a and b are between 0 and 1
a=1/3, b=1/2
\((\frac{1}{3})^2< (\frac{1}{2})^2<\frac{1}{3}<\frac{1}{2}\)
Possible

II. \(a < b < a^2 < b^2\) …… This will be the case (i), when a and b are > 1
a=2, b=3
\(2< 3<2^2<3^2\)
Possible

III. \(b < a < a^2 < b^2\) …… This will be the case (ii), when a and b are <-1
a=-2, b=-3
\(-3< -2<(-2)^2<(-3)^2\)
Possible


All three possible.


E
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Re: If a × b is a positive number & b^2 > ab, then which of the [#permalink] New post   16 Jul 2021, 11:21
Question has asked which of the following can be true
So test the cases with
B,A
1/2,1/3 case 1
3,2 case 2
-4,-2 case 3
Option E is correct

archana21 wrote:
If a × b is a positive number and b^2 > ab, then which of the following can be true?

I. a^2 < b^2 < a < b

II. a < b < a^2 < b^2

III. b < a < a^2 < b^2

A) I only
B) II only
C) III only
D) I & III only
E) I, II & III


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archana21
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Re: If a × b is a positive number & b^2 > ab, then which of the [#permalink] New post   16 Jul 2021, 13:17
chetan2u wrote:
archana21 wrote:
If a × b is a positive number and b^2 > ab, then which of the following can be true?

I. a^2 < b^2 < a < b

II. a < b < a^2 < b^2

III. b < a < a^2 < b^2

A) I only
B) II only
C) III only
D) I & III only
E) I, II & III



a*b is positive => Both a and b have same sign.
\(b^2>ab……….b^2-ab>0……….b(b-a)>0\)
This means if b>0, then b-a>0……b>a>0…(i)
If b<0, then b-a<0 or b<a………b<a<0…(ii)

In both the above cases, we can say that |b|>|a|

I. \(a^2 < b^2 < a < b\)…… This will be the case (i), when a and b are between 0 and 1
a=1/3, b=1/2
\((\frac{1}{3})^2< (\frac{1}{2})^2<\frac{1}{3}<\frac{1}{2}\)
Possible

II. \(a < b < a^2 < b^2\) …… This will be the case (i), when a and b are > 1
a=2, b=3
\(2< 3<2^2<3^2\)
Possible

III. \(b < a < a^2 < b^2\) …… This will be the case (ii), when a and b are <-1
a=-2, b=-3
\(-3< -2<(-2)^2<(-3)^2\)
Possible


All three possible.


E


If b(b-a)>0 --> for "b", this only means that b >0, right? Could you please explain how you derived that b<0, given that b(b-a)>0?
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Re: If a × b is a positive number & b^2 > ab, then which of the [#permalink] New post   25 Feb 2023, 01:17
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Re: If a × b is a positive number & b^2 > ab, then which of the [#permalink]
25 Feb 2023, 01:17
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