stmt 1: |a| = b => b >= 0. INSUFFICIENT as we cannot say if a^2*b >0, it can be equal to 0.

stmt 2: ab<0. INSUFFICIENT.

Either a<0 or b<0. If a<0 a^2*b >0 and if b<0, a^2*b <0.
combining stmt 1 and 2. b>0 and a<0.
Since, a^2 >0, a^2*b > 0. SUFFICIENT.

Therefore, answer: C
OA?

if abs(a)=b then b is always >0
Answer would be A if a and b were not 0
But we never know
So (1) is not suff
(2) is not suff neither

Therefore C

According to me, the answer is C.

1) |a| = b, which means that b should be positive.
2) says that one of a or b should be negative.

From 1 we know that "b" is positive hence "a" should be negative and we know that a^2*b < 0.

1. Insufficient: because a can be either 0 or any other number (+ve or -ve)
2. Insufficient: because ab < 0 gives us that one of them is -ve but we don't know which.

Combined: Sufficient: because from 1 & 2, b can't be -ve so a should be -ve and so a^2*b is > 0
It is C.

1: insufficient a = b = 1 or a = b = 0 (b >=0)
2: insufficient a = 1, b = -1 or a = -1, b = 1 (a, b are not equal to 0)

1&2: since b >= 0 and b is not equal to zero -> b > 0, a is not equal to 0 -> a^2*b > 0 -> C

With respect to 1 & 2...You're saying neither A nor B can = 0 because anything * 0 = 0 and we're told in 2) that ab <0 (which is not zero). Then with 1, b = |a| and the absolute value of A cannot be nagative. So we know from the 2 statements that B is not 0, and it is not negative. That makes a be negative and not zero (not zero from #2).

Thanks. That helps me out a bunch. +1 for you.

1: a can be 0. Insuff.

2: b could be -. INsuff.

Together, suff b/c a is not 0.

Q. Is a^2*b > 0?
1) |a|=b
2) ab<0

From 1)

b must be possitive. a's value doesn't matter because a^2 is anyways going to be positive. So far 1) is suff but ... we dont know if a is a non zero or not. hence 1) is not suff.

From 2)

either a or b has to be negative. And neither a or b is zero.
if a is -ve and b is +ve then a^2*b > 0
if a is +ve and b is -ve then a^2*b < 0

therefore not sufficient

TOGETHER:

a and b not equal to zero.
b is +ve
a is -ve

Sufficient.

ANSWER: C

Allen, please help me understand this :
a^2*b = (a^2)* b ?
or
a^2*b = a^(2* b) ?

1. If a^(2b) then answer is C

2. If (a^2)b then answer is E

Please tell us which is the case - 1 or 2

I've decided that when I get a DS like this, I'm going to consider a few numbers. I'm going to start with the following set:

{-2, -1, -0.5, 0, 0.5, 1, 2}

That should give an good indication as to how the inequality acts with both + and - numbers.

C

1. a>0,b>0 yes
a<0,b> 0 yes
a= b =0 no

insuff

2.a>0 b<0 no
a<0 b>0 yes

insuff

from 1 and 2
a< 0 b >0
Suff

C

IMO for both cases viz. a^(2b) or (a^2)b the answer is C.

the solution:
a != 0 and b != 0 (a&b not equal to 0) from ab<0
& b>0 from |a|=b ---1
=> a<0
=> a^(2b)>0 ---2
& a^2>0 ---3

hence for the case
1. a^(2b) >0 from ---2
2. (a^2)b > 0 from ---1 & 3

Please correct me if i made a mistake.

assuming we are working with a^(2b) > 0, and the following assumptions

B is not negative, B = |A|
A is negative since AB< 0

use -2, and -.5 for A. and 2,.5 for B respectively.

A = -2: (-2)^(2*2) = 16
A = -.5: (-.5)^(2*.5) = -.5

answers on both side of 0.....get E

To prove, we need:
a) find out if a or b or both are zero?
b) Whether b is +ve or -ve.

1) b = |a| means b is non-negative i.e. either +ve or 0 - Insufficient.

2) ab<0 means a & b are not 0. but either a or b is -ve

Combining: we can say that b is +ve and a is -ve.
C.

Hello Experts!
Kindly share your views and methods on this question.
Thanks
Celestial

Hi Celestial09,

The question is a classic example where students miss out on the concept that \(|x| => 0\) and not \(|x| > 0\). Let's analyze the question statement first to see what exactly we need to find.

Analyze the given information in the question
The question asks us if \(a^2*b > 0\). We know that \(a^2 => 0\). To answer the question, we need to know for sure if \(a\) and \(b\) are not equal to 0 along with the sign of \(b\). Please note that sign of \(a\) does not make any difference as \(a^2\) will never be negative. With this understanding, let's evaluate the statements now.

Analyze statement-I independently
St-I tells us that \(|a| = b\). Since \(|a| => 0\), it means that \(b => 0\). Using this information, \(|a| = b = 0\) or \(|a| = b > 0\). Since the statement does not tell us for sure if \(a\) and \(b\) are not equal to 0 we can't say if \(a^2 * b > 0\).

Analyze statement-II independently
St-II tells us that \(ab < 0\) i.e. \(a\), \(b\) have the opposite signs. So,

i. If \(a > 0\), then \(b < 0\). This would mean \(a^2 * b < 0\)
ii. If \(a < 0\), then \(b > 0\). This would mean that \(a^2 * b > 0\) as \(a^2\) is positive irrespective of the sign of \(a\).

Thus, the statement does not tell us for sure if \(a^2 * b > 0\).

However, the statement tells us about an important nature of both \(a\) and \(b\). Since product of \(a\) and \(b\) is not equal to 0, we can say that neither of \(a\) or \(b\) is equal to zero. This nature of \(a\) and \(b\) would be helpful when we combine the analysis from st-I & II

Combine analysis from st-I & II
St-I tells us that \(|a| = b => 0\) and st-II tells us that \(b < 0\) or \(b > 0\) and \(a\) not equal to 0.

Combining both the statements, we can say that \(|a| = b > 0\). Since none of \(a\) and \(b\) is equal to \(0\) and \(b > 0\), we can safely say that \(a^2 > 0\) and \(b > 0\) i.e. \(a^2* b > 0\).

Hence combining both the statements is sufficient to answer our question.

Hope its clear!

Regards
Harsh
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Is a^2*b > 0?

Since a^2 is a non-negative value (0 or positive), then for a^2*b to be positive a must not be 0 and b must be positive.

(1) |a| = b. Absolute value of a number (|a|) is also a non-negative value (0 or positive), thus this statement implies that \(b\geq{0}\). Since both a and b can be 0, then this statement is NOT sufficient.

(2) ab < 0. This statement rules out any of the unknowns being 0 but b can be positive as well as negative. Not sufficient.

(1)+(2) Since from (2) \(b\neq{0}\) then from (1) \(b>0\). So, we have that neither of a and b is 0 and b is positive, therefore a^2*b > 0. Sufficient.

Answer: C.

Hope it's clear.
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