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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.
_________________

------------------------------------ J Allen Morris **I'm pretty sure I'm right, but then again, I'm just a guy with his head up his a$$.

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

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.

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.

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.

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