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Official Solution:
Is \(q > 0\)?
(1) \(p*q^3*r^4 < 0\)
Since \(r^4\) in the inequality above is positive, we can divide by it to obtain \(p*q^3 < 0\). This implies that \(p\) and \(q\) have opposite signs. However, this information is not sufficient to determine if \(q > 0\).
(2) \(p*q^2*s^6 > 0\)
Both \(q^2\) and \(s^6\) in the inequality above are positive, so we can divide by them to get \(p > 0\). However, from this statement, the only thing we know about \(q\) is that it's not 0. This information is not sufficient to determine if \(q > 0\).
(1)+(2) From (1), we know that \(p\) and \(q\) have opposite signs, and from (2), we know that \(p\) is positive. Therefore, \(q\) must be negative, which provides a definite NO answer to the question. Sufficient.
Answer: C
Is \(q > 0\)?
(1) \(p*q^3*r^4 < 0\)
Since \(r^4\) in the inequality above is positive, we can divide by it to obtain \(p*q^3 < 0\). This implies that \(p\) and \(q\) have opposite signs. However, this information is not sufficient to determine if \(q > 0\).
(2) \(p*q^2*s^6 > 0\)
Both \(q^2\) and \(s^6\) in the inequality above are positive, so we can divide by them to get \(p > 0\). However, from this statement, the only thing we know about \(q\) is that it's not 0. This information is not sufficient to determine if \(q > 0\).
(1)+(2) From (1), we know that \(p\) and \(q\) have opposite signs, and from (2), we know that \(p\) is positive. Therefore, \(q\) must be negative, which provides a definite NO answer to the question. Sufficient.
Answer: C
