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31 Jul 2025, 08:48
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If \(a < b < c\), and \(c^4 < b^4 < a^4\), which of the following could be true?
I. \(abc > 0\)
II. \(a^3b^5c^7 < 0\)
III. \(a^2b^4c^6 = 0\)
A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III
I. \(abc > 0\)
II. \(a^3b^5c^7 < 0\)
III. \(a^2b^4c^6 = 0\)
A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III
31 Jul 2025, 08:48
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Official Solution:
If \(a < b < c\), and \(c^4 < b^4 < a^4\), which of the following could be true?
I. \(abc > 0\)
II. \(a^3b^5c^7 < 0\)
III. \(a^2b^4c^6 = 0\)
A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III
Taking the fourth root of \(c^4 < b^4 < a^4\) gives \(|c| < |b| < |a|\).
Given that \(a < b\) and \(|b| < |a|\), we can deduce that \(a\) must be negative. That's because for \(a\) to be further from zero than \(b\) (\(|b| < |a|\)) and still be less than \(b\) (\(a < b\)), \(a\) must be negative.
Given that \(b < c\) and \(|c| < |b|\), we can deduce that \(b\) must also be negative. That's because for \(b\) to be further from zero than \(c\) (\(|c| < |b|\)) and still be less than \(c\) (\(b < c\)), \(b\) must be negative.
So both \(a\) and \(b\) must be negative numbers. However, \(c\) can be negative, 0, or positive. For example, if \(a = -3\) and \(b = -2\), then \(c\) can be -1, 0, or 1.
Since \(c\) can be any of those, and each option contains \(c\), then each option could be true, making option E correct.
Still, if interested, let's analyze the options, keeping in mind that we are asked to determine which of the following COULD be true:
I. \(abc > 0\)
\(ab\) will be positive, and since \(c\) could be positive, \(abc\) could be positive.
II. \(a^3b^5c^7 < 0\)
\(a^3b^5\) will be positive, and if \(c\) is negative, then \(a^3b^5c^7\) could be negative.
III. \(a^2b^4c^6 = 0\)
Since \(c\) can be 0, \(a^2b^4c^6\) can also be 0.
Answer: E
If \(a < b < c\), and \(c^4 < b^4 < a^4\), which of the following could be true?
I. \(abc > 0\)
II. \(a^3b^5c^7 < 0\)
III. \(a^2b^4c^6 = 0\)
A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III
Taking the fourth root of \(c^4 < b^4 < a^4\) gives \(|c| < |b| < |a|\).
Given that \(a < b\) and \(|b| < |a|\), we can deduce that \(a\) must be negative. That's because for \(a\) to be further from zero than \(b\) (\(|b| < |a|\)) and still be less than \(b\) (\(a < b\)), \(a\) must be negative.
Given that \(b < c\) and \(|c| < |b|\), we can deduce that \(b\) must also be negative. That's because for \(b\) to be further from zero than \(c\) (\(|c| < |b|\)) and still be less than \(c\) (\(b < c\)), \(b\) must be negative.
So both \(a\) and \(b\) must be negative numbers. However, \(c\) can be negative, 0, or positive. For example, if \(a = -3\) and \(b = -2\), then \(c\) can be -1, 0, or 1.
Since \(c\) can be any of those, and each option contains \(c\), then each option could be true, making option E correct.
Still, if interested, let's analyze the options, keeping in mind that we are asked to determine which of the following COULD be true:
I. \(abc > 0\)
\(ab\) will be positive, and since \(c\) could be positive, \(abc\) could be positive.
II. \(a^3b^5c^7 < 0\)
\(a^3b^5\) will be positive, and if \(c\) is negative, then \(a^3b^5c^7\) could be negative.
III. \(a^2b^4c^6 = 0\)
Since \(c\) can be 0, \(a^2b^4c^6\) can also be 0.
Answer: E
