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16 Sep 2014, 00:57
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E
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
85%
(hard)
Question Stats:
45% (02:01) correct
55%
(01:42)
wrong
based on 146
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If \(a^2 - b^2 = b^2 - c^2\), is \(a = |b|\)?
(1) \(b = |c|\)
(2) \(b = |a|\)
(1) \(b = |c|\)
(2) \(b = |a|\)
16 Sep 2014, 00:58
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Official Solution:
If \(a^2 - b^2 = b^2 - c^2\), is \(a = |b|\)?
(1) \(b = |c|\).
Squaring the above gives \(b^2 = c^2\). Substituting this into the equation given in the stem yields:
\(a^2 - b^2 = b^2 - b^2\);
\(a^2 - b^2 = 0\);
\(a^2 = b^2\);
\(|a| = |b|\).
If \(a\) is non-negative, then \(|a| = |b|\) implies \(a = |b|\). However, if \(a\) is negative, then \(|a| = |b|\) implies \(-a = |b|\). For example, if \(a = b = c = 1\), the answer is YES; but if \(a = -1\) and \(b = c = 1\), the answer is NO. Not sufficient.
(2) \(b = |a|\).
The above implies that \(b\) is non-negative, hence \(|b| = b\), and the question becomes whether \(a=b\). If \(a\) is also non-negative, then \(b = |a|\) gives \(a=b\), and in this case, the answer to the question is YES. However, if \(a\) is negative, then \(b=|a|\) gives \(b=-a\), and in this case, the answer to the question is NO. For example, if \(a=b=c=1\), the answer to the question is YES, but if \(a=-1\) and \(b=c=1\), the answer to the question is NO. Not sufficient.
(1)+(2) When combining the statements, we know that \(|a| = |b| = |c|\), with \(b\) being non-negative. We can rephrase the question as: is \(a=b\)? The issue is that we still don't know the sign of \(a\). If \(a\) is also non-negative, the answer to the question is YES. However, if \(a\) is negative, the answer to the question is NO. We can consider the same examples as above: if \(a=b=c=1\), the answer to the question is YES, but if \(a=-1\) and \(b=c=1\), the answer to the question is NO. Not sufficient.
Answer: E
If \(a^2 - b^2 = b^2 - c^2\), is \(a = |b|\)?
(1) \(b = |c|\).
Squaring the above gives \(b^2 = c^2\). Substituting this into the equation given in the stem yields:
\(a^2 - b^2 = b^2 - b^2\);
\(a^2 - b^2 = 0\);
\(a^2 = b^2\);
\(|a| = |b|\).
If \(a\) is non-negative, then \(|a| = |b|\) implies \(a = |b|\). However, if \(a\) is negative, then \(|a| = |b|\) implies \(-a = |b|\). For example, if \(a = b = c = 1\), the answer is YES; but if \(a = -1\) and \(b = c = 1\), the answer is NO. Not sufficient.
(2) \(b = |a|\).
The above implies that \(b\) is non-negative, hence \(|b| = b\), and the question becomes whether \(a=b\). If \(a\) is also non-negative, then \(b = |a|\) gives \(a=b\), and in this case, the answer to the question is YES. However, if \(a\) is negative, then \(b=|a|\) gives \(b=-a\), and in this case, the answer to the question is NO. For example, if \(a=b=c=1\), the answer to the question is YES, but if \(a=-1\) and \(b=c=1\), the answer to the question is NO. Not sufficient.
(1)+(2) When combining the statements, we know that \(|a| = |b| = |c|\), with \(b\) being non-negative. We can rephrase the question as: is \(a=b\)? The issue is that we still don't know the sign of \(a\). If \(a\) is also non-negative, the answer to the question is YES. However, if \(a\) is negative, the answer to the question is NO. We can consider the same examples as above: if \(a=b=c=1\), the answer to the question is YES, but if \(a=-1\) and \(b=c=1\), the answer to the question is NO. Not sufficient.
Answer: E
04 Jul 2015, 10:33
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My doubt may sound silly but this is confusing me a lot.
what is the difference between
1. \(|a| = |b|\).
2. \(a = |b|\).
3. \(|a| = b\).
please explain the concept.
what is the difference between
1. \(|a| = |b|\).
2. \(a = |b|\).
3. \(|a| = b\).
please explain the concept.
