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12 Jan 2026, 09:46
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\(a\), \(b\), and \(c\) are distinct non-zero integers, and the standard deviation of \(\{a, b, c\}\) is greater than the standard deviation of \(\{|a|, |b|, |c|\}\). If the range of \(\{a, b, c\}\) is 4, which of the following could be true?
I. The median of \(\{a, b, c\}\) is -2
II. The product of \(a\), \(b\), and \(c\) is a prime number
III. The mode of \(\{|a|, |b|, |c|\}\) is 1
A. I only
B. II only
C. III only
D. I and III only
E. I, II, and III
I. The median of \(\{a, b, c\}\) is -2
II. The product of \(a\), \(b\), and \(c\) is a prime number
III. The mode of \(\{|a|, |b|, |c|\}\) is 1
A. I only
B. II only
C. III only
D. I and III only
E. I, II, and III
12 Jan 2026, 09:46
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Official Solution:
\(a\), \(b\), and \(c\) are distinct non-zero integers, and the standard deviation of \(\{a, b, c\}\) is greater than the standard deviation of \(\{|a|, |b|, |c|\}\). If the range of \(\{a, b, c\}\) is 4, which of the following could be true?
I. The median of \(\{a, b, c\}\) is -2
II. The product of \(a\), \(b\), and \(c\) is a prime number
III. The mode of \(\{|a|, |b|, |c|\}\) is 1
A. I only
B. II only
C. III only
D. I and III only
E. I, II, and III
Since the standard deviation of \(\{a, b, c\}\) is greater than that of \(\{|a|, |b|, |c|\}\), \(\{a, b, c\}\) must contain both positive and negative integers. This is because if all numbers were positive or all negative, the standard deviations of \(\{a, b, c\}\) and \(\{|a|, |b|, |c|\}\) would be the same.
This fact, together with the range being 4, implies that the least and greatest integers in the set can be \((-3, 1)\), \((-2, 2)\), or \((-1, 3)\). So we can have the following sets:
\(\{-3, -2, 1\}\)
\(\{-3, -1, 1\}\)
\(\{-2, -1, 2\}\)
\(\{-2, 1, 2\}\)
\(\{-1, 1, 3\}\)
\(\{-1, 2, 3\}\)
Let's evaluate the options:
I. The median of \(\{a, b, c\}\) is -2
This could be true if we have the set \(\{-3, -2, 1\}\).
II. The product of \(a\), \(b\), and \(c\) is a prime number
This could be true if we have the set \(\{-3, -1, 1\}\).
III. The mode of \(\{|a|, |b|, |c|\}\) is 1
This could be true if we have the set \(\{-3, -1, 1\}\) or \(\{-1, 1, 3\}\).
Thus, all three options could be true.
Answer: E
\(a\), \(b\), and \(c\) are distinct non-zero integers, and the standard deviation of \(\{a, b, c\}\) is greater than the standard deviation of \(\{|a|, |b|, |c|\}\). If the range of \(\{a, b, c\}\) is 4, which of the following could be true?
I. The median of \(\{a, b, c\}\) is -2
II. The product of \(a\), \(b\), and \(c\) is a prime number
III. The mode of \(\{|a|, |b|, |c|\}\) is 1
A. I only
B. II only
C. III only
D. I and III only
E. I, II, and III
Since the standard deviation of \(\{a, b, c\}\) is greater than that of \(\{|a|, |b|, |c|\}\), \(\{a, b, c\}\) must contain both positive and negative integers. This is because if all numbers were positive or all negative, the standard deviations of \(\{a, b, c\}\) and \(\{|a|, |b|, |c|\}\) would be the same.
This fact, together with the range being 4, implies that the least and greatest integers in the set can be \((-3, 1)\), \((-2, 2)\), or \((-1, 3)\). So we can have the following sets:
\(\{-3, -2, 1\}\)
\(\{-3, -1, 1\}\)
\(\{-2, -1, 2\}\)
\(\{-2, 1, 2\}\)
\(\{-1, 1, 3\}\)
\(\{-1, 2, 3\}\)
Let's evaluate the options:
I. The median of \(\{a, b, c\}\) is -2
This could be true if we have the set \(\{-3, -2, 1\}\).
II. The product of \(a\), \(b\), and \(c\) is a prime number
This could be true if we have the set \(\{-3, -1, 1\}\).
III. The mode of \(\{|a|, |b|, |c|\}\) is 1
This could be true if we have the set \(\{-3, -1, 1\}\) or \(\{-1, 1, 3\}\).
Thus, all three options could be true.
Answer: E
