Re M20-35
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16 Sep 2014, 01:09
Official Solution:
What is the value of \(x\)?
Remember, the median of a list with an odd number of elements is simply the middle element when the list is arranged in ascending or descending order. Conversely, the median of a list with an even number of elements is the average of the two middle elements when the list is arranged in ascending or descending order.
(1) The median of list \(\{x, -1, 1, 3, -x\}\) is 0.
Given that this list consists of five elements, which is an odd number, the median must indeed be the middle term, and hence, one of the elements of the list. Therefore, \(x=0\). This results in the list \(\{-1, 0, 0, 1, 3\}\). Sufficient.
(2) The median of list \(\{x, -1, 1, 3, -x\}\) is \(\frac{x}{2}\).
Again, this list consists of five elements, an odd number, so the median must be the middle term. This means the median could be either \(x\), \(-x\), or 1. Note that -1 and 3 cannot be the median because, for -1 to be the middle element, both \(x\) and \(-x\) would have to be less than -1, which is not possible. Similarly, for 3 to be the middle element, both \(x\) and \(-x\) would have to be greater than 3, which is also not possible. If either \(x=\frac{x}{2}\) or \(-x=\frac{x}{2}\), then \(x=0\) and the list becomes \(\{-1, 0, 0, 1, 3\}\). If \(1=\frac{x}{2}\), then \(x=2\) and the list becomes \(\{-2, -1, 1, 2, 3\}\). Therefore, \(x\) can be either 0 or 2. Not sufficient.
Answer: A