PriyankaPalit7 wrote:
Is \(d>1\)?
(1) Median of \(d\), \(-d\), \(\frac{1}{d}\) is \(d\).
(2) Median of \(d\), \(d^2\), \(d^3\) is \(d\)
Since we have three numbers, and since the median is just the middle of 3, then translating out statements into equations is easy.
We'll do this and simplify, a Precise approach.
(1) Since d is the median then 1/d ≤ d ≤ -d or -d ≤ d ≤ 1/d. In the first case d ≤ -d implies d ≤ 0 so it is definitely smaller than 1. In the second case d ≤ 1/d implies that d ≤ 1 (just multiply by d) so it is once again smaller than 1.
Then the answer is NO: Sufficient.
(2) Our two options are d^2 ≤ d ≤ d^3 or d^3 ≤ d ≤ d^2. In the first case d^2 ≤ d implies d(d - 1) ≤ 0 so d is between 0 and 1. In the second case d^3≤d^2 implies d^2(d-1) ≤ 0 so, since d^2 is nonnegative, then d - 1 ≤ 0 and therefore d ≤ 1.
Then the answer is once again NO: Sufficient.
(D) is our answer.
Note that another solution would be to notice that d = 1 works in both cases, and then to see that if d > 1 then 1/d must be the median of (-d, 1/d, d) and d^2 must be the median of (d, d^2, d^3) in contradiction to the given data. This is a more 'Logical' approach, as it revolves around seeing the logical patterns behind the number choices.
-d ≤ d ≤ 1/d. "In the second case d ≤ 1/d implies that d ≤ 1 (just multiply by d) so it is once again smaller than 1."
Here you just multiplied both sides by d assuming it is >0. If d<0, your derivation just doesn't hold. Try -2 for example: