Set X:-

#elements=2n+1, where \(n\geq{1}\) (since set X contains consecutive odd integers with mean as an odd integer)

Since 37 is the mean, hence set X contains odd numbers of elements with at least 37 as one of the member.

Set Y:-

#elements=2n+1, where \(n\geq{5}\)
Since 37 is the mean, hence set Y contains odd numbers of elements with at least 37 as one of the member.

Set Z:-

Z=X U Y

Now let's evaluate each statement:-

I. SD of set Z will be same as SD of set X when set X and set Y are equal. In all other cases, SD of Z will be different from SD of X. Hence this statement is not true.
II.SD of set Z will be same as SD of set Y when set X and set Y are equal. In all other cases, SD of Z will be different from SD of Y. Hence this statement is not true.
Note:- 2 sets equally spaced , there SD depend on #elements they possess. Greater the no of elements in the set, greater is the SD.
III. Here there are 3 cases, viz,
1. Set Z=Set X=Set Y
2. Set Z= Set X or Set Y
3. Set Z= Set X and Set Y
In the above cases, since both Set X and Y have mean 37. Hence, set Z will always has a mean of 37. hence this statement is correct.

Answer Option. C

Hi HisHo,

First of all, please notice that the question stem is a "MUST BE TRUE". In these type of questions, you need to validate the data at all possible circumstances.

You have found out one possible case where (II) is valid. But there are many cases as highlighted in the explanation where (II) is not valid( When set X and Y are not equal).
Hope it helps.
_________________

Regards,

PKN

Rise above the storm, you will find the sunshine