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 Let x1, x2,....., x11 be 11 distinct positive integers. If we replace the largest of these integers by the median of the other 10 integers, then 

(a)  the median remains the same

(b)  the mean increases

(c) the median decreases

(d) the mean remains the same

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The correct option is (C).

Let x1​,x2​,x3​,x4​,...........x11​ are distinct positive integers

Suppose the increasing order of the numbers are

⇒  x1​,x2​,x3​,x4​,x5​,x6​,x7​,x8​,x9​,x10​,x11

x11​ is largest number and the median is x6​.

Now, the median of first 10 numbers = (x5​ + x6)/2​​

⇒  Let a =  (x5​ + x6)/2

Now, we have to replace largest number x11​ by the median of first 10 numbers i.e. a.

So, new increasing order will be

⇒  x1​, x2​, x3​, x4​, x5​, a , x6​, x7​, x8​, x9​, x10

And new median will be a i.e. (x5​ + x6)/2

Which is lesser than the median of first eleven number x6​

⇒  So, median decreases.

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