Question

 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

Answer 1

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 x_6​

⇒  So, median decreases.