KVPY SA Exam 2015 Question Paper

Question : 65 of 80

Marks: +1, -0

The average incomes of the people in two villages are P and Q , respectively. Assume that P ≠ Q . A person moves from the first village to the second village. The new average incomes are P' and Q' , respectively. Which of the following is not possible?

P'>P and Q'>Q
P’ > P and Q' < Q
P' = P and Q' = Q
P' < P and Q' < Q

Validate

Solution:

{x_1+ x_2+ .... x _n}/n=P

{y_1+y_2+ .... y _m}/m=Q

{x_1+ x_2+ .... x _n}=nP

{y_1+y_2+ .... y _m} =mQ

Now if a person moves from Ist village to IInd village then

{x_1+ x_2+ .... x _{n-1}}/(n-1)=P'

⇒

nP-x_n=(n-1)P'

⇒P'(n/{n-1})P-(n/{n-1})

And

{y_1+y_2+ .... y _m+x_n}/{m+1}=Q'

⇒ mQ + xn = (m + 1)Q'

∴ If

P' = P ⇒ P(1-n/{n-1})=x_n/{n-1}

⇒

{-P}/{n-1}=-x_n/{n-1}⇒P=x_n

& when Q = Q'
⇒

x_n = Q'= Q

In that case P = Q (which is not true)
∴ (C) P' = P and Q' = Q is not possible

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