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JEE Main & Advanced AIEEE Solved Paper-2009

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    Directions: Questions No. 87 are Assertion - Reason type questions. Each of these questions contains two statements: Statement-1 (Assertion) and Statement-2 (Reason). Each of these questions also have four alternative choices, only one of which is the correct answer. You have to select the correct choice. Statement 1: The variance of first n even natural numbers is \[\frac{{{n}^{2}}-1}{4}\] Statement 2: The sum of first n natural numbers is \[\frac{n(n+1)}{2}\] and the sum of squares of first n natural numbers is\[\frac{n(n+1)(2n+1)}{6}\]     AIEEE  Solved  Paper-2009

    A) Statement-1 is true, Statement-2 is true, Statement-2 is a correct explanation for statement-1

    B) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for statement-1.

    C) Statement-1 is true, statement-2 is false.

    D) Statement-1 is false, Statement-2 is true

    Correct Answer: D

    Solution :

    \[\overline{x}=\frac{2+4+6+....2n}{n}=n+1\] variance\[({{\sigma }^{2}})=\frac{\sum\limits_{i=1}^{n}{{{({{x}_{i}}-\overline{x})}^{2}}}}{n}=\frac{\sum\limits_{i=1}^{n}{{{(2i-(n+1))}^{2}}}}{n}\] \[=\frac{4\sum\limits_{i=1}^{n}{{{i}^{2}}}+\sum\limits_{i=1}^{n}{{{(n+1)}^{2}}-4(n+1)}\sum\limits_{i=1}^{n}{i}}{n}={{n}^{2}}-1\] 


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