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MHT CET202313 May 2023Evening ShiftMathematicsStatisticsActual

Variance of first 2 n natural numbers is

Options

  1. A4 n^2+1 12
  2. B(2 n -1)^2 12
  3. Cn ^2 3 -1
  4. D4 n^2-1 12

Correct answer

D. 4 n^2-1 12

Step-by-step solution

aligned ^2= 1 2 n [1^2+2^2+3^2+ . & .+(2 n)^2 ] & - ( 1+2+3+ +2 n 2 n )^2 aligned aligned & = 1 2 n [ 2 n(2 n+1)(4 n+1) 6 ]- [ 1 2 n 2 n(2 n+1) 2 ]^2 & = (2 n+1)(4 n+1) 6 - ( 2 n+1 2 )^2 & = 2 n+1 2 ( 4 n+1 3 - 2 n+1 2 ) & = 2 n+1 2 ( 2 n-1 6 ) & = 4 n^2-1 12 aligned

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