QJEEM19Macb98: JEE Main 2019 Mathematics - Class 11 JEEM19Macb98 Solution: The mean and variance of seven observations are 8 and 16,…

Given mean ( $\mu$ ) = 8

variance ( ${\sigma ^2}$ ) = 16

No of observations (N) = 7

Let the two unknown observation = x and y

We know,

${\sigma ^2} = {{\sum {x_i^2} } \over N} - {\mu ^2}$ = 16

$\Rightarrow$ ${{{2^2} + {4^2} + {{10}^2} + {{12}^2} + {{14}^2} + {x^2} + {y^2}} \over 7} - {\left( 8 \right)^2}$ = 16

$\Rightarrow$ x² + y² = 100 ........(1)

We know,

$\mu$ = ${{\sum {{x_i}} } \over N}$ = 8

$\Rightarrow$ ${{2 + 4 + 10 + 12 + 14 + x + y} \over 7}$ = 8

$\Rightarrow$ x + y = 14 ...........(2)

As (x + y)² = x² + y² + 2xy

$\Rightarrow$ (14)² = 100 + 2xy

$\Rightarrow$ 196 = 100 + 2xy

$\Rightarrow$ xy = 48