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સંખ્યાઓ $a, b, 8, 5, 10 $ નો મધ્યક $6$ અને વિચરણ $6.80 $ હોય તો નીચે આપેલ પૈકી કઇ એક $a $ અને $b $ માટે શક્ય કિંમત છે ?
$a = 0, b = 7$
$a = 5, b = 2$
$a = 1, b = 6$
$a = 3, b = 4$
Solution
$\,\,\because \,\,\,\frac{{\Sigma {x_i}}}{n}\,\, = \,\,\bar x$
$ \Rightarrow \,\,\,\frac{{a\,\, + \,\,b\,\, + \,\,8\,\, + \,\,5\,\, + \,\,10}}{5}\,\,\, = \,\,6\,\,$
$ \Rightarrow \,\,a\,\, + \,\,b\,\, = \,\,7\,\,…….\,\,(1)$
અને $\,\frac{{\Sigma x_i^2}}{n}\,\, – \,\,{{\bar x}^2}\,\, = \,\,{\sigma ^2}\,\,\,\,\,\,$
$ \Rightarrow \,\,\frac{{{a^2}\, + \,\,{b^2}\, + \,\,64\,\, + \,\,25\,\, + \,\,100}}{5}\,\, – \,\,36\,\, = \,\,6.8$
$ \Rightarrow \,\,{a^2}\, + \,\,{b^2}\, = \,\,25\,\,\,\,\,……..\,\,(2)$
સમીકરણ $(1)$ અને $(2) $ ને ઉકેલતા ${\text{a}}\,\, = \,\,{\text{4,}}\,\,{\text{b}}\,\, = \,\,{\text{3}}$ અથવા $\,{\text{a}}\,\, = \,\,{\text{3,}}\,\,\,{\text{b}}\,\, = \,\,{\text{4}}$
