QJEEM16Ma7835: JEE Main 2016 Mathematics - Class 11 JEEM16Ma7835 Solution: If the standard deviation of the numbers 2, 3, a and 11 is 3.5, …

Question

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If the standard deviation of the numbers 2, 3, a and 11 is 3.5, then which of the following is true?

Accepted Answer

The formula for standard deviation (S.D)

= \sqrt {{{\sum {x_i^2} } \over n} - {{\left( {{{\sum {{x_i}} } \over n}} \right)}^2}}

Where

\sum {x_i^2 = }

Sum of square of the numbers

= {2^2} + {3^2} + {a^2} + {11^2}

= 4 + 9 + {a^2} + 121

= 134 + {a^2}

\sum {{x_i}} =

Sum of numbers

= 2 + 3 + a + 11

= 16 + a

\therefore\,\,\,

SD = \sqrt {{{134 + a{}^2} \over 4} - {{\left( {{{16 + a} \over 4}} \right)}^2}}

\Rightarrow \sqrt {{{134 + {a^2}} \over 4} - {{\left( {{{16 + a} \over 4}} \right)}^2}} = 3.5 = {7 \over 2}

(given)

\Rightarrow {{134 + {a^2}} \over 4} - {\left( {{{16 + a} \over 4}} \right)^2} = {{49} \over 4}

\Rightarrow 4\left( {134 + {a^2}} \right) - \left( {256 + 32a + {a^2}} \right) = 4 \times 49

\Rightarrow 3{a^2} - 32a + 84 = 0

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