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KEAM 2016 Math Paper

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Question : 82 of 120

Marks: +1, -0

If the mean of the numbers a, b, 8,5,10 is 6 and their variance is 6.8, then ab is equal to

6
7
12
14
25

Solution:

Given,

\;\frac{a+b+8+5+10}{5}=6

\Rightarrow \;\; a+b+23=30

a+b=7

....(i)

and

\;\frac{\sum (x_i-\overline{x})^{2}}{n}=\sigma^{2}

\;\frac{(a-6)^{2}+(b-6)^{2}+4+1+16}{5}=\;\frac{34}{5}

\Rightarrow (a-6)^{2}+(b-6)^{2}+21=34

\Rightarrow \;\; a^{2}+b^{2}-84+93=34

[ \because a+b=7]

\Rightarrow \;\; a^{2}+b^{2}=25

....(ii)

From Eq. (i),

(a+b)^{2}=(7)^{2}

\Rightarrow \;\; a^{2}+b^{2}+2 a b=49

\Rightarrow \;\; 25+2 a b=49

[from Eq. (ii)

\Rightarrow \;\; 2 a b=49-25

\Rightarrow \;\; 2 a b=24 \Rightarrow a b=12

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