Question

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

• A. 6
• B. 7
• C. 12
• D. 14
• E. 25

Answer 1

Answer: (C)

Given, $\frac{a+b+8+5+10}{5}=6$
$\Rightarrow a+b+23=30$
$\Rightarrow a+b=7......(i)$
and $\frac{\Sigma {\left(x_i-\bar{x}\right)}^2}{n}={\sigma }^2$
$\frac{(a-6{)}^2+(b-6{)}^2+4+1+16}{5}=\frac{34}{5}$
$\Rightarrow (a-6{)}^2+(b-6{)}^2+21=34$
$\Rightarrow a^2+b^2-84+93=34[\because a+b=7]$
$\Rightarrow a^2+b^2=25$
From Eq. (i), $(a+b{)}^2=(7{)}^2....(ii)$
$\Rightarrow a^2+b^2+2ab=49$
$\Rightarrow 25+2ab=49[$ from $Eq.(ii)]$
$\Rightarrow 2ab=49-25$
$\Rightarrow 2ab=24\Rightarrow ab=12$