Question:
If the mean and the standard deviation of the data $3,5,7, \mathrm{a}, \mathrm{b}$ are 5 and 2 respectively, then $\mathrm{a}$ and $\mathrm{b}$ are the roots of the equation :
Correct Option: , 2
Solution:
Mean $=5$
$\frac{3+5+7+a+b}{5}=5$
$a+b=10$.....(i)
S.d. $=2 \Rightarrow \sqrt{\frac{\sum_{i=1}^{5}\left(x_{i}-\bar{x}\right)^{2}}{5}}=2$
$(3-5)^{2}+(5-5)^{2}+(7-5)^{2}+(a-5)^{2}+(b-5)^{2}=20$
$\Rightarrow 4+0+4+(a-5)^{2}+(b-5)^{2}=20$
$a^{2}+b^{2}-10(a+b)+50=12$
$(a+b)^{2}-2 a b-100+50=12$
$a b=19$ ..........(ii)
Equation is $x^{2}-10 x+19=0$