Question

If the mean and the standard deviation of the data $3,5,7, a , b$ are $5$ and $2$ respectively, then a and $b$ are the roots of the equation :

• A. $2 x^{2}-20 x+19=0$
• B. $x^{2}-10 x+19=0$
• C. $x^{2}-10 x+18=0$
• D. $x^{2}-20 x+18=0$

Answer 1

Answer: (B)

Mean $=5$
$\frac{3+5+7+a+b}{5}=5$
$a+b=10 \dots$(i)
S.d. $=2 \Rightarrow \sqrt{\frac{\displaystyle\sum_{i=1}^{5}\left(x_{i}-\bar{x}\right)^{2}}{5}}=2$
$(3-5)^{2}+(5-5)^{2}+(7-5)^{2}+(a-5)^{2}+(b-5)^{2}=20$
$\Rightarrow 4+0+4+(a-5)^{2}+(b-5)^{2}=20$
$a^{2}+b^{2}-10(a+b)+50=12$
$(a+b)^{2}-2 a b-100+50=12$
$a b=19 \dots$(ii)
Equation is $x^{2}-10 x+19=0$