If the mean and the standard deviation

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:

1. (1) $x^{2}-10 x+18=0$

2. (2) $2 x^{2}-20 x+19=0$

3. (3) $x^{2}-10 x+19=0$

4. (4) $x^{2}-20 x+18=0$

Correct Option: , 3

Solution:

Mean $=\frac{3+5+7+a+b}{5}=5 \Rightarrow a+b=10$

Variance $=\frac{3^{2}+5^{2}+7^{2}+a^{2}+b^{2}}{5}-(5)^{2}=4$

$\Rightarrow a^{2}+b^{2}=62$

$\Rightarrow(a+b)^{2}-2 a b=62$

$\Rightarrow a b=19$

Hence, $a$ and $b$ are the roots of the equation,