Question

Consider the equation $(m^2+1)x^2-3x+(m^2+1{)}^2=0.$ Let $p$ be the least value of product of roots and $q$ be the greatest value of sum of roots of the equation. Then the sum of an infinitely decreasing G.P. whose first term is equal to $p+2$ and the common ratio is $\frac{2}{q},$ is

• A. $9$
• B. $6$
• C. $14$
• D. $8$

Answer 1

The correct option is A $9$

$(m^2+1)x^2-3x+(m^2+1{)}^2=0$
Sum of roots $=\frac{3}{m^2+1}$
$\therefore (\text{Sum}{)}_{max}=3$ (for $m=0)$

Product of roots $=m^2+1$
$(\text{Product}{)}_{min}=1$ (for $m=0)$

Given, $(\text{Sum}{)}_{max}=q$
and $(\text{Product}{)}_{min}=p$
Then, first term of G.P. $=p+2=1+2=3$
and common ratio $=\frac{2}{q}=\frac{2}{3}$
$\therefore S_{\infty }=\frac{3}{1-\frac{2}{3}}=9$