Question

If the root of $4x^2+5\lambda =(5\lambda +1)x$ differ by unity, then the sum of the possible values of $\lambda$ is

• A. $-6$
• B. $\frac{4}{5}$
• C. $\frac{14}{5}$
• D. None of these

Answer 1

Answer: (C)

$\frac{14}{5}$

Let $\alpha ,\beta$ be the roots of $4x^2-(5\lambda +1)x+5\lambda =0$
$\alpha +\beta =\frac{(5\lambda +1)}{4}$
$\alpha \beta =\frac{5\lambda }{4}$
Now as given
$|\alpha -\beta |=1$
Using
${(\alpha -\beta )}^2={(\alpha +\beta )}^2-4\alpha \beta$
$\Rightarrow {\left(\frac{(5\lambda +1)}{4}\right)}^2-4\frac{5\lambda }{4}=1\Rightarrow 5{\lambda }^2-14\lambda -3=0\Rightarrow \lambda =-\frac{1}{5},\lambda =3$
Hence sum of values of $\lambda$ is
$-\frac{1}{5}+3=\frac{14}{5}$