Question

The value of $\lambda$ for which one root of the equation $x^2+(1-2\lambda )x+({\lambda }^2-\lambda -2)=0$ is greater than $3$ and the other is less than $3$ is given by

• A. $\lambda <2$
• B. $2<\lambda <5$
• C. $\lambda >5$
• D. $\lambda >1$

Answer 1

The correct option is B $2<\lambda <5$

The graph of given quadratic opens upwards. For one root to be greater than $3$ and other less than $3$, $f\left(3\right)<0$.
This gives,
$\Rightarrow 3^2+(1-2\lambda )3+{\lambda }^2-\lambda -2<0\Rightarrow {\lambda }^2-7\lambda +10<0\Rightarrow (\lambda -5)(\lambda -2)<0\Rightarrow \lambda <5,\lambda <2\therefore 2<\lambda <5$

The correct option is B.