Question

If $x^2-2(4\lambda -1)x+(15{\lambda }^2-2\lambda -7)>0\forall x\in R,$ then $\lambda$ satisfies the interval

• A. $(2,4)$
• B. $(0,1)$
• C. $(-1,-2)$
• D. None of these

Answer 1

The correct option is A $(2,4)$

The sign of quadratic expression for all $x\in R$ is same as that of the coefficient of $x^2$.
Here coefficient of $x^2=1>0$ provided $b^2-4ac<0$
$\Rightarrow 4{(4\lambda -1)}^2-4\left(1\right)(15{\lambda }^2-2\lambda -7)<0$
$\Rightarrow {(4\lambda -1)}^2-(15{\lambda }^2-2\lambda -7)<0$
$\Rightarrow {\lambda }^2-6\lambda +8<0\Rightarrow (\lambda -2)(\lambda -4)<0$
$\Rightarrow 2<\lambda <4\therefore \lambda \in (2,4)$
Hence, choice (A) is correct answer.