Question

For all $x \in R$, if $\lambda$ $x^{2} - 9\lambda x + 5\lambda + 1 > 0$, then $\lambda$ lies in the interval

• A. $\left(-\frac{4}{61},\,0\right)$
• B. $\left[0,\,\frac{4}{61}\right]$
• C. $[0,\,\frac{4}{61})$
• D. None of these

Answer 1

Answer: (C)

We have, $\lambda x^{2} - 9\lambda x + 5\lambda + 1$
$= \lambda\left(x^{2} - 9x\right) + 5\lambda + 1$
$= \lambda \left(x^{2} - 9x+\frac{81}{4}\right)+\left(5\lambda-\frac{81}{4}\lambda + 1\right)$
$= \lambda \left(x-\frac{9}{2}\right)^{2} + \left(1-\frac{61\lambda}{4}\right) > 0$ if $\lambda \ge 0$ and $1- \frac{61\lambda }{4} > 0$
(For $\lambda = 0$, the given expression is $1$)
i.e., if $\lambda \ge 0$ and $\lambda < \frac{4}{61}$ i.e., if $0 \le \lambda < \frac{4}{61}$.