Question

If $\lambda =\sqrt{4+\sqrt{\frac{37b-81b^4-3}{b}}}$ and $f(x)=(x-p+5)(x-p)$ where $p,\lambda \in R$ and $b>0$ then

• A. sum of integral values of $\lambda$ is 9
• B. sum of integral values of $\lambda$ is 5
• C. complete set of values of $p$ for which $f(\lambda )<0$ for all possible values of $\lambda$ is $(3,7)$
• D. complete set of values of $p$ for which $f(\lambda )<0$ for all possible values of $\lambda$ is $(4,7)$

Answer 1

Answer: (B), (C)

$\lambda =\sqrt{4+\sqrt{\frac{37b-81b^4-3}{b}}}\le 3(81b^3+\frac{3}{b}\ge 12$ Using A.M. $\ge$ G.M. $)$
$\Rightarrow 2\le 1\le 3$
$f(x)=(x-(p-5))(x-p)$
$f(\lambda )<0$
$\Rightarrow p-5<2\le 1\le 3<p\Rightarrow p\in (3,7)$