Question

If $\lambda=\sqrt{4+\sqrt{\frac{37 b-81 b^4-3}{b}}}$ 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)

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