The set of values of 'a' for which th | vedclass

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Question

$f(x)=x^{2}-(a+2) x-(a+3)$

for $f(\mathrm{x})$ to be negative for atleast one positive

$\mathrm{x}$, following cases may be there $-$

Case$-I

Vedclass · Accepted Answer

$\left( { - 3,\infty } \right)$

Vedclass · Answer

$f(x)=x^{2}-(a+2) x-(a+3)$ for $f(\mathrm{x})$ to be negative for atleast one positive $\mathrm{x}$, following cases may be there $-$ Case$-I$ $\quad f(0)<0$ $-(a+3)<0 \Rightarrow a+3>0$ $\Rightarrow a>-3$ Case$-II$ $\quad(1) D>0$ $(a+2)^{2}+4(a+3)>0$ $a^{2}+8 a+16>0$ $\Rightarrow(a+4)^{2}>0$ $\Rightarrow a \in R-\{-4\}$ $(2)$ $f(0) \geq 0$ $-(a+3) \geq 0$ $\Rightarrow a+3 \leq 0$ $\Rightarrow a \leq-3$ $(3)$ $-\frac{b}{2 a}>0 \Rightarrow a>-2$ intersection of $(1),(2) \&(3)$ is a $\in \phi$ $ herefore a \in(-3, \infty)$

The set of values of $'a'$ for which the inequality ${x^2} - (a + 2)x - (a + 3) < 0$ is satisfied by atleast one positive real $x$ , is

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