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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
$\left[ { - 3,\infty } \right)$
$\left( { - 3,\infty } \right)$
$\left( { - \infty , - 3} \right)$
$\left( { - \infty , 3} \right]$
Solution
$f(x)=x^{2}-(a+2) x-(a+3)$
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 $\in \phi$
$\therefore a \in(-3, \infty)$
