Functions Question 2
Question 2 - 2024 (01 Feb Shift 2)
If the domain of the function $f(x)=\frac{\sqrt{x^{2}-25}}{\left(4-x^{2}\right)}+\log _{10}\left(x^{2}+2 x-15\right)$ is $(-\infty, \alpha) U[\beta, \infty)$, then $\alpha^{2}+\beta^{3}$ is equal to :
(1) 140
(2) 175
(3) 150
(4) 125
Show Answer
Answer (3)
Solution
$f(\mathrm{x})=\frac{\sqrt{\mathrm{x}^{2}-25}}{4-\mathrm{x}^{2}}+\log _{10}\left(\mathrm{x}^{2}+2 \mathrm{x}-15\right)$
Domain : $\mathrm{x}^{2}-25 \geq 0 \Rightarrow \mathrm{x} \in(-\infty,-5] \cup[5, \infty)$
$4-\mathrm{x}^{2} \neq 0 \Rightarrow \mathrm{x} \neq{-2,2}$
$\mathrm{x}^{2}+2 \mathrm{x}-15>0 \Rightarrow(\mathrm{x}+5)(\mathrm{x}-3)>0$
$\Rightarrow \mathrm{x} \in(-\infty,-5) \cup(3, \infty)$
$\therefore \mathrm{x} \in(-\infty,-5) \cup[5, \infty)$
$\alpha=-5 ; \beta=5$
$\therefore \alpha^{2}+\beta^{3}=150$
