Question

Let $f\left(x\right)=\frac{\sqrt{sgn \left(\alpha x^{2} + \alpha x + 1\right)}}{cot^{- 1} \left(x^{2} - \alpha \right)}$ . If $f\left(x\right)$ is continuous for all $x\in R$ , then number of integers in the range of $\alpha $ is

• A. $0$
• B. $4$
• C. $5$
• D. $6$

Answer 1

Answer: (B)

$\because $ $f\left(x\right)$ is continuous for all $x\in R$
$\therefore $ $\alpha x^{2}+\alpha x+1>0\forall x\in R$
(i) If $\alpha =0$ then it is true.
(ii) If $\alpha \neq 0$ then $D < 0$
$\alpha ^{2}-4\alpha < 0\Rightarrow 0 < \alpha < 4$
$\therefore $ Integral values of $\alpha =0,1,2,3$
$\therefore $ Number of integral values of $\alpha =4$