Question

Let $f\left(x\right)=\frac{\sqrt{sgn(\alpha x^2+\alpha x+1)}}{{\cot }^{-1}(x^2-\alpha )}$. If $f\left(x\right)$ is continuous for all $x\in R$, then number of integer in the range of $\alpha$, is
[Note : sgn k denotes signum function of k.]

• A. $0$
• B. $4$
• C. $5$
• D. $6$

Answer 1

The correct option is C $5$

Given: $f\left(x\right)=\frac{\sqrt{sgn(\alpha x^2+\alpha x+1)}}{{cot}^{-1}(x^2-\alpha )}andf\left(x\right)iscontinousforallx\epsilon R$

To find: Number of integer in the range of $\alpha$

Sol: $\alpha x^2+\alpha x+1\to isaquadratic$

Whose discriminant$=x^2-4\alpha$

Now, for $f\left(x\right)$ to be a valid function

$sgn(\alpha x^2+\alpha x+1)\ge 0forallx\epsilon R$

$⟹\alpha x^2+\alpha x+1\ge 0⟹Discriminant\le 0$

$⟹\alpha (\alpha -4)\le 0$

$⟹\alpha \in [0,4]$

$\therefore \alpha \in [0,4]$

$⟹5$ integers are in the range of $\alpha$

Hence, the correct answer is 5