The number of real roots


The number of real roots of the equation $\tan ^{-1} \sqrt{x(x+1)}+\sin ^{-1} \sqrt{x^{2}+x+1}=\frac{\pi}{4}$ is :

  1. 1

  2. 2

  3. 4

  4. 0

Correct Option: , 4


$\tan ^{-1} \sqrt{x^{2}+x}+\sin ^{-1} \sqrt{x^{2}+x+1}=\frac{\pi}{4}$

For equation to be defined,

$x^{2}+x \geq 0$

$\Rightarrow \quad x^{2}+x+1 \geq 1$

$\therefore \quad$ only possibility that the equation is defined

$x^{2}+x=0 \quad \Rightarrow x=0 ; x=-1$

None of these values satisfy

$\therefore$ No of roots $=0$

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