Question

The number of real solutions of $\left(cot\right)^{- 1} \sqrt{x \left(x + 4\right)}+\left(cos\right)^{- 1} ⁡ \sqrt{x^{2} + 4 x + 1}=\frac{\pi }{2}$ is equal to

• A. $0$
• B. $1$
• C. $2$
• D. Infinite

Answer 1

Answer: (C)

Let, $f\left(x\right)=\left(cot\right)^{- 1} \sqrt{x \left(x + 4\right)}+\left(cos\right)^{- 1} ⁡ \sqrt{x^{2} + 4 x + 1}$
For $f\left(x\right)$ to be defined, we have
$x^{2}+4x\geq 0$ and $0\leq x^{2}+4x+1\leq 1$
Which is only possible if
$x^{2}+4x=0\Rightarrow x=0,-4$
Also these values satisfies the equation
$\therefore $ Number of solutions $=2$