First slide

Properties of ITF

Question

The number of real solutions of $\tan^{- 1} \sqrt{x (x + 1)} + \sin^{- 1} \sqrt{x^{2} + x + 1} = \frac{π}{2}$ is

Moderate

A

0
B

1
C

2
D

infinite

Solution

$\begin{array}{l} \tan^{- 1} \sqrt{x (x + 1)} + \sin^{- 1} \sqrt{x^{2} + x + 1} = \frac{π}{2} \\ \Rightarrow \cos^{- 1} \frac{1}{\sqrt{1 + {(x^{2} + x)}^{2}}} + \sin^{- 1} \sqrt{x^{2} + x + 1} = \frac{π}{2} \\ \Rightarrow \cos^{- 1} \frac{1}{\sqrt{1 + {(x^{2} + x)}^{2}}} = \frac{π}{2} - \sin^{- 1} \sqrt{x^{2} + x + 1} \\ = \cos^{- 1} \sqrt{x^{2} + x + 1} \\ \Rightarrow \frac{1}{\sqrt{1 + {(x^{2} + x)}^{2}}} = \sqrt{x^{2} + x + 1} \\ \Rightarrow (x^{2} + x + 1) [{(x^{2} + x)}^{2} + 1] = 1 \\ \Rightarrow {(x^{2} + x)}^{3} + {(x^{2} + x)}^{2} + (x^{2} + x) = 0 \\ \Rightarrow x^{2} + x = 0 \Rightarrow x = - 1, 0 \end{array}$

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