Inverse Circular Functions 1 Question 1
1. The number of real solutions of
$$ \tan ^{-1} \sqrt{x(x+1)}+\sin ^{-1} \sqrt{x^{2}+x+1}=\frac{\pi}{2} \text { is } $$
(a) zero
(b) one
(c) two
(d) infinite
Fill in the Blank
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Answer:
Correct Answer: 1. (c)
Solution:
- Given function is
$$ \tan ^{-1} \sqrt{x(x+1)}+\sin ^{-1} \sqrt{x^{2}+x+1}=\frac{\pi}{2} $$
Function is defined, if
(i) $x(x+1) \geq 0$, since domain of square root function.
(ii) $x^{2}+x+1 \geq 0$, since domain of square root function.
(iii) $\sqrt{x^{2}+x+1} \leq 1$, since domain of $\sin ^{-1}$ function.
From (ii) and (iii), $0 \leq x^{2}+x+1 \leq 1 \cap x^{2}+x \geq 0$
| $\Rightarrow$ | $0 \leq x^{2}+x+1 \leq 1 \cap x^{2}+x+1 \geq 1$ |
|---|---|
| $\Rightarrow$ | $x^{2}+x+1=1$ |
| $\Rightarrow$ | $x^{2}+x=0$ |
| $\Rightarrow$ | $x(x+1)=0$ |
| $\Rightarrow$ | $x=0, x=-1$ |
