Inverse Circular Functions 1 Question 1

$A \rightarrow p ; B \rightarrow q ; C \rightarrow p ; D \rightarrow s$$A \rightarrow p ; B \rightarrow q ; C \rightarrow p ; D \rightarrow s$

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

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Answer:

Correct Answer: 1. (c)

Solution:

  1. 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), $0 \leq x^{2}+x+1 \leq 1 \cap x^{2}+x \geq 0$

$\Rightarrow$ $0 \leq x^{2}+x+1 \leq 1 \cap x^{2}+x+1 \geq 1$
$\Rightarrow$ $x^{2}+x+1=1$
$\Rightarrow$ $x^{2}+x=0$
$\Rightarrow$ $x(x+1)=0$
$\Rightarrow$ $x=0, x=-1$


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