Question

Let $f(x)=\sqrt{\cos ^{-1} \sqrt{1-x^2}-\sin ^{-1} x}$
Statement-1 : Range of $f ( x )$ is $[0, \sqrt{\pi}]$
Statement-2 : The value of the function increases as $x$ increases in its domain.

• A. Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1.
• B. Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1.
• C. Statement-1 is true, statement-2 is false.
• D. Statement-1 is false, statement-2 is true

Answer 1

Answer: (C)

For $x \geq 0 f ( x )=0$ and for $x <0$,
$f(x)=\sqrt{-2 \sin ^{-1} x}$
$\Rightarrow$ range of $f ( x )$ is $[0, \sqrt{\pi}]$
$\therefore$ Statement- 1 is true but Statement- 2 is false