Question

The function $f\left(x\right)={\sin }^{-1}(x^2-2x+2)$ is defined at $x=a$ and $f\left(a\right)=b.$ Then

• A. $b$ is an irrational number
• B. $a$ is a negative integer
• C. $b$ is a rational number
• D. $a$ is a positive integer

Answer 1

Answer: (A), (D)

$a$ is a positive integer

$f\left(x\right)={\sin }^{-1}(x^2-2x+2)$
Domain of ${\sin }^{-1}\left(x\right)$ is $[-1,1]$

$x^2-2x+2=(x-1{)}^2+1\ge 1$
Clearly, the function $f\left(x\right)$ is defined only when $x=1$
Substitute $x=1$ in the above expression, we get
$f\left(1\right)={\sin }^{-1}\left(1\right)=\frac{\pi }{2}$
$\Rightarrow a=1,b=\frac{\pi }{2}$