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$f(x) = \left\{ {\begin{array}{*{20}{c}}
{x + 2,}&{if\,\,x\,\, < \,\,1}\\
{0,}&{if\,\,\,x = 1}\\
{x - 2,}&{if\,\,x\,\, > \,\,1}
\end{array}} \right.$
$\sin ^{-1}(a x)+\cos ^{-1}(y)+\cos ^{-1}(b x y)=\frac{\pi}{2} .$
Match the statements in Column $I$ with the statements in Column $II$ and indicate your answer by darkening the appropriate bubbles in the $4 \times 4$ matrix given in the $ORS$.
| Column $I$ | Column $II$ |
| $(A)$ If $a=1$ and $b=0$, then ( $x, y$ ) | $(p)$ lies on the circle $x^2+y^2=1$ |
| $(B)$ If $a=1$ and $b=1$, then $(x, y)$ | $(q)$ lies on $\left(x^2-1\right)\left(y^2-1\right)=0$ |
| $(C)$ If $a=1$ and $b=2$, then ( $x, y)$ | $(r)$ lies on $y=x$ |
| $(D)$ If $a=2$ and $b=2$, then $(x, y)$ | $(s)$ lies on $\left(4 x^2-1\right)\left(y^2-1\right)=0$ |