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| Column $I$ | Column $II$ |
| $(A)$ The set $\left\{\operatorname{Re}\left(\frac{2 i z}{1-z^2}\right): z\right.$ is a complex number, $\left.|z|=1, z \neq \pm 1\right\}$ is | $(p)$ $(-\infty,-1) \cup(1, \infty)$ |
| $(B)$ The domain of the function $f(x)=\sin ^{-1}\left(\frac{8(3)^{x-2}}{1-3^{2(x-1)}}\right)$ is is | $(q)$ $(-\infty, 0) \cup(0, \infty)$ |
| $(C)$ If $f(\theta)=\left|\begin{array}{ccc}1 & \tan \theta & 1 \\ -\tan \theta & 1 & \tan \theta \\ -1 & -\tan \theta & 1\end{array}\right|$, then the set $\left\{f(\theta): 0 \leq \theta<\frac{\pi}{2}\right\}$ is | $(r)$ $[2, \infty)$ |
| $(D)$ If $f(x)=x^{3 / 2}(3 x-10), x \geq 0$, then $f(x)$ is increasing in | $(s)$ $(-\infty,-1] \cup[1, \infty)$ |
| $(t)$ $(-\infty, 0] \cup[2, \infty)$ |
$I.$ $f$ has a zero in $(0,1)$
$II.$ $f$ is monotone in $(0,1)$ Then,
$f(x)=\sin x-e^{x} \,\,\,\, \text { if } x \leq 0$
$\quad\quad\quad a+[-x] \,\,\,\, \text { if } 0\,<\,x\,<\,1$
$\quad\quad\quad 2 x-b \,\,\,\,\,\,\,\, \text { if } \geq 1$
where $[\mathrm{x}]$ is the greatest integer less than or equal to $\mathrm{x}$. If $\mathrm{f}$ is continuous on $\mathrm{R}$, then $(\mathrm{a}+\mathrm{b})$ is equal to:
$\begin{bmatrix}0&\text{i}\\\text{i}&0\end{bmatrix}$