If $$I=\int \frac{d x}{\sin (x-a) \sin (x-b)}$$, then I is given by

Let $$\mathrm{P}(x)$$ be a polynomial of degree 2, with $$\mathrm{P}(2)=-1, \mathrm{P}^{\prime}(2)=0, \mathrm{P}^{\prime \prime}(2)=2$$, then $$\mathrm{P}(1.001)$$ is

If $$y=\sqrt{(x-\sin x)+\sqrt{(x-\sin x)+\sqrt{(x-\sin x) \ldots.}}}$$, then $$\frac{\mathrm{d} y}{\mathrm{~d} x}=$$

Let $$\mathrm{f}(x)=5-|x-2|$$ and $$\mathrm{g}(x)=|x+1|, x \in \mathrm{R}$$ If $$\mathrm{f}(x)$$ attains maximum value at $$\alpha$$ and $$\mathrm{g}(x)$$ attains minimum value at $$\beta$$, then $$\lim _\limits{x \rightarrow-\alpha \beta} \frac{(x-1)\left(x^2-5 x+6\right)}{x^2-6 x+8}$$ is equal to