Let $$\mathrm{f}(x)=\log (\sin x), 0 < x < \pi$$ and $$\mathrm{g}(x)=\sin ^{-1}\left(\mathrm{e}^{-x}\right), x \geq 0$$. If $$\alpha$$ is a positive real number such that $$\mathrm{a}=(\mathrm{fog})^{\prime}(\alpha)$$ and $$\mathrm{b}=(\mathrm{fog})(\alpha)$$, then

Five students are to be arranged on a platform such that the boy $$B_1$$ occupies the second position and such that the girl $$G_1$$ is always adjacent to the girl $$G_2$$. Then, the number of such possible arrangements is

If the volume of the parallelopiped is $$158 \mathrm{~cu}$$. units whose coterminus edges are given by the vectors $$\bar{a}=(\hat{i}+\hat{j}+n \hat{k}), \bar{b}=2 \hat{i}+4 \hat{j}-n \hat{k}$$ and $$\bar{c}=\hat{i}+n \hat{j}+3 \hat{k}$$, where $$n \geq 0$$, then the value of $$n$$ is

Let $$\mathrm{f}(x)$$ be positive for all real $$x$$. If $$\mathrm{I}_1=\int_\limits{1-\mathrm{h}}^{\mathrm{h}} x \mathrm{f}(x(1-x)) \mathrm{d} x$$ and $$\mathrm{I}_2=\int_\limits{1-\mathrm{h}}^{\mathrm{h}} \mathrm{f}(x(1-x)) \mathrm{d} x$$, where $$(2 h-1)>0$$, then $$\frac{I_1}{I_2}$$ is