The value of $$\int(1-\cos x) \cdot \operatorname{cosec}^2 x d x$$ is

If $$\cos ^{-1} \sqrt{\mathrm{p}}+\cos ^{-1} \sqrt{1-\mathrm{p}}+\cos ^{-1} \sqrt{1-\mathrm{q}}=\frac{3 \pi}{4}$$, then $$\mathrm{q}$$ is

The slope of the tangent to a curve $$y=\mathrm{f}(x)$$ at $$(x, \mathrm{f}(x))$$ is $$2 x+1$$. If the curve passes through the point $$(1,2)$$, then the area (in sq. units), bounded by the curve, the $$\mathrm{X}$$-axis and the line $$x=1$$, is

$$A$$ rod $$A B, 13$$ feet long moves with its ends $$A$$ and $$B$$ on two perpendicular lines $$O X$$ and $$O Y$$ respectively. When $$A$$ is 5 feet from $$O$$, it is moving away at the rate of $$3 \mathrm{feet} / \mathrm{sec}$$. At this instant, $$\mathrm{B}$$ is moving at the rate