If $$y=\sin \left(2 \tan ^{-1} \sqrt{\frac{1+x}{1-x}}\right)$$ then $$\frac{\mathrm{d} y}{\mathrm{~d} x}$$ is equal to

The magnitude of the projection of the vector $$2 \hat{\mathbf{i}}+ 3\hat{\mathbf{j}}+\hat{\mathbf{k}}$$ on the vector perpendicular to the plane containing the vectors $$\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$$ and $$\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$$ is

$$\matrix{ {f(x) = a{x^2} + bx + 1,} & {if} & {\left| {2x - 3} \right| \ge 2} \cr { = 3x + 2,} & {if} & {{1 \over 2} < x < {5 \over 2}} \cr } $$

is continuous on its domain, then $$a+b$$ has the value

If $$\bar{a}=\hat{\boldsymbol{i}}+\hat{\boldsymbol{j}}+\hat{\boldsymbol{k}}, \bar{b}=\hat{\boldsymbol{i}}-\hat{\boldsymbol{j}}+\hat{\boldsymbol{k}}$$ and $$\bar{c}=\hat{\boldsymbol{i}}-\hat{\boldsymbol{j}}-\hat{\boldsymbol{k}}$$ are three vectors then vector $$\bar{r}$$ in the plane of $$\bar{a}$$ and $$\bar{b}$$, whose projection on $$\bar{c}$$ is $$\frac{1}{\sqrt{3}}$$, is given by