Let $$R$$ be a relation on $$Z \times Z$$ defined by $$(a, b) R(c, d)$$ if and only if $$a d-b c$$ is divisible by 5. Then $$R$$ is

If $$f(x)=\left\{\begin{array}{cc}2+2 x, & -1 \leq x < 0 \\ 1-\frac{x}{3}, & 0 \leq x \leq 3\end{array} ; g(x)=\left\{\begin{array}{cc}-x, & -3 \leq x \leq 0 \\ x, & 0 < x \leq 1\end{array}\right.\right.$$, then range of $$(f o g)(x)$$ is

Let $$O$$ be the origin and the position vectors of $$A$$ and $$B$$ be $$2 \hat{i}+2 \hat{j}+\hat{k}$$ and $$2 \hat{i}+4 \hat{j}+4 \hat{k}$$ respectively. If the internal bisector of $$\angle \mathrm{AOB}$$ meets the line $$\mathrm{AB}$$ at $$\mathrm{C}$$, then the length of $$O C$$ is

In a $$\triangle A B C$$, suppose $$y=x$$ is the equation of the bisector of the angle $$B$$ and the equation of the side $$A C$$ is $$2 x-y=2$$. If $$2 A B=B C$$ and the points $$A$$ and $$B$$ are respectively $$(4,6)$$ and $$(\alpha, \beta)$$, then $$\alpha+2 \beta$$ is equal to