If a circle passes through points $$(4,0)$$ and $$(0,2)$$ and its centre lies on $$\mathrm{Y}$$-axis. If the radius of the circle is $$r$$, then the value of $$r^2-r+1$$ is

If $$\mathrm{A}=\left[\begin{array}{ll}\mathrm{i} & 1 \\ 1 & 0\end{array}\right]$$ where $$\mathrm{i}=\sqrt{-1}$$ and $$\mathrm{B}=\mathrm{A}^{2029}$$, then $$\mathrm{B}^{-1}=$$

The solution of the differential equation $$\frac{\mathrm{d} y}{\mathrm{~d} x}+\frac{y}{x}=\sin x$$ is

Let $$f:[-1,2] \rightarrow[0, \infty)$$ be a continuous function such that $$\mathrm{f}(x)=\mathrm{f}(1-x), \forall x \in[-1,2]$$

Let $$\mathrm{R}_1=\int_{-1}^2 x \mathrm{f}(x) \mathrm{d} x$$ and $$\mathrm{R}_2$$ be the area of the region bounded by $$y=\mathrm{f}(x), x=-1, x=2$$ and the $$\mathrm{X}$$-axis, then $$\mathrm{R}_2$$ is