The equation of the tangent to the curve $\left(1+x^2\right) y=2-x$, where it crosses the X -axis, is

If $y=\alpha \log x+\beta x^3-x$ has extreme values at $x=-1$ and $x=1$, then $\alpha$ and $\beta$ are respectively

The length of the perpendicular drawn from the origin on the normal to the curve $x^2+2 x y-3 y^2=0$ at the point $(2,2)$ is

If $\mathrm{f}(x)=\log (1+x)-\frac{2 x}{2+x}$ then $\mathrm{f}(x)$ is increasing in