If the tangent of the curve $x y+a x+b y=0$ at $(1,1)$ makes an angle $\tan ^{-1} 2$ with $X$-axis, then $\frac{a b}{a+b}=$

If the displacement $S$ of a particle travelling along a straight line in $t$ seconds is given by $S=2 t^3+2 t^2-2 t-3$, then the time taken (in second) by the particle to change its direction is

If the function $f(x)=x^3+b x^2+c x-6$ satisfies all the conditions of Rolle's theorem in $[1,3]$ and $f^{\prime}\left(\frac{2 \sqrt{3}+1}{\sqrt{3}}\right)=0$, then $b c=$

If the surface area of a spherical bubble is increasing at the rate of $4 \mathrm{sq} . \mathrm{cm} / \mathrm{sec}$, then the rate of change in its volume (in cubic $\mathrm{cm} / \mathrm{sec}$ ) when its radius is 8 cms is