The integral $\int\limits_0^1 \cot ^{-1}\left(1+x+x^2\right) d x$ is equal to :

From a month of 31 days, 3 different dates are selected at random. If the probability that these dates are in an increasing A.P. is equal to $\frac{a}{b}$, where $a, b \in$ N and $\operatorname{gcd}(a, b)=1$, then $a+b$ is equal to $\_\_\_\_$

Let $f(x)=\left\{\begin{array}{cc}e^{x-1} & , x<0 \\ x^2-5 x+6 & , x \geq 0\end{array}\right.$ and $g(x)=f(|x|)+|f(x)|$. If the number of points where $g$ is not continuous and is not differentiable are $\alpha$ and $\beta$ respectively, then $\alpha+\beta$ is equal to $\_\_\_\_$

Let $\mathrm{A}, \mathrm{B}$ be points on the two half-lines $x-\sqrt{3}|y|=\alpha, \alpha>0$ at a distance of $\alpha$ from their point of intersection $P$. The line segment $A B$ meets the angle bisector of the given half-lines at the point $Q$. If $P Q=\frac{9}{2}$ and $R$ is the radius of the circumcircle of $\triangle \mathrm{PAB}$, then $\frac{\alpha^2}{R}$ is equal to $\_\_\_\_$