Consider the curve $C_1$ given by

$$ y=e^{-x} \quad \text { for } x \in[0,10 \pi], $$

and the curve $C_2$ given by

$$ y=e^{-x}(\sin x+\cos x) \quad \text { for } x \in[0,10 \pi] . $$

Let $n$ be the total number of points of intersection of the curves $C_1$ and $C_2$.

Suppose that $\alpha_1, \alpha_2, \ldots, \alpha_n \in[0,10 \pi]$ are the $x$-coordinates of the points of intersection of the curves $C_1$ and $C_2$ such that

$$ \alpha_1<\alpha_2<\cdots<\alpha_n . $$

The value of $n$ is $\_\_\_\_$ .

Let

$$ \alpha=\frac{1}{\sin 60^{\circ} \sin 61^{\circ}}+\frac{1}{\sin 62^{\circ} \sin 63^{\circ}}+\cdots+\frac{1}{\sin 118^{\circ} \sin 119^{\circ}} $$

Then the value of

$$ \left(\frac{\operatorname{cosec} 1^{\circ}}{\alpha}\right)^2 $$

is _____________.

Consider an obtuse angled triangle $A B C$ in which the difference between the largest and the smallest angle is $\frac{\pi}{2}$ and whose sides are in arithmetic progression. Suppose that the vertices of this triangle lie on a circle of radius 1.

$$ \text { Then the inradius of the triangle } A B C \text { is } $$ :

Let $\alpha$ and $\beta$ be real numbers such that $-\frac{\pi}{4}<\beta<0<\alpha<\frac{\pi}{4}$.

If $\sin (\alpha+\beta)=\frac{1}{3}$ and $\cos (\alpha-\beta)=\frac{2}{3}$, then the greatest integer less than or equal to

$$ \left(\frac{\sin \alpha}{\cos \beta}+\frac{\cos \beta}{\sin \alpha}+\frac{\cos \alpha}{\sin \beta}+\frac{\sin \beta}{\cos \alpha}\right)^{2} $$ is