Consider the ellipse $E$ given by $\frac{x^2}{18}+\frac{y^2}{12}=1$. Let $H$ be the hyperbola whose eccentricity is the reciprocal of the eccentricity of $E$ and whose foci are the same as that of $E$. Let $P$ and $Q$ be the points of intersection of $H$ and the parabola $\sqrt{5} y=x^2$ in the first quadrant. Let $d$ be the distance between $P$ and $Q$.

If $a$ and $b$ are the integers such that $d^2=a+b \sqrt{5}$, then the value of $a-b$ is $\_\_\_\_$ .

For a real number $\alpha$, let $[\alpha]$ denote the greatest integer less than or equal to $\alpha$. For a finite set $S$, let $|S|$ denote the number of elements in the set $S$.

Consider the functions $f:(-3,3) \rightarrow(-\infty, \infty)$ and $g:(-3,3) \rightarrow(-\infty, \infty)$ defined by

$$ f(x)=\left[x^3\right] \log _e\left(1+\sin ^2(\pi(x-[x]))\right) $$

and

$$ g(x)=x^3 \sin ^2\left(\pi \log _e(1+x-[x])\right) . $$

Let

$$ A=\{x \in(-3,3): f \text { is discontinuous at } x\} $$

and

$$ B=\{x \in(-3,3): g \text { is discontinuous at } x\} . $$

Then the value of $|A|+2|B|-|A \cap B|$ is $\_\_\_\_$ .

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 $\_\_\_\_$ .

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

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

Let $\beta$ be the area of the region enclosed between the curves $C_1, C_2$, and the lines $x=\alpha_1$ and $x=\alpha_4$. Then the value of

$$ -\frac{1}{\pi} \log _e\left(\beta-2 e^{-\frac{\pi}{2}}\right) $$

is $\_\_\_\_$ .