Let $\mathrm{A}=\{1,4,7\}$ and $\mathrm{B}=\{2,3,8\}$. Then the number of elements, in the relation $R=\left\{\left(\left(a_1, b_1\right),\left(a_2, b_2\right)\right) \in((A \times B) \times(A \times B)): a_1+b_2\right.$ divides $\left.a_2+b_1\right\}$ is $\_\_\_\_$ .

From the point $(-1,-1)$, two rays are sent making angles of $45^{\circ}$ with the line $x+y=0$. These rays get reflected from the mirror $x+2 y=1$. If the equations of the reflected rays are $\mathrm{a} x+\mathrm{b} y=9$ and $c x+d y=7, a, b, c, d \in \mathbf{Z}$, then the value of $a d+b c$ is $\_\_\_\_$ .

If $S=\left\{\theta \in[-\pi, \pi]: \cos \theta \cos \frac{5 \theta}{2}=\cos 7 \theta \cos \frac{7 \theta}{2}\right\}$, then $n(S)$ is equal to $\_\_\_\_$ .

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a function such that $f(x)+3 f\left(\frac{\pi}{2}-x\right)=\sin x, x \in \mathbf{R}$. Let the maximum value of $f$ on $\mathbf{R}$ be $\alpha$. If the area of the region bounded by the curves $g(x)=x^2$ and $h(x)=\beta x^3, \beta>0$, is $\alpha^2$, then $30 \beta^3$ is equal to $\_\_\_\_$ .