I3.1 If $a=\frac{\sin 15^{\circ}}{\cos 75^{\circ}}+\frac{1}{\sin ^{2} 75^{\circ}}-\tan ^{2} 15^{\circ}$, find $a$.
I3.2 If the lines $a x+2 y+1=0$ and $3 x+b y+5=0$ are perpendicular to each other, find $b$.
I3.3 The three points $(2, b),(4,-b)$ and $\left(5, \frac{c}{2}\right)$ are collinear. Find $c$.
I3.4 If $\frac{1}{x}: \frac{1}{y}: \frac{1}{z}=3: 4: 5$ and $\frac{1}{x+y}: \frac{1}{y+z}=9 c: d$, find $d$.
$$
\begin{aligned}
a & =\frac{\sin 15^{\circ}}{\sin 15^{\circ}}+\sec ^{2} 15^{\circ}-\tan ^{2} 15^{\circ} \\
& =1+1=2 \\
- & \frac{a}{2} \times\left(-\frac{3}{b}\right)=-1 \\
\Rightarrow b & =-3
\end{aligned}
$$

The three points are $(2,-3),(4,3)$ and $\left(5, \frac{c}{2}\right)$, so their slopes are equal.
$$
\begin{array}{l}
\frac{3-(-3)}{4-2}=\frac{\frac{c}{2}-3}{5-4} \\
\Rightarrow \frac{c}{2}-3=3 \\
\Rightarrow c=12 \\
x: y: z=\frac{1}{3}: \frac{1}{4}: \frac{1}{5} \\
=\frac{20}{60}: \frac{15}{60}: \frac{12}{60} \\
=20: 15: 12 \\
x=20 k, y=15 k, z=12 k \\
\frac{1}{x+y}: \frac{1}{y+z}=\frac{1}{20 k+15 k}: \frac{1}{15 k+12 k} \\
=27: 35 \\
\text { = 108: } 140=9 c: d \\
\Rightarrow d=140 \\
\end{array}
$$