I4.1 Solve the equation $2^{a+1}+2^{a}+2^{a-1}=112$ in $a$.
I4.2 If $a$ is one root of the equation $x^{2}-b x+35=0$, find $b$.
I4.3 If $\sin \theta=\frac{-b}{15}$, where $180^{\circ}<\theta<270^{\circ}$, and $\tan \theta=\frac{c}{3}$, find $c$.
I4.4 The probability of getting a sum of $c$ in throwing two dice is $\frac{1}{d}$. Find $d$.
$$
\begin{array}{l}
2^{a} \cdot\left(2+1+\frac{1}{2}\right)=112 \\
2^{a}=32 \\
a=5
\end{array}
$$

Method 2
$$
\begin{array}{l}
112=64+32+16=2^{6}+2^{5}+2^{4} \\
a=5
\end{array}
$$

One root of $x^{2}-b x+35=0$ is 5
$$
\begin{array}{l}
\Rightarrow 5^{2}-5 b+35=0 \\
\Rightarrow b=12 \\
\sin \theta=-\frac{12}{15}=-\frac{4}{5} \\
\Rightarrow \tan \theta=\frac{4}{3} \\
\Rightarrow c=4 \\
\mathrm{P}(\text { sum }=4)=\mathrm{P}((1,3),(2,2),(3,1)) \\
\quad=\frac{3}{36}=\frac{1}{12}=\frac{1}{d} \\
\Rightarrow d=12
\end{array}
$$