I1.1 Find $a$ if $2 t+1$ is a factor of $4 t^{2}+12 t+a$.
I1.2 $\sqrt{K}$ denotes the nonnegative square root of $K$, where $K \geq 0$. If $b$ is the root of the equation $\sqrt{a-x}=x-3$, find $b$.

I1.3 If $c$ is the greatest value of $\frac{20}{b+2 \cos \theta}$, find $c$.
I1.4 A man drives a car at $3 c \mathrm{~km} / \mathrm{h}$ for 3 hours and then $4 c \mathrm{~km} / \mathrm{h}$ for 2 hours. If his average speed for the whole journey is $d \mathrm{~km} / \mathrm{h}$, find $d$.
Let $f(t)=4 t^{2}+12 t+a$
$$
\begin{array}{l}
f\left(-\frac{1}{2}\right)=4\left(-\frac{1}{2}\right)^{2}+12\left(-\frac{1}{2}\right)+a=0 \\
a=5 \\
(\sqrt{5-x})^{2}=(x-3)^{2} \\
\Rightarrow 5-x=x^{2}-6 x+9 \\
\Rightarrow x^{2}-5 x+4=0 \\
\Rightarrow x=1 \text { or } 4
\end{array}
$$

When $x=1$, LHS $=2 \neq-1=$ RHS
When $x=4$, LHS $=1=$ RHS.
$$
\begin{array}{l}
\therefore x=b=4 \\
\frac{20}{b+2 \cos \theta}=\frac{20}{4+2 \cos \theta}=\frac{10}{2+\cos \theta} \\
c=\text { the greatest value }=\frac{10}{2-1}=10
\end{array}
$$

Total distance travelled $=(30 \times 3+40 \times 2) \mathrm{km}=170 \mathrm{~km}$
$$
d=\frac{170}{3+2}=34
$$