I1.1 If the interior angles of a $P$-sided polygon form an Arithmetic Progression and the smallest and the largest angles are $20^{\circ}$ and $160^{\circ}$ respectively. Find the value of $P$.

I1.2 In $\triangle A B C, A B=5, A C=6$ and $B C=p$. If $\frac{1}{Q}=\cos 2 A$, find the value of $Q$.
I1.3 If $\log _{2} Q+\log _{4} Q+\log _{8} Q=\frac{R}{2}$, find the value of $R$.
I1.4 If the product of the numbers $R$ and $\frac{11}{S}$ is the same as their sum, find the value of $S$.
$$
\begin{array}{l}
\text { Sum of all interior angles }=\frac{P}{2}\left(20^{\circ}+160^{\circ}\right)=180^{\circ}(P-2) \\
90 P=180 P-360 \\
\Rightarrow P=4
\end{array}
$$
(Hint: $\cos 2 A=2 \cos ^{2} A-1$ )
$$
\begin{array}{l}
\cos A=\frac{6^{2}+5^{2}-4^{2}}{2 \times 6 \times 5}=\frac{3}{4} \\
\cos 2 \mathrm{~A}=2 \cos ^{2} A-1 \\
=2 \times\left(\frac{3}{4}\right)^{2}-1=\frac{1}{8} \\
Q=8 \\
\frac{R}{2}=\log _{2} 8+\log _{4} 8+\log _{8} 8 \\
=3+\frac{3}{2}+1=\frac{11}{2} \\
R=11 \\
11 \times \frac{11}{S}=11+\frac{11}{S} \\
\Rightarrow \frac{110}{S}=11 \\
S=10 \\
\end{array}
$$