I2.1 If $0^{\circ} \leq \theta<360^{\circ}$, the equation in $\theta: 3 \cos \theta+\frac{1}{\cos \theta}=4$ has $p$ roots. Find $p$.
I2.2 If $x-\frac{1}{x}=p$ and $x^{3}-\frac{1}{x^{3}}=q$, find $q$.
I2.3 A circle is inscribed in an equilateral triangle of perimeter $q \mathrm{~cm}$. If the area of the circle is $k \pi \mathrm{cm}^{2}$, find $k$.

I2.4 Each interior angle of a regular polygon of $k$ sides is $m^{\circ}$. Find $m$.
$$
\begin{array}{l}
3 \cos ^{2} \theta+1=4 \cos \theta \\
\Rightarrow 3 \cos ^{2} \theta-4 \cos \theta+1=0 \\
\Rightarrow \cos \theta=\frac{1}{3} \text { or } 1 \\
p=3 \\
x-\frac{1}{x}=3 ;\left(x-\frac{1}{x}\right)^{2}=9 \\
\Rightarrow x^{2}+\frac{1}{x^{2}}=11 \\
q=x^{3}-\frac{1}{x^{3}} \\
=\left(x-\frac{1}{x}\right)\left(x^{2}+1+\frac{1}{x^{2}}\right) \\
=3(11+1)=36
\end{array}
$$

Let the equilateral triangle be $A B C$, the centre of the inscribed circle is $O$, which touches the triangle at $D$ and $E$, with radius $r \mathrm{~cm}$
$$
\begin{array}{l}
\text { Perimeter }=36 \mathrm{~cm} \\
\Rightarrow \text { Each side }=12 \mathrm{~cm} \\
\angle A C B=60^{\circ}(\angle \mathrm{s} \text { of an equilateral } \triangle \text { ) } \\
\angle O D C=90^{\circ} \text { (tangent } \perp \text { radius) } \\
\angle O C D=30^{\circ} \text { (tangent from ext. pt.) } \\
C D=6 \mathrm{~cm} \text { (tangent from ext. pt.) } \\
r=6 \tan 30^{\circ}=2 \sqrt{3}
\end{array}
$$

Area of circle $=\pi(2 \sqrt{3})^{2} \mathrm{~cm}^{2}=12 \pi \mathrm{cm}^{2}$
$$
k=12
$$

Angle sum of 12 -sides polygon $=180^{\circ}(12-2)=1800^{\circ}$
Each interior angle $=m^{\circ}=1800^{\circ} \div 12=150^{\circ}$
$$
m=150
$$