I1.1 Let $a$ be a real number and $\sqrt{a}=\sqrt{7+\sqrt{13}}-\sqrt{7-\sqrt{13}}$. Find the value of $a$.
I1.2 In Figure 1, the straight line $\ell$ passes though the point $(a, 3)$, and makes an angle $45^{\circ}$ with the $x$-axis. If the equation of $\ell$ is $x+m y+n=0$ and $b=|1+m+n|$, find the value of $b$.

I1.3 If $x-b$ is a factor of $x^{3}-6 x^{2}+11 x+c$, find the value of $c$.
I1.4 If $\cos x+\sin x=-\frac{c}{5}$ and $d=\tan x+\cot x$, find the value of $d$.
$\begin{array}{l}(\sqrt{a})^{2}=(\sqrt{7+\sqrt{13}}-\sqrt{7-\sqrt{13}})^{2} \\ a=7+\sqrt{13}-2 \sqrt{7^{2}-\sqrt{13}^{2}}+7-\sqrt{13} \\ \quad=14-2 \sqrt{36}=2 \\ \ell: \frac{y-3}{x-2}=\tan 45^{\circ} \\ y-3=x-2 \\ x-y+1=0, m=-1, n=1 \\ b=|1-1+1|=1 \\ f(x)=x^{3}-6 x^{2}+11 x+c \\ f(1)=1-6+11+c=0 \\ c=-6 \\ \cos x+\sin x=\frac{6}{5} \\ (\cos x+\sin x)^{2}=\frac{36}{25} \\ 1+2 \sin x \cos x=\frac{36}{25} \\ 2 \sin x \cos x=\frac{11}{25} \Rightarrow \sin x \cos x=\frac{11}{50} \\ d=\tan x+\cot x=\frac{\sin x}{\cos x}+\frac{\cos x}{\sin x}=\frac{\sin ^{2} x+\cos ^{2} x}{\sin x \cos _{x}}=\frac{1}{\sin x \cos x}=\frac{50}{11}\end{array}$