Let $\overrightarrow{a_k}=\left(\tan \theta_k\right) \hat{i}+\hat{j}$ and $\overrightarrow{b_k}=\hat{i}-\left(\cot \theta_k\right) \hat{j}$, where $\theta_k=\frac{2^{k-1} \pi}{2^n+1}$, for some $n \in \mathbb{N}, n>5$. Then the value of $\frac{\sum\limits_{k=1}^n\left|\overrightarrow{a_k}\right|^2}{\sum\limits_{k=1}^n\left|\overrightarrow{b_k}\right|^2}$ is

Let $\cos (\alpha+\beta)=-\frac{1}{10}$ and $\sin (\alpha-\beta)=\frac{3}{8}$, where $0<\alpha<\frac{\pi}{3}$ and $0<\beta<\frac{\pi}{4}$. If $\tan 2 \alpha=\frac{3(1-r \sqrt{5})}{\sqrt{11}(s+\sqrt{5})}, r, s \in N$, then $r+s$ is equal to $\_\_\_\_$ .

$$ \text { If } \frac{\cos ^2 48^{\circ}-\sin ^2 12^{\circ}}{\sin ^2 24^{\circ}-\sin ^2 6^{\circ}}=\frac{\alpha+\beta \sqrt{5}}{2} \text {, where } \alpha, \beta \in \mathbb{N} \text {, then } \alpha+\beta \text { is equal to ___________} $$