Consider the parabola $\mathrm{P}: y^2=4 k x$ and the ellipse $\mathrm{E}: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. Let the line segment joining the points of intersection of P and E , be their latus rectums. If the eccentricity of E is $e$, then $e^2+2 \sqrt{2}$ is equal to $\_\_\_\_$ .

If $\mathrm{A}=\frac{\sin 3^{\circ}}{\cos 9^{\circ}}+\frac{\sin 9^{\circ}}{\cos 27^{\circ}}+\frac{\sin 27^{\circ}}{\cos 81^{\circ}}$ and $\mathrm{B}=\tan 81^{\circ}-\tan 3^{\circ}$, then $\frac{\mathrm{B}}{\mathrm{A}}$ is equal to

$\_\_\_\_$ .

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

The number of points, at which the function $f(x)=\max \left\{6 x, 2+3 x^2\right\}+|x-1| \cos \left|x^2-\frac{1}{4}\right|, x \in(-\pi, \pi)$, is not differentiable, is

$\_\_\_\_$ .