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

$\_\_\_\_$ .

In a screw gauge when the circular scale is given five complete rotations it moves linearly by 2.5 mm . If the circular scale has 100 divisions, the least count of screw gauge is $\_\_\_\_$ mm.

The increase in the pressure required to decrease the volume ( $\Delta V$ ) of water is $6.3 \times 10^7 \mathrm{~N} / \mathrm{m}^2$. The percentage decrease in the volume is $\_\_\_\_$ .

(Bulk modulus of water $=2.1 \times 10^9 \mathrm{~N} / \mathrm{m}^2$.)