If $$Z_1=2+i$$ and $$Z_2=3-4 i$$ and $$\frac{\overline{Z_1}}{\overline{Z_2}}=a+b i$$, then the value of $$-7 a+b$$ is (where $$i=\sqrt{-1}$$ and $$a, b \in R)$$

Let $$\alpha \in\left(0, \frac{\pi}{2}\right)$$ be fixed. If the integral $$\int \frac{\tan x+\tan \alpha}{\tan x-\tan \alpha} \mathrm{d} x=\mathrm{A}(x) \cos 2 \alpha+\mathrm{B}(x) \sin 2 \alpha+\mathrm{c},$$ (where $$\mathrm{c}$$ is a constant of integration), then functions $$\mathrm{A}(x)$$ and $$\mathrm{B}(x)$$ are respectively

Two adjacent sides of a parallelogram are $$2 \hat{i}-4 \hat{j}+5 \hat{k}$$ and $$\hat{i}-2 \hat{j}-3 \hat{k}$$, then the unit vector parallel to its diagonal is

A water tank has a shape of inverted right circular cone whose semi-vertical angle is $$\tan ^{-1}\left(\frac{1}{2}\right)$$. Water is poured into it at constant rate of 5 cubic meter/minute. The rate in meter/ minute at which level of water is rising, at the instant when depth of water in the tank is $$10 \mathrm{~m}$$ is