Equations with Solutions at Boundary Values

Q. The number of real solutions of the equation
${\sin }^{-1}(\sum _{i=1}^{\infty }{x}^{i+1}-x\sum _{i=1}^{\infty }{\left(\frac{x}{2}\right)}^i)=\frac{\pi }{2}-{\cos }^{-1}(\sum _{i=1}^{\infty }{(-\frac{x}{2})}^i-\sum _{i=1}^{\infty }(-x{)}^i)$
lying in the interval $(-\frac{1}{2},\frac{1}{2})$ is .
(Here, the inverse trigonometric functions ${\sin }^{-1}x$ and
${\cos }^{-1}x$ assume values in $[-\frac{\pi }{2},\frac{\pi }{2}]$ and $[0,\pi ]$, respectively.)

Q. The minimum value of $f\left(x\right)=|x-1|+|x-2|+|x-3|$ is

1. $2$
2. $6$
3. $3$
4. $1$

Q. If equation in variable $\theta$, $3\tan (\theta -\alpha )=\tan (\theta +\alpha )$, (where $\alpha$ is constant), has no real solution, then $\alpha$ can be (wherever $\tan (\theta -\alpha )$ and $\tan (\theta +\alpha )$ both are defined)

1. $\frac{\pi }{15}$
2. $\frac{5\pi }{18}$
3. $\frac{5\pi }{12}$
4. $\frac{17\pi }{18}$

Q. If equation in variable $\theta$, $3\tan (\theta -\alpha )=\tan (\theta +\alpha )$, (where $\alpha$ is constant), has no real solution, then $\alpha$ can be (wherever $\tan (\theta -\alpha )$ and $\tan (\theta +\alpha )$ both are defined)

1. $\frac{\pi }{15}$
2. $\frac{5\pi }{18}$
3. $\frac{5\pi }{12}$
4. $\frac{17\pi }{18}$