Shortest Distance between Two Skew Lines

Q.

if normal to the curve $y=f\left(x\right)$is parallel to $x$-axis then correct statement is

1. $\frac{dy}{dx}=0$

2. $\frac{dy}{dx}=1$

3. $\frac{dx}{dy}=0$

4. None of these

Q. Let $z_1$ and $z_2$ be two distinct complex numbers and let $z=(1-t)z_1+t{z}_2$ for some real number $t$ with $0<t<1$. If arg$\left(\omega \right)$ denotes the principal argument of a non-zero complex number $\omega$, then

1. $|z-z_1|+|z+z_2|=|z_1-z_2|$
2. arg$(z-z_1)$ = arg$(z-z_2)$
3. $\left|\begin{array}{cc}z-z_1& \overline{z}-\overline{z_1}\\ z_2-z_1& \overline{z_2}-\overline{z_1}\end{array}\right|=0$
4. arg$(z-z_1)$ = arg$(z_2-z_1)$

Q. The shortest distance between the lines
$\frac{x+1}{7}=\frac{y+1}{-6}=\frac{z+1}{1}$ and
$\frac{x-3}{1}=\frac{y-5}{-2}=\frac{z-7}{1}$ is

1. 11
2. 12
3. $2\sqrt{29}$
4. $2\sqrt{30}$

Q. The shortest distance between the lines
$\frac{x-3}{3}=\frac{y-8}{-1}=\frac{z-3}{1}$ and
$\frac{x-3}{-3}=\frac{y-7}{2}=\frac{z-6}{4}$

1. $\sqrt{10}$
2. $2\sqrt{20}$
3. $3\sqrt{30}$
4. $4\sqrt{40}$

Q. Let $z_1$ and $z_2$be nth roots of unity which subtend a right angle at the origin, then n must be of the form (where, k is an integer)

1. 4k+1
2. 4k+2
3. 4k+3
4. 4k