Space geometry

Determine whether any of the lines $p$, $q$ or $r$ defined by the parametric equations given below passes through the coordinate origin. $$\begin{array}{rlrlrl}p:x& =-2+4t,& q:x& =-5-5s,& r:x& =3-6u,\\ y& =1-2t,& y& =2-2s,& y& =-\frac{1}{2}+u,\\ z& =-3+3t;t\in \mathbb{R}& z& =5+5s;s\in \mathbb{R}& z& =2-4u;u\in \mathbb{R}\end{array}$$

We are given points $A=[-2;3;0]$, $B=[6;1;6]$ and $C=[1;0;4]$. Find the parametric equations of a line $p$ that passes through the point $C$ and through the midpoint of the line segment $AB$.

Given points $A=[-4;1;4]$ and $B=[4;-3;0]$, determine which of the following parametric equations does not define the ray $AB$.

Find the missing coordinates of the points$M=[2;m;0]$ and $N=[0;3;n]$ so that they lie on a plane $\rho$ defined by the following parametric equations: $$\begin{array}{rl}\rho :x& =4+2s,\\ y& =-1-2t,\\ z& =1+t+s;t,s\in \mathbb{R}\end{array}$$ Choose the option in which values of both $m$ and $n$ are correct.