Analytical Space Geometry

1003124001

Level: 
A
We are given a straight line \( q=\left\{[3t,2-2t,1+t]\text{, }t\in\mathbb{R}\right\} \) and four points \( A=[-6,6,-1] \), \( B=[-3,0,0] \), \( C=[0,2,1] \) and \( D=[3,0,2] \). Out of the given points select all that lie on the straight line \( q \). (Choose the corresponding option.)
\( A \), \( C \), \( D \)
\( B \), \( C \), \( D \)
\( B \), \( C \)
\( A \), \( B \), \( C \)

1003124002

Level: 
A
From the given options choose the parametric equations which describe a straight line \( p \) passing through the points \( A=[-2,0,1] \) and \( B=[2,0,-3] \).
\( \begin{aligned} p\colon x&=2-t, \\ y&=0, \\ z&=-3+t,\ t\in\mathbb{R} \end{aligned} \)
\( \begin{aligned} p\colon x&=2+4t, \\ y&=0, \\ z&=-3+4t,\ t\in\mathbb{R} \end{aligned} \)
\( \begin{aligned} p\colon x&=2, \\ y&=0, \\ z&=-3+t,\ t\in\mathbb{R} \end{aligned} \)
\( \begin{aligned} p\colon x&=2-2t, \\ y&=0, \\ z&=-3+t,\ t\in\mathbb{R} \end{aligned} \)

1003124003

Level: 
A
Find the missing coordinates of the point \( B=[x_B, y_B,-3] \) lying on a straight line \( p \) defined by the parametric equations \[\begin{aligned} p\colon x&=-1+\frac14m,\\ y&=2+m,\\ z&=5-m,\ m\in\mathbb{R}.\end{aligned} \]
\( B=[1,10,-3] \)
\( B=[-3,-6,-3] \)
\( B=[1,3,-3] \)
\( B=[-3,6,-3] \)

1003124004

Level: 
A
Find the value of a parameter \( a\in\mathbb{R} \) so that the point \( B=[1,4,5] \) lies on the straight line \( p \) defined by the parametric equations \[\begin{aligned}p\colon x&=-1+m,\\ y&=2+am,\\ z&=3+m,\ m\in\mathbb{R}. \end{aligned}\]
\( a=1 \)
\( a=-1 \)
\( a=2 \)
no such value of \( a \) exists

1003124005

Level: 
A
Find the value of a parameter \( a\in\mathbb{R} \) so that the point \( C=[2,0,6] \) lies on the straight line \( p \) defined by the parametric equations \[\begin{aligned}p\colon x&=-1+m,\\ y&=a+m,\\ z&=3+m,\ m\in\mathbb{R}.\end{aligned}\]
\( a=-3 \)
\( a=0 \)
\( a=-1 \)
no such values of \(a \) exists

1003124006

Level: 
A
Find the value of a parameter \( a\in\mathbb{R} \) so that the point \( D=[-2,1,1] \) lies on the straight line \( p \) defined by the parametric equations \[\begin{aligned}p\colon x&=1+m,\\ y&=-2+m,\\ z&=a+m,\ m\in\mathbb{R}. \end{aligned}\]
no such values of \(a \) exists
\( a=-1 \)
\( a=0 \)
\( a = 1\)

1003164401

Level: 
A
Let a straight line \( p \) be defined by parametric equations: \begin{align*} x&=-1+2t, \\ y&=2+t, \\ z&=5-t,\ t\in\mathbb{R}. \end{align*} Find the coordinates of the intersection point \( M \) of the line \( p \) with the \( xy \)-coordinate plane.
\( M=[9,7,0] \)
\( M=[0,0,5] \)
\( M=[-1,2,0] \)
\( M=[0,0,-1] \)

1003164402

Level: 
A
Let a straight line \( p \) be defined by parametric equations: \begin{align*} x&=-1+2t, \\ y&=2+t, \\ z&=5-t,\ t\in\mathbb{R}. \end{align*} Find the coordinates of the intersection point \( M \) of the line \( p \) with the \( xz \)-coordinate plane.
\( M=[-5,0,7] \)
\( M=[0,2,0] \)
\( M=[-1,0,5] \)
\( M=[2,0,-1] \)

1003164403

Level: 
A
Let a straight line $p$ be defined by parametric equations: \begin{align*} x&=-1+t, \\ y&=2+3t, \\ z&=5-t,\ t\in\mathbb{R}. \end{align*} Find the coordinates of the intersection point \( M \) of the line \( p \) with the \( yz \)-coordinate plane.
\( M=[0,5,4] \)
\( M=[-1,0,0] \)
\( M=[0,3,-1] \)
\( M=[1,0,0] \)

1003164404

Level: 
A
Let a straight line \( p \) be defined by parametric equations: \begin{align*} x&=3+t, \\ y&=2-t, \\ z&=4,\ t\in\mathbb{R}. \end{align*} Find the coordinates of the intersection point \( M \) of the line \( p \) with the \( xy \)-coordinate plane.
There is no such point \( M \).
\( M=[0,0,4] \)
\( M=[-3,2,0] \)
\( M=[1,-1,0] \)