Analytic geometry in a space

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 \( M \).
\( M=[0;0;4] \)
\( M=[-3;2;0] \)
\( M=[1;-1;0] \)