Space geometry

1103189004

Level: 
B
We are given the point \( A=[2;-1;-4] \) and planes \( \rho \) by \( x-y+3z-5=0 \) and \( \sigma \) by \( 2x-y-z-8=0 \). Find the general form of the equation of the plane \( \alpha \) which passes through the point \( A \) and is perpendicular to both planes (see the picture).
\( \alpha\colon 4x+7y+z+3=0 \)
\( \alpha\colon -2x+5y-3z-3=0 \)
\( \alpha\colon 4x-7y+z+3=0 \)
\( \alpha\colon 2x-5y+3z+3=0 \)

1103189003

Level: 
B
Find the general form of the equation of the plane \( \beta \) that passes through the straight line \( p \) given by parametric equations \begin{align*} x&=1+2t, \\ y&=-2t, \\ z&=1+t;\ t\in\mathbb{R}, \end{align*} and is perpendicular to the plane \( \alpha \) given by \( x+3y-z-7=0 \) (see the picture).
\( \beta\colon x-3y-8z+7=0 \)
\( \beta\colon 2x-2y+z-3=0 \)
\( \beta\colon x-3y-8z-7=0 \)
\( \beta\colon 2x-2y+z+3=0 \)

1103189002

Level: 
B
Find the general form of the equation of the plane \( \beta \) that passes through the points \( M=[-1;1;-3] \) and \( N=[0;2;-1] \) and is perpendicular to the plane \( \alpha \) given by \( 3x-y+2=0 \) (see the picture).
\( \beta\colon x+3y-2z-8=0 \)
\( \beta\colon x+3z+10=0 \)
\( \beta\colon x+3z+3=0 \)
\( \beta\colon x+3y-2z+8=0 \)

1103189001

Level: 
B
Find the general form of the equation of the plane \( \alpha \) that is perpendicular to the straight line \( p \) given by: \begin{align*} x&=7+t, \\ y&=2t, \\ z&=4-t;\ t\in\mathbb{R}, \end{align*} and passes through the point \( A=[1;0;4] \). Consequently, find the coordinates of the point \( B \) which is the point of intersection of \( p \) and \( \alpha \) (see the picture).
\( \alpha\colon x+2y-z+3=0;\ B=[6;-2;5] \)
\( \alpha\colon x+2y-z-3;\ B=[6;-2;5] \)
\( \alpha\colon x+2y-z-3=0;\ B=[8;2;3] \)
\( \alpha\colon x+2y-z+3=0;\ B=[8;2;3] \)

1003188907

Level: 
A
We are given two intersecting planes \( x-6y+9z-4=0 \) and \( x-2y+3z-4=0 \). Find the parametric equations of their line of intersection \( p \).
\( \begin{aligned} p\colon x&=4, \\ y&=\phantom{4+}\ 3t, \\ z&=\phantom{4+}\ 2t;\ t\in\mathbb{R} \end{aligned} \)
\( \begin{aligned} p\colon x&=4+t , \\ y&=\phantom{4+}\ 3t , \\ z&=\phantom{4+}\ 2t;\ t\in\mathbb{R} \end{aligned} \)
\( \begin{aligned} p\colon x&=4, \\ y&=\frac32+3t, \\ z&=1+2t;\ t\in\mathbb{R} \end{aligned} \)
\( \begin{aligned} p\colon x&=4+t, \\ y&=\frac32+3t, \\ z&=1+2t;\ t\in\mathbb{R} \end{aligned} \)

1003188906

Level: 
A
Let there be planes \( \alpha \), \( \beta \), \( \gamma \) and \( \delta \) defined by their general equations: \[ \begin{aligned} &\alpha\colon \frac23x-4y+6z-\frac83=0; \\ &\beta\colon x-2y+3z-4=0; \\ &\gamma\colon 2x-12y+18z-4 =0; \\ &\delta\colon x-6y+9z-4 =0. \end{aligned} \] Out of the following statements, select the one that is not true.
\( \alpha \parallel\delta\text{, }\alpha\neq\delta \)
Planes \( \beta \) and \( \delta \) are intersecting.
\( \gamma\parallel\delta\text{, }\gamma\neq\delta \)
Planes \( \alpha \) and \( \beta \) are intersecting.
\( \alpha = \delta \)

1003188905

Level: 
A
Determine the relative position of the plane \( \rho \) with general equation \( 5x-4y+z-4=0 \) and the straight line \( p \) with parametric equations: \[ \begin{aligned} x&=-1+t,\\ y&=2-2t,\\ z&=3+t;\ t\in\mathbb{R}. \end{aligned} \]
\( p \) is intersecting \( \rho \)
\( p\parallel \rho\text{, } p\not{\!\!\subset}\rho \)
\( p \subset \rho \)

1003188904

Level: 
A
Determine the relative position of the plane \( \rho \) with general equation \( 7x-2y+z-2=0 \) and the straight line \( p \) with parametric equations: \[ \begin{aligned} x&=3+t, \\ y&=-5-2t, \\ z&=3-11t;\ t\in\mathbb{R}. \end{aligned} \]
\( p\parallel \rho\text{, }p\not{\!\!\subset}\rho \)
\( p \subset \rho \)
\( p \) is intersecting the plane \( \rho \)

1003188903

Level: 
A
Determine the relative position of the plane \( \rho \) with general equation \( 2x-y+z-2=0 \) and the straight line \( p \) with parametric equations: \[ \begin{aligned} x&=2-t, \\ y&=5-2t, \\ z&=3;\ t\in\mathbb{R}. \end{aligned} \]
\( p \subset \rho \)
\( p\parallel\rho\text{, }p\not{\!\!\subset} \rho \)
\( p \) is intersecting the plane \( \rho \)