1003061206
Level:
A
Find the value of a parameter \( a \), such that the line \( ax-6y-15=0 \) is parallel to the line \( y=-\frac23 x+4 \).
\( a=-4 \)
\( a=4 \)
\( a=-\frac23 \)
\( a=-2 \)
Plane geometry
1003061208
Level:
A
We are given a straight line \( q=\left\{[1+3t;2-2t]\text{, }t\in\mathbb{R}\right\} \). Specify the value of a parameter \( a \) such that the line \( 5x+ay+1=0 \) is parallel to \( q \).
\( a=7.5 \)
\( a=2.5 \)
\( a=-7.5 \)
\( a=-2.5 \)
1003061304
Level:
A
Determine the relative position of the lines
\( p\colon4x-3y+9=0 \) and
\[ \begin{aligned} q\colon x&=6+3t, \\
y&=11+4t, \end{aligned} \]
where \( t\in\mathbb{R}\).
identical lines, \( p=q \)
parallel different lines, \( p\parallel q;\ p\neq q \)
intersecting lines, \( p\cap q=\{[0;3]\} \)
intersecting lines, \( p\cap q=\{[6;11]\} \)
1003061305
Level:
A
Determine the relative position of the lines \( p\colon 4x+6y-5=0 \) and \( q\colon y=-\frac23 x-6 \).
parallel different lines, \( p\parallel q;\ p\neq q \)
identical lines, \( p=q \)
intersecting lines, \( p\cap q=\left\{\left[0;\frac54\right]\right\} \)
intersecting lines, \( p\cap q=\left\{\left[0;\frac56\right]\right\} \)
1003061306
Level:
A
Determine the relative position of the lines \( p\colon 2x-3y+7=0 \) and
\[ \begin{aligned} q\colon x& =2+t, \\
y& = -3t, \end{aligned} \]
where \( t\in\mathbb{R} \).
intersecting lines, \( p\cap q=\left\{\left[1;3\right]\right\} \)
identical lines, \( p=q \)
parallel different lines, \( p\parallel q;\ p\neq q \)
intersecting lines, \( p\cap q=\left\{\left[7;7\right]\right\} \)
1103061201
Level:
A
From the following list, choose parametric equations, that do not represent the straight line passing through the points \( A \) and \( B \) (see the picture).
$\begin{aligned}
p\colon x&=2+4t, \\
y&=6+2t;\ t\in\mathbb{R}
\end{aligned}$
$\begin{aligned}
p\colon x&=2+2t, \\
y&=1+t;\ t\in\mathbb{R}
\end{aligned}$
$\begin{aligned}
p\colon x&=6+4t, \\
y&=3+2t;\ t\in\mathbb{R}
\end{aligned}$
$\begin{aligned}
p\colon x&=2-2t, \\
y&=1-t;\ t\in\mathbb{R}
\end{aligned}$
$\begin{aligned}
p\colon x&=4+4t, \\
y&=2+2t;\ t\in\mathbb{R}
\end{aligned}$
1103061202
Level:
A
A straight line \( p \) is given by the point \( A \) and the normal vector \( \vec{n} \) (see the picture). Find its equation in general form.
\( p\colon 2x-5y-6=0 \)
\( p\colon 2x+5y-6=0 \)
\( p\colon 5x-2y-15=0 \)
\( p\colon 5x+2y-15=0 \)
1103061203
Level:
A
A straight line \( p \) is given by the point \( A \) and the direction angle \( \varphi \) (see the picture). Choose the equation of the line \( p \) in the slope-intercept form.
\( p\colon y=-\sqrt3x+3 \)
\( p\colon y=\sqrt3x+3 \)
\( p\colon y=1.7x+3 \)
\( p\colon y=-1.7x+3 \)
1103061204
Level:
A
From the following list choose the equation of a straight line that passes through the given point \( K \) and is not parallel to the given line \( m \) (see the picture).
\( g\colon y=-\frac32x+\frac{13}2 \)
\( b\colon 2x-3y-13=0 \)
\( f\colon y=\frac23x-\frac{13}3 \)
$\begin{aligned}
q\colon x&=5+3t, \\
y&=-1+2t;\ t\in\mathbb{R}
\end{aligned}$
1103061205
Level:
A
From the following list choose the equation of a straight line that passes through the given point \( K \) and is not perpendicular to the given line \( m \) (see the picture).
\( r\colon y=\frac23x-\frac{13}3 \)
\( p\colon 3x+2y-13=0 \)
\( s\colon y=-\frac32x+\frac{13}2 \)
$\begin{aligned}
q\colon x&=5+2t, \\
y&=-1-3t;\ t\in\mathbb{R}
\end{aligned}$