2010008909
Level:
C
We are given a triangle $KLM$, where $K=[2;3;-1]$, $L= [-1;5;3]$, and $M=[3;4;2]$. Find the height to the side $KM$.
$3\sqrt2$
$2\sqrt3$
$3\sqrt5$
$2\sqrt5$
Analytical space geometry
2010008908
Level:
C
We are given skew lines $a$ and $b$.
\begin{align*}
a\colon x&= -1-2t, & b\colon x&= 1-3s, \\
y&= -2+3t, & y&=2s, \\
z&= -4+2t;\ t\in\mathbb{R}, & z&= 2-2s;\ s\in\mathbb{R}.
\end{align*}
Find parametric equations of a straight line $p$, that is intersecting both lines $a$ and $b$ and lying in the plane $2x+3y-z-8=0$.
$\begin{aligned}
p\colon x&=-9+r, \\
y&=10+r, \\
z&=4+5r;\ r\in\mathbb{R}
\end{aligned}$
$\begin{aligned}
p\colon x&=-9-2r, \\
y&=10-2r, \\
z&=4+10r;\ r\in\mathbb{R}
\end{aligned}$
$\begin{aligned}
p\colon x&=-9-10r, \\
y&=10+9r, \\
z&=4-r;\ r\in\mathbb{R}
\end{aligned}$
$\begin{aligned}
p\colon x&=-9+2r, \\
y&=10+2r, \\
z&=4-2r;\ r\in\mathbb{R}
\end{aligned}$
2010008907
Level:
A
Find the real number \( m \) so that the line \( KL \), where \( K=[8;2;-2] \) and \( L=[3;-2;m] \), is parallel to the plane \( 4x-2y+3z-5=0 \).
\( m=2\)
\( m=-2\)
\( m=6\)
\( m=-6\)
2010008906
Level:
A
We are given two intersecting planes \(2x - 3y + 5z - 9 = 0\) and \(3x - y + 2z - 1 = 0\). Find the parametric equations of their line of intersection \(p\).
\( \begin{aligned}
p\colon x&=-1-t, \\
y&=-2+ 11t, \\
z&=1+ 7t;\ t\in\mathbb{R}
\end{aligned} \)
\( \begin{aligned}
p\colon x&=-1-11t, \\
y&=-2+ 11t, \\
z&=1+ 7t;\ t\in\mathbb{R}
\end{aligned} \)
\( \begin{aligned}
p\colon x&=-1+t, \\
y&=-2+ 11t, \\
z&=1- 11t;\ t\in\mathbb{R}
\end{aligned} \)
\( \begin{aligned}
p\colon x&=-1-11t, \\
y&=-2+ 11t, \\
z&=1- 11t;\ t\in\mathbb{R}
\end{aligned} \)
2010008905
Level:
A
Determine the relative position of the plane \( \sigma \) with general equation \( x-2y+3z-1=0 \) and the straight line \( p \) with parametric equations:
\[ \begin{aligned}
x&=4, \\
y&=5+3t, \\
z&=2+2t;\ t\in\mathbb{R}.
\end{aligned} \]
\( p\parallel\sigma,\ p\not{\!\!\subset} \sigma \)
\( p \subset \sigma \)
\( p \) is intersecting the plane \( \sigma \)
2010008904
Level:
A
We are given points \( K=[4;0;3] \), \( L=[1;-3;2] \) and \( M=[2;2;0] \). From the following list, choose the parametric equations which represent a plane \( \sigma \) defined by the points \( K \), \( L \), and \( M \).
$\begin{aligned}
\sigma\colon x&=1+3r+s, \\
y&=-3+3r+5s, \\
z&=2+r-2s;\ r,s\in\mathbb{R}
\end{aligned}$
$\begin{aligned}
\sigma\colon x&=1-3r-s, \\
y&=-3+3r-5s, \\
z&=2+r+2s;\ r,s\in\mathbb{R}
\end{aligned}$
$\begin{aligned}
\sigma\colon x&=1-3r+s, \\
y&=-3-3r+5s, \\
z&=2+r-2s;\ r,s\in\mathbb{R}
\end{aligned}$
$\begin{aligned}
\sigma\colon x&=1+3r+s, \\
y&=-3+3r-5s, \\
z&=2-r+2s;\ r,s\in\mathbb{R}
\end{aligned}$
2010008903
Level:
A
Given points \( P=[3;-4;1] \) and \( Q=[-1;3;6] \), determine which of the following parametric equations defines the ray \( QP\).
$\begin{aligned}
\mapsto QP\colon x&=-1-4t, \\
y&=3+7t, \\
z&=6+5t;\ t\in (-\infty;0]
\end{aligned}$
$\begin{aligned}
\mapsto QP\colon x&=3-4t, \\
y&=-4+7t, \\
z&=1+5t;\ t\in[ -1;\infty)
\end{aligned}$
$\begin{aligned}
\mapsto QP\colon x&=3+4t, \\
y&=-4-7t, \\
z&=1-5t;\ t\in[ 0;\infty)
\end{aligned}$
$\begin{aligned}
\mapsto QP\colon x&=-1+4t, \\
y&=3-7t, \\
z&=6-5t;\ t\in (-\infty;1]
\end{aligned}$
2010008902
Level:
A
Given points \( A=[-2;5;1] \) and \( B=[3;-1;2] \), determine which of the following parametric equations defines the ray \( AB \).
$\begin{aligned}
\mapsto AB\colon x&=3+5t, \\
y&=-1-6t, \\
z&=2+t;\ t\in [ -1;\infty)
\end{aligned}$
$\begin{aligned}
\mapsto AB\colon x&=-2+5t, \\
y&=5-6t, \\
z&=1+t;\ t\in(-\infty;1]
\end{aligned}$
$\begin{aligned}
\mapsto AB\colon x&=3-5t, \\
y&=-1+6t, \\
z&=2-t;\ t\in(-\infty;0]
\end{aligned}$
$\begin{aligned}
\mapsto AB\colon x&=-2-5t, \\
y&=5+6t, \\
z&=1-t;\ t\in [ 0;\infty)
\end{aligned}$
2010008901
Level:
A
Given points \( K=[-3;1;5] \) and \( L=[1;-5;4] \), determine which of the following parametric equations does not define the ray \( KL \).
$\begin{aligned}
\mapsto KL\colon x&=-3+4t, \\
y&=1-6t, \\
z&=5-t;\ t\in(-\infty;0]
\end{aligned}$
$\begin{aligned}
\mapsto KL\colon x&=-3+4t, \\
y&=1-6t, \\
z&=5-t;\ t\in [ 0;\infty)
\end{aligned}$
$\begin{aligned}
\mapsto KL\colon x&=-3-8t, \\
y&=1+12t, \\
z&=5+2t;\ t\in(-\infty;0]
\end{aligned}$
$\begin{aligned}
\mapsto KL\colon x&=-3+8t, \\
y&=1-12t, \\
z&=5-2t;\ t\in [ 0;\infty)
\end{aligned}$
2010005008
Level:
A
Determine whether the following planes \(\alpha \) and
\(\beta\) are parallel, identical or intersecting.
\[
\begin{aligned}[t] \alpha \colon &x = 1-m+2n, &
\\&y =2m-n,
\\&z = 2-m+n;\ m,n\in \mathbb{R},
\\ \end{aligned}\qquad \beta \colon x-y-3z+5 = 0
\]
identical
intersecting
parallel, not identical