9000107504
Level:
B
Find the angle between the lines
\[
p\colon 2x - 3y + 1 = 0;\quad q\colon 3x + 2y - 3 = 0.
\]
\(90^{\circ }\)
\(60^{\circ }\)
\(0^{\circ }\)
\(30^{\circ }\)
Plane geometry
9000107506
Level:
B
Find \(\cos \varphi \) where
\(\varphi \) is the angle
between the lines \(p\)
and \(q\).
\[
p\colon y = 2x - 11;\quad q\colon y = \frac{1}
{4}x
\]
\(\frac{6\sqrt{85}}
{85} \)
\(\frac{1}
{\sqrt{22}}\)
\(\frac{\sqrt{6}}
{85} \)
\(\frac{\sqrt{17}}
{30} \)
9000107505
Level:
B
Find \(\cos \varphi \) where
\(\varphi \) is the angle
between the lines \(p\)
and \(q\).
\[
\begin{aligned}[t] p\colon x& = 1 + 4t, &
\\y & = 3 - 3t;\ t\in \mathbb{R};
\\ \end{aligned} \quad q\colon x + y - 3 = 0
\]
\(\frac{7\sqrt{2}}
{10} \)
\(- \frac{7}
{5\sqrt{2}}\)
\(\frac{\sqrt{2}}
{5} \)
\(\frac{\sqrt{2}}
{10} \)
9000107507
Level:
B
Find \(\mathop{\mathrm{tg}}\nolimits \varphi \) where
\(\varphi \) is the angle
between the lines \(p\)
and \(q\).
\[
\begin{aligned}[t] p\colon x& = 1 + t, &
\\y & = 3 + 2t;\ t\in \mathbb{R};
\\ \end{aligned}\quad q\colon y = 1
\]
\(2\)
\(\frac{1}
{2}\)
\(- 1\)
\(0\)
9000107508
Level:
B
Find \(\cos \varphi \) where
\(\varphi \) is the angle
between the lines \(p\)
and \(q\).
\[
\begin{aligned}[t] p\colon x& = t, &
\\y & = -3;\ t\in \mathbb{R};
\\ \end{aligned}\quad q\colon y = 1
\]
\(1\)
\(\frac{1}
{\sqrt{2}}\)
\(0\)
\(\frac{\sqrt{10}}
{10} \)
9000107510
Level:
A
In the following list identify a line parallel to the line
\(q\).
\[
\begin{aligned}q\colon x& = t, &
\\y & = 1 + 5t;\ t\in \mathbb{R}
\\ \end{aligned}
\]
\(p\colon - 5x + y - 13 = 0\)
\(p\colon x + 5y - 1 = 0\)
\(p\colon x - 5 = 0\)
\(p\colon 10x + 2y - 1 = 0\)
9000106806
Level:
C
Given points \(A = [0;5]\),
\(B = [6;1]\),
\(C = [7;9]\),
find the direction vector of the line passing through the point
\(A\) and perpendicular
to the segment \(BC\)
(i.e. the line which contains the altitude of the triangle
\(ABC\) through
the point \(A\)).
\((8;-1)\)
\((1;8)\)
\((1;9)\)
\((-9;1)\)
9000106807
Level:
C
Consider the points \(A = [0;5]\),
\(B = [6;1]\),
\(C = [7;9]\) and the
triangle \(ABC\).
Find the direction vector of the line which is the perpendicular bisector of the side
\(b\)
(i.e. the line through the midpoint of the side
\(AC\) which is perpendicular
to the segment \(AC\)).
\((4;-7)\)
\((7;4)\)
\((7;9)\)
\((7;-9)\)
9000106808
Level:
C
Consider the points \(A = [0;5]\),
\(B = [6;1]\),
\(C = [7;9]\) and the
triangle \(ABC\).
Find the direction vector of the line which is the bisector of the angle
\(ACB\)
(i.e. the line which splits the internal angle at the point
\(C\) into
two angles with equal measures).
\((2;3)\)
\((6;-4)\)
\((7;9)\)
\((7;8)\)
9000106002
Level:
A
In the following list identify a vector in the direction of the following parametric
line.
\[\begin{aligned}
x =\ &t - 1, & &
\\y =\ &t - 2;\ t\in \mathbb{R}\text{.} & &
\end{aligned}\]
\(\left (1;1\right )\)
\(\left (1;2\right )\)
\(\left (-1;-2\right )\)
\(\left (1;-1\right )\)