1003020807
Level:
A
Find the set of all values of a parameter \( a\in\mathbb{R} \) such that the distance of the points \( [a;3;-2] \) and \( [1;1;4] \) equals \( 7 \).
\( \{4;-2\} \)
\( \{4;2\} \)
\( \emptyset \)
\( \{0\} \)
Points and vectors
1003020806
Level:
A
Let \( ABC \) be a triangle with \( A = [1 ; 2 ; -3] \), \( B = [-3 ; 3 ; -2] \) and \( C = [-1 ; 1 ; -1] \). Find its perimeter.
\( 6+3\sqrt2 \)
\( 18 \)
\( 6 + 2\sqrt3 \)
\( 9\sqrt2 \)
1103020805
Level:
A
We are given the points \( A = [1;1;4] \), \( C = [0;4;7] \) and \( D = [2;0;5] \) as seen in the picture.
What are the coordinates of a point \( B \), if \( ABCD \) is a parallelogram?
\( B = [-1;5;6] \)
\( B = [3;-3;2] \)
\( B = [-2;4;3] \)
\( B = [-3;3;-2] \)
1103020804
Level:
A
In the parallelogram \( ABCD \) shown in the picture, \( G \) is the midpoint of \( CD \), \( F \) is the midpoint of \( BC \) and \( \overrightarrow{u}=\overrightarrow{CG} \), \( \overrightarrow{v}=\overrightarrow{CF} \), \( \overrightarrow{a}=\overrightarrow{AD} \) and \( \overrightarrow{b}=\overrightarrow{AC} \).
Express vectors \( \overrightarrow{a} \) and \( \overrightarrow{b} \) as a linear combination of vectors \( \overrightarrow{u} \) and \( \overrightarrow{v} \).
\( \overrightarrow{a}=-2\overrightarrow{v};\ \overrightarrow{b}=-2\overrightarrow{u}-2\overrightarrow{v} \)
\( \overrightarrow{a}=\overrightarrow{b}+2\overrightarrow{u};\ \overrightarrow{b}=-2\overrightarrow{u}+2\overrightarrow{v} \)
\( \overrightarrow{a}=\overrightarrow{b}-2\overrightarrow{u};\ \overrightarrow{b}=-\sqrt2\overrightarrow{u}-\sqrt2\overrightarrow{v} \)
\( \overrightarrow{a}=-2\overrightarrow{v};\ \overrightarrow{b}=2\overrightarrow{u}+2\overrightarrow{v} \)
1103020803
Level:
A
Find the coordinates of the centroid of the triangle \( ABC \). Triangle \( ABC \) is shown in the picture.
\( T=\left[1; \frac13\right] \)
\( T=[1;0] \)
\( T=[1;1] \)
\( T=\left[\frac43; 0\right] \)
1003020802
Level:
A
\( S \) is the midpoint of \( AB \). \( A = [5; -7] \), \( S = [0; -9] \). Find the coordinates of \( B \).
\( B = [-5; -11] \)
\( B = [5; -18] \)
\( B = \left[-\frac52; -\frac{11}2\right] \)
\( B = [5; 11] \)
1103020801
Level:
A
Find the coordinates of the midpoints of the line segments \( AB \), \( BC \), \( AC \). For coordinates of the points \( A \), \( B \) and \( C \), see the picture.
\( S_{AB}=\left[-\frac12;1 \right]\text{, }\ S_{BC}=[4;2 ]\text{, }\ S_{AC}=\left[\frac12; 4\right] \)
\( S_{AB}=\left[-\frac32;2 \right]\text{, }\ S_{BC}=[1;3 ]\text{, }\ S_{AC}=\left[\frac52; 4\right] \)
\( S_{AB}=\left[\frac12;1 \right]\text{, }\ S_{BC}=[4;2 ]\text{, }\ S_{AC}=\left[-\frac12; 4\right] \)
\( S_{AB}=\left[1;-\frac12 \right]\text{, }\ S_{BC}=[2;4 ]\text{, }\ S_{AC}=\left[4;\frac12\right] \)
1103020808
Level:
A
Let \( ABC \) be a triangle. In the picture, the midpoint of the side \( BC \) and the centroid of the triangle are indicated. Out of the following vector relations select the one that is not true.
\( \overrightarrow{ST}= \frac12 \overrightarrow{AT} \)
\( \overrightarrow{AT}= \frac23\overrightarrow{AS} \)
\( \overrightarrow{ST} = -\frac13\overrightarrow{AS} \)
\( \overrightarrow{SA}= -3\overrightarrow{TS} \)
1003030605
Level:
B
Let \( \overrightarrow{a}=(3;-5) \) and \( \overrightarrow{b}=(6;-10) \). Find all the vectors \( \overrightarrow{c} \) such that
\[ \overrightarrow{a}\cdot\overrightarrow{c}=11\ \text{ and }\ \overrightarrow{b}\cdot\overrightarrow{c}=22\text{ .} \]
\( \overrightarrow{c}=(2+5k;-1+3k);\ k\in\mathbb{R} \)
\( \overrightarrow{c}_1=(7;2);\ \overrightarrow{c}_2=(-7;-2) \)
\( \overrightarrow{c}=(2k;-k);\ k\in\mathbb{R} \)
\( \overrightarrow{c}_1=(2;-1);\ \overrightarrow{c}_2=(-2;1) \)
1003030604
Level:
B
Let \( \overrightarrow{a}=(2;- 3) \) and \( \overrightarrow{b}=(3;-2) \). Find all the vectors \( \overrightarrow{c} \) such that
\[ \overrightarrow{a}\cdot\overrightarrow{c}=8\ \text{ and }\ \overrightarrow{b}\cdot\overrightarrow{c}=27. \]
\( \overrightarrow{c}=(13;6) \)
\( \overrightarrow{c_1}=(13;6);\ \overrightarrow{c_2}=(-13;-6) \)
\( \overrightarrow{c}=(13k;6k);\ k\in\mathbb{R} \)
\( \overrightarrow{c}=(-13;-6) \)