1003020204
Level:
A
Assuming \( x\in\mathbb{Z} \), find the solution set of the equation.
\[ 3-6x+3\cdot\left\{x-\left[2-(x+1)\right]\right\}=0\]
\( \mathbb{Z} \)
\( \emptyset \)
\( \{1\} \)
\( \{0\} \)
A
1003020203
Level:
A
Find the solution set of the following equation.
\[ \frac{x+5}2-\frac{x+1}4=\frac{22+2x}8\]
\( \emptyset \)
\( \mathbb{R} \)
\( \{4\} \)
\( \{-4\} \)
1003020202
Level:
A
Assuming \( x\in\mathbb{N} \), find the solution set of the equation.
\[ 3x-\left[1+2\cdot(3x-2)\right]=9\]
\( \emptyset \)
\( \mathbb{N} \)
\( \{-2\} \)
\( \left\{\frac{14}9\right\} \)
1003020201
Level:
A
Find the solution set of the following equation.
\[ 2-\frac{2-x}3=\frac x2+\frac{8-x}6 \]
\( \mathbb{R} \)
\( \emptyset \)
\( \{0\} \)
\( \{4\} \)
1103030705
Level:
A
Let there be a triangle KLM and vectors \( \overrightarrow{a} \), \( \overrightarrow{c} \) in the coordinate system. Triangle \( KLM \) and vectors \( \overrightarrow{a} \), \( \overrightarrow{c} \) are
given in the coordinate system shown in the picture. Point T is the centroid of the triangle KLM. Express vector \( \overrightarrow{x} \), where \( \overrightarrow{x}=\overrightarrow{KT} \) as a linear combination of \( \overrightarrow{a} \) and \( \overrightarrow{c} \) and evaluate \( \left|\overrightarrow{x}\right| \).
\( \overrightarrow{x}=\frac13 \overrightarrow{a}+\frac13 \overrightarrow{c} \), \( \left|\overrightarrow{x}\right|=5 \)
\( \overrightarrow{x}=\frac23 \overrightarrow{a}+\frac23 \overrightarrow{c} \), \( \left|\overrightarrow{x}\right|=10 \)
\( \overrightarrow{x}=\frac12 \overrightarrow{a}+\frac12 \overrightarrow{c} \), \( \left|\overrightarrow{x}\right|=\frac{15}2 \)
\( \overrightarrow{x}=\frac14 \overrightarrow{a}+\frac14 \overrightarrow{c} \), \( \left|\overrightarrow{x}\right|=\frac{225}{12} \)
1103030704
Level:
A
We are given points \( A = [2;1] \), \( B = [4;-1] \), and \( T = [6;2] \), where point \( T \) is the centroid of triangle \( ABC \). Find the length of the median of triangle \( ABC \) to side \( AC \).
\( |t_b|=\frac{\sqrt{117}}2 \)
\( |t_b|=\frac{\sqrt{45}}2 \)
\( |t_b|=\frac{\sqrt{153}}2 \)
\( |t_b|=\sqrt{117} \)
1103030703
Level:
A
We are given points \( A = [2;1] \), \( B = [4;-1] \), and \( T = [6;2] \), where point \( T \) is the centroid of triangle \( ABC \). Find the coordinates of \( C \), which is the vertex of \( ABC \).
\( C = [12;6] \)
\( C = [8;4] \)
\( C = [9;6] \)
\( C = [8;5] \)
1103030702
Level:
A
We are given points \( A = [1;-1;2] \), \( B = [0;5;-3] \), \( S = [2;0;5] \). Point \( S \) is the centre of a parallelogram \( ABCD \). Evaluate the length of \( AD \).
\( |AD|=\sqrt{146} \)
\( |AD|=\sqrt{114} \)
\( |AD|=\sqrt{44} \)
\( |AD|=\sqrt{172} \)
1103030701
Level:
A
We are given points \( A = [1;-1;2] \), \( B = [0;5;-3] \), \( S = [2;0;5] \). Point \( S \) is the centre of a parallelogram \( ABCD \). Find the coordinates of vertices \( C \) and \( D \).
\( C = [3;1;8]; D = [4;-5;13] \)
\( C = [4;-5;13]; D = [3;1;8] \)
\( C = [1;1;3]; D = [2;-5;8] \)
\( C = [-3;-1;-8]; D = [-4;5;-13] \)
1103024506
Level:
A
Given the graph of the function \( f \), find \( \lim\limits_{x\rightarrow-1}f(x) \).
\( 2 \)
\( 1 \)
\( -1 \)
This limit does not exist.