2010006907
Level:
B
Find all the values of \(x\)
such that the following infinite series is convergent.
\[
1 + 2x - 3 + (2x-3)^{2} + (2x - 3)^{3}+\cdots
\]
\(x\in (1;2)\)
\(x\in (-\infty ;-1)\)
\(x\in (1;+\infty )\)
\(x\in \mathbb{R}\)
B
2010006906
Level:
B
Solve the following equation.
\[
1 + 3x + 9x^{2} + \cdots = 2
\]
\(x = \frac{1}
{6}\)
\(x = -\frac{1}
{3}\)
\(x = \frac{1}
{3}\)
The equation has no solution.
2010006904
Level:
B
We are given the equation
\[ \sum\limits_{n=0}^{\infty}\frac{(x+2)^n}{3^n}=\frac{x+3}{2x+1} \]
with the unknown \( x \) being a real number. What is the set of all its solutions?
\( \{ 0\} \)
\( \{ -8;0 \} \)
\( \{ \} \)
\( \{ -6;2 \} \)
\( \{ -8 \} \)
2010006903
Level:
B
What is the sum of the repeating numbers \( 0.\overline{45} \) and \( 0.2\overline{81} \)?
\( \frac{81}{110} \)
\( \frac{59}{110} \)
\( \frac{110}{81} \)
\( \frac{36}{110} \)
\( \frac{81}{100} \)
2010006802
Level:
B
We are given an arithmetic sequence \( (a_n)^{\infty}_{n=1}\), where \(a_3=328\), \(a_5=320\). Find \( n\), so that \(a_n=0\).
\(85\)
\(79\)
\(83\)
\(81\)
2010006801
Level:
B
We are given an arithmetic sequence \( (a_n)^{\infty}_{n=1}\), where \(a_2=429\), \(a_5=420\). Find \(n\), so that \(a_n=0\).
\(145\)
\(141\)
\(144\)
\(142\)
2010006704
Level:
B
Determine all the values of the parameter
\(c\in \mathbb{R}\)
so that the following system has two solutions in
\(\mathbb{R}\times \mathbb{R}\).
\[ \begin{alignedat}{80}
&x^{2} & + &2y^{2} & = 6 & & & & & &
\\ &x & + &y & = c & & & & & &
\\\end{alignedat}\]
\(|c| < 3\)
\(|c| =3\)
\(|c| > 3\)
\(|c| \in \mathbb{R}\)
2010006703
Level:
B
Identify a true statement related to the solution of the following system in
\(\mathbb{R}\times \mathbb{R}\).
\[ \begin{alignedat}{80}
&x^{2} & + &2 & &y^{2} & - & &4x & = &0 & & & & & & & & & & & &
\\ &x & + & & &y & = &4 & & & & & & & & & & & &
\\\end{alignedat}\]
The system has two solutions.
The system has a unique solution.
The system does not have any solution.
The system has infinitely many solutions.
2010006702
Level:
B
The augmented matrix of a system of three equations with
three unknowns is row equivalent with the following matrix
\(A'\). Find
the solution of the system.
\[
A' = \left(\begin{array}{ccc|c}
2 & 3 & 1 & 7\\
0 & 3 & 4 & 0\\
0 & 0 & 5 & 45
\end{array}\right)
\]
\([17;-12;9]\)
\([12;10;-9]\)
\([-19;12;9]\)
\([7;0;45]\)
2010006614
Level:
B
Assuming \( y\in \mathbb{R}\setminus \{-2;1;2\}\), simplify the expression:
\[ \left( \frac{y+2}{y-1} - \frac{y+5}{y+2}\right)\cdot \left(y+\frac{y}{y-2}\right) \]
\( \frac{9y}{y^2-4}\)
\( \frac{9y}{4-y^2}\)
\( \frac{y(8y-1)}{y^2-4}\)
\( \frac{y(1-8y)}{y^2-4}\)