9000020407
Level:
A
In the following list identify an equation with real solution.
\(- 0.5x^{2} + 2x + 3 = 0\)
\(- x^{2} + 4x - 5 = 0\)
\(2x^{2} - 3x + 3 = 0\)
\(x^{2} - x + 1 = 0\)
A
9000020408
Level:
A
Which of the given equations have at least one root the same?
\[ \begin{aligned}
x^{2} + 8x + 15 & = 0 &\text{(1)}
\\x^{2} - 8x + 15 & = 0 &\text{(2)}
\\x^{2} +\phantom{ 8}x - 12 & = 0 &\text{(3)}
\\x^{2} - 2x -\phantom{ 1}8 & = 0 &\text{(4)}
\\\end{aligned}\]
equations (2) and (3)
equations (1) and (3)
equations (2) and (4)
Such a pair does not exist.
9000020404
Level:
A
Find a number which is a sum of one half of the bigger solution of
\[
x^{2} - 10x + 24 = 0
\]
and a double of the smaller solution of
\[
-x^{2} + 10x - 16 = 0.
\]
\(7\)
\(12\)
\(6\)
\(14\)
9000020006
Level:
A
Identify a true statement referring to the solution of the following equation.
\[
\sqrt{3x - 8} = x - 6
\]
The equation has a unique solution and this solution is an odd number.
The equation has two solutions, the sum of these solutions is divisible by
\(5\).
The equation has a unique solution and this solution is an even number.
The equation does not have a solution in
\(\mathbb{R}\).
9000020007
Level:
A
Identify a true statement referring to the solution of the following equation.
\[
\sqrt{x^{2 } - 4} = x + 1
\]
The equation does not have a solution in
\(\mathbb{R}\).
The equation has a unique negative solution.
The equation has a unique positive solution.
The equation has two solutions.
9000020008
Level:
A
Identify a true statement referring to the solution of the following equation.
\[
6x - 13\sqrt{x} + 6 = 0
\]
Hint: Use the substitution \(y = \sqrt{x}\).
The solutions \(x_{1}\)
and \(x_{2}\)
satisfy \(x_{1} = \frac{1}
{x_{2}} \).
The equation has a unique solution
\(x_{1}\). This solution
satisfies \(x_{1} < 1\).
The equation has a unique solution
\(x_{1}\). This solution
satisfies \(x_{1} > 1\).
The equation does not have a solution in
\(\mathbb{R}\).
9000020002
Level:
A
Find the domain of the following equation.
\[
\sqrt{6 - x} = 11
\]
\((-\infty ;6] \)
\((5;\infty )\)
\((-\infty ;5)\)
\([ - 6;\infty )\)
9000020003
Level:
A
Find the domain of the following equation.
\[
\sqrt{3x + 6} + \sqrt{8 - 2x} = 11
\]
\([ - 2;4] \)
\((-\infty ;-2] \)
\([ - 2;\infty )\)
\([ 4;\infty )\)
9000020001
Level:
A
Find the domain of the following equation.
\[
\sqrt{2x - 5} = 3
\]
\(\left [ \frac{5}
{2};\infty \right )\)
\(\left (\frac{2}
{5};\infty \right )\)
\(\left [ -\frac{5}
{2};\infty \right )\)
\(\left (\infty ; \frac{2}
{5}\right )\)
9000020005
Level:
A
Identify a true statement referring to the solution of the following equation.
\[
\sqrt{2x - 5} = 3
\]
The solution is a prime number.
The solution is an even number.
The solution is a number divisible by the number
\(3\).
The solution is an irrational number.