A

9000020906

Level: 
A
Identify an equation which can be obtained from the following system by eliminating one of the variables. \[ \begin{alignedat}{80} &y^{2} & - &2 &x & + &3 & = 0 & & & & & & & & \\ &x & - & &y & - &1 & = 0 & & & & & & & & \\\end{alignedat}\]
\((y - 1)^{2} = 0\)
\((y + 1)^{2} = 0\)
\((x - 4)^{2} = 0\)
\((x + 2)^{2} = 0\)

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.

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.

9000014808

Level: 
A
Find the intervals of monotonicity of the quadratic function \(f(x) = 2x^{2} + 3\).
The function is increasing on \(\left [ 0;\infty \right )\) and decreasing on \(\left (-\infty ;0\right ] \).
The function is increasing on \(\left (3;\infty \right )\) and decreasing on \(\left (-\infty ;3\right )\).
The function is increasing on \(\left [ -\frac{3} {2};\infty \right )\) and decreasing on \(\left (-\infty ;-\frac{3} {2}\right ] \).
The function is increasing on its domain.