C

2010013202

Level: 
C
One of the roots of the equation \( x^{2} + px - 8 = 0\) with the parameter \(p\in \mathbb{C}\) is \(x_{1} = \sqrt{7} +\mathrm{i}\). Find the second root \(x_{2}\) and the corresponding value of the parameter \(p\).
\(x_{2} = \mathrm{i}-\sqrt{7},\ p = -2\mathrm{i}\)
\(x_{2} = -\mathrm{i}-\sqrt{7},\ p = 2\mathrm{i}\)
\(x_{2} = -\mathrm{i}+\sqrt{7},\ p = 2\mathrm{i}\)
\(x_{2} = -\mathrm{i}-\sqrt{7},\ p = 4\mathrm{i}\)

2010013107

Level: 
C
Let \( z_1 = x^2 + 9y\,\mathrm{i}-10\,\mathrm{i} \) and \( z_2 = 8x-15+ y^2\,\mathrm{i} \). Find all \( [x;y] \in \mathbb{R}\times\mathbb{R} \) such that \( z_1= \overline{z_2} \).
\( [x;y]\in\left\{[3;-10], [3;1], [5;-10], [5;1]\right\} \)
\( [x;y]\in\left\{[-10;3], [1;3], [-10;5], [1;5]\right\} \)
\( [x;y]\in\left\{[3;10], [3;-1], [5;10], [5;-1]\right\} \)
\( [x;y]\in\left\{[-3;-10], [-3;1], [-5;-10], [-5;1]\right\} \)

2010017806

Level: 
C
We want to lift an edge of a large square plate with a side of \(4\,\mathrm{m}\) so that it creates a shelter (see the picture). To what height \(h\) do we have to lift the edge of the plate, if the created shelter shall be of the largest possible volume?
$h=2\sqrt2\,\mathrm{m}$
$h=4\cdot \sqrt{\frac23}\,\mathrm{m}$
$h=\frac43\sqrt3\,\mathrm{m}$
$h=\left( -\frac12 + \sqrt{65}\right)\,\mathrm{m}$

2010017805

Level: 
C
What dimensions (in centimeters) must a glass aquarium in the shape of a cuboid with a square bottom have, so that its volume is \(20\) liters and the surface of the aquarium is as small as possible. (We consider the cuboid without the lid.)
$a\doteq 34.2\,\mathrm{cm}$, $v\doteq 17.1\,\mathrm{cm}$
$a\doteq 27.1\,\mathrm{cm}$, $v\doteq 27.1\,\mathrm{cm}$
$a\doteq 63.2\,\mathrm{cm}$, $v\doteq 5\,\mathrm{cm}$
$a\doteq 13.6\,\mathrm{cm}$, $v\doteq 108.6\,\mathrm{cm}$

2010017804

Level: 
C
Using a \(60\,\mathrm{m}\) long wire mash we shall fence a rectangular garden with two inner walls (see the picture). What will the dimensions \(a\) and \(b\) of the garden be, if there is \(2\,\mathrm{m}\) wide opening in one outside wall and the area of the garden shall be as large as possible? (The wire mesh is used to make inner walls as well.)
$a=7.75\,\mathrm{m}$, $b=15.5\,\mathrm{m}$
$a=7.25\,\mathrm{m}$, $b=16.5\,\mathrm{m}$
$a=7.5\,\mathrm{m}$, $b=16\,\mathrm{m}$
$a=10\,\mathrm{m}$, $b=11\,\mathrm{m}$

2000017706

Level: 
C
Which of the systems has its solution graphed on the number line?
\(\begin{aligned} -5x-4 &>11-2x \\ 8-9x &> 2x-69 \end{aligned}\)
\(\begin{aligned} -5x-4 &>11-2x \\ 8-9x& < 2x-69 \end{aligned}\)
\(\begin{aligned} -5x-4 &< 11-2x\\ 8-9x &< 2x-69 \end{aligned}\)
\(\begin{aligned} -5x-4& < 11-2x\\ 8-9x &> 2x-69 \end{aligned}\)

2000017705

Level: 
C
The interval \( \left[ -\frac{12}{11}; \frac6{23}\right)\) is the solution of a system of two linear inequalities with one unknown. Which of the following systems is it?
\(\begin{aligned} \frac{x}3-\frac{x}4 &> 2x-\frac12 \\ 3x+8 &\geq 2-\frac52x \end{aligned}\)
\(\begin{aligned} \frac{x}3-\frac{x}4 &\geq 2x-\frac12\\ 3x+8 &> 2-\frac52x \end{aligned}\)
\(\begin{aligned} \frac{x}3-\frac{x}4& < 2x-\frac12 \\ 3x+8 &\geq 2-\frac52x \end{aligned}\)
\(\begin{aligned} \frac{x}3-\frac{x}4 &> 2x-\frac12 \\ 3x+8 &\leq 2-\frac52x \end{aligned}\)

2000017704

Level: 
C
Assuming \( x \in \mathbb{R}\), find the solution set of the following system of inequalities. \[\begin{aligned} 2x- [x-(2x+1)]\cdot 3 &> (3+x)-2(1-x)-2x+6 \\ x^2-3\cdot [x-2x(1-x)] &< 5(10-x^2)-2x \end{aligned}\]
\( (1;10)\)
\( \emptyset \)
\( (-10;1)\)
\( \{1;10\}\)