9000022810
Level:
B
Find the solution set of the following quadratic inequality.
\[
-x^{2} + 2x + 3 > 0
\]
\((-1,3)\)
\((-\infty ,-1)\)
\((-\infty ,-1)\cup (3,\infty )\)
\((3,\infty )\)
Quadratic Equations and Inequalities
9000022809
Level:
B
Find the solution set of the following quadratic inequality.
\[
4x^{2} + 4x + 1 < 0
\]
\(\emptyset \)
\(\mathbb{R}\)
\(\left \{\frac{1}
{2}\right \}\)
\(\left \{-\frac{1}
{2}\right \}\)
9000022808
Level:
B
Find all the real values of \(x\)
which ensure that the following expression is negative.
\[
-x^{2} + 4x - 4
\]
\(x\in \mathbb{R}\setminus \{2\}\)
none \(x\)
with this property
\(x = 2\)
\(x\in \mathbb{R}\)
9000022901
Level:
C
An arrow has been shot at the angle
\(60^{\circ }\) at the
speed \(10\, \mathrm{m}\, \mathrm{s}^{-1}\).
Find the time when the height equals to the horizontal distance from the take-off
point. Hint: The position is given by the equations
\(x = v_{0}t\cdot \cos \alpha \),
\(y = v_{0}t\cdot \sin \alpha -\frac{1}
{2}gt^{2}\). Use
\(g = 10\, \mathrm{m}\, \mathrm{s}^{-2}\) as an
acceleration of gravity.
\(\left (\sqrt{3} - 1\right )\, \mathrm{s}\)
\(\left (\sqrt{3} + 1\right )\, \mathrm{s}\)
\(\sqrt{3}\, \mathrm{s}\)
\(\left (\sqrt{2} - 1\right )\, \mathrm{s}\)
\(\left (\sqrt{2} + 1\right )\, \mathrm{s}\)
9000022303
Level:
B
The solution set of one of the following quadratic inequalities is the interval
\((2,5)\).
Determine this inequality.
\(x^{2} - 7x + 10 < 0\)
\(x^{2} + 7x + 10 > 0\)
\(x^{2} - 7x + 10\leq 0\)
\(x^{2} + 7x + 10\geq 0\)
\(x^{2} - 7x + 10 > 0\)
9000020407
Level:
A
In the following list identify an equation with real solution.
\(- 0.5x^{2} + 2x + 3 = 0\)
\(- x^{2} + 4x - 5 = 0\)
\(2x^{2} - 3x + 3 = 0\)
\(x^{2} - x + 1 = 0\)
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
\[
-x^{2} + 10x - 16 = 0.
\]
\(7\)
\(12\)
\(6\)
\(14\)
9000020909
Level:
B
The sum of squares of two consecutive integers is
\(1201\).
Identify these integers.
\(24\)
and \(25\)
\(23\)
and \(24\)
\(25\)
and \(26\)
\(26\)
and \(27\)
9000020409
Level:
B
One of the solutions of the quadratic equation \( x^{2} + bx - 10 = 0\) is \(x_{1} = 5\). Find the second solution \(x_{2}\) and the value of the coefficient \(b\).
\(x_{2} = -2\) and
\(b = -3\)
\(x_{2} = -3\) and
\(b = -2\)
\(x_{2} = 2\) and
\(b = 3\)
\(x_{2} = 3\) and
\(b = 2\)