9000024801
Level:
B
In the following list identify an inequality which does not have a solution.
\(\sqrt{2x - 3} < -6\)
\(\sqrt{x^{2 } - 3x} > 5\)
\(\sqrt{1 + x^{2}} > -10\)
\(\sqrt{2x^{2}} < 4\)
Radical equations and inequalities
9000024804
Level:
B
How many solutions does the inequality
\[
\sqrt{x + 17} > x - 3
\]
have in the set \(\mathbb{N}\)?
Seven solutions in \(\mathbb{N}\).
No solution in \(\mathbb{N}\).
Five solutions in \(\mathbb{N}\).
More than seven solutions in \(\mathbb{N}\).
9000024805
Level:
C
A falling body dropped at a velocity
\(60\, \mathrm{m}\mathrm{s}^{-1}\). Find the
initial height \(h\),
if the relation between the velocity and the initial height
\(h\) is
\(v = \sqrt{2hg}\). Use
\(g = 10\, \mathrm{m}\mathrm{s}^{-2}\) for
acceleration of gravity.
The initial height is between \(150\, \mathrm{m}\)
and \(200\, \mathrm{m}\).
The initial height is smaller than \(100\, \mathrm{m}\).
The initial height is between \(100\, \mathrm{m}\)
and \(150\, \mathrm{m}\).
The initial height is bigger than \(200\, \mathrm{m}\).
9000024809
Level:
B
Find the solution set of the following inequality.
\[
\sqrt{x + 3} > x - 3
\]
\([ -3;6)\)
\( (1;6)\)
\([ -3;3] \)
\( (-\infty ;1)\cup (6;+\infty )\)
9000024810
Level:
C
Find the solution set of the following inequality.
\[
\sqrt{|x|} > x
\]
\(K = (-\infty ;0)\cup (0;1)\)
\(K = (-\infty ;1)\)
\(K =\mathbb{R} ^{+}\)
\(K = (0;1)\)
9000024802
Level:
A
Consider the equation
\[
\sqrt{x^{2 } - 2x + 1} = x + 2
\]
and the equation which arises from this equation by squaring both sides of the
equation, i.e. the equation
\[
\left (\sqrt{x^{2 } - 2x + 1}\right )^{2} = (x + 2)^{2}.
\]
Identify a true statement.
Both equations are equivalent only if
\(x\geq - 2\).
Both equations are equivalent.
Both equations are equivalent only if
\(x\leq - 2\).
None of the above.
9000024803
Level:
A
Removing radical in an equation by squaring both sides may enrich the set of
solutions of this equation and checking the solutions of the new equation in the
original equation may be necessary. Identify a correct conclusion in the particular
case of the following equation.
\[
-\sqrt{x^{2 } - 2x + 1} = x
\]
If we look for the solution in the set
\(\mathbb{R}^{-}\), then
squaring both sides of the equation gives an equivalent equation. The checking of the
solution is not necessary.
If we look for the solution in the set
\(\mathbb{R}^{+}\), then
squaring both sides of the equation gives an equivalent equation. The checking of the
solution is not necessary.
If we look for the solution in the set
\(\mathbb{R}\), then
squaring both sides of the equation gives an equivalent equation. The checking of the
solution is not necessary.
None of the above.
9000024806
Level:
B
In the following list identify the interval which is a subset of the solution set of
the following inequality.
\[
\sqrt{x^{2 } + 2x - 3} > x + 2
\]
\((-\infty ;-3] \)
\(\left (-\frac{7}
{2};+\infty \right )\)
\((1;+\infty )\)
\((-\infty ;-2)\)
9000024807
Level:
C
A body hangs on a string of the length
\(l_{1}\). The length
\(l\) of the spring
defines the period \(T\)
of motion by the relation
\[
T = 2\pi \sqrt{ \frac{l}
{g}},
\]
where \(g\)
is a standard acceleration of gravity. We have to adjust the length of the string such
that the period doubles. Find the new length of the string.
We elongate the string by \(3\cdot l_{1}\),
i.e. \(l_{2} = l_{1} + 3l_{1}\).
The length doubles, i.e. \(l_{2} = 2l_{1}\).
The new length will be half of the original length, i.e.
\(l_{2} = \frac{1}
{2}l_1\).
We shorten the string by \(3\cdot l_{1}\),
i.e. \(l_{2} = l_{1} - 3l_{1}\).
9000024808
Level:
C
In the following list identify a true statement referring to the following equation.
\[
\sqrt{4x^{2 } - \sqrt{8x + 5}} = 2x + 1
\]
The equation has a unique solution, this solution is a negative number.
The equation has two solutions, both solutions have an opposite sign.
The equation has a unique solution, this solution is a positive number.
The equation does not have a solution.