9000104306
Level:
A
Assuming \(a = 0\),
solve the following inequality.
\[
a\left (a - 1\right )x < 1
\]
\(x\in\mathbb{R}\)
\(x\in\mathbb{R}\setminus \{1\}\)
\(x\in\emptyset \)
\( x\in\left \{ \frac{1}
{a\left (a-1\right )}\right \}\)
A
9000104308
Level:
A
Assuming \(a = \frac{1}
{2}\),
solve the following inequality.
\[
2a^{2}x - 1 > ax
\]
\(\emptyset \)
\(\mathbb{R}\)
\(\left ( \frac{1}
{a\left (2a-1\right )};\infty \right )\)
\(\left (-\infty ; \frac{1}
{a\left (2a-1\right )}\right )\)
9000104309
Level:
A
Assuming \(a = -1\),
solve the following inequality.
\[
a^{2}x - 1 < a - ax
\]
\(\emptyset \)
\(\mathbb{R}\)
\(\mathbb{R}\setminus \{- 1\}\)
\(\mathbb{R}\setminus \{ - 1;0\}\)
9000104402
Level:
A
Find a set of the values of the real parameter
\(a\) which
ensure that the following equation has no solution.
\[
2a^{2}x - ax - 2a = -1
\]
\(\left \{0\right \}\)
\(\left \{\frac{1}
{2}\right \}\)
\(\left \{-\frac{1}
{2}\right \}\)
\(\left \{-\frac{1}
{2}; \frac{1}
{2}\right \}\)
9000104403
Level:
A
Find a set of the values of the real parameter
\(a\) which
ensure that the following equation has infinitely many solutions.
\[
3a^{2}x - 2ax + 4 = 6a
\]
\(\left \{\frac{2}
{3}\right \}\)
\(\left \{-\frac{2}
{3}\right \}\)
\(\left \{0\right \}\)
\(\left \{0; \frac{2}
{3}\right \}\)
9000104404
Level:
A
Find a set of the values of the real parameter
\(a\) which
ensure that the following equation has infinitely many solutions.
\[
a^{2}x + 1 = a^{2} + ax
\]
\(\left \{1\right \}\)
\(\left \{-1;1\right \}\)
\(\left \{0\right \}\)
\(\left \{-1\right \}\)
9000104405
Level:
A
Find a set of the values of the real parameter
\(a\) which
ensure that the following equation has a unique solution.
\[
a^{3}x + 3 = 3a^{2}x + a
\]
\(\mathbb{R}\setminus \left \{0;3\right \}\)
\(\left \{0\right \}\)
\(\left \{0;3\right \}\)
\(\mathbb{R}\setminus \left \{3\right \}\)
9000104501
Level:
A
Consider equation
\[
\frac{x - 3}
{a} = \frac{a - x}
{3} + 2
\]
with an unknown \(x\in \mathbb{R}\)
and a real parameter \(a\in \mathbb{R}\setminus \{0\}\).
Identify a statement which is not true.
For \(a\mathrel{\in }\{ - 3;0\}\) we
have \(x = \frac{1}
{a+3}\).
For \(a\mathrel{\notin }\{ - 3;0\}\) we
have \(x = a + 3\).
If \(a = -3\),
then the equation has infinitely many solutions.
9000101604
Level:
A
Expand the polynomial \(\left (2x^{2} + 4x\right )^{2} -\left (4x - 2x^{2}\right )^{2}\).
\(32x^{3}\)
\(0\)
\(32x^{3} - 8x\)
\(32x^{3} - 32x^{2} + 8x\)
9000101001
Level:
A
Determine whether two lines $p$ and $q$ are identical, parallel, intersecting or skew.
\[\begin{aligned}
p\colon x & = 1 + t, & &
\\y & = 2 - t, & &
\\z & = 1 - t;\ t\in \mathbb{R} & &
\end{aligned}\] \[\begin{aligned}
q\colon x & = 2s, & &
\\y & = -1, & &
\\z & = 2 - 2s;\ s\in \mathbb{R} & &
\end{aligned}\]
intersecting lines
skew lines
identical lines
parallel lines, not identical