9000104303
Level:
B
Assuming \(a < 3\),
solve the following inequality.
\[
ax - 3\geq 3x - a
\]
\(\left (-\infty ;-1\right ] \)
\(\left (-\infty ;-1\right )\)
\(\left (-1;\infty \right )\)
\(\mathbb{R}\)
B
9000105409
Level:
B
Find the distance between the point \([7;3]\)
and the focus of the parabola \(x^{2} - 8x + 8y + 8 = 0\).
\(5\)
\(9\sqrt5\)
\(3\)
\(\sqrt{13}\)
9000104304
Level:
B
Assuming \(a < 0\),
solve the following inequality.
\[
\frac{x}
{a}\geq 1
\]
\(\left (-\infty ;a\right ] \)
\(\left (-\infty ;a\right )\)
\(\left [ a;\infty \right )\)
\(\left (a;\infty \right )\)
9000104305
Level:
B
Assuming \(a > -1\),
solve the following inequality.
\[
\frac{2x}
{a + 1} - 1 < 0
\]
\(\left (-\infty ; \frac{a+1}
{2} \right )\)
\(\left (-\frac{a+1}
{2} ; \frac{a+1}
{2} \right )\)
\(\left \{\frac{a+1}
{2} \right \}\)
\(\left (\frac{a+1}
{2} ;\infty \right )\)
9000104307
Level:
B
Assuming \(a\in \left (0;2\right )\),
solve the following inequality.
\[
a\left (a - 2\right )x > 1
\]
\(\left (-\infty ; \frac{1}
{a\left (a-2\right )}\right )\)
\(\left ( \frac{1}
{a\left (a-2\right )};\infty \right )\)
\(\emptyset \)
\(\left \{ \frac{1}
{a\left (a-2\right )}\right \}\)
9000101805
Level:
B
Find the vector \(\vec{v}\) such that
the length of this vector is \(5\)
and \(\vec{v}\) is perpendicular
to the vector \(\vec{u} = (-1;0.75)\).
\(\vec{v} = (3;4)\)
\(\vec{v} = (3;-4)\)
\(\vec{v} = (4;-3)\)
\(\vec{v} = (5;0)\)
9000104310
Level:
B
Assuming \(a\in \left (0;1\right )\),
solve the following inequality.
\[
2a\left (1 - a\right )x > 3
\]
\(\left ( \frac{3}
{2a\left (1-a\right )};\infty \right )\)
\(\left (- \frac{3}
{2a\left (1-a\right )};\infty \right )\)
\(\left (- \frac{3}
{2a\left (1-a\right )}; \frac{3}
{2a\left (1-a\right )}\right )\)
\(\left (-\infty ; \frac{3}
{2a\left (1-a\right )}\right )\)
9000101806
Level:
B
Find the value of the parameter \(a\)
which ensures that the vectors \(\vec{u} = (3;a;-2)\)
and \(\vec{v} = (-6;4;a - 3)\)
are perpendicular.
\(a = 6\)
\(a = 12\)
\(a = -6\)
\(a = 3\)
9000105401
Level:
B
The parabola \(P\colon x^{2} - 6x - 4y + 5 = 0\) has
two \(x\)-intercepts.
Find their distance.
\(4\)
\(6\)
\(8\)
\(10\)
9000101807
Level:
B
Given points \(A = [1;1]\),
\(B = [5;2]\) and
\(C = [8;7]\), find the
angle \(\measuredangle ABC\).
\(135^{\circ }\)
\(26.5^{\circ }\)
\(30^{\circ }\)
\(60^{\circ }\)