2010006009
Level:
B
Find the distance between the point \([5,5]\)
and the focus of the parabola \(y^{2} +12x + 6y -15 = 0\).
\(10\)
\(8\)
\(3\sqrt{10}\)
\(2\sqrt{41}\)
B
2010006008
Level:
B
Parabola is a set of points that are equidistant from the point (focus)
and the line (directrix). Find the equation of the directrix of the parabola
\(x^{2} + 4x +8y-20= 0\).
\(y-5 = 0\)
\(y-1 = 0\)
\(x = 0\)
\(x+4 = 0\)
2010005909
Level:
B
Which of the given points is one of the vertices of the hyperbola \(25y^{2} - 4x^{2} - 24x + 50y - 111 = 0\)?
\([-3,1]\)
\([3,1]\)
\([-3,4]\)
\([3,4]\)
2010005907
Level:
B
Find the vertex of the following parabola.
\[
y^{2} + 12x - 6y - 15 = 0
\]
\([2,3]\)
\([-2,3]\)
\([2,-3]\)
\([-2,-3]\)
2010005902
Level:
B
Find the distance between the points of intersection of the given hyperbola with the given straight line $q$.
\[
H\colon \frac{\left (y+6\right )^{2}}
{10} -\frac{\left (x-5\right )^{2}}
{6} = 1,\quad q\colon y+1 = 0
\]
\(6\)
\(8\)
\(10\)
\(12\)
2010005901
Level:
B
Find the distance between the points where the
\(y\)-axis
intersects the following hyperbola.
\[
H\colon \frac{\left (y+3\right )^{2}}
{36} -\frac{\left (x+4\right )^{2}}
{9} = 1
\]
\(20\)
\(16\)
\(10\)
\(8\)
2010005707
Level:
B
Which of the given numbers belongs to the interval \( [ -5,5 ]\)?
\( \left(\sqrt2\right)^4-\left(\sqrt3\right)^4 \)
\( 2\left(\sqrt{0{.}1}\right)^2\cdot\left(\sqrt2\right)^{10} \)
\( \left(2\sqrt{5}\right)^2-\left(\sqrt3\right)^6 \)
\( 2\left(\sqrt{0{.}1}\right)^4+\left(\sqrt2\right)^8 \)
2010005706
Level:
B
What is the value of \( \left(3^{\sqrt6+\sqrt2}\right)^{\sqrt6-\sqrt2} \)?
\( 81\)
\( 3^{8} \)
\( 3^{\sqrt{32}} \)
\( 3^{2\sqrt{6}} \)
2010005705
Level:
B
Writing the expression \( \frac{16\cdot \sqrt[4]{4}\cdot \sqrt[3]{\frac12}}{\sqrt{8}\cdot\sqrt[3]{2}\cdot 4\cdot \sqrt[6]{16}} \) in the form of a power of \( 2 \), we will get:
\( 2^{-\frac13} \)
\( 2^{\frac13} \)
\( 2^{-\frac23} \)
\( 2^{\frac56} \)
2010005704
Level:
B
Writing the expression \( \frac{(0.25)^{-2}\cdot (x \colon y^2)^{-2} }{(2y)^4\cdot x^{-2}}\), \( x\neq0\), \( y\neq0 \) in a simplified form, we will get:
\( 1\)
\( y^{-8} \)
\( x^{-4} \)
\( \frac14 \)