9000105504
Level:
B
Find the distance between the points where the
\(y\)-axis
intersects the following hyperbola.
\[
H\colon \frac{\left (x - 4\right )^{2}}
{10} -\frac{\left (y - 5\right )^{2}}
{15} = 1
\]
\(6\)
\(2\)
\(4\)
\(8\)
B
9000105502
Level:
B
Find the distance between the points where the
\(x\)-axis
intersects the following hyperbola.
\[
H\colon \frac{\left (x - 1\right )^{2}}
{10} -\frac{\left (y - 3\right )^{2}}
{6} = 1
\]
\(10\)
\(14\)
\(12\)
\(8\)
9000105503
Level:
B
Find the distance between the points where the
\(y\)-axis
intersects the following hyperbola.
\[
H\colon \frac{\left (x - 4\right )^{2}}
{8} -\frac{\left (y - 3\right )^{2}}
{1} = 1
\]
\(2\)
\(4\)
\(6\)
\(8\)
9000106304
Level:
B
Find the third coordinate of the point
\(B = [2;0;?]\) using the fact that this
point is in the plane \(\alpha \)
defined by the equation
\[
\alpha \colon 2x + y - z - 5 = 0.
\]
Use the point \(B\) to
find the angle \(\varphi \)
between the plane \(\alpha \)
and the line \(AB\),
where \(A = [0;0;1]\).
\(\varphi = 60^{\circ }\)
\(\varphi = 45^{\circ }\)
\(\varphi = 30^{\circ }\)
\(\varphi = 75^{\circ }\)
9000106306
Level:
B
Find the general equation of the plane which is perpendicular to the plane
\(\alpha \)
\[
\alpha \colon 2x + y - z - 5 = 0
\]
and contains the line \(AB\),
where \(A = [0;0;1]\)
and \(B\) is a
point in \(\alpha \)
defined by it's first two coordinates
\[
B = [2;0;?].
\]
\(x - y + z - 1 = 0\)
\(x + y - z + 1 = 0\)
\(2x - y + z - 1 = 0\)
\(- 2x + y - z + 1 = 0\)
9000106308
Level:
B
In the following list identify a pair of planes such that the distance of planes from the plane $\alpha$ is the same as the distance between the point $A=[0;0;1]$ and the plane \(\alpha \).
\[
\alpha \colon 2x + y - z - 5 = 0
\]
\(\begin{aligned}[t] 2x + y - z +\phantom{ 1}1& = 0&
\\2x + y - z - 11& = 0
\\ \end{aligned}\)
\(\begin{aligned}[t] 2x + y - z +\phantom{ 1}1& = 0&
\\2x + y - z - 10& = 0
\\ \end{aligned}\)
\(\begin{aligned}[t] 2x + y - z +\phantom{ 1}1& = 0&
\\2x + y - z - 12& = 0
\\ \end{aligned}\)
\(\begin{aligned}[t] 2x + y - z + 1& = 0&
\\2x + y - z - 9& = 0
\\ \end{aligned}\)
9000105501
Level:
B
Find the distance between the points where the
\(x\)-axis
intersects the following hyperbola.
\[
H\colon \frac{\left (x - 3\right )^{2}}
{20} -\frac{\left (y - 2\right )^{2}}
{5} = 1
\]
\(12\)
\(14\)
\(10\)
\(8\)
9000106301
Level:
B
Find the line $k$ which is perpendicular to the plane
\(\alpha \)
\[
\alpha \colon 2x + y - z - 5 = 0
\]
and passes through the point \(A = [0;0;1]\).
\(\begin{aligned}[t] x& =\phantom{ 1 -} 2t, &
\\y& =\phantom{ 1 -}\ t,
\\z& = 1 - t;\ t\in \mathbb{R}
\\ \end{aligned}\)
\(\begin{aligned}[t] x& =\phantom{ -}2 + 2m, &
\\y& =\phantom{ -}1 +\phantom{ 2}m,
\\z& = -1 -\phantom{ 2}m;\ m\in \mathbb{R}
\\ \end{aligned}\)
\(\begin{aligned}[t] x& =\phantom{ -}2k, &
\\y& =\phantom{ -2}k,
\\z& = -\phantom{2}k;\ k\in \mathbb{R}
\\ \end{aligned}\)
\(\begin{aligned}[t] x& =\phantom{ -}2, &
\\y& =\phantom{ -}1,
\\z& = -1 + u;\ u\in \mathbb{R}
\\ \end{aligned}\)
9000105509
Level:
B
Find the distance between the points of intersection of the given hyperbola with the given straight line.
\[
H\colon \frac{\left (x - 6\right )^{2}}
{10} -\frac{\left (y - 2\right )^{2}}
{6} = 1;\quad p\colon x - 11 = 0
\]
\(6\)
\(12\)
\(10\)
\(8\)
9000106302
Level:
B
The plane \(\alpha \)
has equation
\[
\alpha : 2x + y - z - 5 = 0.
\]
The line \(k\) passes
through the point \(A = [0;0;1]\) and
is perpendicular to \(\alpha \).
Find the intersection \(S\)
of the line \(k\) and
the plane \(\alpha \).
\(S = [2;1;0]\)
\(S = [2;0;1]\)
\(S = [-2;1;0]\)
\(S = [-2;0;1]\)