9000121704
Level:
A
Find the measure of the angle \(CAB\)
in an isosceles triangle \(ABC\)
with the interior angle \( BCA\)
of the measure \(120^{\circ }\).
\(30^{\circ }\)
\(60^{\circ }\)
\(15^{\circ }\)
\(45^{\circ }\)
A
9000117402
Level:
A
Determine whether the following planes
\(\rho \) and
\(\sigma \) are
parallel, identical or intersecting.
\[
\begin{aligned}[t] \rho \colon &x = 2 + u - v, &
\\&y = 1 + 2u + 4v,
\\&z = -1 + 3u + 3v,\ u,v\in \mathbb{R},
\\ \end{aligned}\qquad \begin{aligned}[t] \sigma \colon &x = 2 + r - s, &
\\&y = 7 + 2r + 4s,
\\&z = 5 + 3r + 3s,\ s,t\in \mathbb{R}.
\\ \end{aligned}
\]
identical
parallel, not identical
intersecting
9000120310
Level:
A
The base of a rectangular box \(ABCDEFGH\)
has sides \(|AB| = 6\, \mathrm{cm}\) and
\(|BC| = 8\, \mathrm{cm}\). The angle between
the solid diagonal \(AG\)
and the base \(ABC\)
is \(60^{\circ }\).
Find the volume of the box.
\(480\sqrt{3}\, \mathrm{cm}^{3}\)
\(960\, \mathrm{cm}^{3}\)
\(288\sqrt{3}\, \mathrm{cm}^{3}\)
\(160\sqrt{3}\, \mathrm{cm}^{3}\)
\(240\, \mathrm{cm}^{3}\)
9000117403
Level:
A
Determine whether the following planes
\(\rho \) and
\(\sigma \) are
parallel, identical or intersecting.
\[
\begin{aligned}[t] \rho \colon &x = -u + v, &
\\&y = u + 2v,
\\&z = -u - v,\ u,v\in \mathbb{R},
\\ \end{aligned}\qquad \sigma \colon x-2y-3z+1 = 0
\]
parallel, not identical
identical
intersecting
9000121006
Level:
A
In the cube \(ABCDEFGH\) find the
angle between the lines \(BG\)
and \(GD\).
\(60^{\circ }\)
\(45^{\circ }\)
\(36.87^{\circ }\)
\(53.13^{\circ }\)
9000121007
Level:
A
In the cube \(ABCDEFGH\) find the
angle between the lines \(S_{BE}S_{AH}\)
and \(HC\), where
\(S_{BE}\) and
\(S_{AH}\) are centers of
the segments \(BE\)
and \(AH\),
respectively.
\(60^{\circ }\)
\(26.57^{\circ }\)
\(45^{\circ }\)
\(53.13^{\circ }\)
9000121009
Level:
A
In the cube \(ABCDEFGH\) find the
angle between the lines \(S_{HD}S_{FC}\)
and \(AB\), where
the points \(S_{HD}\)
and \(S_{FC}\) are
centers of \(HD\)
and \(FC\),
respectively.
\(26.57^{\circ }\)
\(45^{\circ }\)
\(54.74^{\circ }\)
\(60^{\circ }\)
9000121010
Level:
A
In the cube \(ABCDEFGH\) find the
angle between the lines \(ES_{CG}\)
and \(AG\), where
\(S_{CG}\) is the center
of the \(CG\).
\(54.74^{\circ }\)
\(19.47^{\circ }\)
\(35.26^{\circ }\)
\(60^{\circ }\)
9000120303
Level:
A
Identify a valid relation involving the angle
\(\alpha \)
defined as an angle between a solid diagonal and a face diagonal through the same
vertex in a cube.
\(\mathop{\mathrm{tg}}\nolimits \alpha = \frac{\sqrt{2}}
{2} \)
\(\sin \alpha = \frac{\sqrt{3}}
{2} \)
\(\cos \alpha = \frac{\sqrt{5}}
{3} \)
\(\mathop{\mathrm{cotg}}\nolimits \alpha = \sqrt{3}\)
\(\alpha = 45^{\circ }\)
9000117404
Level:
A
Determine whether the following planes are parallel, identical or intersecting.
\[\begin{aligned}
\rho \colon \frac{3}
{8}x + \frac{1}
{2}y -\frac{2}
{3}z - 1 = 0,\qquad \sigma \colon \frac{3}
{4}x + y -\frac{4}
{3}z - 2 = 0 & &
\end{aligned}\]
identical
parallel, not identical
intersecting