9000121703
Level:
A
Consider a triangle \(ABC\) with
sides of the lengths \(4\, \mathrm{cm}\),
\(4\, \mathrm{cm}\) and
\(4\, \mathrm{cm}\). Identify the type
of the triangle \(ABC\).
equilateral
isosceles
right
obtuse
A
9000120309
Level:
A
The sides of a rectangular box shown in the picture are \(a = 3\, \mathrm{cm}\),
\(b = 4\, \mathrm{cm}\), and
\(c = 12\, \mathrm{cm}\). The space diagonal
is \(u_{t}\) and the longest
face diagonal is \(u_{s}\).
Find the ratio \(u_{t} : u_{s}\).
\(13\sqrt{10} : 40\)
\(13 : \sqrt{153}\)
\(13 : 12\)
\(4\sqrt{10} : 5\)
\(4\sqrt{10} : 13\)
9000121704
Level:
A
Find the measure of the angle \(CAB\)
in an isosceles triangle \(ABC\)
with the interior angle \( BCA\)
of the measure \(120^{\circ }\).
\(30^{\circ }\)
\(60^{\circ }\)
\(15^{\circ }\)
\(45^{\circ }\)
9000121705
Level:
A
Consider an isosceles triangle \(ABC\) with sides \(AC\) and \(BC\) of equal length. The measure of the angle \( BAC\) is \(40^{\circ }\). \(X\) is the point of intersection between the line $AB$ and the line through the vertex \(C\) perpendicular to it. Find the measure of the angle \( BCX\).
\(50^{\circ }\)
\(80^{\circ }\)
\(100^{\circ }\)
\(40^{\circ }\)
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.
\(480\sqrt{3}\, \mathrm{cm}^{3}\)
\(960\, \mathrm{cm}^{3}\)
\(288\sqrt{3}\, \mathrm{cm}^{3}\)
\(160\sqrt{3}\, \mathrm{cm}^{3}\)
\(240\, \mathrm{cm}^{3}\)
9000120307
Level:
A
The lengths of a side, base diagonal and solid diagonal through the vertex
\(A\) in a rectangular
box \(ABCDEFGH\)
are \(|AB| = 6\, \mathrm{cm}\),
\(|AC| = 10\, \mathrm{cm}\),
\(|AG| = 15\, \mathrm{cm}\). Find
the volume of the box.
\(240\sqrt{5}\, \mathrm{cm}^{3}\)
\(900\, \mathrm{cm}^{3}\)
\(300\sqrt{5}\, \mathrm{cm}^{3}\)
\(600\sqrt{2}\, \mathrm{cm}^{3}\)
\(240\sqrt{2}\, \mathrm{cm}^{3}\)
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.
\(26.57^{\circ }\)
\(45^{\circ }\)
\(54.74^{\circ }\)
\(60^{\circ }\)
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