C

9000120308

Level: 
C
The height \(v\) of a regular hexagonal prism is a double of its side \(a\). The volume of the prism is \(648\sqrt{3}\, \mathrm{cm}^{3}\). Use this information to find the length of the longest solid diagonal in the prism.
\(12\sqrt{2}\, \mathrm{cm}\)
\(10\sqrt{6}\, \mathrm{cm}\)
\(12\sqrt{6}\, \mathrm{cm}\)
\(6\sqrt{10}\, \mathrm{cm}\)
\(\sqrt{432}\, \mathrm{cm}\)

9000120304

Level: 
C
The side of a regular hexagonal prism \(ABCDEFA'B'C'D'E'F'\) is \(a = 3\, \mathrm{cm}\) and the height \(v = 8\, \mathrm{cm}\). Find the length of the diagonal \(AD'\).
\(10\, \mathrm{cm}\)
\(\sqrt{73}\, \mathrm{cm}\)
\(\sqrt{82}\, \mathrm{cm}\)
\(2\sqrt{8}\, \mathrm{cm}\)
\(2\sqrt{6}\, \mathrm{cm}\)

9000120305

Level: 
C
The side of a regular hexagonal prism \(ABCDEFA'B'C'D'E'F'\) shown in the picture is \(a = 3\, \mathrm{cm}\) and the height is \(v = 8\, \mathrm{cm}\). Find the angle between the diagonal \(AD'\) and the base plane \(ABC\) (round your result to the nearest degree).
\(53^{\circ }\)
\(37^{\circ }\)
\(45^{\circ }\)
\(61^{\circ }\)
\(72^{\circ }\)

9000106905

Level: 
C
The motion with a constant deceleration is described by the relation \[ s = v_{0}t -\frac{1} {2}at^{2}. \] Consequently, the graph which shows the distance as a function of time is part of a parabola. Find the vertex equation of this parabola, if \(v_{0} = 8\, \mathrm{m}/\mathrm{s}\) and \(a = 4\, \mathrm{m}/\mathrm{s}^{2}\).
\(-\frac{1} {2}(s - 8) = (t - 2)^{2}\)
\(\frac{1} {2}(s + 4) = (t + 2)^{2}\)
\(2(s + 8) = (t + 2)^{2}\)
\(- 2(s + 4) = (t + 2)^{2}\)

9000106806

Level: 
C
Given points \(A = [0;5]\), \(B = [6;1]\), \(C = [7;9]\), find the direction vector of the line passing through the point \(A\) and perpendicular to the segment \(BC\) (i.e. the line which contains the altitude of the triangle \(ABC\) through the point \(A\)).
\((8;-1)\)
\((1;8)\)
\((1;9)\)
\((-9;1)\)

9000106807

Level: 
C
Consider the points \(A = [0;5]\), \(B = [6;1]\), \(C = [7;9]\) and the triangle \(ABC\). Find the direction vector of the line which is the perpendicular bisector of the side \(b\) (i.e. the line through the midpoint of the side \(AC\) which is perpendicular to the segment \(AC\)).
\((4;-7)\)
\((7;4)\)
\((7;9)\)
\((7;-9)\)

9000106808

Level: 
C
Consider the points \(A = [0;5]\), \(B = [6;1]\), \(C = [7;9]\) and the triangle \(ABC\). Find the direction vector of the line which is the bisector of the angle \(ACB\) (i.e. the line which splits the internal angle at the point \(C\) into two angles with equal measures).
\((2;3)\)
\((6;-4)\)
\((7;9)\)
\((7;8)\)

9000106805

Level: 
C
Given points \(A = [0;5]\), \(B = [6;1]\), \(C = [7;9]\), find the direction vector of the line passing through the point \(A\) and the midpoint of the segment \(BC\) (i.e. the median of the triangle \(ABC\) through the vertex \(A\)).
\((1;0)\)
\((1;8)\)
\((1;9)\)
\((6.5;5)\)