9000108708
Level:
C
Find the volume of the parallelepiped \(ABCDEFGH\)
where \(A = [1;0;0]\),
\(B = [2;0;0]\),
\(D = [3;-2;0]\),
\(E = [2;1;5]\).
\(10\)
\(12\)
\(15\)
\(20\)
Points and vectors
9000108704
Level:
B
Consider a pair of vectors \(\vec{u} = (1;0;-1)\)
and \(\vec{v} = (2;-1;1)\). Find all the
vectors \(\vec{w}\) which are
perpendicular to both \(\vec{u}\)
and \(\vec{v}\) and
satisfy \(\left |\vec{w}\right | = 2\).
\(\vec{w} = \left (\frac{2\sqrt{11}}
{11} ; \frac{6\sqrt{11}}
{11} ; \frac{2\sqrt{11}}
{11} \right )\),
\(\vec{w} = \left (-\frac{2\sqrt{11}}
{11} ;-\frac{6\sqrt{11}}
{11} ;-\frac{2\sqrt{11}}
{11} \right )\)
\(\vec{w} = (-1;-3;-1)\),
\(\vec{w} = (1;3;1)\)
\(\vec{w} = \left (-\frac{1}
{2};-\frac{3}
{2};-\frac{1}
{2}\right )\),
\(\vec{w} = \left (\frac{1}
{2}; \frac{3}
{2}; \frac{1}
{2}\right )\)
\(\vec{w} = \left (\frac{2\sqrt{2}}
{3} ; \frac{3\sqrt{2}}
{2} ; \frac{2\sqrt{2}}
{3} \right )\),
\(\vec{w} = \left (-\frac{2\sqrt{2}}
{3} ;-\frac{3\sqrt{2}}
{2} ;-\frac{2\sqrt{2}}
{3} \right )\)
9000108702
Level:
B
A square has its center in \([1;4]\)
and a vertex in \([-1;2]\).
Find the remaining vertices of the square.
\([3;6]\),
\([-1;6]\),
\([3;2]\)
\([3;6]\),
\([-1;5]\),
\([3;1]\)
\([3;6]\),
\([-2;6]\),
\([4;2]\)
\([3;6]\),
\([-1;5]\),
\([3;2]\)
9000108701
Level:
B
Find all vectors which are perpendicular to the vector
\(\vec{u} = (3;4)\) and have the
length equal to \(1\).
\(\left (\frac{4}
{5};-\frac{3}
{5}\right )\),
\(\left (-\frac{4}
{5}; \frac{3}
{5}\right )\)
\(\left (\frac{4}
{7};-\frac{3}
{7}\right )\),
\(\left (-\frac{4}
{7}; \frac{3}
{7}\right )\)
\(\left ( \frac{1}
{\sqrt{10}};- \frac{3}
{\sqrt{10}}\right )\),
\(\left (- \frac{1}
{\sqrt{10}}; \frac{3}
{\sqrt{10}}\right )\)
\(\left (\frac{4}
{5}; \frac{3}
{5}\right )\),
\(\left (-\frac{4}
{5};-\frac{3}
{5}\right )\)
9000108804
Level:
B
The point \(A = [3;2]\) is rotated
about the center \(B = [1;1]\)
by \(60^{\circ }\). Find
the coordinate of its final position. Consider both clockwise and counterclockwise
direction.
\(\left [2\pm \frac{\sqrt{3}}
{2} ; \frac{3}
{2} \mp \sqrt{3}\right ]\)
\(\left [1\pm \frac{\sqrt{3}}
{2} ; \frac{1}
{2} \mp \sqrt{3}\right ]\)
\(\left [2\pm \frac{\sqrt{2}}
{2} ; \frac{3}
{2} \mp \sqrt{2}\right ]\)
\(\left [1\pm \frac{\sqrt{2}}
{2} ; \frac{1}
{2} \mp \sqrt{2}\right ]\)
9000108703
Level:
B
Given points \(A = [1;3]\),
\(C = [4;3]\),
\(B = [x;2]\), find the value of the
parameter \(x\) which ensures
that the vector \(AB\) is
perpendicular to the vector \(AC\).
\(1\)
\(2\)
\(- 1\)
\(0\)
9000108705
Level:
B
Given vectors \(\vec{u} = (1;y;3)\) and
\(\vec{v} = (1;2;1)\), find the value of
the parameter \(y\)
which ensures that the vectors are perpendicular.
\(- 2\)
\(1\)
\(2\)
\(0\)
9000108707
Level:
C
Find the area of the parallelogram \(ABCD\)
where \(A = [1;3;2]\),
\(B = [3;4;6]\),
\(D = [2;5;8]\).
\(\sqrt{77}\)
\(8\)
\(2\sqrt{13}\)
\(\sqrt{94}\)
9000108801
Level:
B
Find the angle between the vector \(\vec{u} = (1;\sqrt{2})\)
and \(\vec{v} = (3;-1)\).
Round to the nearest degree.
\(73^{\circ }\)
\(42^{\circ }\)
\(57^{\circ }\)
\(64^{\circ }\)
9000108802
Level:
B
Given the points \(A = [1;2]\),
\(B = [2;6]\) and
\(C = [3;-1]\), find the interior
angles of the triangle \(ABC\).
Round to the nearest degree.
\(22^{\circ }\),
\(26^{\circ }\),
\(132^{\circ }\)
\(26^{\circ }\),
\(45^{\circ }\),
\(109^{\circ }\)
\(22^{\circ }\),
\(48^{\circ }\),
\(110^{\circ }\)
\(17^{\circ }\),
\(31^{\circ }\),
\(132^{\circ }\)