B

9000108706

Level: 
B
Find all vectors which are parallel to the vector \(\vec{u} = (3;-1)\) and have the length equal to \(1\).
\(\left (\frac{3\sqrt{10}} {10} ;-\frac{\sqrt{10}} {10} \right )\), \(\left (-\frac{3\sqrt{10}} {10} ; \frac{\sqrt{10}} {10} \right )\)
\((0;-1)\), \((0;1)\)
\((-3;1)\), \((3;-1)\)
\(\left (\frac{3} {4};-\frac{1} {4}\right )\), \(\left (-\frac{3} {4}; \frac{1} {4}\right )\)

9000115602

Level: 
B
Complete the following statement: „The number is divisible by three if and only if ...”
the sum of its digits is divisible by three.
the number constituted from the last two digits is divisible by three.
the sum of its digits is odd.
the last digit of this number is \(3,\ 6\) or \(9\).

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 \(\left |\vec{w}\right | = 2\).
\(\vec{w} = \left (\frac{2\sqrt{11}} {11} ; \frac{6\sqrt{11}} {11} ; \frac{2\sqrt{11}} {11} \right )\), \(\vec{w} = \left (-\frac{2\sqrt{11}} {11} ;-\frac{6\sqrt{11}} {11} ;-\frac{2\sqrt{11}} {11} \right )\)
\(\vec{w} = (-1;-3;-1)\), \(\vec{w} = (1;3;1)\)
\(\vec{w} = \left (-\frac{1} {2};-\frac{3} {2};-\frac{1} {2}\right )\), \(\vec{w} = \left (\frac{1} {2}; \frac{3} {2}; \frac{1} {2}\right )\)
\(\vec{w} = \left (\frac{2\sqrt{2}} {3} ; \frac{3\sqrt{2}} {2} ; \frac{2\sqrt{2}} {3} \right )\), \(\vec{w} = \left (-\frac{2\sqrt{2}} {3} ;-\frac{3\sqrt{2}} {2} ;-\frac{2\sqrt{2}} {3} \right )\)

9000115603

Level: 
B
Complete the following statement: „The number is divisible by four if and only if ...”
the number constituted from the last two digits is divisible by four.
the sum of its digits is divisible by four.
the last digit of this number is \(4\).
the last digit of this number is even.

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.
\(\left [2\pm \frac{\sqrt{3}} {2} ; \frac{3} {2} \mp \sqrt{3}\right ]\)
\(\left [1\pm \frac{\sqrt{3}} {2} ; \frac{1} {2} \mp \sqrt{3}\right ]\)
\(\left [2\pm \frac{\sqrt{2}} {2} ; \frac{3} {2} \mp \sqrt{2}\right ]\)
\(\left [1\pm \frac{\sqrt{2}} {2} ; \frac{1} {2} \mp \sqrt{2}\right ]\)