2010016008
Level:
A
If \(\log_2 a=b\) then the value of \(\log_8 a\) is equal to:
\(\frac{b}3\)
\(\frac{b}2\)
\(3b\)
\(4b\)
A
2010016007
Level:
A
If \(\log_3 a=b\) then the value of \(\log_9 a\) is equal to:
\(\frac{b}2\)
\(2b\)
\( \frac2{b}\)
\(9b\)
2010015807
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 shortest
face diagonal is \(u_{s}\).
Find the ratio \(u_{s} : u_{t}\).
\(5 : 13\)
\(13 : 5\)
\(13\sqrt{10}:40\)
\(4\sqrt{10}:13\)
2010015805
Level:
A
A cuboid has sides \(a = 6\, \mathrm{cm}\) and
\(b = 8\, \mathrm{cm}\), and the space diagonal
\(u = 11\, \mathrm{cm}\). Find the length of the side \(c\) (see the picture).
\( \sqrt{21}\,\mathrm{cm} \)
\( \sqrt{221}\,\mathrm{cm} \)
\( 21\,\mathrm{cm} \)
\( 10\,\mathrm{cm} \)
2010015707
Level:
A
Given vectors \(\vec{a} = (-1;2;1)\),
\(\vec{b} = (0;-1;1)\), and
\(\vec{c} = (-2;0;1)\), find the length
of the vector \(\vec{u} =\vec{ a} - 2\vec{b} + \vec{c}\).
\(|\vec{u}| = 5\)
\(|\vec{u}| = \sqrt{10}\)
\(|\vec{u}| = 3\)
\(|\vec{u}| = 1\)
2010015706
Level:
A
Given the vectors \(\vec{a}\),
\(\vec{b}\), and
\(\vec{c}\), find
\(\vec{a} - 3\vec{b} +\vec{ c}\).
\((10;5)\)
\((-8;5)\)
\((-8;-1)\)
\((7;5)\)
2010015705
Level:
A
We are given points \( A = [2;1] \), \( B = [7;2] \), and \( T = [4;3] \), where point \( T \) is the centroid of triangle \( ABC \). Find the coordinates of \( C \), which is the vertex of \( ABC \).
\( C = [3;6] \)
\( C = [4;8] \)
\( C = [3.5;7] \)
\( C = [5;6] \)
2010015704
Level:
A
Given the vectors \( \overrightarrow{a} \), \( \overrightarrow{b} \), and \( \overrightarrow{c} \) shown in the picture, express the vector \( \overrightarrow{c} \) as a linear combination of vectors \( \overrightarrow{a} \) and \( \overrightarrow{b} \).
\( \overrightarrow{c} = -\overrightarrow{a}-2\overrightarrow{b} \)
\( \overrightarrow{c} = -\overrightarrow{a} + \frac12 \overrightarrow{b} \)
\( \overrightarrow{c} = -2\overrightarrow{a} - \overrightarrow{b} \)
\( \overrightarrow{c} = 2\overrightarrow{a} + \frac32 \overrightarrow{b} \)
2010015703
Level:
A
The picture shows a rectangular cuboid \( ABCDEFGH \). In the cuboid find the vector that is the sum of \( \overrightarrow{AB} + \overrightarrow{AH} + \overrightarrow{EG} + \overrightarrow{FA} + \overrightarrow{HE} \).
\( \overrightarrow{AC} \)
\( \overrightarrow{FH} \)
\( \overrightarrow{AG} \)
\( \overrightarrow{BH} \)
2010015701
Level:
A
\( S \) is the midpoint of \( AB \). \( B = [3; -5] \), \( S = [0; -7] \). Find the coordinates of \( A \).
\( A = [-3; -9] \)
\( A = [1.5; -6] \)
\( A = [3; -12] \)
\( A = [-3; -2] \)