1003044605
Level:
B
Identify the solution of the equation.
\[ 3^{2x-2}-244\cdot3^{x-5}=-3^{-3} \]
\( x_1=-3;\ x_2=2 \)
\( x_1=-2;\ x_2=3 \)
\( x_1=\frac1{27};\ x_2=9 \)
\( x_1=-9;\ x_2=-\frac1{27} \)
B
1003044604
Level:
B
The following exponential equation has two solutions \( x_1 \) and \( x_2 \). Find the product of \( x_1 \) and \( x_2 \).
\[ 6^x+6^{2-x}=37 \]
\( 0 \)
\( 2 \)
\( 36 \)
\( 37 \)
1003044603
Level:
B
How many solutions does the following equation have?
\[ 81^x-8\cdot9^x=9 \]
Exactly one solution
Exactly two solutions
No solutions
Infinitely many solutions
1003044602
Level:
B
Identify the solution of the equation.
\[ 25^x+5=6\cdot5^x \]
\( x_1=0;\ x_2=1 \)
\( x=1 \)
\( x_1=1;\ x_2=5 \)
\( x_1=-1;\ x_2=-5 \)
1003044601
Level:
B
Identify the solution of the equation.
\[ 4^{2x}-20\cdot4^x=-64\]
\( x_1=1;\ x_2=2 \)
\( x_1=16;\ x_2=4 \)
\( x_1=-16;\ x_2=-4 \)
\( x_1=1;\ x_2=4 \)
1103025306
Level:
B
The base \( ABCD \) of a square pyramid \( ABCDV \) has an edge of \( 6\,\mathrm{cm} \). The height of the pyramid is \( 3\sqrt2\,\mathrm{cm} \). Find the distance between the point \( A \) and the line \( BV \) (see the picture).
\( 3\sqrt3\,\mathrm{cm} \)
\( 3\sqrt2\,\mathrm{cm} \)
\( \frac32\sqrt3\,\mathrm{cm} \)
\( 3\sqrt6\,\mathrm{cm} \)
1103025305
Level:
B
The base \( ABCD \) of a square pyramid \( ABCDV \) has an edge of \( 4\,\mathrm{cm} \). The height of the pyramid is \( 6\,\mathrm{cm} \). Find the distance between the point \( A \) and the point \( S_{VC} \), where \( S_{VC} \) is the midpoint of the edge \( VC \).
\( 3\sqrt{3}\,\mathrm{cm} \)
\( 4\sqrt{3}\,\mathrm{cm} \)
\( \sqrt{10}\,\mathrm{cm} \)
\( 3\sqrt{10}\,\mathrm{cm} \)
1103025304
Level:
B
The base \( ABCD \) of a square pyramid \( ABCDV \) has an edge of \( 8\,\mathrm{cm} \). The height of the pyramid is \( 9\,\mathrm{cm} \). Find the distance between the line \( S_{VA}S_{VD} \) and the line \( BC \). The points $S_{VA}$ and $S_{VD}$ are the midpoints of $VA$ and $VD$, respectively.
\( 7.5\,\mathrm{cm} \)
\( \frac23\sqrt{97}\,\mathrm{cm} \)
\( \frac{\sqrt{97}}2\,\mathrm{cm} \)
\( \sqrt{17}\,\mathrm{cm} \)
1103025303
Level:
B
The base \( ABCD \) of a square pyramid \( ABCDV \) has an edge of \( 6\,\mathrm{cm} \). The height of the pyramid is \( 4\,\mathrm{cm} \). Find the distance between the line \( S_{VA}S_{VC} \) and the line \( AC \). The points $S_{VA}$ and $S_{VC}$ are the midpoints of $VA$ and $VC$, respectively.
\( 2\,\mathrm{cm} \)
\( 2.5\,\mathrm{cm} \)
\( \frac{\sqrt{52}}2\,\mathrm{cm} \)
\( 4\,\mathrm{cm} \)
1103025302
Level:
B
The base \( ABCD \) of a square pyramid \( ABCDV \) has an edge of \( 6\,\mathrm{cm} \). The height of the pyramid is \( 4\,\mathrm{cm} \). Find the distance between the line \( S_{VB}S_{VC}\) and the line \( BC \). The points $S_{VB}$ and $S_{VC}$ are the midpoints of $VB$ and $VC$, respectively.
\( 2.5\,\mathrm{cm} \)
\( 2\,\mathrm{cm} \)
\( \frac{\sqrt{52}}2\,\mathrm{cm} \)
\( \frac{25}2\,\mathrm{cm} \)