1003044501
Level:
A
Identify the solution of the equation.
\[ 2^{x^2-x-3}=8. \]
\( x_1=3 \), \( x_2=-2 \)
\( x_1=-3\), \( x_2=2 \)
\( x_1=0 \), \( x_2=1 \)
\( x_1=0 \), \( x_2=-1 \)
Exponential equations and inequalities
1003044703
Level:
B
The following exponential equation has two solutions \( x_1 \) and \( x_2 \). Solution \( x_1 \) is a prime number. Find the sum of \( 3x_1 \) and \( 2x_2 \).
\[ 3^{3-x}\cdot4^{x-1}+3^x\cdot4^{2-x}=21 \]
\( 8 \)
\( 7 \)
\( 0 \)
\( 3 \)
1003044702
Level:
B
How many solutions does the following equation have?
\[ 2^x\cdot3^{1-x}-2^{1-x}\cdot3^x=1 \]
Exactly one solution, \( x=0 \)
Exactly one solution, \( x=-1 \)
Exactly two solutions, \( x_1=-1\), \( x_2=0 \)
Exactly two solutions, \( x_1=-\frac23\), \( x_2=1 \)
1003044701
Level:
B
Identify the solution of the following equation.
\[10\cdot25^x+10\cdot4^x=29\cdot10^x \]
\( x_1=-1;\ x_2=1 \)
The equation has no solution.
\( x_1=\frac25;\ x_2=\frac52 \)
\( x_1=-\frac52;\ x_2=-\frac25 \)
1003044607
Level:
B
Identify the solution of the equation.
\[ 16\cdot16^{\sqrt{2x+5}}-65\cdot4^{\sqrt{2x+5}}+4=0 \]
\( x=-2 \)
\( x_1=-\frac12;\ x_2=2 \)
\( x_1=\frac1{16};\ x_2=4 \)
\( x_1=-4;\ x_2=-\frac1{16} \)
1003044606
Level:
B
How many solutions does the following equation have?
\[ 7^{\sqrt x}+7^{1-\sqrt x}=8 \]
Exactly two solutions
Exactly one solution
No solutions
Infinitely many solutions
1003044605
Level:
B
Identify the solution of the equation.
\[ 3^{2x-2}-244\cdot3^{x-5}=-3^{-3} \]
\( x_1=-3;\ x_2=2 \)
\( x_1=-2;\ x_2=3 \)
\( x_1=\frac1{27};\ x_2=9 \)
\( x_1=-9;\ x_2=-\frac1{27} \)
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.
\[ 25^x+5=6\cdot5^x \]
\( x_1=0;\ x_2=1 \)
\( x=1 \)
\( x_1=1;\ x_2=5 \)
\( x_1=-1;\ x_2=-5 \)