1003064203
Level:
C
Find the solution of the following inequality.
\[ \sqrt{2^{2x}-2^{3+x} } < 2^x-2^3 \]
No solution
\( x\in(-\infty;3) \)
\( x\in(8;\infty) \)
\( x\in[3;\infty) \)
Exponential equations and inequalities
1003064202
Level:
C
Find the solution of the following inequality.
\[ 2^x\leq 3-2^{1-x} \]
\( x\in[0; 1] \)
\( x\in[1; 2] \)
\( x\in[-2; -1] \)
No solution
1003064201
Level:
C
How many solutions does the following inequality in the set of positive integers have?
\[ 16^x-3\cdot4^x-4\leq0 \]
Exactly one solution
Infinitely many solutions
Exactly two solutions
No solutions
1003044507
Level:
A
How many solutions does the following equation have?
\[ 2^{x+2}+3\cdot3^{x+2}=3^{x+3}+2\cdot2^{x+1} \]
Infinitely many solutions
Exactly one solution
No solutions
Exactly two solutions
1003044506
Level:
A
Which of the given numbers is the product of all the solutions of the following exponential equations?
\begin{align*} 3^x+3^{x+2}&=30\\ 3^{x+1}+3^{x-1}&=\frac{10}9 \end{align*}
\( -1 \)
\( 0 \)
\( 1 \)
\( 2 \)
1003044505
Level:
A
Identify the solution of the equation.
\[ 2^{x-1}-7^{x-3}=2^{x-2}+2^{x-3}\]
\( x=3 \)
The equation has no solution.
\( x=-3 \)
\( x=0 \)
1003044504
Level:
A
How many solutions does the following equation have?
\[ 5^x+5^{x+1}+5^{x+2}+5^{x+3}=156 \]
Exactly one solution
No solutions
Infinitely many solutions
Exactly two solutions
1003044503
Level:
A
Identify the solution of the equation.
\[ 3^{x-3}-3^{x-2}=2 \]
The equation has no solution.
\( x=3 \)
\( x=-3 \)
\( x=2 \)
1003044502
Level:
A
The following exponential equation has solutions \( x_1 \) and \( x_2 \). Find the sum of \( x_1 \) and \( x_2 \).
\[ \frac{4^{x^2 }}{16^{x+1}} =16\cdot4^x \]
\( 3 \)
\( -3 \)
\( -1 \)
\( 2 \)
1003044501
Level:
A
Identify the solution of the equation.
\[ 2^{x^2-x-3}=8. \]
\( x_1=3 \), \( x_2=-2 \)
\( x_1=-3\), \( x_2=2 \)
\( x_1=0 \), \( x_2=1 \)
\( x_1=0 \), \( x_2=-1 \)