2000000502
Level:
B
Find the solution of the following equation.
\[
11^{x-3}=13^{3-x}
\]
\(x=3\)
\(x=-3\)
\(x=11\)
\(x=13\)
Exponential Equations and Inequalities
2000000504
Level:
B
The equation \(2^x=6-3m\) with the unknown variable \(x\) and the parameter \(m\) has a solution if and only if:
\( m \in (-\infty,2)\)
\( m \in (-\infty,-2)\)
\( m \in (2,\infty)\)
\( m \in (-\infty,4)\)
2000000506
Level:
B
Which of the equations listed below has no solution?
\(5^x +3=2\)
\(\frac{1}{2^x}-12=11\)
\(10^{x+1} +3=4.23\)
\(\left (\sqrt{3}\right)^x +2=3\)
2000003001
Level:
B
Which of the given numbers is the solution of the equation
\( 3^{2x} + \frac{4}{27} = 2\cdot 3^{2x+1}-9^x\)?
\( x=-\frac{3}{2} \)
\( x=1 \)
\( x=\frac{3}{2} \)
\( x=-1\)
2000010602
Level:
B
For which values of the parameter \(k\) has the equation \(|2^x-3|=k\) two solutions the product of which is a negative number?
\( k \in (2,3)\)
\( k \in [ 2,3 ] \)
\( k \in (-\infty,2) \cup (3,+\infty)\)
2000010603
Level:
B
Find the coordinates of the point of intersection of the graphs of functions \( f(x)=\left(\frac35\right)^x\) and \(g(x)=\left(\frac{\sqrt{15}}{5}\right)^{x-1}\).
\( \left[-1,\frac53\right]\)
\( \left[-3,\frac{25}9\right]\)
The graphs of functions \(f\) and \(g\) have no points in common.
2000010606
Level:
B
For which values of parameter \(p\) is the function \(f(x)=(p^2-4p+3)^x\) an increasing exponential function?
\(p \in \left(-\infty,2-\sqrt{2}\right) \cup \left(2+\sqrt{2},\infty\right)\)
\(p \in \left(2-\sqrt{2},2+\sqrt{2}\right)\)
\(p \in \left(2-\sqrt{2},1\right) \cup \left(3,2+\sqrt{2}\right)\)
200001601
Level:
B
For which values of parameter \(m\in \mathbb{R}\) does the equation \(\left(\frac15\right)^x+m^2-9m+18=0\) have a solution?
\( m \in (3,6)\)
\( m \in (-\infty,3)\cup (6,\infty)\)
\( m=3 \lor m=6\)
200001602
Level:
B
Find all the values of parameter \(m\in \mathbb{R}\) such that the sum of the roots of the equation
\[2^{(m+1)x^2-4mx+\frac32}=\sqrt{2}\]
is greater than \(2\).
\( m \in (-\infty,-1)\cup (1,\infty)\)
\( m \in (-1,1)\)
\( m \in (1,\infty)\)
200001603
Level:
B
Find the solution of the equation \(2^2 \cdot 2^4\cdot 2^6 \cdot \dots \cdot 2^{2x}=\left(\frac{1}{32}\right)^{-x-1} \).
\( x=5\)
\( x=2.5\)
\( x=-1\)