Exponential equations and inequalities

9000003709

Level: 
C
Solve the following inequality. \[ \left (\frac{2} {3}\right )^{2-3x} < \frac{2^{x+1}} {3^{x+1}} \]
\(\left (-\infty ; \frac{1} {4}\right )\)
\(\left (-\frac{1} {4};\infty \right )\)
\((-\infty ;4)\)
\(\left (\frac{1} {4};\infty \right )\)
\((4;\infty )\)
\(\left (-\infty ;-\frac{1} {4}\right )\)

9000003708

Level: 
B
Consider the exponential equation \[ 4^{x+2} - 5\cdot 4^{x+1} + 4^{x-1} + 240 = 0 \] with \(x\in \mathbb{R}\). In the following list identify a true statement on this equation.
The equation has a unique solution. This solution is a positive integer.
The equation has a unique solution. This solution is a negative number.
The equation does not have a solution.
The equation has two solutions.
Zero is a solution of this equation.
The equation has a unique solution. This solution is a negative integer.