Exponential equations and inequalities

200001604

Level: 
B
Let \( A= \left\{ x \in \mathbb{R}\colon \left(\frac{\sqrt{2}}2\right)^{5x} < 8 \cdot 4^{3-2x}\right\}\) and \( B=\{x \in \mathbb{R}\colon 2^x-4\cdot 2^{-x}>3\}\). Find \(A \cap B\).
\(A \cap B=(2;6)\)
\(A \cap B=(-\infty;-1)\cup(4;6)\)
\(A \cap B=(-\infty;-1)\cup(2;6)\)

2000010606

Level: 
B
For which values of parameter \(p\) is the function \(f(x)=(p^2-4p+3)^x\) an increasing exponential function?
\(p \in \left(-\infty;2-\sqrt{2}\right) \cup \left(2+\sqrt{2};\infty\right)\)
\(p \in \left(2-\sqrt{2};2+\sqrt{2}\right)\)
\(p \in \left(2-\sqrt{2};1\right) \cup \left(3;2+\sqrt{2}\right)\)

2000010605

Level: 
C
The patient took a single dose of \(50\ \mathrm{mg}\) of the drug. Within \(3\) hours \(40\%\) of the dose was excreted from his body. The mass \(m\) (mg) of the drug in the body after time \(t\) (hours) is given by the formula \(m(t)=m_0a^t\), where \(m_0\) (mg) is the initial mass and \(a\) is a constant. Calculate how much medicine the patient had in his body after \(12\) hours.
\(6.48\ \mathrm{mg}\)
\(1.28\ \mathrm{mg}\)
\(4.8\ \mathrm{mg}\)

2000010604

Level: 
C
\(10\ \mathrm{mg}\) of a \(320\ \mathrm{mg}\) sample of a radioactive element remained after \(20\) days. Calculate the half-life \(T\) (days) of this element if you know that the dependence of its mass \(m\) (mg) on time \(t\) (days) is given by the formula \(m(t)=m_0\left(\frac12\right)^{\frac{t}{T}}\), where \(m_0\) (mg) is the initial mass.
\(T=4\)
\( T=32\)
\( T=16\)

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.

2000010601

Level: 
C
The graph of the function \(f(x)=a^x+b~\) ( \(a>0\), \(a\neq1\) ) has been moved \(4\) units to the right and two units down. The shifted graph intersects the \(x\)-axis at the point \([4;0]\) and passes through the point \([8;3]\). Find \(a\) and \(b\) and solve the inequality \(f(x)\leq 5\).
\( a=\sqrt{2}\), \(b=1\), \( x \in ( -\infty;4]\)
\( a=\sqrt[4]{3}\), \(b=2\), \( x \in ( -\infty;4]\)
\( a=\sqrt{2}\), \(b=-4\), \( x \in ( -\infty;9]\)