2000000503
Level:
C
The set of all solutions of the inequality \(4^{\frac{2}{x+3}}>1\) is the interval:
\( (-3;\infty)\)
\( (-\infty;3)\)
\( (3;\infty)\)
\( (-\infty;-3)\)
Exponential equations and inequalities
2000000507
Level:
C
Find the solution set of the following inequality.
\[
16^x -4^x\leq 0
\]
\( (-\infty; 0] \)
\( [ 0; \infty) \)
\( [ 1; \infty) \)
\( (-\infty; 1] \)
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 \(f(x)\leq 5\).
\( a=\sqrt{2}\), \(b=1\), \( x \in ( -\infty;4]\)
\( a=\sqrt[4]{3}\), \(b=2\), \( x \in ( -\infty;4]\)
\( a=\sqrt{2}\), \(b=-4\), \( x \in ( -\infty;9]\)
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.
\(T=4\)
\( T=32\)
\( T=16\)
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.
\(6.48\ \mathrm{mg}\)
\(1.28\ \mathrm{mg}\)
\(4.8\ \mathrm{mg}\)
2010008206
Level:
C
Solve the following inequality.
\[ 3^{x-1}-3^{x+2}-3^{x+1}\geq -105\]
\(x \in (-\infty;2 ] \)
\(x \in [ 2;+\infty) \)
\(x \in [ 3;+\infty) \)
\(x \in (-\infty;3 ] \)
2010008207
Level:
C
Solve the following inequality.
\[ 2^{x-2}-2^{x+1}-2^{x+4}\leq -142\]
\(x \in [ 3;+\infty) \)
\(x \in [ 2;+\infty) \)
\(x \in (-\infty;2 ] \)
\(x \in (-\infty;3 ] \)
2010008208
Level:
C
Solve the following inequality.
\[ \left(\frac15\right)^{2x}- 3\cdot \left(\frac15\right)^x-10 >0\]
\(x \in (-\infty;-1 ) \)
\(x \in (-1;+\infty) \)
\(x \in (-2;-1) \)
\(x \in (-\infty;0) \)
2010008209
Level:
C
Solve the following inequality.
\[ \left(\frac12\right)^{2x}- 7\cdot \left(\frac12\right)^x-8 < 0\]
\(x \in (-3;+\infty) \)
\(x \in (-3;0) \)
\(x \in (-\infty;-1) \)
\(x \in (-\infty;-3) \)
2010008214
Level:
C
How many solutions in the set of positive integers does the following inequality have?
\[ 2^{3x}\cdot 7^{x-2}\leq 4^{x+1}\]
\( 2\)
\( 3\)
\( 1\)
\( 4\)