1003102411
Level:
B
Without a calculator evaluate the following expression and select the correct value.
\[ \frac{\log\sqrt6}{\log6} \]
\( \frac12 \)
\( 2 \)
\( \frac{\sqrt6}6 \)
\( \sqrt6-6 \)
B
1003102409
Level:
B
Without a calculator evaluate the following expression and select the correct value.
\[ \log_64+\log_69 \]
\( 2 \)
\( 36 \)
\( 13 \)
\( 6 \)
1003090506
Level:
B
The interest rate on deposits in a certain bank initially equaled \( 5\% \) per year but then it was decreased by \( 20\% \). By how many percentage points has the interest rate decreased?
by \( 1\) percentage point
by \( 1.2 \) percentage point
by \( 2 \) percentage points
by \( 2.1 \) percentage point
1003090505
Level:
B
The interest rate on deposits in a certain bank initially equaled \( 5\% \) per year but then it was increased by \( 1.2 \) percentage point. What was the percentage increase of the interest rate?
\( 24\% \)
\( 6\% \)
\( 6.2\% \)
\( 20\% \)
1003109305
Level:
B
Solve the following equation in the set of complex numbers. (Solve the equation by substitution.)
\[ (2x + 3)^4 - 256 = 0 \]
\( \left\{-\frac72;\frac12;-\frac32\pm2\mathrm{i} \right\} \)
\( \left\{-\frac72;\frac12;\frac32\pm2\mathrm{i} \right\} \)
\( \left\{\frac72;-\frac12;\frac32\pm2\mathrm{i} \right\} \)
\( \left\{\frac72;-\frac12;-\frac32\pm2\mathrm{i} \right\} \)
1003034107
Level:
B
By computing \( 4^{11}\cdot4^{-11}\) we get:
\( 1 \)
\( 0 \)
\( 4^{22} \)
\( 16^{-121} \)
1003083004
Level:
B
Find the real number \( a \) such that the following system of equations has no solution.
\[ \begin{aligned} \frac25x-\frac a4y&=4 \\ -\frac x4 + \frac{5y}8&=\frac52 \end{aligned}\]
\( 4 \)
\( -\frac52 \)
Such a real number \( a \) does not exist.
\( -4 \)
1003102505
Level:
B
Find the set of all complex roots of the following equation.
\[ \left(x^2- 2x + 5\right)\left(x^2 + 6x + 10\right) = 0 \]
\( \{1\pm2\mathrm{i}; -3\pm\mathrm{i} \} \)
\( \{-1\pm2\mathrm{i}; -3\pm\mathrm{i} \} \)
\( \{-1\pm2\mathrm{i}; 3\pm\mathrm{i} \} \)
\( \{1\pm2\mathrm{i}; 3\pm\mathrm{i} \} \)
1003102504
Level:
B
Find the set of all complex roots of the following quadratic equation.
\[ x^2 + 4x + 8 = 0 \]
\( \left\{ 2\sqrt2\left(\cos\frac{3\pi}4+\mathrm{i}\cdot\sin\frac{3\pi}4\right); 2\sqrt2\left(\cos\frac{5\pi}4+\mathrm{i}\cdot\sin\frac{5\pi}4\right) \right\} \)
\( \left\{ 2\left(\cos\frac{3\pi}4+\mathrm{i}\cdot\sin\frac{3\pi}4\right); 2\left(\cos\frac{5\pi}4+\mathrm{i}\cdot\sin\frac{5\pi}4\right) \right\} \)
\( \left\{ 2\sqrt2\left(\cos\frac{\pi}4+\mathrm{i}\cdot\sin\frac{\pi}4\right); 2\sqrt2\left(\cos\frac{7\pi}4+\mathrm{i}\cdot\sin\frac{7\pi}4\right) \right\} \)
\( \left\{ 2\left(\cos\frac{\pi}4+\mathrm{i}\cdot\sin\frac{\pi}4\right); 2\left(\cos\frac{7\pi}4+\mathrm{i}\cdot\sin\frac{7\pi}4\right) \right\} \)
1003076707
Level:
B
How many values of the angle \( \alpha \in\left[0^{\circ}; 360^{\circ}\right] \) satisfy the equation
\( \sin \alpha = \cos\alpha \)?
\( 2 \)
\( 1 \)
\( 0 \)
\( 3 \)