1003118201
Level:
C
Suppose that \( \sqrt[3]2\approx 1.25 \). Give the approximate value of \( \left(\frac1{81}\right)^{-\sqrt[3]2}\). Do not use a calculator.
\( 243 \)
\( 729 \)
\( \frac19 \)
\( 0 \)
C
1003263405
Level:
C
Find a true statement about the function \( f(x)=\sin x+\frac12\cos2x \) on the interval \( [0;\pi] \).
The function has global minima at the points \( x=0 \), \( x=\frac{\pi}2 \) and \( x=\pi \).
The only global minimum of \( f \) on this interval is at the point \( x=\frac{\pi}2 \).
The only global maximum of \( f \) on this interval is at the point \( x=\frac{\pi}6 \).
The function \( f \) has no global minimum on this interval.
1003263404
Level:
C
Find the global extrema of the following function on the interval \( [-1;3] \).
\[ f(x)=x^2\cdot \mathrm{e}^{-x} \]
the global minimum at \( x=0 \), the global maximum at \( x=-1 \)
the global minimum at \( x=0 \), the global maximum at \( x=2 \)
the global minimum at \( x=3 \), the global maximum at \( x=-1 \)
the global minimum at \( x=-1 \), the global maximum at \( x=0 \)
1003263403
Level:
C
Find the global extrema of the following function on the interval \( [0;3] \).
\[ f(x)=2x^3-3x^2-12x \]
the global minimum at \( x=2 \), the global maximum at \( x=0 \)
the global minimum at \( x=2 \), the global maximum at \( x=-1 \)
the global minimum at \( x=0 \), the global maximum at \( x=2 \)
the global minimum at \( x=3 \), the global maximum at \( x=0 \)
1103263402
Level:
C
The graph of \( f \) is given in the figure. Choose which of the following statements about the function \( f \) are true.
\[
\begin{array}{l}
\text{A: The global minimum of } f \text{ on the interval } (-3;3) \text{ is at } x=0. \\
\text{B: The global maxima of } f \text{ on the interval } [-3;3] \text{ are at } x=-2 \text{ and } x=2. \\
\text{C: On } (-2;3] \text{ there is the global minimum of } f \text{ at } x=3 \text{ and the global maximum of } f \text{ at } x=2. \\
\text{D: The function } f \text{ has no global minimum on the interval } (-3;3). \\
\text{E: The function } f \text{ has no global maximum on the interval } (-3;3) .
\end{array}
\]
The only true statements are:
B, C, D
C, D, E
A, B, C
A, B
C, D
A, E
1103263401
Level:
C
The graph of \( f \) is given in the figure. Choose which of the following statements about the function \( f \) are true.
\[
\begin{array}{l}
\text{A: The global maximum of } f \text{ on the interval }[-4;4] \text{ is at } x=4. \\
\text{B: The only global minimum of } f \text{ on the interval } [-4;4] \text{ is at } x=2. \\
\text{C: On } (-2;3] \text{ there is the global minimum of } f \text{ at } x=2 \text{ and the global maximum of } f \text{ at } x=-2. \\
\text{D: The function } f \text{ has no global maximum on the interval } [-3;4). \\
\text{E: The function } f \text{ has no global minimum on the interval } [-4;2).
\end{array}
\]
The only true statements are:
A, D
B, C
B, D, E
A, D, E
A, B, E
C, D
1103107014
Level:
C
Let \( ABCDEFA'B'C'D'E'F' \) be a regular hexagonal prism with the base edge length of \( 4\,\mathrm{cm} \) and the height of \( 8\,\mathrm{cm} \). Find the angle between the line \( BA’ \) and the plane \( AEE’ \) (see the picture). Round the result to two decimal places.
\( 26.57^{\circ} \)
\( 63.43^{\circ} \)
\( 30^{\circ} \)
\( 22.5^{\circ} \)
1103107013
Level:
C
Let \( ABCDEFA'B'C'D'E'F' \) be a regular hexagonal prism with the base edge length of \( 4\,\mathrm{cm} \) and the height of \( 8\,\mathrm{cm} \). Find the angle between the plane \( BCC' \) and the plane \( CDD' \) (see the picture).
\( 60^{\circ} \)
\( 120^{\circ} \)
\( 90^{\circ} \)
\( 72^{\circ} \)
1103107012
Level:
C
Let \( ABCDEFA'B'C'D'E'F' \) be a regular hexagonal prism with the base edge length of \( 4\,\mathrm{cm} \) and the height of \( 8\,\mathrm{cm} \). Find the angle between the plane \( ADD' \) and the plane \( CDD' \) (see the picture).
\( 60^{\circ} \)
\( 45^{\circ} \)
\( 90^{\circ} \)
\( 72^{\circ} \)
1103107011
Level:
C
Let \( ABCDEFA'B'C'D'E'F' \) be a regular hexagonal prism with the base edge length of \( 4\,\mathrm{cm} \) and the height of \( 8\,\mathrm{cm} \). Find the angle between the line \( FC' \) and the base plane \( ABC \) (see the picture).
\( 45^{\circ} \)
\( 60^{\circ} \)
\( 30^{\circ} \)
\( 72^{\circ} \)