1003134704
Level:
B
The first term of a geometric sequence is \( 3 \). Find the common ratio so that the sum of the first \( 3 \) terms is minimal.
\( -\frac12 \)
\( -5 \)
\( 0 \)
\( 2 \)
\( -1 \)
B
1003134703
Level:
B
The first term of a geometric sequence is different from zero and \( s_3=-3s_2 \). (\( s_n \) is the sum of the first n terms.) Find the common ratio.
\( -2 \)
\( 2 \)
\( \frac12 \)
\( 0 \)
\( -3 \)
1003134702
Level:
B
The sum of the first three terms of a geometric sequence is \( \frac78 \) and the common ratio is \( \frac12 \). Find the sum of all the terms from the \( 3 \)rd to the \( 5 \)th of the sequence.
\( \frac7{32} \)
\( \frac{31}{32} \)
\( \frac3{32} \)
\( \frac78 \)
\( \frac58 \)
1003134701
Level:
B
The sum of the first and the second term of a geometric sequence is \( 1 \), the sum of the third and the fourth term is \( 4 \) and the common ratio is negative. Find the first term.
\( -1 \)
\( 2 \)
\( -2 \)
\( 3 \)
\( \frac13 \)
1103021513
Level:
B
The distance of the chord \( AB \) from the centre of the circle is equal to \( 2/3 \) of its radius. Find the measure of the angle \( SAB \). (See the picture.) Round the result to two decimal places.
\( 41.81^{\circ} \)
\( 48.19^{\circ} \)
\( 33.69^{\circ} \)
\( 56.31^{\circ} \)
1003118706
Level:
B
Consider a truncated cone with the diameters of its bases \( 2\,\mathrm{cm} \) and \( 10\,\mathrm{cm} \), and with the height \( 4\,\mathrm{cm} \). Which of the following formulas cannot be used to calculate the volume of such truncated cone?
\( V=\pi\int\limits_0^4(5-x)\,\mathrm{d}x \)
\( V=\pi\int\limits_0^4(5-x)^2\mathrm{d}x \)
\( V=\frac{\pi}3\cdot4\cdot(25+5+1) \)
\( V=\frac{\pi}3\cdot25\cdot5-\frac{\pi}3\cdot1\cdot1 \)
1003118705
Level:
B
Peter and Jane both calculated the volume of a solid of revolution using a definite integral. Both chose a solid obtained by rotation of a line segment about the \( x \)-axis. The endpoints of Peter’s line segment are \( [0;1] \) and \( [5;4] \), the endpoints of Jane’s line segment are \( [0;3] \) and \( [5;0] \). Finally, they compared their calculated volumes. Which of the following statements is true?
Peter’s solid is \( 20\pi \) bigger.
Jane’s solid is \( 20\pi \) bigger.
Both solids have the same volume.
The difference between Peter’s solid and Jane’s solid is \( 10\pi \).
1103118704
Level:
B
Which of the given equations defines the line that together with \( x=0 \) and \( x \)-axis bounds the right triangle, if by rotation of this triangle about the \( x \)-axis the cone of height \( 10 \) is obtained as indicated in the picture?
\( y=-0.4x+4 \)
\( y=-2.5x+10 \)
\( y=4x+10 \)
\( y=10x+4 \)
1003118703
Level:
B
A right trapezoid is bounded by \( y=ax+1 \), \( x=0 \), \( x=6 \), and by the \( x \)-axis. Rotating the trapezoid around the \( x \)-axis we get a truncated cone. Find the value of the parameter \( a > 0 \) so that the volume of the truncated cone is \( 26\pi \).
\( a=\frac13 \)
\( a=\frac12 \)
\( a=3 \)
\( a=2 \)
1003118702
Level:
B
It is possible to calculate the volume of a sphere with the radius of \( 3 \) using a definite integral. Which of the following formulas is not correct?
\( \int\limits_{-3}^3\left(9-x^2\right)\,\mathrm{d}x \)
\( \pi\int\limits_{-3}^3\left(9-x^2\right)\,\mathrm{d}x \)
\( 2\pi\int\limits_{0}^3\left(9-x^2\right)\,\mathrm{d}x \)
\( \pi\int\limits_{-3}^3\left(-\sqrt{9-x^2}\right)^2\,\mathrm{d}x \)