1003170604
Level:
C
Find the sum of the first ten natural powers of the number \( 2 \).
\( 2\,046 \)
\( 2\,048 \)
\( 2\,047 \)
\( 2\,000 \)
\( 1\,536 \)
Geometric sequences
1003170605
Level:
C
Suppose we insert number \( 10\,530 \) between \( 2 \) unknown numbers such that they form \( 3 \) consecutive terms of a geometric sequence and their sum is \( 31\,707 \). What is the smaller of the two unknown numbers?
\( 9\,477 \)
\( 10\,500 \)
\( 9\,832 \)
\( 10\,034 \)
\( 5\,265 \)
1003170606
Level:
C
Which of the following numbers cannot be a term of the geometric sequence given by its second term \( a_2=360 \) and the common ratio \( q = \frac35 \)?
\( 250 \)
\( 216 \)
\( 600 \)
\( \frac{1\,944}{25} \)
\( \frac{648}5 \)
1103170607
Level:
C
Let there be an equilateral triangle with a side length of \( 16\,\mathrm{cm} \). Connecting the centres of its sides with line segments we form another equilateral triangle. If we add three more triangles using the same procedure, what is the sum of their perimeters?
\( 93\,\mathrm{cm} \)
\( 72\,\mathrm{cm} \)
\( 144\,\mathrm{cm} \)
\( 31\,\mathrm{cm} \)
\( 90\,\mathrm{cm} \)
1103170608
Level:
C
Let there be an equilateral triangle with a side length of \( 16\,\mathrm{cm} \). Connecting the centres of its sides with line segments we form another equilateral triangle. If we add two more triangles using the same procedure, what is the sum of their areas?
\( 85\sqrt3\,\mathrm{cm}^2 \)
\( 128\sqrt3\,\mathrm{cm}^2 \)
\( \frac{341}4\sqrt3\,\mathrm{cm}^2 \)
\( 90\,\mathrm{cm}^2 \)
\( 148\sqrt3\,\mathrm{cm}^2 \)
2010005507
Level:
C
A motorbike depreciates (loses value) at the rate of \(12\, \%\)
per year. After how many full years will the value of the motorbike decrease to less than one third of its initial value?
\(9\)
\(6\)
\(7\)
\(8\)
\(10\)
2010005508
Level:
C
The half-life of Copper-60 is approximately \( 24 \) minutes. How long will it take before its weight decreases from \( 1\,024\,\mathrm{g} \) to \( 8\,\mathrm{g} \)?
\( 2 \) hours \( 48 \) minutes
\( 3 \) hours \( 9 \) minutes
\( 3 \) hours \( 30 \) minutes
\( 2 \) hours \( 27 \) minutes
\( 3 \) hours \( 51 \) minutes
2010005509
Level:
C
How many numbers do we need to insert between the numbers \( 5 \) and \( 640 \) so that the inserted numbers with the given two numbers are consecutive terms of a geometric sequence? The sum of all numbers inserted must be \( 630 \).
\( 6 \)
\( 4 \)
\( 3 \)
\( 5 \)
\( 7 \)
2110014005
Level:
C
Between the roots of the quadratic equation \(4x^2-35x+54=0\) insert two numbers, so that the numbers with the roots together form four consecutive terms of a geometric sequence. This part of the geometric sequence is shown in one of the graphs. Choose the graph.
2110014006
Level:
C
Between the roots of the quadratic equation \(9x^2-35x+24=0\) insert two numbers, so that the numbers with the roots together form four consecutive terms of a geometric sequence. This part of the geometric sequence is shown in one of the graphs. Choose the graph.
