2010015404
Level:
C
The average age in the group of six persons is \(48\) years. The average age of the first five of them is \(51\) years. How old is the sixth person?
\(33\) years
\(32.5\) years
\(39.5\) years
\(43\) years
C
2010015305
Level:
C
In a triangle \( ABC \), \( a=15\,\mathrm{cm} \), \( c=8\,\mathrm{cm} \) and the measure of the angle \( CAB \) is \( 120^{\circ} \). Which of the following numbers gives as accurately as possible the measure of the angle \( BCA \)?
\( 27.51^{\circ} \)
\( 16.12^{\circ} \)
\( 30.13^{\circ} \)
\( 12.45^{\circ} \)
2010015306
Level:
C
The angles \( \alpha \), \(\beta \), \( \gamma \) of a right-angled triangle \( ABC \) are in the ratio \( 1:2:3 \) (See the picture.). From the following ratios of sides select the one that is equal to \( 1:\sqrt3 \).
\( a:b \)
\( b:a\)
\( c:b \)
\( a:c \)
2010015304
Level:
C
Which of the following formulas applies to the given triangle \( ABC \)? (See the picture.)
\( c^2 = a^2+b^2 -2ab\cos\gamma \)
\( c^2 = a^2+b^2 +2ab\cos\gamma \)
\( c^2 = a^2+b^2 -2ab\cos\alpha \)
\( c^2 = a^2+b^2 -2ab\sin\beta \)
2010015303
Level:
C
Triangle \( ABC \) is given (See the picture.). Choose the correct statement, if \(r\) is a radius of its circumcircle.
\( \frac{b}{\sin\beta} = 2r \)
\( \frac{a}{\sin \alpha}= \frac{\sin\beta}{b}\)
\( c \sin \alpha = b \sin \gamma \)
\( \frac{a}{\sin \alpha}= r\)
2010015207
Level:
C
The lengths of the sides of a triangle are \( 4\,\mathrm{cm} \), \( 6\,\mathrm{cm} \) and \( 8\,\mathrm{cm} \). Calculate cosine of its greatest interior angle.
\( -\frac14\)
\( 104.47^{\circ}\)
\( 28.96^{\circ}\)
\(\frac78\)
2010015206
Level:
C
The lengths of the sides in a triangle are \( a \), \( b \), \( c \) and the opposite angles are \( \alpha \), \( \beta \), \( \gamma \). Give the measure of \( \beta \) if \( b^2=a^2+c^2+ac\sqrt3 \).
\( 150^{\circ}\)
\( 30^{\circ}\)
\( 60^{\circ}\)
\( 120^{\circ}\)
2010015003
Level:
C
\( ABCD \) is a rhombus, the measure of the angle \( DAB \) is \(70^{\circ}\) and the shorter diagonal \( u = 50\,\mathrm{cm} \). Determine the height \(v\) of the rhombus. Round the result to two decimal places.
\( 40.96\,\mathrm{cm} \)
\( 28.68\,\mathrm{cm} \)
\( 71.41\,\mathrm{cm} \)
\( 46.98\,\mathrm{cm} \)
2110014806
Level:
C
Identify the graph of the function \( f(x)=-\sqrt{|x|};\ x\in [ -9;9] \).
2010014805
Level:
C
Choose the function that describes the dependence of a cube surface \( S \) on the length of its space diagonal \( u \).
\( S=2u^2;\ u\in(0;\infty) \)
\( S=6u^2;\ u\in(0;\infty) \)
\( S=3u^2;\ u\in(0;\infty) \)
\( S=18u^2;\ u\in(0;\infty) \)