1003085502
Level:
B
Determine the value of \( a \) (\(a\in\mathbb{R}\)) for which the function \( f(x)=\frac{ax}{x^2+a} \) is not continuous at \( x = -1 \).
\( -1 \)
\( 1 \)
\( 0 \)
\( 1\), \( -1 \)
B
1003085501
Level:
B
Decide which of the following functions are continuous at \( x = 1 \).
\[\begin{aligned} f_1(x)&=\frac{x^2+1}{x-1} \\
f_2(x)&=\sqrt{x-1} \\
f_3(x)&=\log x \\
f_4(x)&=\mathrm{tg}(x-1)
\end{aligned}\]
The only such functions are:
\( f_3 \), \( f_4 \)
\( f_2 \), \( f_3 \), \( f_4 \)
\( f_2 \), \( f_3 \)
\( f_3 \)
1003085410
Level:
B
Veronica and Joseph are on their romantic hot air balloon ride. When proposing in \( 1\,500\,\mathrm{m} \) altitude, Joseph accidentally drops a ring over the edge of the balloon basket. The ring perishes in the ground direction the same as does Veronicas enthusiasm for getting married. In how many seconds does the ring hit the ground? (Ignore air resistance and round the result to the nearest integer.)
(Note: Distance \( d \) (in meters) travelled by an object falling for time \( t \) (in seconds) is given by \( d=\frac12\,\mathrm{gt}^2 \), where \( g \) is gravitational acceleration, \( g = 9.81\,\mathrm{m/s^2} \).)
\( 17 \)
\( 15 \)
\( 20 \)
\( 21 \)
1003085409
Level:
B
The area of the rectangle is \( 1\,269\,\mathrm{cm}^2 \). Its length is by \( 20\,\mathrm{cm} \) longer than its width. Find the sum of the length and the width of the rectangle.
\( 74\,\mathrm{cm} \)
\( 35\,\mathrm{cm} \)
\( 27\,\mathrm{cm} \)
\( 57\,\mathrm{cm} \)
1003085407
Level:
B
If the area of the square is increased by \( 64\% \), by what percentage the length of its side is increased? Round the result to an integer.
\( 28 \)
\( 8 \)
\( 32 \)
\( 16 \)
1003085402
Level:
B
The product of two consecutive even numbers is \( 2 808 \). Find their sum.
\( 106 \)
\( 104 \)
\( 102 \)
\( 108 \)
1003085401
Level:
B
On their birthday students bring candies for their classmates. The birthday person gives a candy to every other student, not to himself or herself. During a year, \( 650 \) candies have been given away. Find out how many students are there in the class. (Note: All students’ birthdays were on school days.)
\( 26 \)
\( 25 \)
\( 27 \)
\( 24 \)
1003102414
Level:
B
From the given expressions choose the one equivalent to \( \log\left( 8\cdot\sqrt[3]{75} \right) \), if \( \log2=a\), \( \log3=b \) and \( \log5=c \).
\( 3a+\frac13 b+\frac23 c \)
\( 3a+\frac13 b+\frac13 c \)
\( 4a+\frac13 b+\frac23 c \)
\( a+\frac13 b+\frac23 c \)
1003102413
Level:
B
Let \( x \), \( y \), \( z\in (0;\infty) \). Find the equivalent form of the following expression.
\[ \log\sqrt{\frac{xz^2}{y^{16}}} \]
\( \frac12\log x-8\log y+\log z \)
\( \frac12\log x+8\log y-\log z \)
\( 8\log x+\frac12\log y-\log z \)
\( \log x-16\log y+2\log z \)
1003102412
Level:
B
If \( a \), \( b \), \( c\in(0;\infty) \) then the expression \( \log_5a-\frac23 \log_5 b+3\log_5c \) is equivalent to:
\( \log_5\frac{ac^3}{\sqrt[3]{b^2}} \)
\( \log_5\frac{a\sqrt[3]{b^2}}{c^3} \)
\( \log_5\frac{3ac}{\frac23 b} \)
\( \log_5\frac{\frac23 ab}{3c} \)