1103124602
Level:
C
Let \( f(x)=\frac{x^2-x-6}{x^2-9} \). One of the following pictures shows a part of the graph of \( f \). Choose the picture.
C
1003118307
Level:
C
Identify which of the following functions has the maximum at \( x=-\frac12 \).
\( m(x)=-\left|\frac{4x+2}{x-2}\right| \)
\( g(x)=\left|-\frac{5x+10}{2x-1}\right| \)
\( f(x)=-\left|\frac{2x+1}{4x+2}\right| \)
\( h(x)=-\left|\frac{x+1}{2x-2}\right| \)
1003118306
Level:
C
Find the true statement about the function \( f(x)=\left|\frac{4x-4}{2x-1}\right| \).
The domain of the function \( f \) is the set \( \left(-\infty;\frac12\right)\cup\left(\frac12;\infty\right) \).
The range of the function \( f \) is the set \( [0;2)\cup(2;\infty) \).
The function \( f \) has the minimum at \( x=4 \).
The function \( f \) is an injective (one-to-one) function.
1003118305
Level:
C
Find the false statement about the function \( f(x)=\left|\frac1{2-3x}-3\right| \).
The domain of the function \( f \) is the set \( \left(-\infty;\frac32\right)\cup\left(\frac32;\infty\right) \).
The range of the function \( f \) is the interval \( \left[0;\infty\right) \).
The function \( f \) has the minimum at \( x=\frac59 \).
The function \( f \) is bounded below.
1003118304
Level:
C
Identify which of the following functions is bounded.
\( h(x)=\frac{3x-6}{2x-4} \)
\( f(x)=\frac{3x-6}{2x} \)
\( g(x)=3-\frac6{2x} \)
\( m(x)=\left|\frac{4x-3}{2x-6}\right| \)
1003047704
Level:
C
Find the limit. \[ \lim\limits_{n\to\infty}\frac{1^2+2^2+\dots+n^2}{n^2+7n-3} \]
Hint: \( 1^2+2^2+\cdots +n^2=\frac16 n(n+1)(2n+1) \).
\( \infty \)
\( 0 \)
\( \frac13 \)
\( -\frac13 \)
\( \frac16 \)
1003047703
Level:
C
Find the limit \( \lim\limits_{n\to\infty}\frac{\frac12+\frac14+\dots+\frac1{2^n}}{\frac13+\frac19+\dots+\frac1{3^n}} \).
\( 2 \)
\( \frac23 \)
\( \infty \)
\( 0 \)
\( \frac32 \)
1003047702
Level:
C
Find the limit:
\[ \lim\limits_{n\to\infty}\left(\frac13+\frac19+\dots+\frac1{3^n} \right) \].
\( \frac12 \)
\( \frac13 \)
\( \frac32 \)
\( \infty \)
\( \frac23 \)
1003047701
Level:
C
Find the limit \( \lim\limits_{n\to\infty}\frac{3+9+\dots+3^n}{4+16+\dots+4^n } \).
\( 0 \)
\( \frac32 \)
\( \infty \)
\( 1 \)
\( \frac34 \)
1003085408
Level:
C
A swimming pool can be filled by two pipes in \( 5 \) hours. It takes \( 24 \) hours longer to fill the pool by only the first pipe than by only the second one. In how many hours does the first pipe fill the pool, in how many hours does the second one? Find the sum of both times.
\( 36 \) hours
\( 20 \) hours
\( 18 \) hours
\( 32 \) hours