1003047704
Level:
C
Find the limit. \[ \lim\limits_{n\to\infty}\frac{1^2+2^2+\dots+n^2}{n^2+7n-3} \]
Hint: \( 1^2+2^2+\cdots +n^2=\frac16 n(n+1)(2n+1) \).
\( \infty \)
\( 0 \)
\( \frac13 \)
\( -\frac13 \)
\( \frac16 \)
C
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
1003085405
Level:
C
Little Red Riding Hood ran (at instantaneous speed) through the forest to see her grandmother, who lives in a cottage, which is \( 4\,\mathrm{km} \) distant. If she ran by \( 4\,\mathrm{km/h} \) faster, she would meet her grandmother \( 10 \) minutes sooner. What was Little Red Riding Hood’s speed?
\( 8\,\mathrm{km/h} \)
\( 12\,\mathrm{km/h} \)
\( 10\,\mathrm{km/h} \)
\( 6\,\mathrm{km/h} \)
1003085404
Level:
C
The surface area of a cuboid is \( 11\,776\,\mathrm{cm}^2 \). Its dimensions are in the ratio \( 3:5:1 \). Find the volume of the cuboid.
\( 61\,440\,\mathrm{cm}^3 \)
\( 15\,360\,\mathrm{cm}^3 \)
\( 30\,720\,\mathrm{cm}^3 \)
\( 21\,722\,\mathrm{cm}^3 \)
1003102415
Level:
C
Let \( a \in(0;\infty) \). The expression \( \log_4a-\log_{16}a-\log_{\frac14}a \) is equivalent to:
\( \frac32 \log_4a \)
\( -\frac12\log_4a \)
\( \log_4a \)
\( 0 \)
1003102410
Level:
C
If \( x\in(0;1)\cup(1;\infty) \) then the product \( \left(\log_x3\right)\left(\log_5x\right) \) can be written as:
\( \log_53 \)
\( \log_35 \)
\( \log_{5x}(3x) \)
\( \log_x3+\log_5x \)
1003102407
Level:
C
If \( \log_ab=100 \) and \( \log_4a=10 \) then the value of \( b \) is:
\( 2^{2000} \)
\( 2^{1000} \)
\( 2^{1002} \)
\( 2^{200} \)