2010005406
Level:
B
Find the limit.
\[
\lim\limits_{n\rightarrow\infty}\frac{3^{n+1}-5^{n+1}}{3^{n-1}-5^{n-1}}
\]
\( 25 \)
\( \frac1{25} \)
\( 0 \)
\( \infty \)
\( 1 \)
B
2010005404
Level:
B
Choose the sequence with the limit equal to \( -3 \).
\( \left(\left(\frac13\right)^n-3\right)_{n=1}^{\infty} \)
\( \left(3^n-3\right)_{n=1}^{\infty} \)
\( \left(3-3^n\right)_{n=1}^{\infty} \)
\( \left(3-\left(\frac13\right)^n\right)_{n=1}^{\infty} \)
\( \left((-3)^n-3\right)_{n=1}^{\infty} \)
2010005403
Level:
B
Find the following limit.
\[
\lim _{n\to \infty }\frac{4^{n}}
{3^{n}-4^n}
\]
\(-1\)
\(0\)
\(\infty \)
\(1 \)
2010005006
Level:
B
Find the angle between the line \(q\)
and the plane \(\sigma \).
\[
\sigma \colon 2x-z+4 = 0;\qquad \qquad \begin{aligned}[t] q\colon x& = 5r, &
\\y & = -3+2r,
\\z & = -2;\ r\in \mathbb{R}
\\ \end{aligned}
\]
Round your answer to the nearest minute.
\(56^{\circ }09'\)
\(56^{\circ }08'\)
\(33^{\circ }51'\)
\(33^{\circ }52'\)
2010005005
Level:
B
Given points \(C = [-2;3;-1]\),
\(D= [1;2;-3]\), find the angle
between the line \(CD\)
and the line \(p\).
\[
\begin{aligned}p\colon x& = 2 -s, &
\\y & = 3,
\\z & = 2s;\ s\in \mathbb{R}
\\ \end{aligned}
\]
Round your answer to the nearest minute.
\(33^{\circ }13'\)
\(56^{\circ }47'\)
\(90^{\circ }\)
\(146^{\circ }47'\)
2010005004
Level:
B
Find the distance between two parallel planes
\(\rho \) and
\(\sigma \).
\[ \begin{aligned}
\rho& \colon 2x - 0.5y - 4z - 4 = 0,\\ \sigma &\colon 4x - y - 8z -2 = 0
\end{aligned}\]
\(\frac{2}
{3}\)
\(\frac{11}
{9}\)
\(\frac{10}
{9}\)
\(\frac{4}
{3}\)
2010004915
Level:
B
Consider the geometric sequence with \(a_{2} = 50\)
and \(a_{3} = 10\).
Find the sum of the first four terms.
\(312\)
\(62.4\)
\(66\)
\(315\)
\(312.4\)
2010004914
Level:
B
Identify the real number \(x\)
which converts the numbers \(a_{1} = 3^{x-6}\),
\(a_{2} = 1\) and
\(a_{3} = 3^{x}\) into
three consecutive terms of a geometric series.
\(x = 3\)
\(x = 1\)
\(x =\log 3\)
\(x = 10\)
\(x = 100\)
2010004913
Level:
B
At which term does the geometric sequence \( (8\, ;\ 10 \, ;\ 12.5 \, ; \ 15.625 \, ; \ \dots)\) exceed \(80\)? (Use a calculator.)
\(12\)
\(11\)
\(13\)
\(14\)
2010004912
Level:
B
At which term does the geometric sequence \( (10\, ;\ 12 \, ;\ 14.4 \, ; \ 17.28 \, ; \ \dots)\) exceed \(100\)? (Use a calculator.)
\(14\)
\(13\)
\(12\)
\(15\)