1003108705
Level:
A
We are given an infinite geometric series:
\[ 4+\frac83+\frac{16}9+\frac{32}{27}+\dots\text{ .} \]
What is the value of its common ratio?
\( \frac23 \)
\( \frac13 \)
\( \frac43 \)
\( \frac32 \)
\( \frac34 \)
A
1003108704
Level:
A
Expand the given sum by listing individual terms.
\[ \sum\limits_{n=2}^6(4n-5) \]
\( 3+7+11+15+19 \)
\( -1+3+7+11+15 \)
\( 4 + 8 +12 +16 +20 \)
\( 3+ 8 + 11 + 15 + 19 \)
\( 8 - 5 + 12 + 16 + 20 \)
1003108703
Level:
A
Use summation notation to express the given infinite series.
\[ \frac3{x^3}+\frac3{x^2}+\frac3x+3+3x+\dots \]
\( \sum\limits_{n=1}^{\infty}3\cdot x^{n-4} \)
\( \sum\limits_{n=1}^{\infty}3\cdot x^{n-3} \)
\( \sum\limits_{n=1}^{\infty}3\cdot x^{n+3} \)
\( \sum\limits_{n=1}^{\infty}3\cdot x^{n+4} \)
\( \sum\limits_{n=1}^{\infty}3\cdot x^{n} \)
1003108702
Level:
A
Use summation notation to express the given infinite series.
\[ -1+2-4+8-16+\dots \]
\( \sum\limits_{n=1}^{\infty} (-1)^n\cdot2^{n-1} \)
\( \sum\limits_{n=1}^{\infty} (-1)^{n-1}\cdot2^{n-1} \)
\( \sum\limits_{n=1}^{\infty} (-1)^{n+1}\cdot2^{n-1} \)
\( \sum\limits_{n=1}^{\infty} (-1)^n\cdot2^{n+1} \)
\( \sum\limits_{n=1}^{\infty} (-1)^n\cdot2^{n} \)
1003108701
Level:
A
Use summation notation to express the given infinite series.
\[1+\frac12+\frac14+\frac18+\dots \]
\( \sum\limits_{n=1}^{\infty} \frac1{2^{n-1}} \)
\( \sum\limits_{n=1}^{\infty} \frac1{2^{n}} \)
\( \sum\limits_{n=1}^{\infty} \frac1{2^{n+1}} \)
\( \sum\limits_{n=1}^{\infty} \frac1{2^{2n}} \)
\( \sum\limits_{n=1}^{\infty} \frac1{2^{2n-1}} \)
1003188908
Level:
A
Find the real number \( k \) so that the line \( AB \), where \( A=[6;k;5] \) and \( B=[3;-1;2] \), is parallel to the plane \( 2x-3y+9z-4=0 \).
\( k=10 \)
\( k=12 \)
\( k=8 \)
\( k=9 \)
1003188907
Level:
A
We are given two intersecting planes \( x-6y+9z-4=0 \) and \( x-2y+3z-4=0 \). Find the parametric equations of their line of intersection \( p \).
\( \begin{aligned}
p\colon x&=4, \\
y&=\phantom{4+}\ 3t, \\
z&=\phantom{4+}\ 2t;\ t\in\mathbb{R}
\end{aligned} \)
\( \begin{aligned}
p\colon x&=4+t , \\
y&=\phantom{4+}\ 3t , \\
z&=\phantom{4+}\ 2t;\ t\in\mathbb{R}
\end{aligned} \)
\( \begin{aligned}
p\colon x&=4, \\
y&=\frac32+3t, \\
z&=1+2t;\ t\in\mathbb{R}
\end{aligned} \)
\( \begin{aligned}
p\colon x&=4+t, \\
y&=\frac32+3t, \\
z&=1+2t;\ t\in\mathbb{R}
\end{aligned} \)
1003188906
Level:
A
Let there be planes \( \alpha \), \( \beta \), \( \gamma \) and \( \delta \) defined by their general equations:
\[ \begin{aligned}
&\alpha\colon \frac23x-4y+6z-\frac83=0; \\
&\beta\colon x-2y+3z-4=0; \\
&\gamma\colon 2x-12y+18z-4 =0; \\
&\delta\colon x-6y+9z-4 =0.
\end{aligned} \]
Out of the following statements, select the one that is not true.
\( \alpha \parallel\delta\text{, }\alpha\neq\delta \)
Planes \( \beta \) and \( \delta \) are intersecting.
\( \gamma\parallel\delta\text{, }\gamma\neq\delta \)
Planes \( \alpha \) and \( \beta \) are intersecting.
\( \alpha = \delta \)
1003188905
Level:
A
Determine the relative position of the plane \( \rho \) with general equation \( 5x-4y+z-4=0 \) and the straight line \( p \) with parametric equations:
\[ \begin{aligned}
x&=-1+t,\\
y&=2-2t,\\
z&=3+t;\ t\in\mathbb{R}.
\end{aligned} \]
\( p \) is intersecting \( \rho \)
\( p\parallel \rho\text{, } p\not{\!\!\subset}\rho \)
\( p \subset \rho \)
1003188904
Level:
A
Determine the relative position of the plane \( \rho \) with general equation \( 7x-2y+z-2=0 \) and the straight line \( p \) with parametric equations:
\[ \begin{aligned}
x&=3+t, \\
y&=-5-2t, \\
z&=3-11t;\ t\in\mathbb{R}.
\end{aligned} \]
\( p\parallel \rho\text{, }p\not{\!\!\subset}\rho \)
\( p \subset \rho \)
\( p \) is intersecting the plane \( \rho \)