C

1003170603

Level: 
C
Between the roots of the equation \( 9x^2+130x-75=0 \) insert two numbers so that the roots and the new numbers form \( 4 \) consecutive terms of a geometric sequence. What is the smaller of the two inserted numbers?
\( -\frac53 \)
\( \frac59 \)
\( -\frac59 \)
\( \frac53 \)
\( -5 \)

1003170602

Level: 
C
The lengths of a cuboid edges form \( 3 \) consecutive terms of a geometric sequence. The volume of the cuboid is \( 140\,608\,\mathrm{cm}^3 \), the sum of its shortest and its longest edge is \( 221\,\mathrm{cm} \). Find the length of its shortest edge.
\( 13\,\mathrm{cm} \)
\( 52\,\mathrm{cm} \)
\( 4\,\mathrm{cm} \)
\( 208\,\mathrm{cm} \)
\( 0.25\,\mathrm{cm} \)

1003170601

Level: 
C
How many numbers do we need to insert between the numbers \( 6 \) and \( 1\,458 \) so that the inserted numbers with the given two numbers are consecutive terms of a geometric sequence? The sum of all numbers inserted must be \( 720 \).
\( 4 \)
\( 6 \)
\( 3 \)
\( 5 \)
\( 7 \)

1103212206

Level: 
C
A cube \( ABCDEFGH \) with an edge length of \( 2 \) is placed in a coordinate system (see the picture). Let \( p \) be a line of intersection of planes \( \alpha \) and \( \beta \), where \( \alpha \) is passing through \( C \), \( F \) and \( H \) and \( \beta \) is passing through \( A \), \( F \) and \( H \). Find the parametric equations of the line \( p \) and calculate the angle \( \varphi \) between planes \( \alpha \) and \( \beta \) . Round \( \varphi \) to the nearest minute.
\( \begin{aligned} p\colon x&=t, & \varphi&\doteq 70^{\circ}32' \\ y&=t, & &\\ z&=2;\ t\in\mathbb{R}, & & \end{aligned} \)
\( \begin{aligned} p\colon x&=2t, & \varphi&\doteq 90^{\circ} \\ y&=2t, & & \\ z&=2+2t;\ t\in\mathbb{R}, & & \end{aligned} \)
\( \begin{aligned} p\colon x&=t, & \varphi&\doteq 90^{\circ}\\ y&=t, & & \\ z&=2;\ t\in\mathbb{R}, & & \end{aligned} \)
\( \begin{aligned} p\colon x&=2t, & \varphi&\doteq 70^{\circ}32' \\ y&=2t, & & \\ z&=2t;\ t\in\mathbb{R}, & & \end{aligned} \)

1103212205

Level: 
C
A cube \( ABCDEFGH \) with an edge length of \( 2 \) is placed in a coordinate system (see the picture). Find the distance between parallel planes \( \alpha \) and \( \beta \), where \( \alpha \) is passing through \( B \), \( D \) and \( G \) and \( \beta \) is passing through \( A \), \( F \) and \( H \).
\( |\alpha\beta|=\frac{2\sqrt3}3 \)
\( |\alpha\beta|=\frac{4\sqrt3}3 \)
\( |\alpha\beta|=\frac{3\sqrt3}2 \)
\( |\alpha\beta|=\frac{3\sqrt3}4 \)

1103212204

Level: 
C
A cube \( ABCDEFGH \) with an edge length of \( 2 \) is placed in a coordinate system (see the picture). Let the point \( M \) be the centre of the edge \( EF \). Find the general form equation of the plane \( \rho \) passing through the points \( B \), \( D \), and \( G \) and calculate the distance of \( M \) from the plane \( \rho \).
\( \rho\colon x-y+z=0;\ |M\rho|=\sqrt3 \)
\( \rho\colon x-y+z+2=0;\ |M\rho|=\sqrt3 \)
\( \rho\colon x-y+z+2=0;\ |M\rho|=2\sqrt3 \)
\( \rho\colon x-y+z=0;\ |M\rho|=2\sqrt3 \)

1103212202

Level: 
C
A straight line \( p \) is given by the points \( M=[4;3;2] \) and \( N=[0;6;7] \) (see the picture). Find the parametric equations of the line \( p' \) that is symmetrical to the line \( p \) in the plane symmetry across the coordinate \( yz \)-plane.
\( \begin{aligned} p'\colon x&=4t, \\ y&=6+3t, \\ z&=7+5t;\ t\in\mathbb{R} \end{aligned} \)
\( \begin{aligned} p'\colon x&=-4t, \\ y&=6+3t, \\ z&=7+5t;\ t\in\mathbb{R} \end{aligned} \)
\( \begin{aligned} p'\colon x&=4t, \\ y&=6-3t, \\ z&=7+5t;\ t\in\mathbb{R} \end{aligned} \)
\( \begin{aligned} p'\colon x&=-4t, \\ y&=6-3t, \\ z&=7+5t;\ t\in\mathbb{R} \end{aligned} \)

1103212203

Level: 
C
A straight line \( p \) is given by the points \( M=[4;3;2] \) and \( N=[8;0;5] \) (see the picture). Find the parametric equations of the line \( p' \) that is symmetrical to the line \( p \) in the plane symmetry across the coordinate \( xz \)-plane.
\( \begin{aligned} p'\colon x&=8+4t, \\ y&=3t, \\ z&=5+3t;\ t\in\mathbb{R} \end{aligned} \)
\( \begin{aligned} p'\colon x&=8+4t, \\ y&=0, \\ z&=5+3t;\ t\in\mathbb{R} \end{aligned} \)
\( \begin{aligned} p'\colon x&=8+4t, \\ y&=-3t, \\ z&=5+3t;\ t\in\mathbb{R} \end{aligned} \)
\( \begin{aligned} p'\colon x&=8-4t, \\ y&=3t, \\ z&=5-3t;\ t\in\mathbb{R} \end{aligned} \)

1103212201

Level: 
C
A straight line \( p \) is given by the points \( M=[4;2;0] \) and \( N=[6;6;7] \) (see the picture). Find the parametric equations of the line \( p' \) that is symmetrical to the line \( p \) in the plane symmetry across the coordinate \( xy \)-plane.
\( \begin{aligned} p'\colon x&=4+2t, \\ y&=2+4t, \\ z&=-7t;\ t\in\mathbb{R} \end{aligned} \)
\( \begin{aligned} p'\colon x&=4+6, \\ y&=2+6t, \\ z&=-7t;\ t\in\mathbb{R} \end{aligned} \)
\( \begin{aligned} p'\colon x&=4+2t, \\ y&=2+4t, \\ z&=7t;\ t\in\mathbb{R} \end{aligned} \)
\( \begin{aligned} p'\colon x&=4+6t, \\ y&=2+6t, \\ z&=7t;\ t\in\mathbb{R} \end{aligned} \)

1003109207

Level: 
C
In \( 2015 \), \( 8\,688 \) choir members sang in the largest gospel choir in Manila. Imagine, the conductor wrote an e-mail on January \( 1 \)st to three choir members. Each of them forwarded the e-mail to other three choir members on the next day and so on... On which day would all members receive the e-mail?
\( 8 \) January
\( 15 \) January
\( 2 \) February
\( 8 \) February
\( 12 \) January