C

1003124304

Level: 
C
Given a function \( f(x)=ax^4+bx \), find real numbers \( a \) and \( b \), such that \( \int\limits_0^1f(x)\,\mathrm{d}x=27 \) and \( \int\limits_{-1}^0f(x)\,\mathrm{d}x=57 \).
\( a=210 \), \( b=-30 \)
\( a=210 \), \( b=30 \)
\( a=75 \), \( b=60 \)
\( a=30 \), \( b=210 \)

1103059607

Level: 
C
Let \( ABCDV \) be a rectangle based-pyramid, where \( V \) is its apex, and let \( XY \) be a line where: \begin{align*} X&\text{ lays on a ray }BA\text{ and }|BA|=|AX|,\\ Y&\text{ lays on the height }SV\text{ and }|SY|=|YV|,\\ S&\text{ is the centre of the pyramid base} \end{align*} (see the picture). The points of intersection of the line \( XY \) with the surface of the pyramid lay:
on the pyramid faces \( ADV \) and \( BCV \)
on the pyramid faces \( DCV \) and \( ABV \)
on the pyramid face \( ADV \) and its edge \( CV \)
on the pyramid edges \( AV \) and \( CV \)

1103059606

Level: 
C
Let \( ABCDV \) be a rectangle based-pyramid, where \( V \) is its apex, and let \( XY \) be a line where: \begin{align*} X&\text{ lays on the side }AV\text{ and }|AX|=|XV|,\\ Y&\text{ lays on a ray }DC\text{ and }|DY|=1.5|DC| \end{align*} (see the picture). The points of intersection of the line \( XY \) with the surface of the pyramid are:
the point \( X \) and a point on the pyramid face \( BCV \)
the point \( X \) and a point on the pyramid face \( DCV \)
the point \( X \) and a point on the pyramid edge \( CV \)
the point \( X \) only

1103059605

Level: 
C
Let \( ABCDEFGH \) be a cube and let \( XY \) be a line where: \begin{align*} X&\text{ lays on a ray }CB\text{ and }|CX|=1.5|BC|,\\ Y&\text{ lays on a ray }EH\text{ and }|EY|=1.5|EH| \end{align*} (see the picture). The points of intersection of the line \( XY \) with the surface of the cube lay:
on the sides \( ABFE \) and \( DCGH \)
on the sides \( EFGH \) and \( ABCD \)
on the side \( ABCD \) and the edge \( HG \)
on the edges \( HG \) and \( AB \)