C

1103059603

Level: 
C
Let \( ABCDEFGH \) be a cube and let \( XY \) be a line where: \begin{align*} X&\text{ lays on a ray }BC\text{ and }|BX|=1.5|BC|,\\ Y&\text{ lays on a ray }HE\text{ and }|HY|=1.5|HE| \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 side \( ABFE \) and the edge \( CG \)
on the edges \( AE \) and \( CG \)
on the sides \( ADHE \) and \( BCGF \)

1103059602

Level: 
C
Let \( ABCDV \) be a rectangle based-pyramid and \( V \) its apex. The pyramid is sliced by a plane \( XYZ \) which is defined by: \begin{align*} X&\text{ is the midpoint of the edge }AD,\\ Y&\in CD\ \wedge\ |DY|=3|CY|,\\ Z&\in BV\ \wedge\ |BZ|=3|VZ| \end{align*} (see the picture). What is the cross-section of the pyramid if we slice it with the plane \( XYZ \)?
a pentagon \( XYKZL \) with points \( K \) and \( L \) lying on the edges \( CV \) and \( AV \)
a triangle \( XYZ \)
a quadrilateral \( XYZL \) with point \( L \) lying on the edge \( AV \)
a quadrilateral \( XYKZ \) with point \( K \) lying on the edge \( CV \)

1103059601

Level: 
C
Let \( ABCDV \) be a rectangle based-pyramid and \( V \) its apex. The pyramid is sliced by a plane \( EFG \) which is defined by: \begin{align*} E&\in BC\ \wedge\ |BE|=2|CE|, \\ F&\in AV\ \wedge\ |AF|=2|VF|, \\ G&\in DV\ \wedge\ |DG|=2|VG| \end{align*} (see the picture). What is the cross-section of the pyramid if we slice it with the plane \( EFG \)?
a trapezium \( BCGF \)
a triangle \( EFG \)
a triangle \( AEV \)
a pentagon \( ABEGF \)

1003124305

Level: 
C
Given a function \( f(x)=ax^6+bx^3+cx+8 \), find real numbers \( a \), \( b \) and \( c \), such that \( \int\limits_0^1f(x)\,\mathrm{d}x=\frac{35}4 \), \( f'(0)=2 \) and \( f'(1)=180 \).
\( a=7 \), \( b=-5 \), \( c=2 \)
\( a=7 \), \( b=5 \), \( c=2 \)
\( a=-7 \), \( b=-5 \), \( c=2 \)
\( a=-7 \), \( b=5 \), \( c=-2 \)

1003124303

Level: 
C
Which of the given values of real numbers \( a \), \( b\in\left(0;\frac{\pi}2\right) \), such as \( a < b \), makes the equality \( \int\limits_a^b \cos x\,\mathrm{d}x=2\cos\frac{\pi}4\cdot\sin\frac{\pi}{12} \) true?
\( a=\frac{\pi}6 \), \( b=\frac{\pi}3 \)
\( a=\frac{\pi}3 \), \( b=\frac{\pi}6 \)
\( a=\frac{\pi}3 \), \( b=\frac{\pi}4 \)
\( a=\frac{\pi}4 \), \( b=\frac{\pi}3 \)

1103124301

Level: 
C
The picture shows graphs of two quadratic functions \( f_1(x) \) and \( f_2(x) \). Find the unknown real positive constant \( a \) (as shown in the picture) such that the value of the definite integral \( \int\limits_{-1}^1 f_1(x)\,\mathrm{d}x \) is greater by \( 8 \) than the value of the definite integral \( \int\limits_{-1}^1 f_2(x)\,\mathrm{d}x \).
\( a = 3 \)
\( a = 1 \)
\( a = 4 \)
\( a = 6 \)