1003027604
Level:
B
How many times is \( \int\limits_{-3}^0 \frac{x^2+x-6}{x-2}\,\mathrm{d}x \) smaller than \( \int\limits_{4}^7 \frac{5x^2-15x-20}{x+1}\,\mathrm{d}x \)?
\( 5 \) times
\( 9 \) times
they are equal
none of the options
B
1003027603
Level:
B
How many times is \( \int\limits_{\frac{\pi}6}^{\frac{\pi}3} \frac{\sin 2x}{\cos x}\,\mathrm{d}x \) bigger than \( \int\limits_{-\frac{\pi}3}^{-\frac{\pi}6} \frac{\sin 2x}{\sin x}\,\mathrm{d}x \)?
they are equal
\( 2 \) times
\( 4 \) times
none of the options
1003027602
Level:
B
How much smaller is \( \int\limits_1^{10}\frac{x+1}{x^2+x}\,\mathrm{d}x \) than \( \int\limits_1^{10}\frac{11-x}{10}\,\mathrm{d}x \)?
more than \( 1 \)
less than \( 1 \)
they are equal
cannot be determined
1003027601
Level:
B
How much bigger is \( \int\limits_0^3\left[(x-3)^2+1\right]\!\mathrm{d}x \) than \( \int\limits_3^6 \log_55\,\mathrm{d}x \)?
\( 9 \)
\( 15 \)
\( 4 \)
none of the options
1103021001
Level:
B
Let \( ABCDEF \) be a regular hexagon with the centre \( S \) and the side of length \( 3\,\mathrm{cm}\).
The point \( G \) is the midpoint of the segment \( AB \).
The vectors \( \vec{u} \), \( \vec{v} \), \( \vec{w} \), \( \vec{z} \) are indicated in the hexagon shown in the picture.
Find the dot product of: \( \vec{v}\cdot\vec{w} \), \( \vec{v}\cdot\vec{z} \) and \( \vec{v}\cdot\vec{u} \).
\( \vec{v}\cdot\vec{w}=9 \), \( \vec{v}\cdot\vec{z} = 0 \), \( \vec{v}\cdot\vec{u}=27 \)
\( \vec{v}\cdot\vec{w}=9 \), \( \vec{v}\cdot\vec{z} = 0 \), \( \vec{v}\cdot\vec{u}=9\sqrt6 \)
\( \vec{v}\cdot\vec{w}=\frac92 \), \( \vec{v}\cdot\vec{z} = 0 \), \( \vec{v}\cdot\vec{u}=9\sqrt6 \)
\( \vec{v}\cdot\vec{w}=\frac92 \), \( \vec{v}\cdot\vec{z} = 1 \), \( \vec{v}\cdot\vec{u}=27 \)
1003024008
Level:
B
Decide which of the given equations describes a hyperbola.
\( 4x^2-9y^2+18y-45=0 \)
\( 4x^2+9y^2-8x-36y+4=0 \)
\( x^2+y^2-2x+4y-4=0 \)
\( x^2+y^2-12x+40=0 \)
\( x^2-2x-4y+1=0 \)
1003024007
Level:
B
Decide whether the equation \( 2x^2-4y^2-6x-12y=0 \) (in \( x \)-\( y \) plane) describes:
a hyperbola
a circle
a parabola
an ellipse
point \( \left[\frac32,\frac{-3}2\right] \)
1003024006
Level:
B
Decide whether the equation \( (x+3)^2+(y+4)^2=5 \) (in \( x \)-\( y \) plane) describes:
a circle with center \( [-3,-4] \)
a circle with center \( [3,4] \)
point \( [-3,-4] \)
a hyperbola
an ellipse with center \( [3,4] \)
1003024005
Level:
B
Decide whether the equation \( 4x^2+4y-8x=1\) (in \( x \)-\( y \) plane) describes:
a parabola
an ellipse
a circle
a hyperbola
point \( \left[0,\frac14\right] \)
1003024004
Level:
B
Decide whether the equation \( 4x^2+9y^2-8x+18y+13=0 \) (in \( x \)-\( y \) plane) describes:
a point
an ellipse
a parabola
a hyperbola
an empty set