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1003024103

Level:

Find the equation of the circle, which passes through the points \( A=[-3;2] \), \( B=[-1;4] \) and \( C=[3;0] \).

\( x^2 + (y-1)^2 = 10 \)

\( (x-1)^2 + y^2 = 10 \)

\( x^2 + (y-1)^2 = \sqrt{10} \)

\( x^2 + y^2 = 10 \)

\( x^2 + (y+1)^2 = 10 \)

1003024203

Level:

Find the tangent equations from the point \( L=[1;-1] \) to the parabola given by \( y^2-4x+2y+9=0 \).

\( x-y-2=0 \), \( x+y=0 \)

\( x+y-2=0 \), \( x+y=0 \)

\( x+y+2=0 \), \( x-y=0 \)

\( x-y+2=0 \), \( -x+y=0 \)

\( -x+y-2=0 \), \( x-y=0 \)

1003024202

Level:

Find the tangent equations from the point \( M=[0;0] \) to the ellipse given by \( x^2+2y^2-8x+4y+12=0\).

\( x-5y=0 \), \( x+y=0 \)

\( 5x-y=0 \), \( x+y=0 \)

\( x+5y=0 \), \( x-y=0 \)

\( -x-5y=0 \), \( -x+y=0 \)

\( 5x+y=0 \), \( x-y=0 \)

1003024201

Level:

Find the tangent equations from the point \( K=[5;0] \) to the circle given by \( (x-1)^2+(y-2)^2=4 \).

\( 4x+3y-20=0 \), \( y=0 \)

\( 3x+4y-20=0 \), \( y=0 \)

\( 4x+3y+20=0 \), \( y=0 \)

\( -4x+3y-20=0 \), \( y=0 \)

\( -3x+4y-20=0 \), \( y=0 \)

A regular hexagonal prism \( ABCDEFA'B'C'D'E'F' \) has the side \( a \) of the length \( 3\,\mathrm{cm} \) and the height \( v \) of the length \( 8\,\mathrm{cm} \). Find the distance between the planes \( AEE' \) and \( BDD' \).

\( 3 \)

\( \sqrt3 \)

\( 2\sqrt3 \)

\( \frac{\sqrt3}2 \)

1103025403

Level:

\( 3\sqrt3 \)

\( 6 \)

\( 6\sqrt3 \)

\( \frac32\sqrt3 \)

1103025402

Level:

A regular hexagonal prism \( ABCDEFA'B'C'D'E'F' \) has the side \( a \) of the length \( 3\,\mathrm{cm} \) and the height \( v \) of the length \( 8\,\mathrm{cm} \). Find the angle between the plane \( ABD' \) and the base plane \( ABC \). Round the result to two decimal places.

\( 57^{\circ} \)

\( 53.13^{\circ} \)

\( 33^{\circ} \)

\( 72.01^{\circ} \)

1103025401

Level:

A regular hexagonal prism \( ABCDEFA'B'C'D'E'F' \) has the side \( a \) of the length \( 3\,\mathrm{cm} \) and the height \( v \) of the length \( 8\,\mathrm{cm} \). Find the angle between the lines \( AD' \) and \( BD' \). Round the result to two decimal places.

\( 17.46^{\circ} \)

\( 72.54^{\circ} \)

\( 16.70^{\circ} \)

\( 20.57^{\circ} \)

1103020106

Level:

In the picture, the shaded region corresponds to the set of points that is the solution to one of the given inequalities. Which inequality is it?

\( y > \frac32x+\frac{13}2 \)

\( y \geq \frac32x+\frac{13}2 \)

\( y < \frac32x+\frac{13}2\)

\( y \leq \frac32x+\frac{13}2\)

1103020105

Level:

Among the pictures below choose the one that shows the graphic solution of the inequality \[ x-\frac{y-5}2 \leq \frac{7+2x}2 \] in \( \mathbb{R}\times\mathbb{R} \).