2010006505
Level:
C
Assuming \(x\in \mathbb{R}\),
find the solution set of the following system.
\[\begin{aligned}
3(x+3) &\leq 3 & &
\\x &\leq 6+4x & &
\end{aligned}\]
\(\{-2\}\)
\([-2;+\infty )\)
\((-\infty ;-2 ]\)
\(\mathbb{R} \)
C
2010006504
Level:
C
How many solutions has the inequality in \( \mathbb{N}\times\mathbb{N} \)? (Hint: Solve the inequality graphically.)
\[ 3y\leq -2x+4 \]
\( 0 \)
\(2\)
\(3\)
infinity
2010006403
Level:
C
Find the number \( x \) if \( 40\% \) of its half is \( 70\% \) of the number \( x -3\).
\( 4\frac15 \)
\( 5\frac14 \)
\( 5 \)
\( 5\frac25 \)
2010006402
Level:
C
One third of a number \( a \) is by \( 75\% \) greater than one seventh of a number \( b \). Choose the true statement.
The number \(a\) is by \(25\%\) smaller than \(b\).
The number \(a\) is by \(75\%\) smaller than \(b\).
The number \(a\) is by \(400\%\) greater than \(b\).
The number \(a\) is by \(75\%\) greater than \(b\).
2010006401
Level:
C
Of the school's \(300\) pupils, \(67\%\) attend the sports club and \(42\%\) attend the music club. \(18\%\) attend both clubs. How many children do not attend any club?
\(27\) children
\(81\) children
\(87\) children
\(30\) children
2010006309
Level:
C
Find the points of intersection of the circle \( x^2+y^2=16 \) and the line \( x-y+4=0 \).
\( [-4;0] \), \( [0;4] \)
\( [0;-4] \), \( [-4;0] \)
\( [4;0] \), \( [0;4] \)
\( [-4;0] \), \( [4;0] \)
\( [4;0] \), \( [0;-4] \)
2010006308
Level:
C
Find the equation of the circle, which passes through the points \( A=[-4;2] \), \( B=[1;3] \) and \( C=[2;-2] \).
\( (x+1)^2+y^2 = 13 \)
\( (x-1)^2+y^2 = 13 \)
\( x^2 + (y-1)^2 = 13 \)
\( x^2 + y^2 = 13x \)
\( x^2 + (y+1)^2 = 13 \)
2010006006
Level:
C
Find the tangent equations from the point \( M=[-2;2] \) to the parabola given by \( x^2+4x-4y+16=0 \).
\( x-y+4=0 \), \( x+y=0 \)
\( x+y-4=0 \), \( x+y=0 \)
\( x+y+4=0 \), \( x-y=0 \)
\( x-y-4=0 \), \( x-y=0 \)
\( x+y-4=0 \), \( x-y=0 \)
2010006005
Level:
C
Find the tangent equations from the point \( N=[0;0] \) to the ellipse given by \( 2x^2+y^2-4x-8y+12=0\).
\( 5x+y=0 \), \( x-y=0 \)
\( 5x-y=0 \), \( x+y=0 \)
\( 5x-y=0 \), \( x-y=0 \)
\( x+5y=0 \), \( x-y=0 \)
\( x-5y=0 \), \( x+y=0 \)
2010006004
Level:
C
The equation of a parabola is given by \( x^2 -8x +3y-2=0 \). Find the equation of a line, which passes through the vertex of this parabola and is parallel to the line \( 2x-5y+8=0 \).
\( -2x+5y-22 = 0 \)
\( 2x-5y-22 = 0 \)
\( 2x-5y-38 = 0 \)
\( 2x-5y+38 = 0 \)
\( -2x+5y+22 = 0 \)