9000149404
Level:
B
Given points \(A = [-3,13]\),
\(K = [0,4]\),
\(L = [-5,-6]\), find the distance
from the point \(A\)
to the line \(KL\).
\(3\sqrt{5}\)
\(3\)
\(5\)
\(\sqrt{5}\)
B
9000149405
Level:
B
Find all the values of the parameter
\(c\) so that the distance from the point \(M = [2,-1]\)
to the line \(3x + 4y + c = 0\) is \(5\).
\(c\in \{ - 27,23\}\)
\(c\in \{25\}\)
\(c\in \{5,25\}\)
\(c\in \{ - 25,25\}\)
9000149406
Level:
B
Given points \(A = [2,-5]\),
\(B = [2,3]\) and
\(C = [-4,-1]\), find the length of the altitude of the triangle \(ABC\) through the point \(C\). Hint: The altitude through the point
\(C\) of a triangle
\(ABC\) is the perpendicular line segment drawn from the vertex \(C\) to the line containing the side \(AB\).
\(6\)
\(\sqrt{2}\)
\(\frac{3}
{2}\)
The points \(A\),
\(B\),
\(C\) do
not define a triangle.
9000149408
Level:
B
On the \(x\)-axis
find the points such that the distance from these points to the line
\(p\colon x - 2y + 2 = 0\) is
\(\sqrt{5}\).
\([3,0]\),
\([-7,0]\)
\([5,0]\)
\(\left [\sqrt{5},0\right ]\),
\(\left [-\sqrt{5},0\right ]\)
\([3,7]\)
9000149410
Level:
B
Find all lines passing through the point
\(A = [-2,-6]\) such that the distance
from the point \([0.0]\)
to these lines is \(2\sqrt{2}\).
\(p_{1}\colon 7x + y + 20 = 0\),
\(p_{2}\colon x - y - 4 = 0\)
\(p\colon 7x - y = 0\)
\(p\colon x + y + 2\sqrt{2} = 0\)
\(p_{1}\colon x - y + 2\sqrt{2} = 0\),
\(p_{2}\colon x + y - 2\sqrt{2} = 0\)
9000149706
Level:
B
Find the center of the following hyperbola.
\[
4x^{2} - 3y^{2} + 8x - 30y - 49 = 0
\]
\([-1,-5]\)
\([-1,5]\)
\([1,-5]\)
\([1,5]\)
9000150102
Level:
B
Evaluate the following integral on \(\mathbb{R}\).
\[
\int 2\sin 2x\, \mathrm{d}x
\]
\(-\cos 2x + c,\ c\in \mathbb{R}\)
\(\cos 2x + c,\ c\in \mathbb{R}\)
\(- 4\cos 2x + c,\ c\in \mathbb{R}\)
\(4\cos 2x + c,\ c\in \mathbb{R}\)
9000149707
Level:
B
Find the center of the following hyperbola.
\[
5x^{2} - 6y^{2} - 30x + 12y + 9 = 0
\]
\([3,1]\)
\([3,-1]\)
\([-3,1]\)
\([-3,-1]\)
9000149402
Level:
B
Find the distance from the origin (i.e. the point
\([0.0]\)) to the
line \(p\colon x + 2y + 5 = 0\).
\(\sqrt{5}\)
\(1\)
The origin is on the line \(p\).
\(8\)
9000149708
Level:
B
Find the vertex of the following parabola.
\[
x^{2} + 8x - 4y + 24 = 0
\]
\([-4,2]\)
\([-4,-2]\)
\([4,2]\)
\([4,-2]\)