C

9000104805

Level: 
C
Find the slope of a line through the center of the hyperbola \[ \frac{(x - 2)^{2}} {4} -\frac{(y + 3)^{2}} {9} = 1 \] which has a unique intersection with this hyperbola.
There is no solution, the line with these properties does not exist.
\(\frac{3} {2}\)
\(-\frac{3} {2}\)
\(\frac{2} {3}\)
\(1\)
\(0\)

9000104809

Level: 
C
Among the following lines (which all pass through the point \([-1;3]\)) identify a line which is tangent to the following hyperbola. \[ (x + 2)\cdot (y - 2) = 1 \]
\(k\colon \ y = -x + 2\)
\(p\colon \ y = 3\)
\(q\colon \ x = -1\)
\(r\colon \ y = x + 4\)
None of the given answers is correct.

9000101708

Level: 
C
Factor the following polynomial. \[ 8x^{3} - 27 \]
\(\left (2x - 3\right )\left (4x^{2} + 6x + 9\right )\)
\(\left (2x - 3\right )\left (4x^{2} - 6x + 9\right )\)
\(\left (2x + 9\right )\left (4x^{2} - 6x + 9\right )\)
\(\left (2x - 3\right )\left (4x^{2} + 6x - 9\right )\)

9000101709

Level: 
C
Factor the following polynomial. \[ 27x^{6}z - 8y^{3}z \]
\(z\left (3x^{2} - 2y\right )\left (9x^{4} + 6x^{2}y + 4y^{2}\right )\)
\(z\left (3x^{2} + 2y\right )\left (9x^{4} + 6x^{2}y - 4y^{2}\right )\)
\(z\left (3x^{2} + 2y\right )\left (9x^{4} - 6x^{2}y + 4y^{2}\right )\)
\(z\left (3x^{2} - 2y\right )\left (9x^{4} + 6x^{2}y^{2} + 4y\right )\)

9000101707

Level: 
C
Factor the following polynomial. \[ x^{6} - 1 \]
\(\left (x - 1\right )\left (x + 1\right )\left (x^{2} + x + 1\right )\left (x^{2} - x + 1\right )\)
\(\left (x - 1\right )\left (x + 1\right )\left (x^{2} + x + 1\right )\left (x^{2} - x - 1\right )\)
\(\left (x - 1\right )\left (x + 1\right )\left (x^{2} + 2x + 1\right )\left (x^{2} - 2x + 1\right )\)
\(\left (x - 1\right )\left (x + 1\right )\left (x^{2} + x - 1\right )\left (x^{2} - x + 1\right )\)

9000090906

Level: 
C
Given lines \(p\) and \(q\), find \(m\in \mathbb{R}\) such that the lines \(p\) and \(q\) are parallel. \[ \begin{aligned}p\colon x& = 1 + t, & \\y & = -3t;\ t\in \mathbb{R}, \\ \end{aligned}\qquad \begin{aligned}q\colon x& = 3 - 2u, & \\y & = 1 + mu;\ u\in \mathbb{R} \\ \end{aligned} \]
\(m = 6\)
\(m = \frac{3} {2}\)
\(m = -\frac{2} {3}\)
does not exist

9000090907

Level: 
C
Given points \(A = [2;m]\) and \(B = [-1;0]\), find \(m\in \mathbb{R}\) such that the line \(p\) is parallel to the line passes through the points \(A\), \(B\). \[ \begin{aligned}p\colon x& = 3 + 2t, & \\y & = 5 - t;\ t\in \mathbb{R} \\ \end{aligned} \]
\(m = -\frac{3} {2}\)
\(m = \frac{3} {2}\)
\(m = -\frac{2} {3}\)
\(m = 2\)
does not exist