C

9000090910

Level: 
C
Given lines \(p\) and \(q\), find \(m\in \mathbb{R}\) such that the line \(p\) is parallel to \(q\). \[ p\colon x+4y-3 = 0,\qquad \begin{aligned}[t] q\colon x& = 1 + mt,& \\y & = 2 - 3t;\ t\in \mathbb{R} \\ \end{aligned} \]
\(m = 12\)
\(m = -\frac{1} {12}\)
\(m = 4\)
\(m = \frac{5} {2}\)
\(m = -1\)

9000090902

Level: 
C
Given the parametric line \(p\), find \(m\in \mathbb{R}\) such that the point \(C = [m;3]\) is on the line \(p\). \[ \begin{aligned}p\colon x& = 1 - t, & \\y & = -3 + 2t;\ t\in \mathbb{R} \\ \end{aligned} \]
\(m = -2\)
\(m = 4\)
\(m = 11\)
\(m = -\frac{11} {3} \)
\(m = \frac{3} {2}\)

9000089001

Level: 
C
Students from a class have a possibility to work in mathematical and physical hobby groups. There are \(31\) students in the class. From the total, \(21\) students are members of mathematical group. Some of the students are members of both groups, but there are \(10\) which are members of just one group. There are \(3\) students which are not members of any of those groups. How many students are members of both mathematical and physical groups?
\(18\)
\(16\)
\(19\)

9000087505

Level: 
C
Assuming \(x\in \mathbb{R}\setminus \left \{-\frac{1} {2}\right \}\), find the quotient of the polynomials. \[ (4x^{3} - 1) : (2x + 1) \]
\(2x^{2} - x + \frac{1} {2} - \frac{\frac{3} {2} } {2x+1}\)
\(2x^{2} + x + \frac{1} {2} - \frac{\frac{3} {2} } {2x+1}\)
\(2x^{2} - x + \frac{1} {4} - \frac{\frac{3} {2} } {2x+1}\)
\(2x^{2} + x + \frac{1} {4} - \frac{\frac{3} {2} } {2x+1}\)

9000087508

Level: 
C
Assuming \(x\in \mathbb{R}\setminus \left \{0, 1, 3\right \}\), find the quotient of the polynomials. \[ (-5x^{4} + 4x^{2} + 3x - 4) : (x^{3} - 4x^{2} + 3x) \]
\(- 5x - 20 + \frac{-61x^{2}+63x-4} {x^{3}-4x^{2}+3x} \)
\(- 5x - 20 + \frac{16x^{2}+23x+36} {x^{3}-4x^{2}+3x} \)
\(- 5x - 10 + \frac{-61x^{2}+63x-4} {x^{3}-4x^{2}+3x} \)
\(- 5x - 10 + \frac{-16x^{2}+23x-36} {x^{3}-4x^{2}+3x} \)

9000087502

Level: 
C
Assuming \(x\in \mathbb{R}\setminus \left \{\pm 1\right \}\), find the quotient of the polynomials: \[ (-2x^{4} - 3x^{2} + 3) : (x^{2} - 1) \]
\(- 2x^{2} - 5 - \frac{2} {x^{2}-1}\)
\(- 2x^{2} - 5 + \frac{2} {x^{2}-1}\)
\(2x^{2} + 5 - \frac{2} {x^{2}-1}\)
\(2x^{2} + 5 + \frac{2} {x^{2}-1}\)

9000087503

Level: 
C
Assuming \(x\in \mathbb{R}\setminus \left \{-\frac{3} {2}\right \}\), find the quotient of the polynomials: \[ (x^{2} + x + 1) : (2x + 3) \]
\(\frac{1} {2}x -\frac{1} {4} + \frac{\frac{7} {4} } {2x+3}\)
\(\frac{1} {2}x -\frac{1} {2} + \frac{\frac{7} {4} } {2x+3}\)
\(x + 2 + \frac{7} {2x+3}\)
\(x - 2 + \frac{7} {2x+3}\)