B

9000022905

Level: 
B
Find the values of a real parameter \(t\) which ensure that the following system has a unique solution. \[ \begin{alignedat}{80} tx & + &y & + &3 & = 0 & & & & & & \\4x & - 2 &y & + &1 & = 0 & & & & & & \\\end{alignedat}\]
\(t\in \mathbb{R}\setminus \{ - 2\}\)
\(t\in \mathbb{R}\)
\(t = -2\)
\(t\in \emptyset \)

9000022906

Level: 
B
Find the values of a real parameter \(t\) which ensure that the following system has a unique solution \([a,b]\) such that both \(a\) and \(b\) are positive real numbers. \[ \begin{alignedat}{80} a & - &tb & = - &2 & & & & & & \\a & + 2 &tb & = &0 & & & & & & \\\end{alignedat}\]
\(t\in \emptyset \)
\(t\in \mathbb{R}^{+}\)
\(t\in \mathbb{R}^{-}\)
\(t = 0\)
\(t\in \mathbb{R}\)

9000021804

Level: 
B
Solve the following inequality. \[ \frac{1} {x - 3}\leq \frac{1} {2 - x} \]
\(x\in (-\infty ,2)\cup \left [ \frac{5} {2},3\right )\)
\(x\in (-\infty ,2)\cup \left [ \frac{5} {3},2\right ] \)
\(x\in \left (-\infty , \frac{5} {2}\right ] \cup \left (3,\infty \right )\)
\(x\in \left [ \frac{5} {2},\infty \right )\)

9000020409

Level: 
B
One of the solutions of the quadratic equation \( x^{2} + bx - 10 = 0\) is \(x_{1} = 5\). Find the second solution \(x_{2}\) and the value of the coefficient \(b\).
\(x_{2} = -2\) and \(b = -3\)
\(x_{2} = -3\) and \(b = -2\)
\(x_{2} = 2\) and \(b = 3\)
\(x_{2} = 3\) and \(b = 2\)