B

9000107505

Level: 
B
Find \(\cos \varphi \) where \(\varphi \) is the angle between the lines \(p\) and \(q\). \[ \begin{aligned}[t] p\colon x& = 1 + 4t, & \\y & = 3 - 3t;\ t\in \mathbb{R}; \\ \end{aligned} \quad q\colon x + y - 3 = 0 \]
\(\frac{7\sqrt{2}} {10} \)
\(- \frac{7} {5\sqrt{2}}\)
\(\frac{\sqrt{2}} {5} \)
\(\frac{\sqrt{2}} {10} \)

9000107507

Level: 
B
Find \(\mathop{\mathrm{tg}}\nolimits \varphi \) where \(\varphi \) is the angle between the lines \(p\) and \(q\). \[ \begin{aligned}[t] p\colon x& = 1 + t, & \\y & = 3 + 2t;\ t\in \mathbb{R}; \\ \end{aligned}\quad q\colon y = 1 \]
\(2\)
\(\frac{1} {2}\)
\(- 1\)
\(0\)

9000108802

Level: 
B
Given the points \(A = [1;2]\), \(B = [2;6]\) and \(C = [3;-1]\), find the interior angles of the triangle \(ABC\). Round to the nearest degree.
\(22^{\circ }\), \(26^{\circ }\), \(132^{\circ }\)
\(26^{\circ }\), \(45^{\circ }\), \(109^{\circ }\)
\(22^{\circ }\), \(48^{\circ }\), \(110^{\circ }\)
\(17^{\circ }\), \(31^{\circ }\), \(132^{\circ }\)

9000108803

Level: 
B
Consider the vector \(\vec{u} = (\sqrt{3};1)\). Find the vector \(\vec{w}\) such that \(\left |\vec{w}\right | = 4\) and the angle between \(\vec{u}\) and \(\vec{w}\) is \(60^{\circ }\). Find all solutions.
\(\vec{w} = (0;4)\), \(\vec{w} = (2\sqrt{3};-2)\)
\(\vec{w} = (0;-4)\), \(\vec{w} = (\sqrt{7};-3)\)
\(\vec{w} = (0;4)\), \(\vec{w} = (\sqrt{7};3)\)
\(\vec{w} = (\sqrt{5};\sqrt{11})\), \(\vec{w} = (2\sqrt{3};-2)\)

9000111807

Level: 
B
In the following list identify a line such that the angle between this line and the plane \[ 2x - y + 3z - 5 = 0 \] is \(30^{\circ }\).
\(\begin{aligned}[t] p\colon x& = 2 + t, & \\y & = 1 + 3t, \\z & = -2t;\ t\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] r\colon x& = -2t, & \\y & = -3 + t, \\z & = 1 - 3t;\ t\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] q\colon x& = 2 + 3t, & \\y & = 3 - 2t, \\z & = 3 + t;\ t\in \mathbb{R} \\ \end{aligned}\)