B

9000107507

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

9000108704

Level: 
B
Consider a pair of vectors \(\vec{u} = (1,0,-1)\) and \(\vec{v} = (2,-1,1)\). Find all the vectors \(\vec{w}\) which are perpendicular to both \(\vec{u}\) and \(\vec{v}\) and satisfy \(\left |\vec{w}\right | = 2\).
\(\vec{w} = \left (\frac{2\sqrt{11}} {11} , \frac{6\sqrt{11}} {11} , \frac{2\sqrt{11}} {11} \right )\), \(\vec{w} = \left (-\frac{2\sqrt{11}} {11} ,-\frac{6\sqrt{11}} {11} ,-\frac{2\sqrt{11}} {11} \right )\)
\(\vec{w} = (-1,-3,-1)\), \(\vec{w} = (1,3,1)\)
\(\vec{w} = \left (-\frac{1} {2},-\frac{3} {2},-\frac{1} {2}\right )\), \(\vec{w} = \left (\frac{1} {2}, \frac{3} {2}, \frac{1} {2}\right )\)
\(\vec{w} = \left (\frac{2\sqrt{2}} {3} , \frac{3\sqrt{2}} {2} , \frac{2\sqrt{2}} {3} \right )\), \(\vec{w} = \left (-\frac{2\sqrt{2}} {3} ,-\frac{3\sqrt{2}} {2} ,-\frac{2\sqrt{2}} {3} \right )\)

9000107509

Level: 
B
In the following list identify a parametric line such that the angle between this line and the line \(q\) is \(0^{\circ }\). \[ q\colon x - 2y + 11 = 0 \]
\(\begin{aligned}[t] p\colon x& = 1 + 4t, & \\y & = 3 + 2t,\ t\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] p\colon x& = 1 + 2t, & \\y & = 2 - t,\ t\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] p\colon x& = 2 - t, & \\y & = 3 + 2t,\ t\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] p\colon x& = t, & \\y & = 1 - 2t,\ t\in \mathbb{R} \\ \end{aligned}\)

9000106302

Level: 
B
The plane \(\alpha \) has equation \[ \alpha : 2x + y - z - 5 = 0. \] The line \(k\) passes through the point \(A = [0,0,1]\) and is perpendicular to \(\alpha \). Find the intersection \(S\) of the line \(k\) and the plane \(\alpha \).
\(S = [2,1,0]\)
\(S = [2,0,1]\)
\(S = [-2,1,0]\)
\(S = [-2,0,1]\)

9000106304

Level: 
B
Find the third coordinate of the point \(B = [2,0,?]\) using the fact that this point is in the plane \(\alpha \) defined by the equation \[ \alpha \colon 2x + y - z - 5 = 0. \] Use the point \(B\) to find the angle \(\varphi \) between the plane \(\alpha \) and the line \(AB\), where \(A = [0,0,1]\).
\(\varphi = 60^{\circ }\)
\(\varphi = 45^{\circ }\)
\(\varphi = 30^{\circ }\)
\(\varphi = 75^{\circ }\)

9000106305

Level: 
B
Find the area of the triangle \(ABS\). Only first two coordinates of the point $B=[2,0,?]$ are given and $B$ lies in the plane $\alpha$ defined by the equation \[ \alpha \colon 2x + y - z - 5 = 0. \] The point \(S\) is the intersection point of the plane \(\alpha \) and the line \(k\) which is perpendicular to \(\alpha \) and passes through the point \(A = [0,0,1]\).
\(\sqrt{3}\)
\(2\)
\(4\)
\(\sqrt{6}\)

9000106306

Level: 
B
Find the general equation of the plane which is perpendicular to the plane \(\alpha \) \[ \alpha \colon 2x + y - z - 5 = 0 \] and contains the line \(AB\), where \(A = [0,0,1]\) and \(B\) is a point in \(\alpha \) defined by it's first two coordinates \[ B = [2,0,?]. \]
\(x - y + z - 1 = 0\)
\(x + y - z + 1 = 0\)
\(2x - y + z - 1 = 0\)
\(- 2x + y - z + 1 = 0\)