9000101703
Level:
B
Factor the following polynomial.
\[
\left (5x - y\right )^{2} -\left (x - y\right )^{2}
\]
\(4x\left (6x - 2y\right )\)
\(x\left (5x - y\right )\)
\(6x\left (6x - 2y\right )\)
\(- 32x^{2}\)
B
9000101102
Level:
B
Find the distance between the point \(A = [1;0;1]\)
and the line \(p\).
\[
\begin{aligned}p\colon x& = 2, &
\\y & = 3t,
\\z & = 1 - t;\ t\in \mathbb{R}
\\ \end{aligned}
\]
\(1\)
\(0\)
\(2\)
\(3\)
9000101106
Level:
B
In the following list identify a point which does not have zero distance from the line
\(m\).
\[
\begin{aligned}m\colon x& = s, &
\\y & = 8 - s,
\\z & = 1 + 3s;\ s\in \mathbb{R}
\\ \end{aligned}
\]
\([2;6;10]\)
\([0;8;1]\)
\([1;7;4]\)
\([8;0;25]\)
9000101103
Level:
B
Find the distance between two parallel lines
\(p\) and
\(q\).
\[
\begin{aligned}p\colon x& = 2, &
\\y & = 3t,
\\z & = 1 - t;\ t\in \mathbb{R}
\\ \end{aligned}\qquad \qquad \begin{aligned}q\colon x& = 3, &
\\y & = 6s,
\\z & = 1 - 2s;\ s\in \mathbb{R}
\\ \end{aligned}
\]
\(1\)
\(2\)
\(3\)
\(4\)
9000100706
Level:
B
Given vectors \(\vec{a} = (-1;2;-3)\),
\(\vec{b} = (0;1;-1)\), find the
vector \(\vec{c}\)
perpendicular to both of them.
\(\vec{c} = (-1;1;1)\)
\(\vec{c} = (-3;0;1)\)
\(\vec{c} = (2;4;2)\)
\(\vec{c} = (-1;-1;1)\)
9000101104
Level:
B
Find the distance between the point \(A = [-1;1;0]\)
and the plane \(\alpha \).
\[
\alpha \colon 12y + 5z - 2 = 0
\]
\(\frac{10}
{13}\)
\(\frac{15}
{13}\)
\(\frac{17}
{13}\)
\(\frac{14}
{13}\)
9000101606
Level:
B
Expand \(\left (x - y\right )^{3} - x\left (x + y\right )^{2}\).
\(- y^{3} - 5x^{2}y + 2xy^{2}\)
\(y^{3} - 5x^{2}y + 2xy^{2}\)
\(- y^{3} - 5x^{2}y - 4xy^{2}\)
\(- y^{3} - 5x^{2}y + 4xy^{2}\)
9000101107
Level:
B
Find the distance between the line \(p\)
and the plane \(\alpha \).
\[
\alpha \colon x-3y+2z-4 = 0,\qquad \qquad \begin{aligned}[t] p\colon x& = 1 + t, &
\\y & = -3t,
\\z & = 2;\ t\in \mathbb{R}
\\ \end{aligned}
\]
\(0\)
\(\frac{5}
{\sqrt{17}}\)
\(2\)
\(1\)
9000101701
Level:
B
Factor the following polynomial.
\[
15xy - 10x - 3y + 2
\]
\(\left (5x - 1\right )\left (3y - 2\right )\)
\(5x\left (3y - 2\right )\)
\(4x\left (3y - 2\right )\)
\(- 5x\left (3y - 2\right )\)
9000101109
Level:
B
The points \(A = [0;5;0]\),
\(B = [5;5;0]\),
\(C = [5;0;0]\),
\(D = [0;0;0]\) define a cube
\(ABCDEFGH\). Find the distance
between the line \(AB\)
and the plane \(EFG\).
\(5\)
\(3\)
\(4\)
\(6\)