9000101606
Level:
B
Expand \(\left (x - y\right )^{3} - x\left (x + y\right )^{2}\).
\(- y^{3} - 5x^{2}y + 2xy^{2}\)
\(y^{3} - 5x^{2}y + 2xy^{2}\)
\(- y^{3} - 5x^{2}y - 4xy^{2}\)
\(- y^{3} - 5x^{2}y + 4xy^{2}\)
B
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 \(EFG\).
\(5\)
\(3\)
\(4\)
\(6\)
9000101705
Level:
B
Factor the following polynomial expression.
\[
16a^{2}b^{2} - 4a^{2}c^{2} - 16b^{2}d^{2} + 4c^{2}d^{2}
\]
\(4\left (a - d\right )\left (a + d\right )\left (2b + c\right )\left (2b - c\right )\)
\(4\left (a + b\right )^{2}\left (2b + c\right )^{2}\)
\(4\left (a - b\right )\left (a + b\right )\left (2b + c\right )\left (2b - c\right )\)
\(4\left (a - c\right )\left (a + c\right )\left (2b + d\right )\left (2b - d\right )\)
9000101110
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 point \(A\)
and the point \(F\).
\(\sqrt{50}\)
\(\sqrt{5}\)
\(5\)
\(10\)
9000101105
Level:
B
Find the distance between two parallel planes
\(\alpha \) and
\(\beta \).
\[ \begin{aligned}
\alpha& \colon x - 1.5y - 3z - 1 = 0,\\ \beta &\colon 2x - 3y - 6z + 3 = 0
\end{aligned}\]
\(\frac{5}
{7}\)
\(\frac{1}
{7}\)
\(\frac{2}
{49}\)
\(\frac{2}
{41}\)
9000101108
Level:
B
Find the distance between the line \(q\)
and the plane \(\beta \).
\[
\beta \colon x+4y+2z-4 = 0,\qquad \qquad \begin{aligned}[t] q\colon x& = 4, &
\\y & = -2t,
\\z & = 1 + 4t;\ t\in \mathbb{R}
\\ \end{aligned}
\]
\(\frac{2}
{\sqrt{21}}\)
\(\frac{4}
{\sqrt{21}}\)
\(0\)
\(1\)
9000101802
Level:
B
Among vectors \(\vec{u} = \left (- \frac{2}
{\sqrt{2}};2\sqrt{2}\right )\),
\(\vec{v} = (-5;10)\),
\(\vec{w} = (2.5;-5)\),
\(\vec{r} = (-3.5;6)\)
find the vector which is not parallel to the vector
\(\vec{a} = (1;-2)\).
\(\vec{r}\)
\(\vec{w}\)
\(\vec{v}\)
\(\vec{u}\)
9000100006
Level:
B
The function \(f(x)= \sqrt{x}\)
is graphed in the picture. Consider the region bounded by the graph of
\(f\) on
\([ 1;\, 4] \), lines
\(x = 1\),
\(x = 4\) and the
\(x\)-axis. Identify
the formula for volume of the solid of revolution obtained by revolving this region about
the \(x\)-axis.
\(V =\pi \int _{ 1}^{4}x\, \mathrm{d}x\)
\(V =\int _{ 1}^{4}x\, \mathrm{d}x\)
\(V =\pi \int _{ 1}^{4}\sqrt{x}\, \mathrm{d}x\)
\(V =\int _{ 1}^{4}\sqrt{x}\, \mathrm{d}x\)